Quantum Electrodynamics: Atoms, Lasers and Gravity [1 ed.] 9789811252259, 9789811252266, 9789811252273

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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QUANTUM ELECTRODYNAMICS: ATOMS, LASERS AND GRAVITY Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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ISBN 978-981-125-225-9 (hardcover) ISBN 978-981-125-226-6 (ebook for institutions) ISBN 978-981-125-227-3 (ebook for individuals)

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Preface

This book is about quantum electrodynamics (QED). In the strict sense of the word, QED is the theory of the quantized electromagnetic field, as opposed to classical electromagnetism. The characteristic element of quantization is the existence of so-called loop corrections which are due to the quantum fluctuations of the electromagnetic field, and of the fermion fields. Quantum fluctuations of the electromagnetic field lead to, e.g., back-reaction of the radiation emitted by a moving electron onto its own trajectory (its energy level). This process is known as the self-energy of the electron; the back-reaction “feedback loop” can also be understood on the basis of a fully quantum theory of the electromagnetic field. Quantum fluctuations of the fermion fields lead to the emergence of short-lived, so-called virtual electron-positron (fermion) pairs [1, 2]; these pairs can be emitted virtually by a photon. Their back-reaction onto the photon propagator (or, loop correction to the photon propagator) leads to a correction to how light travels in vacuum or in a medium. As such, QED is an extension of classical electrodynamics into the quantum domain. However, if we were to restrict the subject of this book to calculations that actually require field quantization, then we would miss some of the most interesting and application-oriented research areas connected with electromagnetism and atoms. Specifically, a large number of very interesting calculations can be done without quantizing the fermion (electron-positron) field, relying only on the quantization of the electromagnetic field. Again, many basic facts can even be inferred on the basis of photon exchange processes (electromagnetic interaction processes) that do not require full quantization of the electromagnetic field. The situation is a little subtle. Let us start by enumerating just the main application areas of QED. (i) QED describes the theory needed in order to infer fundamental constants, and to constrain their conceivable variation with space and time. The spectra of bound atomic systems are explained, and lepton moments (notably, the electron and muon anomalous magnetic moments) can be calculated. (ii) Atomic physics and QED theory are important tools in the description of physical mechanisms of rather cross-disciplinary importance: An example is cosmological hydrogen recombination, which describes vii

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the absorption of free electrons by protons in the Universe, to form hydrogen atoms. An incomplete list of other processes which require a QED description is as follows: For example, we have resonant, radiationless recombination processes (dielectronic recombination), Casimir–Polder long-range interactions among atoms and molecules, and the atom-surface long-range interaction. The modification of the lifetime of atomic states and of the electromagnetic self-energy shift in cavities also profit from the QED formalism. Finally, we have radiation and particle creation processes in strong light fields, in regimes where “matter perturbs light” instead of the opposite. (iii) Last, but certainly not least, QED is important for a number of practically important applications, both within fundamental physics but also, in technologically oriented areas. These include synchrotron radiation, Compton backscattering and the concomitant generation of high-energy photons, and Bhabha scattering (electron-positron scattering) which has been used for the normalization of cross sections at electron-positron colliders such as LEP (CERN). Finally, the Casimir–Polder interaction of atoms in the vicinity of extended objects (plates) has technological relevance in the nanoworld and needs to be described by QED. This book is thus written with the following three main notions in mind. (i) The book should be readable in today’s relatively fast-paced academic environment, where different obligations and continuous interruptions plague both students and researchers on different levels in their daily life. Consequently, the representation is as self-contained as possible in all sections in order to ensure ease of reading. (ii) The book is meant to provide information on different levels. The book is written with both the advanced student as well as the interested senior physicist in mind. Students of field theory may enhance their knowledge by studying applications of field theory in atomic physics; senior atomic physicists may widen their horizon, becoming familiar with quantum-field theoretical techniques, and finally, researchers in bound-state quantum electrodynamics may find a compendium of useful formulas to consider and as reference material. We have tried to write the book so that it is intermediate between a textbook and a monograph, providing both information accompanying, say, a graduate course on quantum electrodynamics, but also, containing information on the status of current research in atomic physics and quantum electrodynamics of bound states. Reference questions (“exercises”) are included for each chapter; these are mostly designed to act as checks for the reader, testing the grasp of a particular section or chapter. The contents of the book can be divided into two parts, the first of which could be entitled as Quantum Mechanics with Field Quantization (Chaps. 2–6). This part of the book is meant to provide a useful complement to a course on quantum mechanics, illustrating the main concepts by a number of interesting applications while expanding on the concepts taught in quantum mechanics by quantizing the electromagnetic field. Most applications relevant to quantum chemistry and to technology can be described using a hybrid approach with a quantization of the electromagnetic field, but without quantization of the fermion field. The formalism of nonrelativistic

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QED (NRQED) is used at a rudimentary level. We need to couple the bound electron to the quantized electromagnetic field and perform a perturbative expansion in electron-photon interactions. The quantized formalism will be contrasted with timeindependent perturbation theory based on interactions with a classical field. The full propagator formalism will not even be necessary at this stage; the quantization of the electromagnetic field will be fully sufficient. In particular, we shall derive the low-energy part of the Lamb shift and include the high-energy part based on formal perturbative operators which will be assumed valid, with a precise derivation given, at a later stage. Also, we will include tiny variations of the Coulomb potential on a phenomenological basis, with a special emphasis on concrete calculations and theoretical predictions. This part will accompany a course on quantum mechanics and will also be interesting for students who wish to broaden the knowledge acquired in quantum mechanics with a perspective on possible applications. Some knowledge of relativistic quantum mechanics, i.e., of the Dirac equation, will also be useful, but not strictly required. The second part is composed of Chaps. 7–19 and could be entitled as the Quantum Field Theory of Bound States. The second part of the book could provide for a useful complement to a graduate course on field theory, illustrating the somewhat abstract concepts on a number of experimentally verifiable calculations concerning atomic physics phenomena. The experimental verification using high-precision spectroscopy can be accomplished in a laboratory of moderate dimensions; the multibillion dollar investment necessary to construct an accelerator is not required. Still, the theoretical description requires all the tools provided by quantum field theory. Measured on this scale, quantum electrodynamics actually is a very gratifying subject: One can be in contact with experimentalists who are able to measure the effects in laboratories close by, and one is able to talk to each in a very friendly and fruitious atmosphere. In that sense, QED in its purest form represents an interesting subfield of physics. Finally, the modification necessary for the description of laser-modified processes (laser-dressed Furry picture) will also be treated. The second part also contains an introduction to the (ordinary) Furry picture, in which one incorporates the Coulomb interaction into the unperturbed Hamiltonian (see Chap. 15). We explain how to apply the vertex calculations from field theory in order to describe bound-state effects. The emphasis is the interplay of dimensional, photon mass, and photon energy regularizations. Also, the matching of the Smatrix element against the effective Hamiltonian will be explained for the α(Zα)5 terms, which correspond to the two-Coulomb–Vertex scattering diagrams involving high-energy photons. An introduction to the Bethe–Salpeter equation (BSEQ) and Nonrelativistic Quantum Electrodynamics (NRQED, an effective theory for bound states) also forms part of this book. The second part may accompany a course on quantum field theory, and it may also be interesting for the phenomenologically interested student who wishes to learn how to apply the formalism to experiments carried out in the laboratory.

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As a monograph with a textbook component, this book tries to bridge areas that have been covered in a number of dedicated books elsewhere in the literature. Topics related to atomic physics and quantum electrodynamics have been discussed in the literature for a number of decades. This introduction would thus be incomplete without a brief survey of existing books in related areas. A necessarily incomplete list of books can be given as follows: V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii [3] have written a classic treatise which was published as volume 4 of the textbook series on physics by L. D. Landau and E. M. Lifshitz (completed after the death of L. D. Landau). A very comprehensive introduction to quantum electrodynamics has been given by A. I. Akhiezer and V. B. Berestetskii [4]. Formal aspects of the problems have been stressed in the book of I. Bialynicki-Birula and Z. Bialynicka-Birula [5] (as usual in the Polish language, the name of the second author, who is the wife of the first author, is being spelled a little differently). The “l” is pronounced like the French h-aspir´e and the notation is reminiscent to the ̵ Furthermore, we should quantum unit of action, or reduced Planck constant, h. mention the monograph of H. A. Bethe and E. E. Salpeter [6], which stresses basic processes of atomic physics, the monograph of W. Heitler [7], which contains a comprehensive introduction to radiation theory, and the book of H. A. Bethe and R. Jackiw [8], which contains a good and easily readable overview of the basic structural and dynamical properties of atoms. Regarding the field-theoretical foundations of quantum electrodynamics, we would like to mention the comprehensive works of J. Schwinger [9], S. S. Schweber [10], and a brief supplemental monograph by R. P. Feynman [11] which contains a number of detailed calculations, as well as the monograph by C. Itzykson and J. B. Zuber [2], which summarizes essentially everything known in field theory up to 1980, in a very concise but comprehensive form. No other book rivals the textbook of W. Greiner and J. Reinhardt [12] in terms of its excellent presentation of scattering processes (Compton, Bhabha, and Møller scattering). The didactic value of the book of Greiner and Reinhardt should not be underestimated. The book of J. M. Jauch and F. Rohrlich [13] gives an elegant introduction into basic concepts of QED as well. The intricacies of angular momentum algebra are treated in the monographs by A. R. Edmonds [14], D. M. Brink, and G. R. Satchler [15], D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii [16]. Foundations of atomic physics, with a special emphasis on the quantum mechanical formulation and basic principles, are discussed in the monographs of H. A. Bethe and E. E. Salpeter [6], mentioned previously, M. E. Rose [17], E. Merzbacher [18], B. H. Bransden and C. J. Joachain [19], and M. Weissbluth [20]. The monographs by W. R. Johnson [21] and by I. P. Grant [22] stress the computational aspects of relativistic many-body atomic physics, whereas the work of I. Lindgren and J. Morrison [23] emphasizes the angular momentum algebra needed for the many-body problem. L. Labzowsky [24] gives an overview of the relativistic many-body formalism. Two very interesting books by J. J. Sakurai [25, 26] on the foundations of quantum mechanics should

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also be mentioned. On the more specialized side, the monograph by M. I. Eides, H. Grotch and V. A. Shelyuto [27] contains a recent comprehensive status report on light atoms. The concepts of field quantization, in the nonrelativistic domain, are discussed in the book of P. W. Milonni [28]. Finally, W. Dittrich and H. Gies [29] have given an account of the modifications of the Maxwell equations in nontrivial background fields. We tried to mention the names of a few researchers who were actively involved in advancements of bound-state QED over the last few decades. The purpose of the book, though, is not to provide an all-inclusive review of all results obtained so far, as the status is evolving and would otherwise constitute a “moving target”. On the occasion, we apologize to all colleagues whose works may have been overlooked in the discussion. This book project was supported by the National Science Foundation under grants PHY–1707489, PHY–2011762, PHY–14030973, PHY– 1710856 and PHY–2110294. Insightful conversations with Professors K. Pachucki, P. J. Mohr, V. A. Yerokhin, C. M. Adhikari, G. W. F. Drake, Y. Itin, I. N´andori and Professor T. W. H¨ ansch are gratefully acknowledged. The authors also express gratitude to Dr. B. J. Wundt for elucidating conversations, and they acknowledge Dr. N. C. James, B. Robinson, D. McNamara, J. Padron, and A. Wetzel for proofreading the manuscript. The Hungarian Academy Institute for Nuclear Physics in Debrecen (ATOMKI) is gratefully acknowledged for hospitality during a Sabbatical stay in the Academic year 2018–2019, when an important part of the book was completed. Rolla, Missouri, and Lancaster, Pennsylvania, May 2021 U. D. Jentschura Ulrich D. Jentschura Missouri University of Science and Technology Department of Physics and LAMOR Rolla, Missouri 65409, USA [email protected] G. S. Adkins Gregory S. Adkins Franklin & Marshall College Department of Physics and Astronomy Lancaster, Pennsylvania 17603, USA [email protected]

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Contents

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Preface 1.

Introduction 1.1 1.2 1.3 1.4 1.5

2.

Accurate Numbers . . . . . . . Fundamental Constants . . . . Overview of Chapter Contents Miscellaneous Remarks . . . . Further Thoughts . . . . . . . .

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From Unit Systems for the Microworld to Field Quantization

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2.3

2.4

2.5

2.6 3.

1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atoms and Field Quantization . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Matter Waves and Hamilton–Jacobi Formalism . . . . . . . 2.2.2 Field Quantization as Second Quantization . . . . . . . . . Unit Systems Scaled to the Microworld . . . . . . . . . . . . . . . . . 2.3.1 Unit Systems and Observation Scales . . . . . . . . . . . . . 2.3.2 Natural Unit System . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Atomic Unit System . . . . . . . . . . . . . . . . . . . . . . . Field Quantization for the Electromagnetic Field . . . . . . . . . . . 2.4.1 Quantization of the Free Electromagnetic Field . . . . . . . 2.4.2 Field Operators and Quantized Hamiltonian . . . . . . . . . 2.4.3 Discretized Formulation of the Field Operators . . . . . . . Interaction Picture and Phase Conventions . . . . . . . . . . . . . . 2.5.1 Field Operators in the Schr¨odinger and Interaction Pictures 2.5.2 Integration Measure and Phase Conventions . . . . . . . . . Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Time-Ordered Perturbations

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3.1 3.2

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Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Ordered Perturbations and Fermi’s Golden Rule . . . . . . . xiii

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3.3

3.4

3.5 4.

Quantum Electrodynamics in Action: Atoms, Lasers and Gravity

3.2.1 Derivation of Fermi’s Golden Rule . . . . . . . . . . . . . . . 3.2.2 Fermi’s Golden Rule and Nuclear Beta Decay . . . . . . . . 3.2.3 Fermi’s Golden Rule and Atomic Decay Rates . . . . . . . Dynamic Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Way 1: Time-Dependent Perturbation Theory . . . . . . . 3.3.2 Way 2: Quantized Fields . . . . . . . . . . . . . . . . . . . . 3.3.3 Way 3: Gell-Mann–Low–Sucher Theorem . . . . . . . . . . Static Stark Effect and Level Width . . . . . . . . . . . . . . . . . . 3.4.1 Stark Shift and Large-Order Perturbation Theory . . . . . 3.4.2 Quantum Electrodynamics and Large-Order Perturbations Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bound-Electron Self–Energy and Bethe Logarithm 4.1 4.2

4.3

4.4

4.5

4.6

4.7

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schr¨ odinger–Coulomb Hamiltonian and Wave Functions . . . . 4.2.1 Spectrum of the Hydrogen Atom and SO(4) Symmetry 4.2.2 Differential Equations and the Hydrogen Atom . . . . . 4.2.3 Schr¨ odinger–Coulomb Bound States . . . . . . . . . . . . 4.2.4 Schr¨ odinger–Coulomb Virial Theorem . . . . . . . . . . 4.2.5 Schr¨ odinger–Coulomb Continuum States . . . . . . . . . 4.2.6 Continuum States for a Repulsive Potential . . . . . . . Schr¨ odinger–Coulomb Green Function in Coordinate Space . . 4.3.1 Green Function and Radial Equation . . . . . . . . . . . 4.3.2 Solution using Whittaker Functions . . . . . . . . . . . . 4.3.3 Solution using Laguerre Polynomials . . . . . . . . . . . 4.3.4 Green Function and Dynamic Polarizability . . . . . . . Schr¨ odinger–Coulomb Green Function in Momentum Space . . 4.4.1 Free Green Function in Momentum Space . . . . . . . . 4.4.2 Toward the SO(4) Symmetry . . . . . . . . . . . . . . . 4.4.3 Four-Dimensional Spherical Harmonics . . . . . . . . . . 4.4.4 Wave Functions in Momentum Space . . . . . . . . . . . 4.4.5 Integral Representation . . . . . . . . . . . . . . . . . . . Modern Ideas and Bound-State Self–Energy . . . . . . . . . . . . 4.5.1 Essence of Renormalization . . . . . . . . . . . . . . . . . 4.5.2 Overlapping Parameter . . . . . . . . . . . . . . . . . . . Bound-State Self–Energy: Low-Energy Part . . . . . . . . . . . . 4.6.1 Length Gauge . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Velocity Gauge . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Calculation of the Bethe Logarithm . . . . . . . . . . . . 4.6.4 Numerical Values of Bethe Logarithms . . . . . . . . . . 4.6.5 Outline of the High-Energy Part . . . . . . . . . . . . . . Applications of the Developed Formalism . . . . . . . . . . . . .

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Contents

4.8 5.

of Self–Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Interatomic and Atom-Surface Interactions 5.1 5.2

5.3

5.4

5.5

5.6 6.

4.7.1 Electric Dipole Decay and Imaginary Part 4.7.2 Magnetic Interactions and Decay Rates . . 4.7.3 Higher-Order Terms . . . . . . . . . . . . . Further Thoughts . . . . . . . . . . . . . . . . . . . .

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Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wigner–Brioullin Perturbation Theory . . . . . . . . . . . . . . . . 5.2.1 Spectral Representation of the Green Function . . . . . . 5.2.2 First-Order and Second-Order Perturbation Theory . . . 5.2.3 Higher-Order Perturbation Theory (Wigner–Brioullin) . Application to the Finite-Size Effect . . . . . . . . . . . . . . . . . 5.3.1 Master Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 First-Order and Second-Order Finite-Size Effect . . . . . Interatomic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Origin of van der Waals and Casimir–Polder Interactions 5.4.2 Calculation of Interatomic Interactions . . . . . . . . . . . 5.4.3 Limit of Large Separation of the Two Atoms . . . . . . . 5.4.4 Nonretarded (van der Waals) Interatomic Interaction . . 5.4.5 Interpolating Formula . . . . . . . . . . . . . . . . . . . . . Atom-Surface Interactions . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Perfectly Conducting Wall, Long Range . . . . . . . . . . 5.5.2 Dipole Interaction . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Multipole Interactions . . . . . . . . . . . . . . . . . . . . . 5.5.4 Interactions with a Dielectric Surface . . . . . . . . . . . . Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Racah–Wigner Algebra 6.1 6.2

6.3

6.4

6.5

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clebsch–Gordan Coefficients . . . . . . . . . . . . . . . . . . 6.2.1 Expansions and Clebsch–Gordan Coefficients . . . 6.2.2 Matrix Elements and Clebsch–Gordan Coefficients Coefficients and Rotations . . . . . . . . . . . . . . . . . . . 6.3.1 Vector Addition or Clebsch–Gordan Coefficients . 6.3.2 Wigner 3j, 6j and 9j Symbols . . . . . . . . . . . . 6.3.3 Gaunt Coefficients . . . . . . . . . . . . . . . . . . . 6.3.4 Representation of Finite Rotations . . . . . . . . . Composed Tensors of Higher Order . . . . . . . . . . . . . . 6.4.1 Construction of the Spin-Angular Function . . . . 6.4.2 Construction of the Vector Spherical Harmonic . . 6.4.3 Spherical Biharmonic . . . . . . . . . . . . . . . . . Applications of Racah–Wigner Algebra . . . . . . . . . . . .

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6.7

6.5.1 Tensor Decomposition of the Light Shift . . . . 6.5.2 Tensorial Decomposition of a Dipole Transition Rydberg Electron and Hydrogenlike Core . . . . . . . . 6.6.1 Physical Foundation . . . . . . . . . . . . . . . . 6.6.2 Angular Algebra . . . . . . . . . . . . . . . . . . Further Thoughts . . . . . . . . . . . . . . . . . . . . . . .

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Free Dirac Equation

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7.1 7.2

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6.6

7.

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Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the Free Dirac Equation . . . . . . . . . . . . . . . . . 7.2.1 Dirac Equation as the Linearized Klein–Gordon Equation 7.2.2 Spinor Lorentz Transformation . . . . . . . . . . . . . . . . . 7.2.3 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Overview of the Symmetry Properties . . . . . . . . . . . . Solutions of the Free Dirac Equation . . . . . . . . . . . . . . . . . . 7.3.1 Plane-Wave Solutions of the Dirac Equation . . . . . . . . . 7.3.2 Dirac Angular Quantum Number . . . . . . . . . . . . . . . 7.3.3 Angular Momenta and Massless Dirac Equation . . . . . . 7.3.4 Angular Momenta and Free Dirac Equation . . . . . . . . . Quantized Dirac Field and Propagators . . . . . . . . . . . . . . . . 7.4.1 Free Dirac Progator in Feynman’s Formulation . . . . . . . 7.4.2 Feynman Propagator and Green Function . . . . . . . . . . 7.4.3 Free Dirac Propagator in the Angular Momentum Basis . Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dirac Equation for Bound States, Lasers and Gravity 8.1 8.2

8.3

8.4

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirac Equation and Coulomb Field . . . . . . . . . . . . . . 8.2.1 Electromagnetic Covariant Derivative . . . . . . . . 8.2.2 Dirac–Coulomb Bound-State Wave Functions . . . 8.2.3 Dirac–Coulomb Continuum-State Wave Functions 8.2.4 Dirac–Coulomb Virial Theorem . . . . . . . . . . . 8.2.5 Dirac–Coulomb Propagator . . . . . . . . . . . . . . Dirac–Volkov Equation for Laser Fields . . . . . . . . . . . 8.3.1 Dirac–Volkov Solutions for Laser Fields . . . . . . 8.3.2 Dirac–Volkov Propagator . . . . . . . . . . . . . . . Dirac Equation with Coupling to Gravitational Fields . . . 8.4.1 Metric and Covariant Derivative . . . . . . . . . . . 8.4.2 Tetrad Basis and Affine Connection Matrix . . . . 8.4.3 Ricci Rotation Coefficients . . . . . . . . . . . . . . 8.4.4 Covariant Derivative of a Spinor . . . . . . . . . . . 8.4.5 Covariant Derivative of the Dirac Matrices . . . . .

265 . . . . . . . . . . . . . . . .

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265 265 265 266 274 279 281 284 284 289 293 293 297 300 301 304

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8.5

8.6 9.

8.4.6 Spin Connection . . . . . . . . . . . . . . . . . . . . 8.4.7 Transformation Properties of Rotation Coefficients Applications of Gravitational Coupling . . . . . . . . . . . . 8.5.1 Dirac–Schwarzschild Hamiltonian . . . . . . . . . . 8.5.2 Dirac Adjoint for Curved Space-Times . . . . . . . 8.5.3 Lagrangian and Charge Conjugation . . . . . . . . Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

305 306 308 308 310 311 313

Electromagnetic Field and Photon Propagators

321

9.1 9.2

321 323 323 325 328 328 331 334 334 335 340 342 342 343 344 346 346 350 356

9.3

9.4

9.5

9.6

9.7

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Orderings, Field Commutators and Green Functions . . . . . 9.2.1 Miscellaneous Fundamental Relations for Green Functions 9.2.2 Distributions and Fourier Transforms . . . . . . . . . . . . . Photon Propagator in Coulomb Gauge . . . . . . . . . . . . . . . . . 9.3.1 Legendre Transformation and Hamiltonian . . . . . . . . . 9.3.2 Matching and Photon Propagator . . . . . . . . . . . . . . . Photon Propagator in Lorenz Gauge . . . . . . . . . . . . . . . . . . 9.4.1 Gauge Invariance and Mass of Photon . . . . . . . . . . . . 9.4.2 Gauge-Fixing Term and Quantization . . . . . . . . . . . . . 9.4.3 Representations of the Photon Propagator . . . . . . . . . . Photon Propagator in General Gauges . . . . . . . . . . . . . . . . . 9.5.1 Construction Principle . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Most General Form and Weyl Gauge . . . . . . . . . . . . . 9.5.3 Gupta–Bleuler Condition . . . . . . . . . . . . . . . . . . . . Wick Theorem and Applications . . . . . . . . . . . . . . . . . . . . . 9.6.1 Time Ordering and Wick Theorem . . . . . . . . . . . . . . 9.6.2 Field Commutators and Current Distributions . . . . . . . Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10. Tree-Level and Loop Diagrams, and Renormalization 10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Tree-Level . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Rutherford Scattering . . . . . . . . . . . 10.2.2 Feynman Rules . . . . . . . . . . . . . . . 10.3 Vertex Correction . . . . . . . . . . . . . . . . . . 10.3.1 Vertex and Pauli–Villars Regularization 10.3.2 Vertex and Form Factors . . . . . . . . . 10.3.3 Vertex and Renormalization . . . . . . . 10.3.4 Detour on Dimensional Regularization . 10.3.5 Detour on Feynman Parameterization . 10.3.6 Vertex and Dimensional Regularization . 10.4 Vacuum Polarization . . . . . . . . . . . . . . . . .

359 . . . . . . . . . . . .

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359 361 361 367 370 370 379 383 390 392 394 400

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10.4.1 Initial Considerations . . . . . . . . . . . . . . . . . . . 10.4.2 Vacuum Polarization and Dimensional Regularization 10.4.3 Vacuum Polarization and Coulomb Potential . . . . . 10.4.4 Vacuum Polarization and Asymptotics . . . . . . . . . 10.5 Self–Energy Operator . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Self–Energy and Pauli–Villars Regularization . . . . . 10.5.2 Self–Energy and Dimensional Regularization . . . . . 10.6 Renormalization of QED . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Bare and Renormalized Lagrangian . . . . . . . . . . . 10.6.2 Renormalization of Vertex and Self–Energy . . . . . . 10.6.3 Renormalization of Vacuum Polarization . . . . . . . . 10.6.4 Compilation of Renormalization Constants . . . . . . 10.6.5 Forest Formula . . . . . . . . . . . . . . . . . . . . . . . 10.7 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

11. Foldy–Wouthuysen Transformation and Lamb Shift 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Leading Relativistic Corrections . . . . . . . . . . . . . . . . . . 11.2.1 Unitary Transformation and Hamiltonian . . . . . . . 11.2.2 Free Dirac Particle . . . . . . . . . . . . . . . . . . . . . 11.2.3 Transformation in the General Case . . . . . . . . . . . 11.2.4 Radiatively Corrected Dirac Hamiltonian . . . . . . . 11.2.5 General Electromagnetic Coupling . . . . . . . . . . . . 11.2.6 General Particle Hamiltonians . . . . . . . . . . . . . . 11.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Coulomb Field Coupling . . . . . . . . . . . . . . . . . . 11.3.2 Magnetic Field Coupling and Electron Moment . . . . 11.3.3 Magnetic Fields and Electrostatic Potentials . . . . . 11.3.4 Transition Current . . . . . . . . . . . . . . . . . . . . . 11.3.5 Nonrelativistic Transition Current and Multipoles . . 11.4 Dirac Form Factor and Bound-State Radiative Energy Shifts 11.4.1 Foldy–Wouthuysen Transformation and Form Factors 11.4.2 High-Energy Part in Photon Mass Regularization . . 11.4.3 High-Energy Part in Photon Energy Regularization . 11.4.4 Self–Energy in Dimensional Regularization . . . . . . 11.4.5 Vacuum Polarization . . . . . . . . . . . . . . . . . . . . 11.5 Foldy–Wouthuysen Transformation and Gravity . . . . . . . . 11.6 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Relativistic Interactions for Many-Particle and Compound Systems

400 402 406 409 411 411 413 413 413 414 416 419 420 422 425

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. . . . . . . . . . . . . . . . . . . . . .

425 426 426 427 429 429 432 434 436 436 439 440 442 447 448 448 449 453 454 456 457 458 461

12.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 12.2 Interatomic Interactions in Covariant Formalism . . . . . . . . . . . 462

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12.3

12.4

12.5

12.6

12.7

12.2.1 General Paradigm of the Matching . . . . . . . . . . 12.2.2 Scattering Amplitudes and Interatomic Interactions From the One-Body to the Two-Body Hamiltonian . . . . . 12.3.1 Relevant Hamiltonians . . . . . . . . . . . . . . . . . . 12.3.2 Matching the Two-Particle Interaction Hamiltonian 12.3.3 Photon Exchange and Breit Hamiltonian . . . . . . . 12.3.4 Breit Hamiltonian for Unequal Particles . . . . . . . Applications and Generalizations for Many-Particle Systems 12.4.1 Application to Many-Electron Systems . . . . . . . . 12.4.2 Single-Particle Hamiltonian for Arbitrary Spin . . . 12.4.3 Interaction Hamiltonian for General Spin . . . . . . Application to Two-Body Bound Systems . . . . . . . . . . . 12.5.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Definition of the Lamb Shift . . . . . . . . . . . . . . 12.5.3 Application to Hydrogenlike Ions . . . . . . . . . . . Compound System in a Homogeneous Magnetic Field . . . . 12.6.1 Power–Zienau Transformation for a Magnetic Field 12.6.2 Reduced-Mass Corrections to the Zeeman Effect . . 12.6.3 Nuclear Magnetic Shielding . . . . . . . . . . . . . . . Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

13. Fully Correlated Basis Sets and Helium 13.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Essential Ingredients for Many-Electron Systems . . . . . 13.2.1 Spin Wave Functions . . . . . . . . . . . . . . . . . 13.2.2 Two-Body Problem: Center-of-Mass Coordinates 13.2.3 Three-Body Problem: Mass Polarization . . . . . 13.3 Relativistic and Radiative Corrections . . . . . . . . . . . 13.3.1 Relativistic Corrections . . . . . . . . . . . . . . . 13.3.2 Derivation of the Helium Lamb Shift . . . . . . . 13.4 Numerical Calculation of the Helium Spectrum . . . . . . 13.4.1 Fully Correlated Basis Sets . . . . . . . . . . . . . 13.4.2 Matrix Elements and Nonorthogonal Basis . . . . 13.4.3 Numerical Results and Technical Issues . . . . . . 13.5 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . .

462 463 468 468 469 471 475 477 477 479 483 484 484 486 487 492 492 497 499 502 507

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. . . . . . . . . . . . .

14. Relativistic Many-Particle Calculations 14.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Many-Particle Systems and Hartree–Fock Approximation . 14.2.1 Nonrelativistic Hartree–Fock Approximation . . . 14.2.2 Implementation and Practical Aspects . . . . . . . 14.3 Relativistic Hartree–Fock Methods . . . . . . . . . . . . . .

507 507 507 510 511 513 513 515 520 520 523 525 529 531

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531 533 533 536 538

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14.3.1 Brown–Ravenhall Disease . . . . . . . . . . . . . . . . 14.3.2 Relativistic Frequency-Dependent Breit Interaction 14.3.3 Relativistic Hartree–Fock Approximation . . . . . . 14.3.4 Relativistic Multi–Configurational Approach . . . . 14.3.5 Relativistic Breit Interaction and Spinors . . . . . . 14.3.6 Subtleties of Nuclear Effects, QED and MCDHF . . 14.3.7 Software Packages and Applications . . . . . . . . . . 14.4 Systems with a Large Number of Electrons . . . . . . . . . . 14.5 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

15. Beyond Breit Hamiltonian and On-Shell Form Factors

557

15.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Furry Picture, Scattering Amplitude and Self–Energy . . . . 15.2.1 Furry Picture . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Matching for the Bound-State Self–Energy . . . . . . 15.3 Binding Correction to the Lamb Shift . . . . . . . . . . . . . 15.3.1 Two-Coulomb-Vertex Scattering Amplitude . . . . . 15.3.2 Forward Scattering Amplitudes . . . . . . . . . . . . 15.3.3 Dispersion Relation and Subtractions . . . . . . . . . 15.3.4 Example Integral . . . . . . . . . . . . . . . . . . . . . 15.3.5 Final Integration and A50 Coefficient . . . . . . . . . 15.4 Higher-Order Terms . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Relativistic Recoil Correction . . . . . . . . . . . . . . . . . . 15.5.1 Retardation and Salpeter Correction . . . . . . . . . 15.5.2 Low-Energy Part and Dipole Approximation . . . . 15.5.3 Middle-Energy Part and Araki–Sucher Distribution 15.5.4 Seagull Part . . . . . . . . . . . . . . . . . . . . . . . . 15.5.5 High-Energy Part . . . . . . . . . . . . . . . . . . . . . 15.5.6 Final Result . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . .

16. Bethe–Salpeter Equation 16.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 General Ideas . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Two-Body Bound States and Green Functions 16.2.2 Questions of Gauge and Transformations . . . . 16.2.3 Reference Kernels and Exact Solutions . . . . . 16.2.4 Bound-State Perturbation Theory . . . . . . . . 16.2.5 Energy Levels at O(Zα)4 (BSEQ Approach) . 16.3 Relativistic Recoil Correction (BSEQ Approach) . . . . 16.3.1 Exploring Integration Regions . . . . . . . . . . 16.3.2 Method of Regions . . . . . . . . . . . . . . . . .

538 539 542 543 545 547 548 551 554

557 558 558 559 561 561 564 566 568 571 572 575 575 577 579 584 587 590 590 593

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593 594 594 604 608 614 618 623 623 627

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16.3.3 Contribution from the Hard Region . . . . . . . . . . . . . 16.3.4 Soft Contribution from Two Transverse Photons . . . . . 16.3.5 Soft Contribution from a Single Transverse Photon . . . 16.3.6 Ultrasoft Contribution from a Single Transverse Photon 16.3.7 Sum of the Contributions . . . . . . . . . . . . . . . . . . . 16.4 Hyperfine Structure of S States (BSEQ Approach) . . . . . . . . 16.5 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

17. NRQED: An Effective Field Theory for Atomic Physics

657

17.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Basics of NRQED . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 NRQED Lagrangian and Feynman Rules . . . . . . . . 17.2.2 Matching Coefficients . . . . . . . . . . . . . . . . . . . 17.2.3 Bethe–Salpeter Equation of NRQED . . . . . . . . . . 17.3 Applications of NRQED . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Energy Levels at O(Zα)4 (NRQED Approach) . . . . 17.3.2 Relativistic Recoil Correction (NRQED Approach) . . 17.3.3 Hyperfine Structure of S States (NRQED Approach) 17.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18. Fermionic Determinants and Effective Lagrangians

657 659 659 662 671 677 677 681 688 688 689 691

18.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Derivation of the Heisenberg–Euler Effective Lagrangian . . . . . 18.2.1 Heisenberg–Euler Lagrangian for Electric Fields . . . . . 18.2.2 Heisenberg–Euler Lagrangian for General Fields . . . . . 18.3 Application of the Heisenberg–Euler Lagrangian . . . . . . . . . . 18.3.1 Modification of the Speed of Light in Background Fields 18.3.2 Effective Lagrangian and Wichmann–Kroll Potential . . . 18.4 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

19. Renormalization-Group Equations 19.1 Orientation and Motivation . . . . . . . . . . . . . . . . . . . . 19.1.1 Renormalization on Different Scales . . . . . . . . . . 19.1.2 Renormalization-Group Equations . . . . . . . . . . . 19.2 Scale Transformations . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 From Poor Man’s Scaling to Nontrivial Scaling . . . 19.2.2 Scale Transformation and Schr¨odinger Hamiltonian 19.2.3 Scale Transformation of the Dirac Hamiltonian . . . 19.3 Renormalization Group Transformations . . . . . . . . . . . . 19.3.1 Functional Equations in the Asymptotic Regime . . 19.3.2 Renormalization Group of Callan and Symanzik . .

631 636 637 641 647 648 654

691 692 692 705 707 707 711 714 715

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715 715 719 721 721 722 724 725 725 731

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19.3.3 Connection of Callan–Symanzik and Gell-Mann–Low 19.4 RG Equations and Applications . . . . . . . . . . . . . . . . . . 19.4.1 Effective Action and Renormalization Group . . . . . 19.4.2 Brodsky–Lepage–Mackenzie Scale Setting . . . . . . . 19.5 Further Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . .

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734 736 736 739 743

Bibliography

747

Index

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Chapter 1

Introduction

1.1

Accurate Numbers

Quantum electrodynamics (QED) is a theory known for the occurrence of long and accurate numbers. At present, QED still remains in a somewhat unique position within the standard model of fundamental interactions, in the sense that it is required to describe both high- and low-energy phenomena. So far, it is the only quantum field theory that has enabled us to infer fundamental constants to better than 8 digits of accuracy, and in the case of the Rydberg constant, 12 decimals (see Refs. [30, 31]). Among other notable achievements, there are three paradigmatic QED effects related to bound-state physics which have been measured to impressive accuracy. They can certainly be called cornerstones for the current belief that QED can describe both light and heavy bound systems (atoms and ions) better than any other theory. Furthermore, these effects have been used to determine various fundamental constants such as the Rydberg constant and the electron mass. The first of these are (i) measurements of atomic transition frequencies which include contributions from QED effects such as the Lamb shift. These transitions entail a change in the principal or angular momentum quantum numbers of the atomic electrons. The second paradigmatic effect is related to (ii) hyperfine-structure transitions which correspond to a spin-flip of the atomic nucleus. These transitions also receive corrections from QED. Finally, (iii) measurements of free-electron and bound-electron g factors have contributed in determining the fine-structure constant and the electron mass. A book on QED would be incomplete without listing some of the impressive experimental results, some of which may have come of age but are far from outdated. Indeed, they have an almost historic character and have advanced physics quite considerably. We should note that the experimental difficulties connected with precise measurements are often underestimated by theorists; the analysis of the systematic effects relevant to a particular apparatus can be full of traps. Likewise, the conceptual and practical difficulties in obtaining accurate theoretical predictions are often underestimated by experimentalists. 1

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In our necessarily incomplete list of QED-related results, we start with the hyperfine structure. The 1S hydrogen hyperfine structure has been measured in Ref. [32] with the result ∆E1S = 1 420 405 751.7667(10) Hz ,

(1.1)

whose uncertainty was, however, later amplified by a factor of three (see Ref. [33]). The value of ∆E1S = 1 420 405 751.768(1) Hz

(1.2)

has been obtained in Ref. [34] as a conservative average of various experimental investigations of comparable accuracy. For the 2S hydrogen hyperfine structure, the most recent experimental result comes from Ref. [35], ∆E2S = 177 556 860(15) Hz .

(1.3)

The latter is in a 1.4 σ disagreement with theory [35–37]. For the hyperfine splitting in muonium, a bound system of a positively charged muon and an electron (µ+ e− ), the experimental result is (see Ref. [38]) −

µe ∆E1S = 4 463 302 765(53) Hz .

(1.4)

Current theory benefits from the availability of nonperturbative self-energy calculations which include the hyperfine effects [39, 40]. The most recent theoretical result is [see Eq. (221) of Ref. [30]] −

µe ∆E1S = 4 463 302 891(272) Hz .

(1.5)

For positronium, the experimental result for the 1S hyperfine splitting interval reads [41–43] e e ∆E1S = 203 389.10(74) MHz . − +

(1.6)

The extreme accuracy of these results implies a certain challenge to theory which provides the result [44–47] e e ∆E1S = 203 391.69(16) MHz , − +

(1.7)

which differs from the experimental result by about three standard deviations. A recent experiment [48] e e ∆E1S = 203 394.2(20) MHz − +

(1.8)

is in better agreement with theory but less precise. We continue with gyromagnetic ratios, or g factors. According to Dirac theory, the g factor of the electron exactly is g = 2. However, taking into account the electron anomaly, which is due to QED effects, we can parameterize the g factor of the electron as g = 2 (1 + κ) ,

(1.9)

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Introduction

where κ is known as the electron anomaly. Throughout this book, we will use Eq. (1.9) for the g factor of a lepton, i.e., the symbol κ is understood to denote the g factor anomaly of the lepton unless otherwise stated or implied by the context. The equality of κ for electron and positron constitutes a test of charge-parity-timeinversion (CPT) symmetry. In an experimental tour-de-force, the following results have been obtained [49], κ = 1 159 652 188.4(4.3) × 10−12 κ = 1 159 652 187.9(4.3) × 10

−12

(electron) ,

(1.10a)

(positron) .

(1.10b)

Recently, these results have been improved [50–52] in a series of equally impressive experiments. The electron g factor is connected with Sommerfeld’s fine-structure constant α. The two leading terms (in the fine-structure constant α) of the theoretical expression relating κ to the quantum electrodynamic expression for the light fermion anomalous magnetic moment (of electron and positron) are given by [53,54] κ=

α 2 α − 0.328 478 444 ( ) + O(α3 ) . 2π π

(1.11)

Recent theoretical efforts have culminated in analytic four-loop calculations [55], as well as numerical five-loop evaluations (see Refs. [56–58] and Table 3 of Ref. [59]). The relation (1.11) constitutes the main input datum for the determination of the fine-structure constant α, which according to Refs. [60, 61] reads as α=

1 e2 ̵ = 137.035 999 084(21) , 4π0 hc

(1.12)

where e is the electron charge (the elementary charge is ∣e∣ = −e). Some of the most accurately measured transitions in simple atomic systems concern atomic hydrogen and deuterium, i.e., one-electron atoms. Indeed, the hyperfine centroid of the 1S–2S transition in hydrogen has been measured as [62–64] ∆E1S−2S = 2 466 061 413 187 035(10) Hz .

(1.13)

The accuracy is 4.2 parts in 1015 . It is experimentally more difficult to measure transitions to higher excited states. For the 1S–3S transition, a recent experimental value from the Paris group reads as [65] F =1→F =1 ∆E1S−3S = 2 922 742 936.729(13) MHz ,

∆E1S−3S = 2 922 743 278.678(13) MHz .

(1.14)

The second entry refers to the hyperfine centroid and is obtained from the first by a subtraction of hyperfine effects. The most recent measurement [66] at the Max–Planck Institute for Quantum Optics gives the result F =1→F =1 ∆E1S−3S = 2 922 742 936.711(17) MHz ,

∆E1S−3S = 2 922 743 278.659(17) MHz , where the latter is the hyperfine centroid.

(1.15)

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Theoretical predictions for other hydrogen transitions that have not yet been measured can be obtained from the web site [67]. For the hyperfine centroids of the 1S–3S and 2S–3S transitions, which are of particular spectroscopic interest, the theoretical predictions are th E1S−3S = 2 922 743 278.671 6(14) MHz , th E1S−3S

= 456 681 865.484 5(14) MHz .

(1.16) (1.17)

The theoretical predictions are the result of decade-long theoretical efforts. A comparison of observed hydrogen and deuterium transition frequencies to theoretical expressions can be used to determine the Rydberg constant R∞ . The most recent value of this constant [60, 61] reads as α2 me c2 = 10 973 731.568 160(21) m−1 . (1.18) 2hc The Rydberg constant is related to transition frequencies in a fictitious hydrogen atom with an infinitely heavy point nucleus and with relativistic effects “switched off”. The Rydberg constant is a wave number; it indicates the number of wavelengths (cycles) per meter. It can be converted to an angular wavenumber k (the ⃗ via multiplication by a factor 2π. The angular wave modulus of the wave vector k) number has units of radians per meter. The Rydberg constant can be converted to a frequency using the formula R∞ =

R∞ c = 3.289 841 960 355(19) × 1015 Hz .

(1.19)

Any book on atomic physics would be incomplete without also indicating the numerical value of the Bohr radius, which reads as [60, 61] ̵ h a0 = = 0.529 177 210 903(80) × 10−10 m . (1.20) αme c As described in Ref. [68], experimental input data from more than one transition are necessary in order to infer fundamental constants (e.g., the Rydberg constant) from measurements. The reason is that there is no one-to-one mapping of a transition to a particular physical constant but rather, adopting a quotation often ascribed to Oppenheimer,1 “understanding hydrogen is understanding all of physics.” Indeed, a “biography of hydrogen” in terms of its discovery potential for all of physics has been given in Ref. [69]. 1.2

Fundamental Constants

QED theory and experiment have often been used in order to constrain possible extensions of the standard model. Tiny conceivable variations of the fundamental forces from their commonly accepted nature would imply small variations in observable phenomena. Because QED effects have been measured very precisely, they can be used to constrain such models. 1 Julius

Robert Oppenheimer (1904–1967).

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One might ask about the foundations of the hypothesis that fundamental constants need to be “constant” through space and time; this hypothesis can ultimately be traced to the Einstein equivalence principle (EEP), which is based on three foundations. Namely, the Einstein equivalence principle affirms (i) the validity of the weak equivalence principle, (ii) local position invariance (LPI) and (iii) local Lorentz invariance (LLI). The weak equivalence principle states that free-fall at a given point in space-time is the same for all physical systems. Newton2 stated that the property of a physical object called “mass” (“inertial mass”) is proportional to the “weight” (which enters the gravitational force law), conjecturing that the two are proportional. LPI implies that the outcome of any local non-gravitational experiment (e.g., a frequency measurement) is independent of when and where in the universe it is performed, and LLI states that the outcome of any local nongravitational experiment is independent of the velocity of the free-falling reference frame in which it is being performed. A conceivable violation of any of these three axioms would induce a positiondependent or time-dependent shift of electromagnetically realized standards of units within the SI system. This could potentially induce a dependence on the location of a laboratory or on the time of measurement. Recently, it has been claimed [70] that a gravitational modification of the vacuum polarization tensor in strong gravitational fields (deep gravitational potentials) could lead to a modification of the speed of light near gravitational centers such as stars or black holes, which would violate the EEP. The mechanism proposed in Ref. [70] involves a gravitational interaction of the virtual electrons and positrons within the quantum electrodynamic “loop” that corrects the photon propagator (i.e., the gravitational correction to “vacuum polarization”). More colloquially speaking, the quantum fluctuations of the fermion field are in turn modified by the influence of the gravitational field onto the fermionic quantum fluctuations. However, in Ref. [71], the claim made in Ref. [70] could be refuted. A relativistically covariant evaluation of the vacuum polarization integral, corrected for gravitational effects, reveals that the speed of light is not altered locally in deep gravitational fields. Two caveats need to be mentioned: (i) The vacuum polarization tensor is modified in a nontrivial way for so-called off-shell photons, which are themselves virtual, as they mediate the Coulomb interaction. (ii) Globally, the speed of light may be altered in the vicinity of strong gravitational centers such as the Sun. Indeed, the Shapiro time delay describes a delay induced by space-time curvature for photons as they propagate through strong gravitational fields. This delay, however, does not affect the outcome of a locally performed experiment, while it has been confirmed with the Cassini spacecraft in superior conjunction on its way to Saturn [72], using Doppler tracking with both X-band (7175 MHz) as well as Ka-band (34 316 MHz) radar. The rather subtle clarification of the effect proposed in Ref. [70] and refuted in Ref. [71] reveals how easy it is to be mistaken in effects whose theoretical 2 Sir

Isaac Newton PRS (1642–1726).

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description requires a combination of quantum electrodynamics and general relativity. Very recently, tiny violations (limitations) of Einstein’s equivalence principle, based on quantum effects, have been discussed in Ref. [73]. An even more intriguing conjecture concerns the possible time variation of fundamental constants. In Ref. [74], Bekenstein3 gave a thorough analysis of the consequences of promoting the fine-structure constant to a scalar field. While the time variation of fundamental constants would entail a violation of the EEP, a number of theoretical models actually allow, or even predict, such variations. One example is given by so-called Kaluza4 –Klein5 theories, where the fine-structure “constant” emerges after an integration over a compactified, extra dimension, and is related to the vacuum expectation value of a scalar quantum field φ [see Eq. (37) of Ref. [75]]. String theories predict interrelations among the fundamental constants [see Eq. (42) of Ref. [75]], e.g., they predict that the fine-structure constant be proportional to the square root of the gravitational constant. Finally, the Higgs mechanism (see Chap. 12 of Ref. [2]) relates fundamental constants such as the masses of elementary particles to the vacuum expectation value of a background field known as the Higgs field, which could in principle change over space and time. In Ref. [74], it has been observed that a part of a charged particle’s mass is the electromagnetic self mass, which also depends on the fine-structure constant. In all of these theories, the fine-structure constant is far from being a fundamental, time-independent constant. An almost historic (from today’s point of view), but very conspicuous, theoretical basis for the variation of fundamental constants is given by Dirac’s large number hypothesis. Let us consider the ratio of the age T of the Universe and the time it takes light to travel a distance equal to the classical electron radius re . One finds that the ratio is of the same order-of-magnitude as the ratio of the electric and gravitational interactions of electrons and protons, re =

̵ αh , me c

T e2 ∼ 1040 ∼ . (re /c) 4π0 G me mp

(1.21)

Dirac conjectured [76] that the equality holds exactly, implying that the gravitational interaction constant G might be inversely proportional to the age T of the Universe. The task of spectroscopy and QED theory is to relate these conjectures to modelindependent, laboratory-based bounds on the temporal variation of constants. A rather illustrative application of QED theory and measurement, constraining various theoretical models, has been given in Ref. [63]. Input data from more than one measurement and more than one transition are needed in order to constrain the variation of fundamental constants. The hydrogen 1S–2S frequency measurement from Eq. (1.13) has been repeated in 1999 and 2003 [62, 63]. Another important input datum is given by the 199 Hg+ 5d10 6s 2 S1/2 (F = 0) ⇔ 5d 9 6s2 2 D5/2 (F ′ = 2) 3 Jacob

David Bekenstein (1947–2015). Franz Eduard Kaluza (1885–1954). 5 Oskar Benjamin Klein (1894–1977). 4 Theodor

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electric quadrupole (“clock”) transition frequency νHg , which was measured at the National Institute of Standards and Technology (NIST) between July 2000 and December 2002 (see Ref. [77]). Using theoretical considerations [63], one may show that the two measurements actually test different combinations of the nuclear magnetic moment of cesium µCs and of the fine-structure constant α. Specifically, in the NIST measurement [77], one investigates the temporal change in the frequency ratio νCs /νHg , which is related to the drift of µCs and α as follows, ∂ νCs ∂ µCs 2.0+0.8+3.2 ln = ln ( α ). ∂t νHg ∂t µB

(1.22)

Here, the quantity µB is an in principle arbitrary normalization which makes the argument of the logarithm dimensionless (the notation is of course inspired by the Bohr magneton). The basis for this assumption lies in the fact that a frequency “measurement” precisely corresponds to a comparison of two frequencies, that of the measured transition and that of the cesium reference standard. The latter in turn is proportional to the Bohr magneton that defines the frequency standard. The 1S–2S experiment [63] tests the drift of ∂ νCs ∂ µCs 2.0+0.8 ln = ln ( α ), ∂t νH ∂t µB

(1.23)

where the subscript H stands for hydrogen in contrast to Hg which denotes mercury. One may solve Eqs. (1.22) and (1.23) for the temporal variations of µCs and α. The results are (see Ref. [63]) ∂ ln α = (−0.9 ± 2.9) × 10−15 yr−1 , ∂t

(1.24)

∂ µCs ln = (0.6 ± 1.3) × 10−14 yr−1 . ∂t µB

(1.25)

and

The drifts are model-independent and consistent with zero. If more than two transitions are used as input data, linear regression must be used in order to solve for the drifts in a model-independent calculation. One may summarize that while some astrophysical observations are claimed to suggest a slight violation of the EEP in the form of time-evolving fundamental constants, no such effects have been detected in laboratory experiments. Recent laboratory experiments yield bounds on the temporal variation of the fine-structure constant (logarithmic time derivative α/α) ˙ smaller than 10−16 per year [78–80]. 1.3

Overview of Chapter Contents

In order to guide the reader in the study of this book, we briefly outline the contents of each chapter. Many phenomena which influence our daily lives are caused by the interaction of the quantized electromagnetic field with atoms. The description of

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atomic (quantum) processes requires a “new” language and a unit system tailored to the physical entities to be described; this is discussed in Chap. 2. One can describe a large number of important phenomena, starting from a nonrelativistic theory of the atom and a quantized theory of the electromagnetic field. These include the dynamic (“ac”) Stark effect and spontaneous atomic decay (see Chap. 3). The approach requires us to become familiar with time-dependent perturbation theory, which is a very useful concept beyond the realm of application discussed here. In many cases, the frequencies of the interacting light are so low that a nonrelativistic treatment on the level of the dipole approximation is fully sufficient. In second-order perturbation theory in the quantized interaction, one can already enter the field of bound-state quantum electrodynamics, on the basis of the calculation of the so-called Bethe6 logarithm (see Chap. 4). In order to proceed ¨ dinger7 –Coulomb8 Green funcwith the calculation, we shall obtain the Schro tion. We will do this by matching homogeneous solutions of the time-independent radial Schr¨ odinger–Coulomb wave equation, at the cusp r1 = r2 of equal radial arguments, using ideas inspired by the treatment of similar problems in classical electrodynamics [81]. Interatomic interactions and interactions of an atom with a dielectric surface require the use of the quantized–field formalism, as well as second-order and fourthorder perturbation theory. This approach is described in Chap. 5. The predominant symmetry that governs the physics of one-electron, and manyelectron, atoms is the rotational symmetry of the Coulomb potential. It has been observed that scalars and vectors, and tensors of second rank transform separately under rotations; this has led to the introduction of so-called Wigner matrices which describe the individual transformation of tensor components under rotations. Also, it has been observed that the tensorial addition of vectors leads to the formation of higher-order tensors; this naturally implies the introduction of vector-addition coefficients, which are otherwise known as Clebsch9 –Gordan10 coefficients. The Racah–Wigner algebra, which describes the formalism, is summarized in Chap. 6. Regarding a fully quantum, and relativistic, field theory of atoms, we start with the free Dirac11 equation in Chap. 7. This equation has originally been obtained as a linearization of the Klein–Gordon12 equation, using matrix algebra which implements a noncommutative multiplication operation. However, the Dirac equation soon turned out to be “more intelligent than its inventor” (this quote has been coined by Dirac himself). Indeed, the Dirac equation was the first equation to describe an “inner” degree of freedom of a particle, in this case, the spin degree of 6 Hans

Albrecht Bethe (1906–2005). Rudolf Josef Alexander Schr¨ odinger (1887–1961). 8 Charles-Augustin de Coulomb (1736–1806). 9 Rudolf Friedrich Alfred Clebsch (1833–1872). 10 Paul Albert Gordan (1837–1912). 11 Paul Adrien Maurice Dirac OM FRS (1902–1984). 12 Walter Gordon (1893–1939). 7 Erwin

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freedom [82,83]. We recall that the electron spin was discovered by Goudsmit13 and Uhlenbeck14 in 1925, in two rather experimentally inspired theoretical papers [84, 85]. A brief, but informative account of the history of the concept of “spin” has been recently given in Ref. [86]. The inclusion of the spin degree of freedom leads to more complicated angular algebra, because the spin needs to be added to the orbital angular momentum. Also, in Chap. 7, we take a look at the quantized Dirac field, and calculate the propagator for a free Dirac particle. The case of an interacting Dirac particle is discussed in Chap. 8. Three situations are of particular significance. First, we treat the Dirac–Coulomb problem, which describes a particle bound to the Coulomb field of a nucleus. The wave function solutions and the propagator are discussed. Second, we consider the Dirac–Volkov problem, which describes a relativistic electron under the influence of a strong laser field which is described as a classical background field. Here, again, the wave function solutions and the propagator are treated. The third problem considered in Chap. 8 is the Dirac equation in curved space-time, for which we present a (hopefully) easily digestible derivation. Up to this point in the current book, we still have not covered, to a full extent, the quantum formalism for the electromagnetic and fermion fields, in fully relativistic form. In Chap. 9, we start this program with the quantized electromagnetic field and discuss field commutators and field propagators. The subject matter goes beyond the contents of Chap. 2, where we only discuss the field operators in the Schr¨ odinger and interaction pictures, but not their time-ordered products, which lead to the propagators. The photon propagator is treated in both Coulomb as well as Lorenz gauges. In Coulomb gauge, the instantaneous Coulomb interaction has to be added (in non-quantized form) to the quantized interaction mediated by the spatial degrees of the freedom of the vector potential operator. Finally, we also consider the generation of photons by a classical, external field configuration, where the photon field is quantized, but the current distribution is kept as a classical entity. In Chap. 10, we catch up on the basic quantum electrodynamic processes, profiting from our exposure to the quantized electromagnetic field (in Chaps. 2 and 9), as well as the quantized fermion field (in Chap. 7). We consider tree-level diagrams involving (free) electron propagators, and loop diagrams (vertex correction and vacuum polarization). The reader is exposed, among other aspects, to the ideas of dimensional regularization and on-shell renormalization. Finally, the forest formula is introduced, which allows one to renormalized nested divergences. The considerations described in Chap. 10 are primarily focused on particle-physics techniques, and therefore are not specific to bound states. They do not require the rather involved angular integrals over spin-angular, and related functions which naturally occur in nonperturbative bound-state calculations. The latter take the Coulomb interaction into account to all orders. 13 Samuel 14 George

Goudsmit (1902–1978). Eugene Uhlenbeck (1900–1988).

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In Chap. 11, we engage in the so-called “Foldy–Wouthuysen program” of identifying the nonrelativistic expansion (the “leading terms” and the “relativistic corrections”) relevant to the description of bound states. In particular, the Foldy– Wouthuysen transformation allows us to separate the particle from the antiparticle degrees of freedom. The approach is based on the observation that the average speed of an electron in a Bohr orbit in a light atom is of the order of v ∼ Zα c, where Z is the nuclear charge number, and α is the fine-structure constant. For not too large Z, one thus has the relation v ≪ c. The binding parameter Zα describes the strength of the coupling of the bound electron to the nucleus. It is thus possible to expand scattering amplitudes, transition matrix elements, and energy shifts in the powers of the dimensionless parameter Zα. In Chap. 11, we begin the related considerations by an expansion about the nonrelativistic limit of the Dirac equation via the Foldy–Wouthuysen transformation for a single particle. In the treatment of the Lamb shift, which also is carried out in Chap. 11, we augment the discussion by considering the effective Dirac equation, corrected by radiative effects previously discussed in Chap. 10 (form factors of the electron). A serious problem arises: The expansion in Zα eventually turns out to be nonanalytic, i.e., it gives rise to powers of Zα and logarithms ln(Zα). We have to approach the formalism at the heart of the Zα-expansion carefully. In Chap. 12, we continue the “Foldy–Wouthuysen program” and use the results obtained in Chap. 11, in particular, for the relativistic corrections to the transition current, in order to calculate the two-particle Breit15 Hamiltonian. This Hamiltonian can be used for many-particle systems and describes the interaction of two charged particles in the non-retardation limit, i.e., in the limit of a static interaction (where the frequency of the exchanged photon vanishes). Corrections of relative order (Zα)2 to the leading Coulomb interaction can be described by the Breit Hamiltonian. A further issue also is given attention in Chap. 12: Namely, when external fields interact with atomic systems, it is necessary to separate the internal degrees of freedom from the external ones, and to identify the generalized correction terms. This poses additional difficulties for many-particle systems, described in Chap. 12. In Chap. 13, the theory of fully correlated basis sets and helium is described. It is well known that the eigenfunctions of many-electron systems cannot be written in closed analytic form. One may try to reduce the problem to the calculation of self-consistent orbitals, which describe the probability density associated with individual electron orbits. However, in this case, one factorizes the dependence on the coordinates of the two electrons and loses the possibility to describe the full correlation among the electrons. If the quantum Hamiltonian cannot be diagonalized in closed analytic form, then one must resort to approximation methods. A fully correlated basis of trial functions, which contain the interelectron distance r12 = ∣⃗ r1 − r⃗2 ∣, leads to a satisfactory answer. In choosing the trial functions (“basis functions”), 15 Gregory

Breit, transcribed from Russian: Grigory Alfredovich Breit-Shneider (1899–1981).

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one must exercise extreme caution. In general, one has to trade suitability in regard to the modeling of the electron correlation against analytic tractability of the basis integrals. The approach has meanwhile been applied with very high accuracy and can be generalized to many-electron systems. It is described in Chap. 13. In Chap. 14, we shall make a certain detour and consider relativistic bound systems with high nuclear charge numbers Z, with an emphasis on the manyelectron case. The binding parameter Zα is treated to all orders in the “nonperturbative” bound-state calculations. Also, at high nuclear charge numbers, the preferred method for the treatment of the electron correlation changes from the socalled LS coupling to jj coupling. The electron correlation term (electron-electron repulsion), which couples the orbital angular momenta, is of order Zα2 mc2 , where m is the electron mass and c is the speed of light. The spin-orbit coupling term for each electron is of order (Zα)4 mc2 , while the spin-spin coupling among electrons is of order α (Zα)3 mc2 . The electron correlation term couples the orbital angular momenta of the electrons together, and eventually leads to LS coupling, while the spin-orbit term couples the orbital angular momentum ` of each electron to the spin s, as summarized in the total angular momentum j. The transition occurs at Z ≈ (1/α)2/3 = 26.58, that is, between iron (Fe, Z = 26) and cobalt (Co, Z = 27) in the periodic table of elements. While a comprehensive discussion of all intricacies of the many-electron problem at high nuclear charge would go beyond the scope of this book, we include a brief survey of existing codes for the treatment of manyelectron systems in Chap. 14. Furthermore, we focus on the angular integrals which are useful for the treatment of photon exchange with tightly bound electrons, where the relativistic corrections can no longer be ignored. In order to go beyond the Breit approximation, one has to include retardation, i.e., one cannot ignore the frequency of the exchanged photon any longer. This is described in Chap. 15, where we carry out example calculations of higher-order effective operators. This includes those related to the α (Zα)5 correction to the Lamb shift. We also investigate the recoil correction (Salpeter16 correction) beyond the Breit terms. This approach allows us to better identify the terms which contribute to a particular energy shift in a given order in the Zα-expansion. A discussion of the intricacies of nonperturbative (in the binding parameter Zα) bound-state calculations, for quantum electrodynamic loop corrections, also is included in Chap. 15. This includes a brief glance at the Furry picture. Any book on quantum electrodynamics would be incomplete without an overview of the Bethe–Salpeter equation, which determines the bound-state energies of two-particle systems and allows one, in principle, to take all quantum electrodynamic loop and recoil corrections into account to all orders. Furthermore, as we see in Chap. 16, the Bethe–Salpeter equation actually can be characterized as the generalization of the Lippmann–Schwinger equation for one-particle states, to two constituent particles. Essentially, the self-energy operator in the Lippmann– 16 Edwin

Ernest Salpeter (1924–2008).

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Schwinger equation is replaced by a two-particle interaction kernel. The theory is based on a sixteen-component, two-particle wave function. The difficulties encountered in a systematic analysis of the Bethe–Salpeter equation are tremendous, and we illustrate its use on the basis of an example calculation, namely, a reevaluation of the Salpeter correction, where we put special emphasis on the need to sum certain classes of diagrams in the low-energy domain, in order to obtain a consistent result. Furthermore, in comparison to our treatment in Chap. 15, we investigate the gauge dependence of the correction, as well as the use of dimensional regularization in its evaluation. A discussion of the leading recoil correction to hyperfine splitting in muonium completes Chap. 16. The latter correction could be characterized as the nuclear-spin-dependent part of the recoil correction. The development of nonrelativistic quantum electrodynamics (NRQED) has been inspired by modern techniques developed within effective field theory. In Chap. 17, we give a corresponding overview. Roughly speaking, one uses the concept that upon a Foldy–Wouthuysen transformation, one can actually separate particle from antiparticle terms in the theory, and promotes this idea to a field theory of its own. That is, one initially pretends that one does not know about the existence of fully relativistic quantum electrodynamics at all. Instead, one tries to write down a theory, i.e., a Lagrangian, in which one uses nonrelativistic fermionic field operators (in particular, one uses different field operators for electrons and positrons). Additional operators are used for the description of the electromagnetic field. The latter are, preferably, written down in Coulomb gauge. Because the theory is inherently nonrelativistic, one also obtains nonrelativistic expressions for the loop corrections. An example is the electron self-energy, for which we have examined the nonrelativistic setting in Chap. 4. The question then is how the high-energy part of the Lamb shift, discussed in Chap. 11, could be incorporated into the framework of NRQED. The answer is based on the observation that the exchange of high-energy photons promotes the virtual electron to a high-energy state, i.e., into a kinematic region which is detached from the applicability of a purely nonrelativistic theory. This implies that it is possible to formulate the effect of the high-energy photons in terms of effective operators, to be added to the NRQED Lagrangian. These operators are not required to lead to renormalizable interactions; the theory is only required to reproduce the result of the full QED theory up to a given order in the αexpansion. Even four-fermion vertices [see Fig. 17.4(k)] are allowed, as we illustrate in a treatment of the Salpeter correction based on NRQED, which supplements the treatments presented in Chaps. 15 and 16. The evaluation of the effective operators by a matching calculation, and the general applicability of these operators for any desired process, are cornerstones of the effective-field-theoretical approach. In Chap. 18, we follow our field-theoretical ambitions and derive the effective Lagrangian of quantum electrodynamics that extends the Maxwell Lagrangian to higher orders in the electromagnetic fields. The derivation augments and contrasts a number of concepts introduced earlier. In contrast to the treatment of photon

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emission by a given classical current distribution (see Chap. 9), we now consider classical electromagnetic fields, but quantize the fermion field. In the sense of Chap. 17, the so-called “QED effective Lagrangian” derived by Heisenberg17 and Euler18 (see Refs. [87, 88]), and Schwinger19 (see Ref. [89]) qualifies itself as an effective operator derived by integrating out the high-energy fermionic degrees of freedom. Very much in line with the concept of NRQED, the fourth-order terms (in the electric and magnetic fields) lead to four-photon (four-boson) vertices which would otherwise constitute a nonrenormalizable interaction. The general applicability of the QED effective action is derived from the fact that one may insert any combination of either stationary or oscillating fields, and make physical predictions. E.g., one derives the correction to the light propagation speed in a strong electromagnetic background field due to virtual excitations of the fermion fields. This constitutes a paradigmatic application of the QED effective action to oscillatory fields. A different application of the QED effective action, this time to a static Coulomb field, complements the discussion: It is given by the derivation of the leading term (for large distances) of the three-Coulomb-vertex correction to the electrostatic interaction of the bound electron and the atomic nucleus, otherwise known as the Wichmann20 –Kroll21 correction [90, 91]. In Chap. 19, we discuss the renormalization group and its relation to quantum electrodynamics. From the discussion of vacuum polarization, we know that the effective strength of the quantum electrodynamic coupling constant grows at small distances, or large momentum scales. The coupling “runs” with the distance, or momentum, scale at which phenomena are observed. The decisive consideration leading to the renormalization group in quantum electrodynamics is as follows. In the limit of small distances, the rate of change of the coupling constant (the finestructure constant) with the momentum scale is independent of the mass of the electron. Instead, it depends only on the value of the coupling at the distance scale in question. A specific functional relation fulfilled by the “running coupling” leads to the introduction of the so-called QED β function which describes the running of the charge with the momentum scale. This leads to the renormalization group (RG) of Gell-Mann22 and Low.23 The RG program is then applied to a number of phenomena of physical interest such as the RG flow of the QED effective action, and the choice of an “optimal” value of the coupling constant for the description of a specific process [so-called Brodsky24 –Lepage25 –Mackenzie26 (BLM) scale 17 Werner

Karl Heisenberg (1901–1976). Heinrich Euler (1909–1941). 19 Julian Seymour Schwinger (1918–1994). 20 Eyvind Hugo Wichmann (1928–2019). 21 Norman Myles Kroll (1922–2004). 22 Murray Gell-Mann (1929–2019). 23 Francis Eugene Low (1921–2007). 24 Stanley Jerome Brodsky (b. 1940). 25 Gerard Peter Lepage (b. 1952). 26 Paul Blanchard Mackenzie (b. 1950). 18 Hans

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setting [92]]. Finally, we investigate possible corollaries from RG transformations, relevant to the high-energy limit of QED. 1.4

Miscellaneous Remarks

One normally does not include remarks on the notation used in a book in the Preface. However, in QED calculations, due to the elevated level of complexity of calculations, one needs to use many symbols, some of which are “overloaded” with different meanings, depending on the context. In order to alleviate at least some conceivable ambiguities, we have used a slightly different font in some cases. For example, we use µ for the reduced mass of a two-body bound system, and µ for the renormalization scale (see Chap. 10). Despite our efforts at disambiguation, µ also appears as the subscript denoting the magnetic projection in the spin-angular function χκ µ (ˆ r). This should not lead to confusion due to the different context in which the symbol appears. Throughout the treatise, κ is the Dirac angular quantum number, defined in Chap. 6, while we denote the anomalous contribution κ to the electron g factor as g = 2(1 + κ). We also note that the notation  is used for the overlapping parameter that separates the low-energy from the high-energy scale in Lamb shift calculations (see Sec. 4.5.2), while we use  for the infinitesimal imaginary part in propagators. Finally, the notation d = 4 − 2ε is used for the non-integer space-time dimension in dimensional regularization (see Chap. 10). 1.5

Further Thoughts

Here are some suggestions for further thought. (1) Covariances. Study some of the recent CODATA papers [30, 60, 61, 93]. They use the method of covariances for the adjustment of fundamental physical constants and combine the results of a number of high-precision experiments in order to determine, in some cases, just one single physical constant. What is a covariance? How does it enter the adjustment of the fundamental constants? (2) Fine-Structure Constant. Read up on different determination of the finestructure constant α, e.g., using photon recoil (see Refs. [94–96]). In particular, convince yourself that the recoil energy ER of the atom upon absorption/ emission of a photon of wave vector k by an atom of mass M is given as ̵ 2 (hk) , (1.26) ER = 2M so that a photon recoil measurement eventually leads to a determination of ̵ h/M . The fine-structure constant α can eventually be determined as α = [2

̵ 1/2 R∞ u M h × × × ] c me u M

where u is the atomic mass unit and R∞ is the Rydberg constant.

(1.27)

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Investigate the following questions: ● How is the Rydberg constant being determined? Study the paper [97] as well as Refs. [30, 60, 93] in order to elucidate the connection of highprecision QED calculations of hydrogen and deuterium energy levels, and the determination of the Rydberg constant, as well as the proton and deuteron radii. ● The primary “other” method for determining the fine-structure constant is based on free-electron g factor measurements [49, 52]. From the boundelectron g factor, using the value of α determined from the free-electron g factor, one infers u/me (see Refs. [60, 98]). So, the value of α determined from the free-electron g factor also influences u/me , and in consequence, the determination of α from photon recoil [see Eq. (1.27)] is not completely independent of the determination of α from the free-electron g factor. Do an approximate calculation to determine if the uncertainty in α, from the current free-electron g factor measurements, could influence the determination of α from the recoil measurement, in an appreciable fashion.

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Chapter 2

From Unit Systems for the Microworld to Field Quantization

2.1

Overview

On the journey toward the microworld described by quantum electrodynamics, we shall meet matter waves in the Hamilton–Jacobi formalism in Sec. 2.2. This episode illustrates the transition from classical mechanics to quantum mechanics. The transition to unit systems pertinent to the microworld is illustrated in Sec. 2.3. We emphasize the distinctions between the natural and the atomic unit systems. Field quantization is discussed in Sec. 2.4, with an emphasis on the electromagnetic field, in the Coulomb gauge. At the end of the chapter, we have the main ingredients at our disposal, in regard to the description of the interaction of atoms with the quantized radiation field. 2.2 2.2.1

Atoms and Field Quantization Matter Waves and Hamilton–Jacobi Formalism

Atoms are described by quantum mechanics. The Schr¨odinger equation describes the propagation (in space and time) of matter waves which correspond to the wave functions of the electrons. Let us consider the transition from the classical theory of an electron orbiting a proton (two oppositely charged objects attracting each other via the Coulomb force) to the quantum mechanical description. A priori, one might think that this transition is equivalent to the transition from ray optics, where light is considered to be a classical “ray”, to wave optics, where light is described by electromagnetic waves. If that analogy were perfect, then quantum mechanics would still be deterministic: Matter waves, which describe the propagation of matter, would correspond to the electromagnetic waves whose propagation is described by the Maxwell equations; matter wave propagation would be deterministic in much the same way as the propagation of electromagnetic waves is deterministic. If we knew the electron’s wave function at time t0 , then we would precisely know what its wave function will be in the future, no matter what interactions the electron went through on its course. What is so fundamentally new about quantum mechanics is the postulate of the 17

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collapse of the wave function upon measurement. This postulate states that after a measurement of a quantum system, the system is in an eigenstate of the quantum mechanical operator that describes the measured physical quantity. This postulate, which incorporates the role of the observer in the outcome of the measurement, implies that after the measurement, the evolution of the wave function of the electron, starts “from scratch” and there is no way to predict the future time development of the system before the measurement was taken. Indeed, all we can say about a quantum mechanical measurement is to give a probability for its outcome; the theory is fundamentally nondeterministic. We investigate the Schr¨ odinger Hamiltonian describing an electron, ̵2 h p⃗ 2 ⃗ 2 + V (⃗ + V (⃗ r) = − ∇ r) , (2.1) H= 2m 2m ̵∇ ⃗ is the momentum operator and V (⃗ where p⃗ = −ih r) is the potential. This equation can be easily motivated: The paradigmatic example of a wave describing a particle is a plane wave of the form exp(i k⃗ ⋅ r⃗), where k⃗ is the wave vector and r⃗ is the ̵ k, ⃗ spatial coordinate. It is an eigenfunction of the operator p⃗ with eigenvalue h 1 which according to de Broglie, describes the relation of the wavelength of the matter wave to the momentum. Furthermore, it is clear that the potential V (⃗ r) should enter the Hamiltonian. It is rather surprising to realize that in comparison to what would be expected of the “wave fronts” (planes of constant classical action) derived from Hamiltonian– Jacobi theory, the Schr¨ odinger equation adds a dissipative term to the propagation of a quantum matter wave. This can be seen as follows. We start from the classical action S0 for a classical particle, L=T −V ,

T=

p⃗ 2 1 ˙ 2 = m r⃗ , 2m 2

S0 = ∫

t 0

dt′ L(⃗ r(t′ ), r⃗˙ (t′ )) ,

(2.2)

where r⃗˙ (t′ ) = d⃗ r(t′ )/dt′ . Because S0 is obtained by integrating the trajectory of the particle from time zero to time t, S0 actually is a function of t and of the endpoint of the trajectory, r⃗(t), and so S0 = S0 (t, r⃗(t)). We find dS0 ∂S0 ∂ r⃗ ∂S0 dS0 ⃗ 0⋅ = L, = + ∇S = + p⃗ ⋅ r⃗˙ , dt dt ∂t ∂t ∂t where the Euler–Lagrange equation ∂L d ∂L = ∂ r⃗(t) dt ∂ r⃗˙ (t)

(2.3)

(2.4)

has been used. Using r⃗˙ = p⃗/m, this can be rewritten as ⃗ 0 )2 ∂S0 p⃗ 2 (∇S = (⃗ p ⋅ r⃗˙ − L) = H = +V = + V (⃗ r) . (2.5) ∂t 2m 2m ⃗ 0 is necessarily perpendicular to the surfaces of constant The gradient vector p⃗ = ∇S classical action S0 . Indeed, the whole point of the Hamilton–Jacobi formalism is −

1 Louis-Victor-Pierre

Ramond, 7` eme duc de Broglie (1892–1987).

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that the classically allowed trajectories are perpendicular to the surfaces of constant action [99]. Likewise, wave fronts of a traveling wave are characterized by a constant phase and move perpendicular to the surfaces where the phase is constant. This suggests the ansatz i (2.6) ψ0 (t, r⃗) = exp ( ̵ S0 (t, r⃗)) h for the matter wave function. This equation for the wave function ψ0 (t, r⃗) would hold if the dynamics of the matter wave were purely governed by the laws of classical mechanics. Let us now formulate the true quantum mechanical wave function ψ(t, r⃗) in terms of an exponential, i ψ(t, r⃗) = exp ( ̵ S(t, r⃗)) , (2.7) h where we drop the subscript zero on both ψ and S. Inserting this ansatz into the Schr¨ odinger equation i ∂t ψ = H ψ, one gets ̵ ⃗ 2 ∂S (∇S) ih ⃗2 S . ∇ (2.8) − = + V (⃗ r) − ∂t 2m 2m This has to be compared with Eq. (2.5), ⃗ 0 )2 ∂S0 (∇S = + V (⃗ r) . (2.9) ∂t 2m The third term on the right-hand side of Eq. (2.8) characterizes a diffusive process in the context of the Brownian motion of the quantum particle [100] and describes the spreading of quantum mechanical wave packets. It may be interpreted as a quantum mechanical correction to the classical equations of motion for matter waves. The ̵ Indeed, the expansion in powers of h ̵ is known as the correction is of order h. semiclassical or Wentzel–Kramers–Brioullin (WKB) expansion [101]. −

2.2.2

Field Quantization as Second Quantization

We have just discussed the transition from classical mechanics to wave mechanics. This is known as first quantization because a fundamentally classical theory (without waves) has been quantized and formulated as a quantum theory (which describes matter waves). Classical electrodynamics describes electromagnetic waves and one might thus consider it to be “first-quantized” in the sense that any wave theory contains aspects otherwise reserved for wave mechanics. The next natural question to ask concerns the nature of second quantization and the quantization of a theory that already describes waves. One may argue that, for the electromagnetic field, first quantization is unnecessary because the field itself is already a spatially distributed quantity which is not necessarily localized at one and only one place, as it would be for a point particle in classical mechanics. In second quantization, one identifies fundamental excitations of the electromagnetic field as “photons” which can be created and annihilated. The wave functions describing the fundamental excitations are multiplied with creation and annihilation operators.

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The same procedure can also be applied to the matter waves which describe electrons. This eventually leads to the “second-quantized” description of the fermion field. Here, we shall first be concerned with the quantized electromagnetic field and describe the atom on the level of first quantization before we finally go on to discuss the quantized fermion field in Chap. 7. The reason is as follows. Typically, one has to quantize a theory when the fundamental energy scale or de Broglie wavelength of a physical process is commensurate with the fundamental excitations of the quantum theory. For the quantized theory of the fermion field, this energy scale is the electron mass scale me c2 ≈ 0.511 MeV, which corresponds to a wavelength of the order of the electron Compton wavelength λe = 2π ¯λe =

h = 2.426 × 10−12 m . me c

(2.10)

The Bohr radius of a typical atom is a0 = 0.529 × 10−10 m = 0.529 ˚ A, where we recall the unit symbol ˚ A for the Angstrom. The Bohr radius is larger than the electron Compton wavelength λe by a factor (2πα)−1 = 21.81, where we recall the value given in Eq. (1.12) for the fine-structure constant. The physics relevant to the quantized fermion field happens at length scales which are a lot smaller than the typical dimension of an atom. In a first approximation, we do not need to quantize the fermion field if we want to describe the interaction of the atom with an external electromagnetic field. In higher orders of approximation, though, one cannot avoid describing the physics that happens at the scale of the electron Compton wavelength; this leads to vacuum polarization and other effects. The fundamental excitations of the electromagnetic field (the photons) can have an arbitrarily small energy, or, expressed differently, an arbitrarily long wavelength. One might think that a quantum description of any electromagnetic interaction is indispensable on all energy scales. This is, however, not the case. The decisive parameter here is the strength of the electromagnetic field. Photons are bosons; they enjoy occupying one and the same quantum state collectively, and in great numbers. One photon more (or less) occupying an already well-populated state (e.g., a laser mode) will not make a large difference. A strong laser field can be described as a classical electromagnetic wave up to an excellent approximation. One has to quantize the electromagnetic field if the number of excitations (the number of photons) is small. In the case of the Casimir2 –Polder3 interaction between macroscopic plates, the geometry of the apparatus imposes boundary conditions that change the shape of all of the photon wave functions collectively. The modification of the zero-point energy of the photon modes then leads to a measurable effect even if no real, and no virtual, photons are present (see Chap. 6 of Ref. [81]). The self-energy of a bound electron involves at least one virtual photon. The correct result for this effect [81] can only be obtained if one quantizes the electromagnetic field. A third example combines both mentioned criteria: An 2 Hendrik 3 Dirk

Brugt Gerhard Casimir (1909–2000). Polder (1919–2001).

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atom tries to emit a single photon, making a quantum jump, and is either hindered or assisted by the specific geometry of the apparatus, e.g., a perfectly conducting cavity. In this case, the quantum description is required for two reasons: (i) because a description of the spontaneous emission of a photon by the atom would be impossible without taking the vacuum fluctuations of the electromagnetic field into account, and (ii) because the modification of the electromagnetic field operators (modification of the photon modes) by the boundary conditions could not be described without quantization. In the formalism to be developed in the following, we shall introduce the funda⃗ and a+ (k), ⃗ which act as particle mental annihilation and creation operators aλ (k) λ creation and annihilation operators for photons of polarization λ and wave vec⃗ They are multiplying the photon wave functions, which are proportional to tor k. exp[−i (ω t − k⃗ ⋅ r⃗)] and represent the wave nature of light. The specific form of the photon field operator is a mathematical manifestation of the wave-particle duality. 2.3 2.3.1

Unit Systems Scaled to the Microworld Unit Systems and Observation Scales

The crossroads of atomic physics and field theory share a number of intricacies and peculiarities. Among these, we notice that three different systems of physical units are currently in use in electrodynamics and in bound-state calculations. All three of these have a raison d’ˆetre; they are specially suited for a particular regime of applications. The SI mksA unit system (commonly referred to as the SI or the SI mksA system) is the basic unit system accepted worldwide for the description of physical processes and recommended by CODATA (Committee on Data for Science and Technology). Here, SI stands for Syst`eme International, while mksA denotes the units of meter, kilogram, second, and Ampere. The distinctive advantage of this unit system is that all physical quantities can immediately be related to physical units as they are read off from ordinary gauged instruments. There is no need for any unit conversion, or for supplementing any prefactors at the end of a calculation. The disadvantage is that a number of formulas become more lengthy than necessary, because of the appearance of additional prefactors. The natural unit system, derived from the Heaviside–Lorentz system, is defined ̵ = h/(2π), so that the numerical values of the reduced Planck constant, denoted as h the speed of light c and the vacuum permittivity 0 attain values of unity, i.e., ̵ = c = 0 = 1. One might ask how this is possible because the Coulomb law, h F = q1 q2 /(4π0 r2 ), relates the force F between two charges q1 and q2 and their mutual distance r. Moreover, q1 and q2 are measured in units of charge (Coulomb in the SI mksA system), whereas r is measured in meters. So, in order for us to be able to set 0 = 1, the units of charge, length and force must be related. This seems counterintuitive since charge is fundamentally different from length and force — but is it? The rationale behind setting 0 = 1 is that the units of length, force, and

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charge can be related if we understand that force is generated by a configuration of two charges at a distance r from each other — as expressed by the Coulomb law. Likewise, we can measure length in terms of the distance traveled by time in a second. If we define our unit of length to be equal to the distance traveled by time in a second, then the speed of light c in our new unit system is one unit of length, divided by one unit of time, and the numerical value of c (the reduced quantity) multiplying this ratio is just one. We now remember that a relation like F = q1 q2 /(4π0 r2 ) allows us to calculate F using a computer that knows nothing about physical units. This is because if q1 and q2 are measured in Coulombs, then 0 furnishes the conversion factor so that F comes out in units of kilogram times meter per second squared. If our base units are chosen so that the numerical value of 0 becomes unity, then we might just as well pretend that we think like a computer, and calculate using the numerical values (the reduced quantities) corresponding to the physical quantities only, dropping the units altogether. Thus, we set c = 1, as ̵ = 0 = 1. Near the end of a calculation, it is the quantity calculated that well as h ̵ c, and 0 that we have neglected tells us how to resupply the missing factors of h, so far. In the natural unit system, one has e2 = 4πα. Because a factor e2 is supplied in every order of QED scattering theory (loop calculations), the natural unit system is well adapted to high-energy processes. In this system, for bound-state theory, the basic unit of energy is m c2 = m, where m is the electron mass. Using the same unit system for bound-state theory as for high-energy QED scattering theory, one obtains uniformity with respect to the prefactors and the general formulas for Feynman diagram calculations. The natural unit system also is very well adapted for identifying the terms in bound-state theory that contribute at a particular order in the fine-structure constant α to a particular atomic property such as the energy of a particular atomic level. Specifically, high-energy virtual photons have an energy ∼ m, the Schr¨ odinger energy and the interelectron interaction are of order α2 m, and relativistic corrections are of order α4 m. Genuine QED energy corrections originating from loop diagrams start at order α5 m in electronic bound systems. In general, if one enters QED bound-state theory from the particle physics side, then the natural unit system appears to be the natural choice for calculations. The atomic unit system is different; it originates from the days when the properties of atoms were being explored by physicists, and the most basic unit of length was naturally identified as the Bohr radius of an atom. In atomic units, all physical quantities (energy, length, velocity) are expressed so that they assume values of order unity for the motion of the electrons in typical atomic systems. For example, the basic unit of energy is known as the Hartree4 energy or simply the “Hartree”, being equal to Eh = α2 m c2 . The ionization potential of hydrogen (the negative of the ground-state binding energy) is equal to one half in atomic units (measured in units of Hartrees). The atomic unit system is very useful for doing calculations 4 Douglas

Rayner Hartree FRS (1897–1956).

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in fundamentally nonrelativistic atomic systems. A particular example would consist in a nonrelativistic calculation of the energy levels of helium or lithium, i.e., a nonrelativistic calculation of a few-body system. Here, the unit system of choice is the atomic unit system; one is interested in atomic level energies and ionization potentials scaled to the systems under investigation. As humans, we have chosen the meter (or the foot) as our basic unit of length because it is adapted to the dimensions of the human body, which first and foremost defines the length scale on which we are operating. A human being measures in between 1.5 and 2.0 meters or in between 5 and 7 feet. The numerical values (the reduced quantities) multiplying the base unit of length are in between 10−1 . . . 101 . Thus, it is natural to choose atomic units so that the typical ionization potential of an atom, or binding energies, are obtained as quantities of the order of one Hartree. In order to figure yet a supplementary analogy, let us mention that in currency trading, the unit of a transaction is the million. Buying “two dollars”, for a currency trader, in reality, means to buy two million dollars. Simply, if the scales involved in a transaction (or, physically, in an interacting system) are different, then it is customary to choose a system of units adapted to the problem at hand. In contrast to the atomic unit system, the base unit of the natural unit system is the reduced Compton wavelength of the electron, which is smaller than the Bohr radius by the fine-structure constant α ≈ 1/137.036. One of the difficulties of the bound-state formalism is to accommodate effects on both of these length scales and to describe them in a such a way that they can be consistently added at the end of a calculation. This matching will be one of the recurrent themes treated in this book. 2.3.2

Natural Unit System

As we enter the quantum world, the fundamental length and energy scales change from mascroscopic to microscopic dimensions. Therefore, it is a good idea to introduce an appropriate system of physical units which is adapted to the physical dimensions. The natural unit system is very well adapted to the quantum world in general, while the atomic unit system leads to additional simplifications for computationally intensive problems where the only relevant degrees of freedom are atomic and all quantities can be scaled to the atomic length and mass scales. We start with the natural unit system. In many works on quantum field theory, ̵ = c = 0 = 1”. This it is simply stated that “natural units are used in which h small digression may serve as a further explanation for the “procedure of going into natural units”. Somewhat non-standard symbols shall be used. It may be surprising to observe that there exists no commonly accepted, standard choice for the natural unit of mass. However, it is natural to choose this scale as m, ˆ where m ˆ is the mass of the lightest stable particle, i.e., the mass of the electron [60], m ˆ = m = me = 9.109 383 56(11) × 10−31 kg.

(2.11)

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ˆ is equal to the reduced Compton wavelength of the The natural unit of length, `, electron, ¯λC , ̵ h `ˆ = ¯λC = = 3.861 592 6764(18) × 10−13 m , (2.12) me c where c is the speed of light in vacuum. Note that the reduced Compton wavelength of the electron is just equal to the Compton wavelength divided by a factor 2π and its designation has nothing to do with the distinction of physical and reduced quantities. The natural unit of time, tˆ, is the reduced Compton wavelength of the electron, ¯λC , divided by a factor of c, ̵ h . (2.13) tˆ = m e c2 Here, we will use the paradigm that within a given unit system, products and ratios of fundamental units yield derived fundamental units. The natural unit of velocity, vˆ, is obtained by dividing the natural unit of length by the natural unit of time, vˆ = `ˆ ⋅ tˆ−1 = c = 299 792 458 m ⋅ s−1 .

(2.14)

Therefore, the speed of light in vacuum has the numerical value of one in natural units, c = 1 vˆ = 1 `ˆ ⋅ tˆ−1 .

(2.15)

−1

The SI mksA unit of action, which is kg ⋅m ⋅s , is obtained by multiplying suitable powers of the SI mksA units of mass, length, and time. The natural unit of action, ˆ is obtained as S, 2

̵ Sˆ = m ˆ ⋅ `ˆ2 ⋅ tˆ−1 = h.

(2.16)

̵ (the Planck constant h divided by 2π) Therefore, the reduced Planck constant h has a numerical value of one in natural units, ̵ = 1 Sˆ = 1 m h ˆ ⋅ `ˆ2 ⋅ tˆ−1 ≡ 1 .

(2.17)

The “reduced quantity” is defined as the “numerical value before the unit symbol”. Within a given unit system, we can eventually drop the unit symbol and identify a physical quantity with its reduced quantity. In natural units, the speed of light is ˆ tˆ ≡ 1. This consideration justifies the identity after the “≡” sign in Eq. (2.17). c = 1 `/ The equivalence amounts to a rescaling of all physical quantities to their natural scales. Instead of working with the physical quantities, one is working with reduced quantities which are real numbers defined as the quotients of the physical quantities and the natural unit in which they are measured [102, 103]. Pedantically, it would be more correct to denote the reduced quantities with an asterisk [102, 103], i.e., to ̵ ∗ = 1 for the result of the operation of dividing the reduced Planck constant write h ̵ by the quantity 1 m h ˆ ⋅ `ˆ2 ⋅ tˆ−1 , which describes its natural unit. However, most authors denote physical and reduced quantities with the same symbol, as we have done in Eq. (2.17).

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Alternatively, in assigning the value 0 = 1 to the vacuum permittivity, we identify the unit of charge with the square root of the unit of force times the unit of distance, according to the formula F = q1 q2 (4π0 r2 )−1 for the Coulomb law. The fact that one can do this is a consequence of the so-called “Buckingham π theorem” (see Ref. [104]), which bears the name of Buckingham.5 This theorem relates the number of equations one has in order to describe a given set of physical processes to the number of dimensionless physical constants that one can derive from the description of the process. It determines the number of dimensionful quantities one ˆ reads as can eliminate in a unit system. So, in natural units, the unit of charge Q √ e ̵c= √ ˆ = 0 h . (2.18) Q 4π α ˆ = 1. As one can If we identify physical and reduced quantities, we can write Q easily verify, in the natural unit system the vacuum permittivity is obtained by multiplying powers of base units: ˆ 2 ⋅ tˆ2 ⋅ m 0 = 1 Q ˆ −1 ⋅ `ˆ−3 .

(2.19)

̵ c, and 0 all have the In view of Eqs. (2.15), (2.17), and (2.19), the constants h, numerical value of one in natural units. However, one can now rescale all physical quantities to their natural scales. That is to say, instead of working with the actual physical quantities, one is working with reduced quantities [102, 103] which are real numbers defined as the quotients of the actual quantities and the natural unit in which they are measured. It is generally recommended to denote the reduced quantities with an asterisk, i.e., to write λ∗ for a wavelength divided by the electron Compton wavelength. We recall once more that it is customary, within the natural unit system, to denote actual and reduced quantities with the same symbol. After the introduction of natural units, the rescaling to natural scales and the identification of physical and reduced quantities, one indeed obtains ̵ = c = 0 = 1. h

(2.20)

After this identification, Eq. (2.18) also implies that e2 = 4πα. One can easily check that this relation is fulfilled irrespective of the mass scale or mass unit m. ˆ Alternatively, we can say that if the natural unit system is defined to be a unit ̵ = c = 0 = 1, then we are free to choose a mass scale m, or, keep system in which h a mass scale m in all formulas. The Syst`eme International with its fundamental units of meter (m), kilogram (kg), second (s), and Ampere (A) is most accurately called the SI mksA system. The recent acceptance of a particular value for the Planck constant in the SI mksA unit system has led to a redefinition of the kilogram as a standard mass and to a redefinition of the SI as a quantum SI unit system [105–109]. The basic idea is as follows: One defines the second as a unit of time; it is fixed by setting the numerical value of the ground state hyperfine splitting of cesium (isotope 133 Cs) to 5 Edgar

Buckingham (1867–1940).

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be equal to νCs = 9 192 631 770 s−1 (exactly). The meter is derived from the unit of time by setting the numerical value of the speed of light to be exactly equal to c = 299 792 458 m/s. In the quantum SI unit system, one compares mechanical and electrical work using a Watt balance, i.e., one “weighs the Watt” (see Ref. [110]). The “weighing of the Watt” leads to a accurate determination of the Planck constant if one measures the voltage using a Programmable Josephson Voltage System (PJVS). Indeed, the Josephson constant KJ = 2e/h = 483 597.8525(30) × 109 Hz/V is very well known [60]. A PJVS generates a voltage V = nV ν0 /KJ , where nV is the number of Josephson junctions and ν0 is the microwave frequency at which the system is being irradiated. In summary, using a Watt balance and knowing the local value of the acceleration due to gravity to high accuracy, one may convert a test mass into an accurately measured voltage V, which, in turn, based on the Josephson effect, is converted into an accurate value of the Planck constant h. Conversely, one may fix the numerical value of the Planck constant equal to exactly h = 6.626 070 15 × 10−34 kg m2 /s,

(2.21)

which was adopted on 16 November 2018 by the International Bureau of Weights and Measures (BIPM). Having a defined value of h, one can map the definition of the kilogram onto a sum of photon frequencies. Namely, one may define that a kilogram is equal to a mass, mkg , whose energy equivalent mkg c2 is equal to the energy sum h ∑i νi of a collection of photons whose sum frequency is 34 ∑ νi = [10 /6.626 070 15] Hz .

(2.22)

i

̵ in the present context is The advantage of defining the (reduced) Planck constant h ˆ tˆ, and Q ˆ involve one mass scale the following. Observe that the natural units m, ˆ `, ̵ and are otherwise related by the constants h, c, and 0 . For different problems, it is often convenient to choose slightly modified natural units with a mass or, equivalently, an energy scale different from the mass of the electron. If all three ̵ c and 0 are defined in the SI mksA unit system, it implies that when constants h, a mass scale is chosen for a generalized natural unit system, all other natural units follow from the mass scale via exact factors. A very important role is played by the Coulomb potential. The name “Coulomb potential” is usually associated with the potential energy of an electron in the electrostatic field of a nucleus with charge number Z. In natural units, it is given by the simple expression V (⃗ r) = V (r) = e (

−Z e2 −4π Zα Zα −Ze )= = =− , 4π 0 r 4π 0 r 4π r r

(2.23)

where we have used Eq. (1.12) expressed in natural units [see Eq. (2.20)]. Here and in the following, we denote the electron charge by e = −∣e∣, so that the charge of the nucleus is −Z e.

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Atomic Unit System

The atomic system is defined through the relations ̵ = me = ∣e∣ = 4π 0 = 1 , h

(2.24)

where we recall that me denotes the electron mass. As a consequence, α=

1 1 e2 ̵ = (4π0 )c = c 4π0 hc

(atomic units) ,

(2.25)

and the Bohr radius is

̵ h 1 = −1 =1 α me c c × 1 × c The atomic unit of energy Eh is the Hartree, a0 =

(atomic units) .

(2.26)

Eh = α2 me c2 .

(2.27)

Eh = 2R∞ h c = 27.211 386 245 988(53) eV .

(2.28)

In SI mksA units, we have [31]

Let us illustrate various unit conversions by way of example. The energy of a bound state in a one-electron atom, including the first relativistic corrections but excluding the reduced-mass effects, can be obtained from the Dirac equation. Without going into details of the derivation, we simply state the result for a quantum level of principal quantum number n and total electron angular momentum j, Enj = m c2 −

1 3 (Zα)2 mc2 (Zα)4 mc2 − ( − ) 2 3 2n n 2j + 1 8 n

(SI mksA units) , (2.29)

where we neglect terms of order α6 m c2 . (Here and in the following, m is the mass of the bound particle, which usually is equal to the electron mass.) The first term on the right-hand side qualifies itself as the electron rest mass converted into an energy. The second term is the Schr¨odinger energy, and the third term is the first ̵ = c = 0 = 1, yet keeping the mass relativistic correction. In natural units with h scale arbitrary, we have from Eq. (2.29), 1 3 (Zα)2 m (Zα)4 m − ( − ) (natural units) . (2.30) 2n2 n3 2j + 1 8 n The leading term takes a simple form and reminds us of the mass-energy equivalence that follows from relativity theory. In atomic units, we have Enj = m −

Enj =

1 Z 2 Z 4 α2 1 3 − − ( − ) 2 2 3 α 2n n 2j + 1 8 n

(atomic units) .

(2.31)

This is obtained by simply dividing the corresponding formula by a factor α2 m. Let us illustrate the unit conversion once more, but this time considering a part of the (Zα)4 mc2 correction which is due to the zitterbewegung term. In SI mksA units, the corresponding Hamiltonian reads ̵3 πh δHZ = Zα δ (3) (⃗ r) (SI mksA units) , (2.32) 2m2 c

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where δ (3) (⃗ r) denotes the Dirac-δ function. The probability density at the origin in a hydrogen eigenstate with quantum numbers n, ` = L, and j (spectroscopic notation nLj , example: nS1/2 for ` = L = 0) is Zαmc 3 1 ( ̵ ) δ`0 h π n3 The energy shift is obtained as ⟨nLj ∣δ (3) (⃗ r)∣nLj ⟩ =

(SI mksA units) .

(2.33)

̵3 πh Zα ⟨nLj ∣δ (3) (⃗ r)∣nLj ⟩ 2m2 c ̵ 3 Zα Zαmc 3 (Z α)4 m c2 πh ( ̵ ) = δ`0 (SI mksA units) . (2.34) = 2m2 c π n3 2 n3 h In atomic units, the expression for the zitterbewegung term converts as follows, ̵3 πh αmc 3 (3) αmc r⃗ δHZ = Zα ( ̵ ) δ ( h ̵ ) 2m2 c h ∆EZ = ⟨nLj ∣δHZ ∣nLj ⟩ =

π Z α4 mc2 (3) π Z α2 (3) δ (⃗ r/a0 ) = δ (⃗ r) 2 2 The probability density at the origin is =

⟨nLj ∣δ (3) (⃗ r)∣ nLj ⟩ =

Z3 δ`0 π n3

(atomic units) .

(atomic units) .

(2.35)

(2.36)

The energy shift reads as follows, ∆EZ = ⟨nLj ∣δHZ ∣ nLj ⟩ =

Z 4 α2 π Z α2 ⟨nLj ∣δ (3) (⃗ r)∣nLj ⟩ = δ`0 2 2 n3

(atomic units) .

(2.37) The quantity ∆EZ obtained is the reduced quantity; one has to multiply it by the atomic energy scale α2 mc2 in order to obtain the actual energy which reproduces Eq. (2.34). In natural units, the zitterbewegung Hamiltonian is ̵3 πh mc 3 (3) mc r⃗ Zα ( ̵ ) δ ( h ̵ ) 2m2 c h π Zα m (3) π Zα m (3) r⃗ = δ ( )= δ (⃗ r) 2 ¯λe 2

δHZ =

(natural units) .

(2.38)

The square of the wave function at the origin reads as ⟨nLj ∣δ (3) (⃗ r)∣ nLj ⟩ =

(Zα)3 δ`0 π n3

(natural units) .

(2.39)

For the energy shift, we have ∆EZ = ⟨nLj ∣δHZ ∣ nLj ⟩ =

π Zα m (Zα)3 (Zα)4 m = δ`0 . 2 π n3 2n3

(2.40)

Again, one may identify this result as the reduced quantity and multiply by c2 to reproduce Eq. (2.34).

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In the remainder of this book, we shall use the SI mksA, the natural, and the atomic system interchangeably. Indeed, all formulations have their advantages and disadvantages. For instance, if we strive to calculate the helium ground-state energy, then the only practical way to set up the calculation is with a computer program where all quantities are scaled to the dimensions of the atom in question, i.e., in atomic units. If we strive to keep a compact notation in writing down a field operator for an electric or magnetic field, in order to carry out a complex calculation while keeping intermediate expressions compact, then the natural unit system is most appropriate. The same is true if we attempt to systematically order the terms of a quantum electrodynamic perturbation series in powers of the finestructure constant; this is most practical in natural units. Finally, if we strive to carry out a model calculation for a process while making sure that we do not miss any important terms in intermediate steps, then it sometimes pays off to fully retain SI mksA units in all intermediate steps of the calculation. Intermediate attempts like the Gaussian system, where one sets 0 = 1/(4π) but keeps most other physical quantities and dimensions unchanged, are not used in the following, and they seem to be disfavored by the scientific community in recent years. Typically, in the Gaussian system, formulas cannot be reduced too much in complexity, but one still loses the tracking of the SI mksA units and incurs several possible sources of confusion in formulas. Examples include the tracking of a factor 4π to an inverse power of 0 or an integration over a solid angle. 2.4 2.4.1

Field Quantization for the Electromagnetic Field Quantization of the Free Electromagnetic Field

We have already mentioned that the process of field quantization can be interpreted in a natural way as “second quantization” (see Sec. 2.2.2). The quantization of the electromagnetic field is quite problematic from a theoretical point of view because it is a massless spin-1 field. Normally, the quantization of a theory proceeds via the correspondence principle: One writes down the Lagrangian, identifies the coordinates, determines the conjugate variables (canonical momenta), and postulates the canonical dispersion relations. However, in a massless spin-1 theory like the Maxwell theory of electrodynamics, as we shall discuss in more detail below, the canonical momentum corresponding to the scalar potential vanishes whereas the conjugate variable of the vector potential is the electric field. One can remedy the problem in various ways, but here, we shall pursue a very direct way which uses a few particular properties of the Coulomb gauge. We recall from classical electrodynamics the Maxwell equations, which read as follows (in natural units)

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⃗ r⃗) = ρ(⃗ ⃗ ⋅ E(t, ∇ r, t) , ⃗ r⃗) = 0 , ⃗ ⋅ B(t, ∇

(2.41a)

(2.41b) ∂ ⃗ r⃗) , ⃗ r⃗) = − B(t, ⃗ × E(t, (2.41c) ∇ ∂t ⃗ r⃗) + J(t, ⃗ r⃗) . ⃗ r⃗) = ∂ E(t, ⃗ × B(t, (2.41d) ∇ ∂t Here, in contrast to classical electrodynamics, we shall order the space-time arguments of the electric and magnetic fields as (t, r⃗), as inspired by the four-vector ⃗ and the magnetic induction field B ⃗ are notation xµ = (t, r⃗). The electric field E ⃗ related to the scalar potential Φ and the vector potential A, ∂ ⃗ ⃗ r⃗) = − ∇Φ(t, ⃗ E(t, r⃗) − A(t, r⃗) , ∂t ⃗ r⃗) = ∇ ⃗ r⃗) . ⃗ × A(t, B(t,

(2.42a) (2.42b)

A vector field A⃗ can be uniquely decomposed into a transverse component A⃗⊥ and ⃗ In Coulomb gauge, ⃗ ⋅ A⃗⊥ = 0 and ∇ ⃗ × A⃗∥ = 0. a longitudinal component A⃗∥ , with ∇ the longitudinal component of the vector potential vanishes, ⃗ r⃗) = 0 , ⃗ ⋅ A(t, ∇

A⃗ (t, r⃗) = A⃗⊥ (t, r⃗) ,

A⃗∥ (t, r⃗) = 0 ,

(2.43)

and the potentials are coupled to the sources as follows (see Chap. 1 of Ref. [81]), ⃗ 2 Φ (t, r⃗) = − ρ (t, r⃗) , ∇

(2.44a)

∂2 ⃗ 2 ) A⃗⊥ (t, r⃗) = J⃗⊥ (t, r⃗) , −∇ (2.44b) ∂t2 ∂ ⃗ Φ (t, r⃗) = J⃗∥ (t, r⃗) . ∇ (2.44c) ∂t In Coulomb gauge, the longitudinal part of the electric field is given as the gradient of the scalar potential, as a consequence of Eqs. (2.42a) and (2.43), (

∂ ⃗∥ (t, r⃗) = −∇Φ(t, ⃗ ⃗ E r⃗) − A⃗∥ (t, r⃗) = −∇Φ(t, r⃗) , ∂t while the transverse part is

(2.45)

⃗⊥ (t, r⃗) = − ∂ A⃗⊥ (t, r⃗) . E (2.46) ∂t In the following, we shall perform a quantization of the electromagnetic field alluding, exclusively, to the transverse components of the vector potential, A⃗ = A⃗⊥ . The scalar potential Φ will not even take part in the quantization process. The justification for this approach in a more general context will be given later in Chap. 9. We also recall from Chap. 8 of Ref. [81] the Lagrangian density of the free electromagnetic field, 1 1 1 ⃗2 ⃗2 L = − F µν Fµν = − (∂µ Aν − ∂ν Aµ ) (∂ µ Aν − ∂ ν Aµ ) = (E −B ) , 4 4 2

(2.47)

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where F µν is the electromagnetic field strength tensor. The quantization procedure in Coulomb gauge, which is especially suited for nonrelativistic problems, is somewhat noncovariant. According to Eq. (2.44a), in the Coulomb gauge, the scalar potential would couple to the charge distribution instantaneously, i.e., in the form of an action-at-a-distance law, were it not for the additional consistency condition (2.44c). The latter forces the (otherwise problematic) longitudinal component of the electric field to be given by a retarded integral (see Sec. 4.2.4 of Ref. [81]). From the Maxwell Lagrangian (2.47), the canonical momenta corresponding to the spatial components A⃗⊥ of the four-vector potential are easily found as follows, Π0 =

∂L = 0, ∂(∂0 A0 )

Πk =

∂L = −∂0 Ak⊥ = E⊥k . ∂(∂0 A⊥,k )

(2.48)

Here, by convention, k = 1, 2, 3 is a spatial index; we use the (flat-)space metric in the convention gµν = diag(1, −1, −1, −1) ,

µ, ν = 0, 1, 2, 3 .

(2.49)

The conjugate variable of the vector potential is obtained as the electric field; it is extremely useful to observe that the conjugate variable of A⊥,k is E⊥k , which implies that the conjugate variable of Ak⊥ is −E⊥k . The Hamiltonian density is obtained from the Lagrangian density by a Legendre transformation, H = H(t, r⃗) = Πk (+

∂Ak⊥ ⃗ 2 − L = 1 (E ⃗2 + B ⃗ 2) − 1 E ⃗2 . )−L=E ⊥ ⊥ ∂t 2 2 ∥

(2.50)

Finally, the Hamiltonian is 1 3 ⃗2 + B ⃗2 − E ⃗2) . (2.51) ∫ d r (E ⊥ ∥ 2 ⃗ 2 have a negative sign; this pheIt may appear as disturbing to see the term E ∥ nomenon finds a natural explanation once the term ρ Φ is added to the Hamiltonian [see Eq. (9.48) and the following discussion in Chap. 9]. We now use the correspondence principle and postulate some fundamental equal-time commutator relations for the quantized theory. Essentially, the paradigm is to reduce the quantization problem, for equal times, to ordinary canonical quantization as known from quantum mechanics, where we would postulate the commutations relation [p, x] = −i for the position operator x and the conjugate momentum operator p = −i∂/∂x. The commutators of the vector potential and of the electric field with themselves vanish, H = ∫ d3 r H(t, r⃗) =

[Ai⊥ (t, r⃗), Aj⊥ (t, r⃗′ )] = 0 ,

[Πi (t, r⃗), Πj (t, r⃗′ )] = 0 .

(2.52)

In analogy to quantum mechanical coordinates and momenta, one might now assume the following commutation relations for the vector potential and its conjugate momenta, ?! [Πi (t, r⃗), A⊥,j (t, r⃗′ )] = − [Πi (t, r⃗), Aj⊥ (t, r⃗′ )] = −i δ ij δ (3) (⃗ r − r⃗′ ) .

(2.53)

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Once again, the sign is motivated by the fact that the conjugate variable of the covariant component A⊥,j is the contravariant field component Πj . However, because we have imposed additional constraints on the vector potential in the form A⃗ = A⃗⊥ [see Eq. (2.43)], Eq. (2.53) is not compatible with the Coulomb gauge condition ⃗ r⃗) = 0. This can be seen by taking the divergence of both sides of Eq. (2.53). ⃗ ⋅ A(t, ∇ The compatibility with the Coulomb gauge condition is restored if we replace the Dirac-δ function on the right-hand side of Eq. (2.53) by the transverse Dirac-δ. It reads as δ ⊥,ij (⃗ r − r⃗′ ), where ′ ki kj d3 k ⃗ (δ ij − ) ei k⋅(⃗r−⃗r ) . (2.54) 3 2 ⃗ (2π) k In coordinate space, it is hard to give the transverse Dirac-δ in a simpler form than ∇i ∇ j ) δ (3) (⃗ r − r⃗′ ) . (2.55) δ ⊥,ij (⃗ r − r⃗′ ) = (δ ij − ⃗2 ∇

δ ⊥,ij (⃗ r − r⃗′ ) = ∫

The transverse Dirac-δ is a tensor of rank two whose divergence with respect to both spatial indices vanishes, ∂ ∇i δ ⊥,ij (⃗ r − r⃗′ ) = ∇j δ ⊥,ij (⃗ r − r⃗′ ) = 0 , ∇i ≡ . (2.56) ∂xi Our notation implies that ∂ x ∂ y ∂ z ⃗ ⋅ V⃗ = ∇j V j = ∇ V + V + V , (2.57) ∂x ∂y ∂z for any vector V⃗ . We shall consistently use superscripts for the “special and spatial” ⃗ Our slightly nonstandard vector components ∇i of the Cartesian three-vector ∇. ⃗ (with notation is motivated by our desire to keep the components of the vector ∇ an upper, contravariant index, see Chap. 6 of Ref. [81]) identical to those we would expect for a Cartesian vector. There is a further certain subtlety in the notation, which we would like to comment on. In this treatise, the spatial components xi (i = 1, 2, 3) of the position operator r⃗ are denoted as 3

r⃗ = x ˆex + y ˆey + z ˆez = x1 ˆex + x2 ˆey + x3 ˆez = ∑ xi ˆei .

(2.58)

i=1

The superscript (rather than subscript) of the Cartesian component stresses the contravariant (rather than covariant) nature of the component (see also Chap. 8). The notation is in part motivated by the desire to avoid confusion between the mth power rm = ∣⃗ r∣m of the radial distance and the mth component xm of the position operator. In Chap. 12, though, we shall slightly change this convention, in view of the complex nature of composite systems, where it becomes necessary to further distinguish position vectors measured in regard to different points of origin. For the time being, the notation (2.58) appears to be a convenient choice. Employing the correspondence principle and the transverse Dirac-δ function, and taking into account that A⃗ is transverse, the modified postulate for the commutator reads [Πi (t, r⃗), A⊥,j (t, r⃗′ )] = − [Πi (t, r⃗), Aj⊥ (t, r⃗′ )] = −i δ ⊥,ij (⃗ r − r⃗′ ) .

(2.59)

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In terms of the electric field operator and the vector potential, the commutation relation reads [Πi (t, r⃗), Aj⊥ (t, r⃗′ )] = [E⊥i (t, r⃗), Aj⊥ (t, r⃗′ )] = i δ ⊥,ij (⃗ r − r⃗′ ) . (2.60) We decompose the field operator into fundamental modes as follows, 2 d3 k ⃗ r) ⃗ r) ⃗ [aλ (k) ⃗ e−i (ω t−k⋅⃗ ⃗ ei (ω t−k⋅⃗ A⃗⊥ (t, r⃗) = ∑ ∫ √ ˆλ (k) + a+λ (k) ], 3 2ω (2π) λ=1 (2.61a) 2 d3 k ⃗ [aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ] , ˆλ (k) A⃗⊥ (x) = ∑ ∫ √ λ 2ω (2π)3 λ=1 ⃗ = (∣k∣, ⃗ k) ⃗ . {k µ=0,1,2,3 } = (ω, k)

(2.61b)

(2.61c) ⃗ Here, λ = 1, 2 denotes the photon polarization and k ⋅ x = k xµ = ω t − k ⋅ r⃗. The decomposition is in fact somewhat arbitrary with respect to prefactors and integration measures because a change in the integration measure can always be reabsorbed in ⃗ a redefinition of the fundamental photon creation and annihilation operators a+λ (k) ⃗ and aλ (k). Later, we will discuss alternative representations. The decomposition of the electromagnetic field operator given in Eq. (2.61) is especially suited for nonrelativistic calculations. In d3 k integrations like the one performed in Eq. (2.61b), it is important to note that the zeroth component, k 0 = ω, of the four-vector k µ needs to be defined in terms of the spatial components; otherwise, the result of the integral could not be expressed solely in terms of t and r⃗. For particles of mass m, the component k 0 is determined as the positive square root √ (2.62) k 0 ≡ ω ≡ k⃗2 + m2 , µ

where m is the particle mass (m = 0 for photons). In the literature on field theory, one finds both integrals of the type ∫ d3 kf (k), as well as ∫ d4 kf (k), where k = (k µ ) is a four-vector. In the first case, the component k 0 is determined by the general condition (2.62), i.e., always positive. In the latter case, the k 0 component is not determined by any additional condition but simply integrated over. Both Eqs. (2.61a) and (2.61b) are examples of the foremost interpretation. ⃗ with For a given wave vector k⃗ of the photon, the two polarization vectors ˆλ (k) λ = 1, 2 fulfill the relations i j 2 i ⃗ ⃗ = δ ij − k k . (2.63) ⃗ ⋅ ˆλ′ (k) ⃗ = δλλ′ , ⃗ = 0, ˆλ (k) k⃗ ⋅ ˆλ (k) jλ (k) ∑ λ (k) k⃗2 λ=1 Using spherical coordinates (r, θ and ϕ), we can express the (originally Cartesian) components of k⃗ as follows, ⃗ (ˆ k⃗ = ∣k∣ ex sin θ cos ϕ + eˆy sin θ sin ϕ + eˆz cos θ) . (2.64) The two polarization vectors may be chosen as ⃗ = eˆx cos θ cos ϕ + eˆy cos θ sin ϕ − eˆz sin θ , eˆ1 (k) (2.65a) ⃗ = − eˆx sin ϕ + eˆy cos ϕ , eˆ2 (k) ⃗ k∣ ⃗ = eˆ1 (k) ⃗ × eˆ2 (k) ⃗ . k/∣

(2.65b) (2.65c)

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⃗ 2 , corresponds The wave equation ◻ A⃗ = 0, with the “quabla” operator ◻ = ∂t2 − ∇ 2 µ 2 2 ⃗ to the equation k = k kµ = ω − k = 0 in Fourier space (where we anticipate the Lorentz-covariant notation and the Einstein summation convention). Inverting (2.61), we can express the fundamental creation and annihilation operators as follows, ⃗

i (ω t−k⋅⃗ r) ⃗ ⋅ (ω A⃗⊥ (t, r⃗) + i ∂ A⃗⊥ (t, r⃗)) ⃗ = ∫ d3 r √e ˆλ (k) aλ (k) ∂t 2 ω (2π)3

exp (i k ⋅ x) ⃗ ⋅ (ω A⃗⊥ (x) + i ∂ A⃗⊥ (x)) , = ∫ d3 r √ ˆλ (k) 3 ∂t 2 ω (2π)

(2.66a)

and ⃗

e−i (ω t−k⋅⃗r) ⃗ = ∫ d3 r √ ⃗ ⋅ (ω A⃗⊥ (t, r⃗) − i ∂ A⃗⊥ (t, r⃗)) a+λ (k) ˆλ (k) 3 ∂t 2 ω (2π) exp (−i k ⋅ x) ⃗ ⋅ (ω A⃗⊥ (x) − i ∂ A⃗⊥ (x)) . = ∫ d3 r √ ˆλ (k) ∂t 2 ω (2π)3

(2.66b)

Here, x = (t, r⃗) is the space-time coordinate. The derivation of Eqs. (2.66a) and (2.66b) is simplified if one observes that the operator i∂t , when acting on the positive-frequency (annihilation) part of A⃗⊥ (t, r⃗), “pulls down” a factor ω from the exponential, whereas it “pulls down” a factor −ω from the negative-frequency part of A⃗⊥ (t, r⃗). Thus, the positive- and negative-frequency parts can be determined by forming appropriate linear combinations. ⃗ and a+ (k) ⃗ suggests that they should be The definition of the operators aλ (k) λ ⃗ by an explicit caltime-independent. It is instructive to verify this fact for aλ (k) culation, because the representation on the right-hand side of Eq. (2.66a) could otherwise suggest a possible time dependence. From Eq. (2.66a), one infers that ⃗

i (ω t−k⋅⃗ r) 2 d ⃗ = i ∫ d3 r √e ⃗ ⋅ (ω 2 A⃗⊥ (t, r⃗) + ∂ A⃗⊥ (t, r⃗)) aλ (k) ˆλ (k) dt ∂t2 2 ω (2π)3 ⃗

ei (ω t−k⋅⃗r) ⃗ ⋅ (ω 2 A⃗⊥ (t, r⃗) + ∇ ⃗ 2 A⃗⊥ (t, r⃗)) = i ∫ d3 r √ ˆλ (k) 2 ω (2π)3 ⃗

ei (ω t−k⋅⃗r) ⃗ ⋅ (ω 2 A⃗⊥ (t, r⃗) + A⃗⊥ (t, r⃗) ∇ ⃗ 2) √ = i ∫ d3 r ˆλ (k) = 0. 2 ω (2π)3

(2.67)

In the transition from the first to the second line, we have used the wave equation ⃗ 2 ) A⃗⊥ (t, r⃗) = 0. In going for the free vector potential operator in the form (∂t2 − ∇ from the second to the third line, we have integrated by parts (twice). Finally, we ⃗ vanishes. ⃗ 2 → −k⃗2 = −ω 2 , verifying that the time derivative of aλ (k) can replace ∇ + ⃗′ ⃗ We can now calculate the commutator [aλ (k), a ′ (k )]. In writing down the λ

integrands, we exploit the time-independence and choose t = t′ . Furthermore, we

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take notice of the fundamental commutators (2.52), ⃗′

′

⃗′

′

⃗

ei (k ⋅⃗r −k⋅⃗r) i ⃗ j ⃗′ √ λ (k) λ′ (k ) 2 (2π)3 ω ω ′ ∂ ∂ × [ω Ai⊥ (t, r⃗) + i Ai⊥ (t, r⃗) , ω ′ Aj⊥ (t, r⃗′ ) − i Aj⊥ (t, r⃗′ )] ∂t ∂t

⃗ a+′ (k⃗′ )] = ∫ d3 r d3 r′ [aλ (k), λ

= ∫ d3 r d3 r′

⃗

ei (k ⋅⃗r −k⋅⃗r) i ⃗ j ⃗′ √ λ (k) λ′ (k ) 2 (2π)3 ω ω ′

× (−i) {ω [E⊥j (t, r⃗′ ), Ai⊥ (t, r⃗)] + ω ′ [E⊥i (t, r⃗), Aj⊥ (t, r⃗′ )]} . (2.68) Using the field commutators from Eq. (2.60), we can reformulate this as ⃗ a+′ (k⃗′ )] = ∫ d3 r d3 r′ [aλ (k), λ

⃗′

′

⃗′

′

⃗

ei (k ⋅⃗r −k⋅⃗r) i ⃗ j ⃗′ √ λ (k) λ′ (k ) (ω + ω ′ ) δ ⊥,ij (⃗ r − r⃗′ ) 2 (2π)3 ω ω ′ ⃗

ei (k ⋅⃗r −k⋅⃗r) i ⃗ j ⃗′ √ λ (k) λ′ (k ) (ω + ω ′ ) = ∫ d rd r 2 (2π)3 ω ω ′ 3

3 ′

× (δ ij δ (3) (⃗ r − r⃗′ ) −

∇i ∇j (3) δ (⃗ r − r⃗′ )) ⃗2 ∇

′ d3 r i (k⃗′ −k)⋅⃗ ⃗ r ⃗ ⋅ ⃗λ′ (k⃗′ )) ω√+ ω = δ (3) (k⃗ − k⃗′ ) δλλ′ . (⃗ e λ (k) 3 (2π) 2 ω ω′ (2.69) √ We observe that (ω + ω ′ ) /(2 ω ω ′ ) → 1 for ω → ω ′ (or k⃗ = k⃗′ ). In the transformations made in Eq. (2.69), we have integrated by parts with respect to both r⃗ and r⃗′ ; ⃗ also has been used. The final result for the commutator the orthogonality k⃗ ⋅ ⃗λ (k) of the fundamental creation and annihilation operators is

=∫

⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ . [aλ (k), λ

(2.70)

We note that its precise form relies on our choice of weight factors under the integral sign in Eq. (2.61). The functional form (2.70) reminds us of the analogy of the electromagnetic oscillators with harmonic oscillator modes of the vacuum. Let us summarize once more the results of the quantization program in Coulomb gauge. We only take into account the two transverse polarization states of the electromagnetic field, not the longitudinal and scalar photons. The scalar potential will not take part in the quantization because we are unable to find a corresponding canonical momentum. This raises the question of how to deal with the Coulomb interaction in the quantized formalism. The answer is as follows: From the quantized formulas in the Coulomb gauge, we obtain the photon propagator (in the Coulomb gauge) after the addition of the instantaneous Coulomb interaction (see Chap. 6 of Ref. [111] and Sec. 9.3 in Chap. 9 here). The propagator looks different from the one obtained in, say, the Lorenz gauge. However, one can show that the difference does not play a role in the calculation of fundamental processes. Indeed, the photon propagator obtained in Coulomb gauge, as a sum of the instantaneous and

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retarded interactions, is related by a gauge transformation to the photon propagator obtained in a covariant gauge (e.g., the Lorenz gauge). The difference in the calculation of fundamental processes vanishes after current conservation has been taken into account (see Chap. 9). Yet another issue, which will be dealt with in greater detail in Chap. 9, concerns the fact that the fully quantized electromagnetic field in Lorenz gauge actually contains two unphysical virtual degrees of freedom (scalar and longitudinal photons). These must later be excluded from the physical spectrum via the Gupta6 – Bleuler7 condition. They are still present in the propagator obtained in Lorenz gauge. Still, we can use both propagators, obtained in Coulomb and in Lorenz gauge, and calculate physical processes. The advantage of the Coulomb gauge is that for low-energy processes, we have two photon polarizations less to sum over, whereas the advantage of the Lorenz gauge is that it yields a manifestly covariant form of the propagator. Also, in Coulomb gauge, it is not necessary to introduce a Gupta–Bleuler condition: The two degrees of freedom taking part in the quantization are the physical ones; the scalar potential is not quantized and the scalar and longitudinal photons never appear.

2.4.2

Field Operators and Quantized Hamiltonian

In classical mechanics, momentum is just a number that can be measured to essentially arbitrary accuracy. In quantum mechanics, momentum is a (Hermitian) operator. In the same light, an electric or magnetic field is a number in classical field theory whereas in quantum field theory, it is given by a Hermitian operator. It is instructive to write down the operators which actually correspond to the electric and magnetic fields; these are Hermitian. The expectation value of these operators in a given state of the photon field is a real quantity while the operators themselves may contain manifestly complex terms. Before discussing the electric and magnetic field operators, we first recall the vector potential operator (2.61) 2 d3 k ⃗ (aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ) . ˆλ (k) A⃗⊥ (t, r⃗) = ∑ ∫ √ λ 3 2ω (2π) λ=1

(2.71)

⃗⊥ = E ⃗ + = −∂ A⃗⊥ /∂t is simply obtained In Coulomb gauge, the electric field operator E ⊥ as the time derivative of the vector potential operator and reads as follows, 3 2 ⃗ i ω (aλ (k) ⃗ e−i k⋅x − a+ (k) ⃗ ei k⋅x ) . ⃗⊥ (t, r⃗) = ∑ ∫ √ d k ˆλ (k) E λ 3 2ω (2π) λ=1 6 Suraj

North Gupta (b. 1924). Bleuler (1912–1992).

7 Konrad

(2.72)

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From Unit Systems for the Microworld to Field Quantization

⃗=B ⃗+ = ∇ ⃗ × A⃗⊥ of the vector The magnetic field operator is obtained as the curl B potential, 3 2 ⃗ (aλ (k) ⃗ e−i k⋅x − a+ (k) ⃗ ei k⋅x ) . ⃗ r⃗) = ∑ ∫ √ d k i (k⃗ × ˆλ (k)) B(t, λ 2ω (2π)3 λ=1

(2.73)

⃗ r⃗) and B(t, ⃗ r⃗) are given by Hermitian operators and contain positiveBoth E(t, frequency terms (proportional to e−ik⋅x ) as well as negative-frequency terms (proportional to eik⋅x ). The positive-frequency as well as negative-frequency parts alone could not act as field operators because these would not correspond to Hermitian operators. Starting from the postulated fundamental commutator relation given in Eq. (2.60), and using the inversion formulas (2.66), we had derived, in Eq. (2.70), ⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ . It is instructive to verify once more the result [aλ (k), λ that this result is consistent with the fundamental commutator of the electric field of the vector potential operator at equal times t = t′ , 2

[E⊥i (t, r⃗), Aj⊥ (t, r⃗′ )] = ∑ ∫ √ λ=1

d3 k

d3 k ′

2

2ω (2π)3

∑∫ √

λ′ =1

2ω ′ (2π)3

⃗ j ′ (k⃗′ ) iλ (k) λ

⃗ r −i k⃗′ ⋅⃗ r′ ⃗ a+′ (k⃗′ )] e−i(ω−ω′ )t eik⋅⃗ × {iω [aλ (k), λ ⃗ r +ik⃗′ ⋅⃗ r′ ⃗ aλ′ (k⃗′ )] ei(ω−ω′ )t e−ik⋅⃗ − iω [a+λ (k), }

=

′ ′ i 2 d3 k i ⃗ j ⃗ ⃗ ⃗ λ (k) λ (k) (ei k⋅(⃗r−⃗r ) + e−i k⋅(⃗r−⃗r ) ) ∑∫ 3 2 λ=1 (2π)

= i δ ⊥,ij (⃗ r − r⃗′ ) .

(2.74)

In the derivation, we have used Eq. (2.54). By virtue of the Heisenberg uncertainty relation, the nonvanishing fundamental commutators of the vector potential and its conjugate momentum imply that the electric field and the vector potential cannot be measured simultaneously with arbitrary accuracy. This statement, though, does not really matter because the gauge-dependent vector potential does not represent a physically measurable quantity. Therefore, it is of prime interest to investigate whether electric and magnetic fields can be measured simultaneously in the quantum world. Indeed, the commutator of the electric and magnetic fields reads as 2

[E⊥i (t, r⃗), B j (t, r⃗′ )] = ∑ ∫ √ λ=1

d3 k 2ω (2π)3

⃗ × {i ω [aλ (k), 2

2

∑∫ √

λ′ =1

a+λ′ (k⃗′ )]

d3 k ′ 2ω ′ (2π)3 ⃗

⃗ (k⃗′ × ˆλ′ (k⃗′ ))j iλ (k) ⃗′

e−i (ω−ω )t ei k⋅⃗r−i k ⋅⃗r + h.c.} , (2.75) ′

′

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where “h.c.” stands for the Hermitian conjugate. It can be simplified to read [E⊥i (t, r⃗), B j (t, r⃗′ )] = =

1 2 d3 k i ⃗ ⃗ ⃗ r −⃗ r′ ) ⃗ j ei k⋅(⃗  (k) (k × ˆλ (k)) + h.c. ∑∫ 2 λ=1 (2π)3 λ ′ d3 k i ⃗ j`m ` m ⃗ i k⋅(⃗ 1 2 ⃗ λ (k)  k λ (k) e r−⃗r ) + h.c. ∑∫ 3 2 λ=1 (2π)

′ 1 d3 k j`m ` im k i k m i k⋅(⃗ ⃗  k (δ − ) e r−⃗r ) + h.c. ∫ 3 2 ⃗ 2 (2π) k ∂ = − i ij` ∇` δ (3) (⃗ r − r⃗′ ) = −i ij` ` δ (3) (⃗ r − r⃗′ ) . (2.76) ∂x

=

This is manifestly different from zero. The presence of the Levi-Civit`a8 tensor ijk implies that, e.g., the x component of the electric field and the y component of the magnetic field cannot be measured simultaneously to arbitrary precision at the same time t. Let us now shift our attention to the Hamiltonian operator HF of the free electromagnetic field. It is time-independent in the absence of source terms. We evaluate it at time t = 0 and take into account that the electric and magnetic field ⃗ + = E, ⃗ B ⃗ + = B), ⃗ operators represent Hermitian operators (E ⊥

⊥

1 3 ⃗ 2 (t = 0, r⃗) + B ⃗ 2 (t = 0, r⃗)) ∫ d r (E ⊥ 2 2 1 2 d3 k d3 k ′ =− ∑ ∫ √ ∑∫ √ 2 λ=1 2ω (2π)3 λ′ =1 2ω ′ (2π)3

HF =

(2.77)

⃗ ⋅ ˆλ′ (k⃗′ ) + (k⃗ × ˆλ (k)) ⃗ ⋅ (k⃗′ × ˆλ′ (k⃗′ )) } × {ω ω ′ ˆλ (k) ⃗

⃗

⃗′

⃗′

⃗ ei k⋅⃗r − a+ (k) ⃗ e−i k⋅⃗r ) (aλ′ (k⃗′ ) ei k ⋅⃗r − a+′ (k⃗′ ) e−i k ⋅⃗r ) . × ∫ d3 r (aλ (k) λ λ The integral over d3 r leads to ⃗′ ⃗r ⃗r ⃗′ 3 ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ − a+λ (k) ) (aλ′ (k⃗′ ) ei k ⋅⃗r − a+λ′ (k⃗′ ) e−i k ⋅⃗r ) ∫ d r (aλ (k)

⃗ aλ′ (−k) ⃗ + a+ (k) ⃗ a+′ (−k)) ⃗ δ (3) (k⃗ + k⃗′ ) = (2π)3 [(aλ (k) λ λ ⃗ aλ′ (k) ⃗ + aλ (k) ⃗ a+′ (k)) ⃗ δ (3) (k⃗ − k⃗′ )] . − (a+λ (k) λ

(2.78)

The Dirac-δ’s lead to the following vector structures in Eq. (2.77), ⃗ ⋅ (k⃗ × ˆλ′ (k)) ⃗ = k⃗2 δλλ′ , (k⃗ × ˆλ (k)) ⃗2

⃗ ⋅ ˆλ′ (−k) ⃗ . ⃗ ⋅ (−k⃗ × ˆλ′ (−k)) ⃗ = − k ˆλ (k) (k⃗ × ˆλ (k))

(2.79a) (2.79b)

We now insert Eqs. (2.78), (2.79a), and (2.79b) into Eq. (2.77); this gives rise to a term in the integrand proportional to ω 2 − k⃗2 = 0, which vanishes. The surviving 8 Tullio

Levi-Civit` a (1873–1941).

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term is the Hamilton operator of the free electromagnetic field, HF = =

1 2 2 d3 k ⃗ aλ′ (k) ⃗ + aλ (k) ⃗ a+′ (k)) ⃗ (ω 2 + k⃗2 ) δλλ′ (a+λ (k) ∑ ∑∫ λ 2 λ=1 λ′ =1 2ω 1 2 3 + ⃗ ⃗ + aλ (k) ⃗ a+′ (k)) ⃗ aλ (k) ∑ ∫ d k ω (aλ (k) λ 2 λ=1 2

⃗ aλ (k) ⃗ + = ∑ ∫ d3 k ω (a+λ (k) λ=1

1 (3) ⃗ δ (0)) . 2

(2.80)

⃗ a+ (k)] ⃗ = δ (3) (k⃗ − k). ⃗ ⃗ is the consequence of the relation [aλ (k), The term δ (3) (0) λ ′ (3) ⃗ ⃗ ⃗ For k = k , the Dirac-δ is evaluated at the origin and the expression δ (0) is ⃗ is undefined and has to be given a physical obtained. A priori, the expression δ (3) (0) interpretation. Let us consider the transition from a continuous photon spectrum to a discrete spectrum with the latter being normalized to a quantization volume V . In a finite normalization volume, we evaluate a discrete sum over the available ⃗ this sum needs to be matched against the integral d3 k over photon wave vectors k; the wave vectors k in the volume V . This integral, in turn, gives rise to a Dirac-δ term δ (3) (⃗0) in the continuum limit. Let us indicate the matching relation [112], ∑→ ⃗ k

V 3 (3) 3 ∫ d k → δ (⃗0) ∫ d k , (2π)3

(2.81)

and discuss its justification. The summation over k⃗ on the left is to be understood ⃗ composed of integers, which are related to the components as a multi-index vector n ⃗ /L. The first replacement “→” simply follows from counting the of k⃗ by k⃗ = 2π n available modes satisfying periodic boundary conditions in a cubic volume V = L3 , which results in one mode per fundamental volume (2π/L) in k-space. This is discussed at various places in the literature; one particular source is Eq. (1.1.39) of Ref. [113]. The second replacement (“→”) in Eq. (2.81) can be justified as follows: The Dirac-δ function in the wave-vector space is ⃗ =∫ δ (3) (k)

R

3

d3 r exp(i k⃗ ⋅ r⃗) . (2π)3

(2.82)

Here, the factor (2π)−3 is due to the definition of the three-dimensional Dirac-δ function. If we restrict the d3 r-integration to a finite volume V , then the Dirac-δ of vanishing argument receives a natural interpretation as δ (3) (k⃗ = ⃗0) ≈ ∫

V

V d3 r exp(i ⃗0 ⋅ r⃗) = . 3 (2π) (2π)3

(2.83)

⃗ in Eq. (2.80) can Applying Eq. (2.83) backwards, the term proportional to δ (3) (0) 1 be identified as a sum over the zero-point energies 2 ω of the “oscillator” modes, H0 =

2 2 1 1 ⃗. ∑ ∑ ω = ∑ ∑ ∣k∣ 2 k⃗ λ=1 2 k⃗ λ=1

(2.84)

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The zero-point Hamiltonian operator is instrumental in the treatment of the Casimir interaction between plates (see Chap. 8 of Ref. [81]). The zero-point energy is relevant for configurations with nontrivial boundary conditions. One cannot always simply shift the reference point of the energy scale and ignore this term. By contrast, in situations with an undistorted vacuum structure, it is indeed permissible to leave out the zero-point energy from the Hamiltonian of the electromagnetic field. In the absence of these subtle effects caused by external conditions, one can indeed subtract the vacuum-expectation value of the Hamiltonian in Eq. (2.80) (which is proportional to the Dirac-δ of vanishing argument) and rewrite the Hamiltonian as the normal-ordered operator 2

⃗ aλ (k) ⃗ . ∶ HF ∶ = HF − ⟨0∣HF ∣0⟩ = ∑ ∫ d3 k ω a+λ (k)

(2.85)

λ=1

The vacuum expectation value ⟨0∣HF ∣0⟩ of HF has been explicitly subtracted. The operator corresponding to the field momentum (Poynting vector operator) is given by 2

⃗ aλ (k) ⃗ . ⃗ × B) ⃗ = ∑ ∫ d3 k k⃗ a+ (k) S⃗ = ∫ d3 r (E λ

(2.86)

λ=1

+ ⃗ The norm of the Fock-space vector ∣1kλ ⃗ ⟩ = aλ (k)∣0⟩, which describes a state of ⃗ is finally calculated as the electromagnetic field with one photon in the mode kλ, follows, 2 (3) ⃗ ⃗ ⃗ + ⃗ ⃗ . ∣∣ ∣1kλ (k − k) ⟨0∣0⟩ = δ (3) (0) (2.87) ⃗ ⟩ ∣∣ = ⟨0∣aλ (k) aλ (k)∣0⟩ = δ √ (3) (⃗ 0), which is of course ill-defined, A normalized state vector is therefore ∣1kλ ⃗ ⟩/ δ but the singular Dirac-δ term again finds a natural explanation within a calculation that uses a finite normalization volume, as outlined in Eq. (2.83). The energy expectation value of a one-photon state ∣1kλ ⃗ ⟩, calculated with the free Hamiltonian (2.85), reads as follows, 2

3 ′ + ⃗′ + ⃗ ⃗′ ⃗ ⟨1kλ ⃗ ∣ ∶ HF ∶ ∣1kλ ⃗ ⟩ = ∑ ∫ d k ω ⟨0∣ [aλ (k), aλ′ (k )] [aλ′ (k ), aλ (k)] ∣0⟩ λ′ =1

2 = ω δ (3) (⃗0) = ω ∣∣ ∣1kλ ⃗ ⟩ ∣∣ .

(2.88)

Again, the somewhat intriguing formally infinite norm of one-photon states disappears as we employ a finite normalization volume V . 2.4.3

Discretized Formulation of the Field Operators

We shall now discuss the transition to the discretized representation with an explicit normalization volume V for the electromagnetic field modes. In Eq. (2.81), we already have encountered terms like δ (3) (⃗0), which can only be interpreted consistently if one transforms the expressions to a finite normalization volume. We will therefore treat alternative, discretized representations of the field operators in the

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current chapter and start with the vector potential operator, whose form is given by 1 1 ⃗ (aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ) , A⃗⊥ (t, r⃗) = ∑ √ √ ˆλ (k) λ 2ω V ⃗ kλ

(2.89)

⃗ The normalization of the discretized annihilation and crewhere again ω = k 0 = ∣k∣. ⃗ and a+ (k) ⃗ remains to be discussed. We observe the different ation operators, aλ (k) λ font used for the discrete-space and continuum-space creation and annihilation operators. The summation over k⃗ is to be understood as a multi-index governing the summations over the allowed quantum numbers n1 , n2 , and n3 , which are related ⃗ /L. Applying the transformation given in to the components of k⃗ as in k⃗ = 2π n d3 k 1 Eq. (2.81), ∫ (2π)3 → V ∑k⃗ , the vector potential operator transforms into √ d3 k V 1 ⃗ (aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ) . (2.90) ⃗ √ √ ˆλ (k) A⊥ (t, r⃗) → ∑ ∫ λ 3 3 (2π) λ 2ω (2π) The normalization volume V is naturally absorbed into the discretized annihilation ⃗ and a+ (k). ⃗ We recall the commutation relation of the and creation operators aλ (k) λ fundamental creation and annihilation operators in continuum space (2.70), ⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ . [aλ (k), λ

(2.91)

The transition to the discretized representation preserves physical units if we identify the peak of the Dirac-δ function with a large normalization volume, inspired by Eq. (2.83), √ √ V V (3) ⃗ ⃗ ′ ⃗ ⃗ . δk⃗k⃗′ aλ (k) → aλ (k) (2.92) δ (k − k ) → 3 (2π) (2π)3 In this case, the discrete-space commutators read as follows, ⃗ a+′ (k⃗′ )] = δ⃗ ⃗′ δλλ′ . [aλ (k), λ kk

(2.93)

⃗ which is different from the one Again, we observe the font in the expression aλ (k) ⃗ used in the continuum-space operator aλ (k). We now reassure ourselves of the consistency of the commutator relation (2.91) with the transition (2.92). On one hand, we have the following relation, using Eq. (2.91), 3 ⃗ a+ (k⃗′ )] = ∫ d3 k δ(k⃗ − k⃗′ ) = 1 , ∫ d k [aλ (k), λ

(2.94)

while on the other hand, the transition to discrete space is mediated by Eq. (2.92), √ 2 ⎛ (2π)3 ⎞ V 3 + ⃗′ ⃗ ⃗ a+ (k⃗′ )] = ∑ δ⃗ ⃗′ = 1 . ( ) [aλ (k), ∫ d k [aλ (k), aλ (k )] → ∑ λ kk (2π)3 ⎝ k⃗ V ⎠ ⃗ k (2.95) The discrete-space version of the normal-ordered field Hamiltonian (2.85) reads as ⃗ aλ (k) ⃗ . ∶ HF ∶ = ∑ ω a+λ (k) ⃗ kλ

(2.96)

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(d) + ⃗ The discrete-space one-photon state ∣1kλ ⃗ ⟩ = aλ (k)∣0⟩, in contrast to Eq. (2.87), is normalized to unity, as an elementary calculation involving Eq. (2.92) shows. Its expectation value on the discrete Hamiltonian (2.96) is (d) (d) ⃗ [aλ (k), ⃗ a+′ (k⃗′ )]∣0⟩ ⟨1 ∣HF ∣1 ⟩ = ∑ ω ⟨0∣[aλ′ (k⃗′ ), a+ (k)] ⃗ kλ

⃗ kλ

λ

⃗ ′ λ′ k

λ

2 = ∑ ω δk⃗2k⃗′ δλλ ′ = ω.

(2.97)

⃗ ′ λ′ k

The discretized electric-field operator is obtained by differentiation of the vector potential and reads as, √ ω ⃗ (aλ (k) ⃗ e−i k⋅x − a+ (k) ⃗ ei k⋅x ) , ⃗⊥ (t, r⃗) = ∑ √i ˆλ (k) (2.98) E λ 2 V ⃗ kλ ⃗ r⃗) = ∇ ⃗ × A⃗⊥ (t, r⃗) can be given as follows, whereas the magnetic field operator B(t, ⃗ (aλ (k) ⃗ e−i k⋅x − a+ (k) ⃗ ei k⋅x ) . ⃗ r⃗) = ∑ √i √1 (k⃗ × ˆλ (k)) B(t, (2.99) λ 2ω V ⃗ kλ The discrete-space version of the commutator of the electric field and the vector potential is a generalization of Eq. (2.75), [E⊥i (t, r⃗), Aj⊥ (t, r⃗′ )] = ∑ ⃗ kλ

′ i ki kj ⃗ (δ ij − ) ei k⋅(⃗r−⃗r ) . V k⃗2

(2.100)

For V → ∞, the latter expression approaches the transverse Dirac-δ function. Finally, we should remember that the discrete as well as the continuum representations of the field operators have important applications. For the derivation of the Lamb shift, the continuum representation is preferable because we have to sum over a continuum of modes. For the ac Stark shift, to be discussed in Sec. 3.3.2, we need the discretized representation because only the laser mode is populated by a large number of photons. The normalization to a finite volume V helps in matching the result with the intensity of the incoming laser radiation. For the treatment of the laser-dressed Lamb shift [114], one needs both forms because both a laser field and the interaction with the continuum of modes of the radiation field need to be described simultaneously. 2.5 2.5.1

Interaction Picture and Phase Conventions Field Operators in the Schr¨ odinger and Interaction Pictures

The last part of the current chapter will be focused on two subtle points, which are normally not treated in detail in quantum field theory books, but which are very useful to know. One of these is connected with the transition from the Schr¨odinger picture to the interaction picture, and another one with certain normalizations and global phases of the field operators. It is known that quantum mechanics can be formulated in the Schr¨odinger and in the interaction picture. Likewise, it is possible to formulate second-quantized

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operators in both pictures. The operators correspond to the quantum mechanical observables, and the field Hamiltonian corresponds to the quantum mechanical Hamilton operator. In general, a Hamiltonian generates time translations, whereas the momentum operator generates spatial translations. Let us consider the expression exp(iHt) A exp(−iHt), where A is an operator and H is the Hamiltonian. If A commutes with H, then the expectation value of A in a particular quantum state is time-independent. If a complete set of commuting observables (including the Hamiltonian) is found, then we can identify completely specified eigenvectors. In the Heisenberg picture of quantum mechanics, the states are timeindependent, whereas the operators evolve with time. By contrast, in the Schr¨ odinger picture, the states evolve, whereas the operators are time-independent. The interaction picture interpolates between these viewpoints: The problem is reformulated so that the quantum states evolve only with the interaction Hamiltonian, as opposed to the complete Hamiltonian. It is useful to observe that the entire canonical formalism of quantum field theory (scattering theory, see Ref. [2]), actually uses the interaction picture without stating this fact explicitly. However, a complementary formulation of field operators in the Schr¨odinger picture is sometimes useful, especially when treating nonrelativistic problems. Let the time evolution of a system be governed by an unperturbed Hamiltonian H0 which, by assumption, carries no explicit time dependence. The operator A, by assumption, carries no explicit time dependence either. Under the action of H0 , a matrix element of the operator A acquires a time dependence according to [⟨a∣A∣b⟩] (t) = ⟨a∣ exp(i H0 t) A exp(−i H0 t)∣b⟩ .

(2.101)

The time evolution is a unitary transformation. In the Schr¨odinger picture, one attributes the time evolution exp(−i H0 t) to the wave function. So, in the Schr¨odinger picture, the unperturbed time evolution of the state vector is given by d ∣b(t)⟩ = e−i H0 t ∣b⟩ . (2.102) i ∣b(t)⟩ = H0 ∣b(t)⟩ , dt In the interaction picture, in contrast to the wave function, the operator A acquires a time dependence according to d i A(t) = [A(t), H0 ] , A(t) = exp(i H0 t) A exp(−i H0 t) . (2.103) dt If A has an additional explicit time dependence, then one has to add a partialderivative term on the right-hand side, i dA(t)/dt = [A(t), H0 ] + i ∂t A(t), but we shall not dwell on this further. Let us now add a further contribution (perturbation) H1 to the Hamiltonian, H = H0 + H1 .

(2.104)

Again, we assume that H1 carries no explicit time dependence. In many practical applications, H1 in itself has no time-dependence, but it is acquired via a timedependent transformation (action of H0 on H1 ). In the Schr¨odinger picture, the equation of motion for the state vector ∣ψ(t)⟩ is d i ∣ψ(t)⟩ = (H0 + H1 ) ∣ψ(t)⟩ . (2.105) dt

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The transition to the interaction picture is done by defining the wave function ∣ψI (t)⟩ according to ∣ψI (t)⟩ = exp(i H0 t) ∣ψ(t)⟩ .

(2.106)

If H1 vanishes, then ∣ψI (t)⟩ is time-independent. Inserting the ansatz (2.106) into (2.105), we obtain the equation of motion for ∣ψI (t)⟩, d ∣ψI (t)⟩ = exp(i H0 t) H1 exp(−i H0 t)∣ψI (t)⟩ = H1 (t) ∣ψI (t)⟩ . (2.107) dt Thus, in the interaction picture, the interaction Hamiltonian is manifestly timedependent even if H1 initially carries no time-dependence, i

HI (t) = exp(i H0 t) H1 exp(−i H0 t) .

(2.108)

Denoting by T the time ordering, we observe that the following form of the timeordered time-evolution operator solves Eq. (2.107), ∣ψI (t)⟩ = T [exp (−i ∫

t 0

dt′ HI (t′ ))] ∣ψI (0)⟩ .

(2.109)

The interaction picture is intermediate between the Schr¨odinger and Heisenberg pictures; in the latter case, the entire time evolution due to both H0 and H1 is ascribed to the operator, and the state vectors are time-independent. We should illustrate these general statements by way of example. For a continuum of field modes, we recall the field Hamiltonian, ⃗ aλ (k) ⃗ , ∶ HF ∶ = ∫ d3 k ω a+λ (k)

⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ . [aλ (k), λ

(2.110)

We identify the time-independent fundamental creation and annihilation opera⃗ = aλ (k, ⃗ t = 0) and tors with the time-dependent ones at time t = 0, i.e., aλ (k) + ⃗ + ⃗ aλ (k) = aλ (k, t = 0). We then start from the Heisenberg equation of motion, ⃗ t) induced by i(dA(t)/dt) = [A(t), H0 ], and calculate the time dependence of aλ (k, ⃗ t = 0) = aλ (k), ⃗ the free normal-ordered Hamiltonian, with the initial condition aλ (k, i

2 d ⃗ t)∣ = [aλ (k, ⃗ t = 0), ∑ ∫ d3 k ′ ω ′ a+′ (k⃗′ ) aλ′ (k⃗′ )] = ω aλ (k) ⃗ , aλ (k, λ dt t=0 λ′ =1

(2.111)

⃗ The equation of motion is immediately where ω ′ = ∣k⃗′ ∣ and we recall that ω = ∣k∣. solved by ⃗ t) = aλ (k) ⃗ e−i ω t , aλ (k,

⃗ t) = aλ (k) ⃗ ei ω t . a+λ (k,

(2.112)

The exponential factor (dependence on time) of the fundamental creation and annihilation operators is thus identified as the time dependence of the operators in the interaction picture, due to the action of the free field Hamiltonian. This consideration allows us to identify the “usual” field operators for the electromagnetic vector potentials given in Eqs. (2.71) and (2.89) as interaction-picture field operators. Conversely, the transformation from the interaction picture to the Schr¨ odinger picture can be accomplished very easily, by setting the time equal to zero.

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Integration Measure and Phase Conventions

A second subtle point concerns the normalizations and phases of field operators, which are not uniformly assigned in the literature. We shall demonstrate that the difference in the integration measures can be absorbed in different conventions for the field commutators. The commutators of the fundamental field operators we have used so far, ⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ , [aλ (k), λ

⃗ a+′ (k⃗′ )] = δ⃗ ⃗′ δλλ′ , [aλ (k), λ kk

(2.113)

display a particularly simple structure with regard to the prefactors. This structure is connected with the integration or summation “measure” used in the field operators (2.71) and (2.89), ∫ √

d3 k 2ω (2π)3

,

1 1 ∑√ √ , V 2ω ⃗ kλ

(2.114)

in the continuum and discrete representations, respectively. Our conventions coincide with those adopted in Ref. [115]. Otherwise, one may point out that the integration measure d3 k d4 k = δ(k 2 ) Θ(k 0 ) , ∫ (2π)3 (2 ω) (2π)3

k 2 = (k 0 )2 − k⃗ 2 = ω 2 − k⃗ 2 ,

(2.115)

has the advantage of being Lorentz-invariant, in contrast to the integration measure in Eq. (2.114). We also recall that by convention, in three-dimensional integrations ⃗ while ω = k 0 is being integrated over in the four-dimensional integral one sets ω = ∣k∣, in Eq. (2.115). We may now transform the fundamental creation and annihilation operators as follows [see Eq. (3.110) of Ref. [2]], ⃗ =√ aλ (k)

1 (2π)3

√

2ω

akλ ⃗ ,

⃗ =√ a+λ (k)

1 (2π)3

√

2ω

a+kλ ⃗ .

(2.116)

Then, 3 (3) ⃗ ⃗ ′ + (k − k ) δλλ′ . [akλ ⃗ , ak ⃗ ′ λ′ ] = (2 ω) (2π) δ

(2.117)

The vector potential operator for the transverse modes reads as A⃗⊥ (t, r⃗) = ∑ ∫ λ

d3 k ⃗ (a⃗ e−i k⋅x + a+⃗ ei k⋅x ) . ˆλ (k) kλ kλ (2π)3 (2 ω)

(2.118)

The operators for the electric and magnetic field follow suit, incorporating the Lorentz-invariant integration measure (2.115). In fully relativistic calculations, it is therefore preferable to use the conventions of Eq. (2.117). In nonrelativistic calculations, the simple structure of the prefactors in Eq. (2.113) is very appealing. It corresponds to the “harmonic-oscillator style” commutator of the discrete-space fundamental commutators given in the rightmost equation in (2.113). Another change in the conventions, which is sometimes useful, concerns the phase of the field operators. In Ref. [113], the authors start formulating the quantized electromagnetic theory from the electric field operator. Naturally, one would

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like to have unit prefactors (no imaginary unit) in front of the fundamental creation and annihilation operators, when considering the “most fundamental” field. This can be achieved by transforming the phases of the fundamental operators according to + akλ a+kλ ⃗ → −i Akλ ⃗ , ⃗ → i Akλ ⃗ .

(2.119)

+ The fundamental commutator is left invariant, i.e., [Akλ ⃗ , Ak ⃗k ⃗ ′ δλλ′ . The ⃗ ′ λ′ ] = δk electric field operator then has a certain similarity to the position operator for a harmonic oscillator, √ ω 1 ⃗ (A⃗ e−i k⋅x + A+⃗ ei k⋅x ) . ⃗ ˆλ (k) (2.120) E⊥ (t, r⃗) = ∑ √ kλ kλ 2 V ⃗ kλ

The vector potential operator now involves a number of imaginary units, 1 1 ⃗ (−i A⃗ e−i k⋅x + i A+⃗ ei k⋅x ) . A⃗⊥ (t, r⃗) = ∑ √ √ ˆλ (k) kλ kλ 2ω V ⃗ kλ

(2.121)

The operator for the magnetic field then reads as ⃗ (A⃗ e−i k⋅x + A+⃗ ei k⋅x ) . ⃗ r⃗) = ∑ √1 √1 (k⃗ × ˆλ (k)) B(t, kλ kλ 2ω V ⃗ kλ

(2.122)

The phase conventions have no consequences for the calculation of physical processes but severely alter the course of intermediate steps in calculations. 2.6

Further Thoughts

Here are some suggestions for further thought. (1) Field Hamiltonian. Verify that the field Hamiltonian given in Eq. (2.77) is time-independent, either by calculating its vanishing time derivative, or by first restoring the time dependence of the field operators and verifying that it cancels. (2) Discretized Field Operator. Repeat all the derivations in Sec. 2.4.2, ranging from Eq. (2.71) to Eq. (2.88), with the discretized version of the field operators given in Sec. 2.4.3. (3) Field Commutator. Based on Eq. (2.60), which we recall for convenience, [E⊥i (t, r⃗), Aj⊥ (t, r⃗′ )] = i δ ⊥,ij (⃗ r − r⃗′ ) ,

(2.123)

show that the commutator of the electric and magnetic fields is [E⊥i (t, r⃗), B j (t, r⃗′ )] = −i jk` ∇k δ ⊥,ij (⃗ r − r⃗′ ) ,

(2.124)

i.e., verify Eq. (2.76). Can one measure the electric and magnetic field at a given point in space simultaneously, and if not, which components of the electric and magnetic fields fail to commute?

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(4) Field Commutator. Complete the missing intermediate steps in Eq. (2.100), i.e., in the derivation of the field commutator [E⊥i (t, r⃗), Aj⊥ (t, r⃗′ )] in the discrete representation. (5) Norm of One-Photon State. Use Eq. (2.93) for the commutator of the discrete-space photon creation and annihilation operators, ⃗ a+′ (k⃗′ )] = δ⃗ ⃗′ δλλ′ . [aλ (k), λ kk

(2.125)

In order to show that the one-photon state in the discrete representation is normalized to unity, 2 ⃗ + ⃗ ∣∣ ∣1kλ ⃗ ⟩ ∣∣ = ⟨0∣aλ (k) aλ (k)∣0⟩ = 1 .

(2.126)

Convince yourself that the problematic infinite norm in the continuum representation, given in Eq. (2.87), 2 (3) ⃗ ⃗ + ⃗ ∣∣ ∣1kλ (0) , ⃗ ⟩ ∣∣ = ⟨0∣aλ (k) aλ (k)∣0⟩ = δ

(2.127)

can be avoided for a finite normalization volume V . (6) Missing Steps. Complete the missing intermediate steps in Eq. (2.111), using the relation [AB, C] = A [B, C] + [A, C]B. Then, repeat the derivation using the discrete-space version of the field operators and Hamiltonian. ̵ 0 and c in the con(7) Missing SI Units. Restore the missing prefactors of h, tinuum and discrete-space versions of the electromagnetic field operators. Your answer should look like √ 3 2 ̵ h d k ⃗r ⃗r ⃗ (aλ (k) ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ ⃗ r) = ∑ ∫ √ ˆλ (k) + a+λ (k) ) , (2.128a) A(⃗ 3 2 ω 0 (2π) λ=1 √ 2 ̵ω d3 k h ⃗r ⃗r ⃗ (i aλ (k) ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ ⃗ E(⃗ r) = ∑ ∫ √ ˆλ (k) − i a+λ (k) ) , (2.128b) 3 2 0 (2π) λ=1 √ 2 ̵ d3 k h ⃗r ⃗r ⃗ (i aλ (k) ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ ⃗ (k⃗ × ˆλ (k)) B(⃗ r) = ∑ √ − i a+λ (k) ), 3 2 ω 0 (2π) λ=1 (2.128c) ⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ . The discrete representation in the where [aλ (k), λ Schr¨ odinger picture should read as √ ̵ h ⃗r ⃗r ⃗ (aλ (k) ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ ⃗ ˆλ (k) + a+λ (k) ), (2.129a) A(⃗ r) = ∑ 2  ω V 0 ⃗ kλ √ ̵ω h ⃗r ⃗r ⃗ (i aλ (k) ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ ⃗ E(⃗ r) = ∑ ˆλ (k) − i a+λ (k) ), (2.129b) 2 0 V ⃗ kλ √ ̵ h ⃗r ⃗r ⃗ (i aλ (k) ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ ⃗ B(⃗ r) = ∑ (k⃗ × ˆλ (k)) − i a+λ (k) ), (2.129c) 2  ω V 0 ⃗ kλ ⃗ a+′ (k⃗′ )] = δ⃗ ⃗′ δλλ′ . where [aλ (k), λ kk

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(8) Dimensional Analysis of QED. Consider the natural unit system introduced in Sec. 2.3.2 and consider the QED Lagrangian in D spatial and one time dimension. Show that charge is dimensionless in natural units only if D = 3. (This consideration has important consequences for dimensional regularization introduced in Chap. 10.)

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Chapter 3

Time-Ordered Perturbations

3.1

Overview

A canonical quantum mechanics course covers a number of problems such as the one-dimensional harmonic oscillator, or first- and second-order perturbation theory. For the current endeavor, we need to recall in more detail a few special chapters, related to atomic physics. This will be necessary in order to tackle a few more problems which are of prime relevance later for all those atomic physics processes, which cannot be described without an understanding of field quantization. The purpose of the current chapter is to first give a coherent description of three aspects which are not normally treated in depth in a canonical quantum mechanics course. Specifically, in Sec. 3.2, we discuss time-ordered perturbation theory, and the derivation of Fermi’s1 Golden Rule. We proceed to a derivation of decay rates of excited atomic states due to the interaction with the quantized radiation field. In Sec. 3.3, the dynamic (ac) Stark shift is calculated using a Schr¨odinger-picture operator for the electromagnetic vector potential. The zero-frequency (static) limit of the dynamic Stark shift (in this limit, called the dc Stark shift) is discussed in Sec. 3.4. Surprisingly, the energy perturbation due to the dc Stark shift allows for an analysis up to very high orders in perturbation theory. Perhaps even more surprisingly, the perturbation series turns out to be a divergent series in large order. As we shall see in Sec. 3.4, this aspect has important consequences and can even be generalized to field theories such as quantum electrodynamics (QED). 3.2

Time-Ordered Perturbations and Fermi’s Golden Rule

3.2.1

Derivation of Fermi’s Golden Rule

Fermi’s golden rule is a nontrivial result that can be derived from time-ordered perturbation theory, and it is rather interesting to consider its derivation in detail. ̵ and c to We temporarily revert to SI mksA units; the number of factors of h be supplemented is not excessive. We start from the time-dependent Schr¨odinger 1 Enrico

Fermi (1901–1954). 49

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equation ̵ ∂ ψ(t, r⃗) . H ψ(t, r⃗) = ih ∂t The Hamiltonian is composed of an unperturbed part and a perturbation, H = H0 + H1 ,

(3.1)

(3.2)

where H1 is the interaction Hamiltonian in the Schr¨odinger picture (see Sec. 2.5.1). The operator H1 is used as defined in Eq. (2.104), not as outlined in Eq. (2.108), where it automatically acquires an explicit time dependence. As we shall see in Sec. 3.2.3, the use of Schr¨ odinger-picture field operators in the description of the interaction with the electromagnetic field makes the calculation of decay rates of atomic states easier. But first, let us dwell on a more general setting; we suppose that a complete set of eigenstates is available, H0 ψm (⃗ r) = Em ψm (⃗ r) ,

⟨ψm ∣ψn ⟩ = δmn .

(3.3)

The unperturbed energy eigenvalues are denoted as Em . The states acquire a timedependence according to ̵

ψ(t, r⃗) = ∑ am (t) ψm (⃗ r) e−i Em t/h .

(3.4)

m

̵ is due only to the action of the free HamiltoThe time-dependence exp(−i Em t/h) nian. Any additional time-dependence due to the interaction is described by the expansion coefficients am (t). One must project onto a complete set of states and take into account Eq. (3.3), establishing that the state evolution is described by the equation ̵ ih

dak (t) ̵ = ∑ ⟨ψk ∣H1 ∣ψm ⟩ am (t) ei (Ek −Em ) t/h . dt m

(3.5)

Time-dependent perturbation theory is based on an expansion of the timedependence of the am (t) in powers of H1 . From this definition, it is immediately (p+1) clear that for a small perturbation H1 , a better approximation ak (t) to ak (t) of (p) order p + 1 is obtained by integrating the approximation ak (t) of order p, ̵ ih

(p+1)

dak

(t)

dt

̵

(p) = ∑ ⟨ψk ∣H1 ∣ψm ⟩ am (t) ei (Ek −Em ) t/h .

(3.6)

m

This is because each time we integrate the right-hand side, one more power of H1 (p) appears in the result for ak (t). The approximation am (t) is of order g p if H1 is proportional to a coupling parameter g. The zeroth-order approximation is obtained by letting H1 → 0 on the right-hand side of Eq. (3.6), (0)

dam (t) = 0, a(0) (3.7) m (t) = δim , dt which implies that we assume the system to be initially prepared in the state ∣ψi ⟩ at time t = 0.

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Time-Ordered Perturbations

The first-order approximation for the final-state amplitude (denoted by the subscript f ) is given as ̵ a (t) = ∑ ∫ ih f

t

(1)

m

=∫

t 0

0

̵

′ i (Ef −Em ) t /h dt′ ⟨ψf ∣H1 ∣ψm ⟩ a(0) m (t ) e ′

̵

dt′ ⟨ψf ∣H1 ∣ψi ⟩ ei (Ef −Ei ) t /h . ′

(3.8)

Integrating Eq. (3.8) to first order, one obtains ̵ a (t) = 2 ⟨ψf ∣H1 ∣ψi ⟩ exp ( ih f (1)

̵ i (Ef − Ei ) t sin [(Ef − Ei ) t/(2h)] ) . ̵ ̵ 2h (Ef − Ei )/h

(3.9)

2

(1)

The time-dependent probability Pf (t) = ∣af (t)∣ for the transition from state ∣ψi ⟩ to state ∣ψf ⟩ then reads as (1)

2

Pf (t) = ∣af (t)∣ = 4

̵ ∣⟨ψf ∣H1 ∣ψi ⟩∣ sin2 [(Ef − Ei ) t/(2h)] . 2 ̵h2 ̵ [(Ef − Ei )/h] 2

(3.10)

Expanding for t → 0, we would tentatively obtain that Pf (t) goes quadratically for small t. However, that picture would be misleading. While the literature is abundant with discussions of Fermi’s golden rule, the details regarding the elimination of the quadratic time-dependence are often obscured. We should first remember that most practically relevant quantum mechanical transitions actually involve a discrete state and a continuum; Fermi’s golden rule first and foremost applies to transitions from discrete states coupled to a continuum. Let us suppose that a bound electron gets ionized upon absorption of an energetic photon. The continuum electron in the final state lies in the positive-energy continuum of the Schr¨ odinger spectrum. Let us also suppose that a bound electron, initially in an excited bound state, makes a transition to the ground state, emitting a photon. While the electron undergoes a transition from one bound state to the other, one might think that neither the initial nor the final state are in a continuum. However, that is not the case. The final state is in a continuum, albeit not in the continuum of the Schr¨ odinger Hamiltonian of the atom, but of the field Hamiltonian of the electromagnetic field. If the final state is in a continuum, then there are quantum states which are energetically displaced only infinitesimally from the actual final state of the process. In the case of a bound-bound transition, the infinitesimally displaced states would be those with near-resonant photons. Furthermore, while not obvious, energy conservation actually holds during a quantum transition from a bound state to a continuum, if the energies of all involved particles are considered. For an atomic deexcitation process, the energy of the initial excited state is equal to the energy of the emitted photon plus the energy of the final state; the latter can be the ground state. Another example can be indicated as follows: During an ionization process, the energy of the initial electron plus the energy of the absorbed photon is equal to the energy of the final continuum electron.

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The essence of Fermi’s golden rule is as follows: We first observe that the ex2 ̵ is highly peaked near Ef ≈ Ei . pression sin2 (ω t) / (ω t) with ω = (Ef − Ei )/h Our assumption of the availability of a continuum of states around the final state forces us to collect the contribution from neighboring states. Energy conservation demands that Ef ≈ Ei , because Ef is the energy of the final state. The energy Ef is calculated taking into account all of its constituent particles. Let us therefore assume that we know the density of states ρ(Ef ) = dn/dEf in an infinitesimal interval dEf around the final state, dn = ρ(Ef ) dEf ,

(3.11)

where dEf measures the interval in the continuum part of the spectrum. Then, the transition rate is obtained as the integral over the available final states, Wf i =

1 ∫ dn Pf (t) . t

(3.12)

This is the essence of the golden rule: For large t, the transition probability Pf (t) is highly peaked near the particular final state ∣ψf ⟩ that fulfills the energy conservation condition. So, we can in fact sum or integrate over all final states dn in the immediate vicinity of the energy-conserving final state, and this integral can be extended to infinite integration limits in the energy domain because as t → ∞, we are in fact picking up a contribution only from a very limited energy region. Furthermore, in this limit, the energy conservation condition has to be fulfilled exactly, and it is thus possible to pull the factor ⟨ψf ∣H1 ∣ψi ⟩ out of the integral in Eq. (3.12), provided the energy conservation condition is fulfilled. The resulting integral ∫

∞ −∞

dω

sin2 (ω t) = πt ω2

(3.13)

is proportional to t, not t2 . Alternatively, one can match the expression to a Dirac-δ function, sin2 (ω t) → π t δ(ω) , t → ∞. (3.14) ω2 Fermi’s golden rule is finally obtained by summing over all the available final states, realizing that the contributing states are those with Ef ≈ Ei because of the peaked structure of sin2 (ω t) /ω 2 near ω = 0, 1 1 ∫ dn Pf (t) = ∫ dEf ρ(Ef ) Pf (t) t t 2 ̵ ∣⟨ψf ∣H1 ∣ψi ⟩∣ sin2 [(Ef − Ei ) t/(2h)] 4 = ∫ dEf ρ(Ef ) 2 2 ̵ ̵ t h [(Ef − Ei )/h]

Wf i =

∞ 4 sin2 (ω t/2) 2 = ̵ ( ∣⟨ψf ∣H1 ∣ψi ⟩∣ ρ(Ef ) ∣ )∫ dω Ef =Ei ht ω2 −∞ 2π 2 = ̵ ( ∣⟨ψf ∣H1 ∣ψi ⟩∣ ρ(Ef ) ∣ ). Ef =Ei h

(3.15)

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We have extracted the value of the integrand in front of the peak as a prefactor. It is easy to convince oneself that the result, Fermi’s golden rule, 2π 2 ), (3.16) Wf i = ̵ ( ∣⟨ψf ∣H1 ∣ψi ⟩∣ ρ(Ef ) ∣ Ef =Ei h has the right physical dimension: Namely, ρ(Ef ) has dimension of inverse energy, because it counts the available states per unit energy interval, per Eq. (3.11). Then, the square of the energy (the dimension of ∣ ⟨ψf ∣H1 ∣ψi ⟩ ∣2 ) is divided by the product ̵ and energy (contained in ρ), and h ̵ has dimension of the product of energy and of h time. So, Wf i has dimension of inverse time, which is correct for a transition rate. 3.2.2

Fermi’s Golden Rule and Nuclear Beta Decay

As an example, let us consider nuclear beta decay. This example will provide us with a straightforward generalization of Fermi’s golden rule, namely, a situation with two particles in the final state, where available final states are degenerate because the transition energy may be distributed among the two particles. In beta decay, a nucleus with charge number Z and neutron number N undergoes a transition (Z, N ) → (Z + 1, N − 1) with a neutron becoming a proton, emitting an electron and an electron antineutrino, n → p + e− + ν¯e .

(3.17)

Beta decay is mediated by weak interactions, whose energy scale is a lot higher than that of a typical nuclear transition. We shall therefore assume that the Hamiltonian matrix element in question is energy-independent and given as GF M ⟨ψf ∣H1 ∣ψi ⟩ = , (3.18) V where GF is the Fermi coupling constant, M is an invariant matrix element, and V is a (large) normalization volume. The matrix element M describes the transition of the nucleus, i.e., the transition of neutron to proton. The recoil momentum and recoil energy of the heavy nucleus will be neglected in the final state of the process. The two emitted particles are the electron and the electron antineutrino. Let the momentum of the electron be p⃗e , and the momentum of the antineutrino be p⃗ν . Energy conservation implies that the initial excitation energy E0 of the nucleus is equal to the energy Ee ≈ c ∣⃗ pe ∣ = c pe of the outgoing electron, plus the energy Eν ≈ c pν of the outgoing antineutrino. Both electron and antineutrino masses will be neglected in the following. Energy conservation implies that E0 = Ee + c pν ≈ c pe + c pν .

(3.19)

The recoil energy of the decaying nucleus will be neglected in the following. The density of two-particle final states in the normalization volume V can be written as follows, d6 n =

V d3 pe V d3 pν ̵ 3 (2π h) ̵ 3, (2π h)

(3.20)

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based on the wisdom that every final quantum state occupies a phase-space volume ̵ [see also Eq. (2.81)]. Because of rotational symmetry, we can write after the h integration over the solid angles Ωe and Ων , 4π 3 2 3 2 (3.21) ∫ d pν = 3 (E0 − Ee ) dEν . ∫ d pe = 4πpe dpe , c Ων Ωe In the normalization volume V , we have V d3 pe V d3 pν V 4πp2e dpe V 4π(E0 − Ee )2 dEν d2 n = ∫ ∫ ̵ 3 Ων (2π h) ̵ 3 = (2π h) ̵ 3 ̵ 3 c3 (2π h) Ωe (2π h) V2 2 2 (3.22) ̵ 6 c3 pe (E0 − Ee ) dpe dEν = dρ(Ee ) dEν , 4π 4 h where the density of electron states is given by V2 dρ(Ee ) dpe = 4 ̵ 6 3 p2e (E0 − Ee )2 dpe . dρ(Ee ) = (3.23) dpe 4π h c The density d2 n is double-differential; the final electron momentum still needs to be integrated over. An increase in the energy of the electron is compensated by a decrease in the energy of the emitted antineutrino, and we have written the doubledifferential density of states so that the dependence on the electron energy is singled out and can be integrated over separately. In the sense of Eq. (3.11), we have for the single-differential density of states, E0 /c dρ(Ee ) , (3.24) dn = dEν ∫ dpe dpe 0 where E0 /c is the highest allowed electron momentum. Now, we use Fermi’s golden rule (3.16) in the form =

2π 2 dn ). ∣ Wf i = ̵ ( ∣⟨ψf ∣H1 ∣ψi ⟩∣ dEν E0 =cpe +cpν h

(3.25)

The volume V 2 cancels against the V −2 factor from the matrix element of H1 given in Eq. (3.18), and we obtain 2

E0 /c 2π GF ∣M∣ 1 Wf i = ̵ ∫ ) V 2 4 ̵ 6 3 p2e (E0 − Ee )2 dpe ( V 4π h c h 0 E0 /c

G2F 2 2 2 ̵ 7 c3 ∣M∣ pe (E0 − Ee ) . 2π 3 h 0 The integral over the available momenta evaluates to E0 /c E5 p2e (E0 − Ee )2 dpe = 0 3 , ∫ 30 c 0 where Ee = c pe . The total decay rate, within our approximations, is thus, 1 G2 ∣M∣2 E 5 Wf i = ∫ dWf i = = F 3 ̵ 6 0̵ , τ 60 π (hc) h =∫

dpe

(3.26)

(3.27)

(3.28)

where τ is the lifetime of the nucleus against beta decay. The functional dependence on E0 has been confirmed in observations of beta decay lifetimes over many orders of magnitude and is commonly referred to as “Sargent’s2 Law”. 2 Bernice

Weldon Sargent (1904–1967).

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3.2.3

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Fermi’s Golden Rule and Atomic Decay Rates

One of the most paradigmatic applications of Fermi’s golden rule within atomic physics consists in the derivation of the decay rate of an excited atomic state against one-photon (electric-dipole) decay to a state of lower energy. We start from the electric field operator given in Eq. (2.129b) in the Schr¨odinger picture, √ ̵ ω⃗ h ⃗r ⃗r k ⃗ (i aλ (k) ⃗ ei k⋅⃗ ⃗ e−i k⋅⃗ ⃗ r) = ∑ ˆλ (k) − i a+λ (k) ). (3.29) E(⃗ 2 V 0 ⃗ kλ ⃗ in contrast to the operators aλ (k), ⃗ (We again recall that the field operators aλ (k), pertain to the discrete representation.) The interaction Hamiltonian that describes the dipole coupling of the electron to the electric field is ⃗ r) ≈ −e r⃗ ⋅ E( ⃗ 0) ⃗ H1 = − e r⃗ ⋅ E(⃗ √ ̵ ω⃗ h k ⃗ (i aλ (k) ⃗ − i a+ (k)) ⃗ , (−e r⃗ ⋅ ˆλ (k)) =∑ λ 2  V 0 ⃗ kλ

(3.30)

where we apply the dipole approximation k⃗ ⋅ r⃗ ≪ 1. For typical atomic transitions, the wavelength of the emitted (visible) light is in the range of some 1000 ˚ A, whereas the atomic dimensions are in the range of a Bohr radius (on the order of 1 ˚ A). The wavelength emitted by the atom is much longer than the dimension of the atom ⃗ r) ≈ E( ⃗ ⃗0), where we assume the atom itself, which justifies the approximation E(⃗ to be located at the origin. The most interesting part of the calculation concerns the proper identification of the quantum states involved in the process. Let ∣ψi ⟩ = ∣φi ⟩ ⊗ ∣0⟩ be the initial excited atomic state, where ∣φi ⟩ is the initial atomic state, and ∣0⟩ is vacuum state of the electromagnetic field (no photons present). The final state is ∣ψf ⟩ = ∣φf ⟩ ⊗ ∣1kλ ⃗ ⟩, + ⟩ = a where ∣ψf ⟩ is the final (energetically lower) state of the process, and ∣1kλ ⃗ ⃗ ∣0⟩ kλ is a one-photon state of wave vector k⃗ and polarization λ. The unperturbed Hamiltonian is the sum of the atomic Hamiltonian and the unperturbed electromagnetic field Hamiltonian, (0) ̵ a+ (k) ⃗ aλ (k) ⃗ , H0 = ∑ Em ∣φm ⟩ ⟨φm ∣ + ∑ hω λ

(3.31)

⃗ kλ

m

(0)

where the unperturbed energies of the atomic states are denoted as Em . Because the energy of the product state (atom ⊗ field) before and after the interaction has (0) to be the same, the photon has to be emitted exactly on resonance, Ei = Ei = Ef = (0) ̵ Ef + h ω. Here, Ei and Ef are the energies of the initial and final product states of (0)

(0)

the atom and field, and Ei and Ef are the energies of the atomic states involved in the process. As already stressed in Sec. 3.2.1, energy conservation Ei = Ef holds provided we count the energies of all involved particles in the initial and final states, and that includes the emitted photons.

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For the interaction, the photon creation operator in the electric-field operator (2.129b) leads to the nonvanishing matrix element √ ̵ω h ⃗ φi ⟩ . ⟨φf ∣⃗ ⟨φf , 1kλ r ⋅ ˆλ (k)∣ (3.32) ⃗ ∣H1 ∣ φi , 0⟩ = i e 2 0 V The density of available final states for the photon is d3 n = d2 ρ(f ) =

V d3 p V d3 k V k 2 dk d2 Ω V k2 2 = = = ̵ 3 (2π)3 ̵ d Ω dEk , (2π h) (2π)3 (2π)3 hc

(3.33a)

V k2 2 ̵ d Ω, (2π)3 hc

(3.33b)

where d2 ρ(f ) is still differential in the solid angle into which the photon is emitted. ̵ k⃗ and the photon energy is Ek = hω ̵ =h ̵ c k where The photon momentum is p⃗ = h ⃗ The differential transition rate (decay rate) is k = ∣k∣. 2π 2 2 d2 Wf i = ̵ ∣⟨φf , 1kλ ⃗ ∣H1 ∣ φi , 0⟩∣ d ρ(f ) h 2 ̵ω 2π h 2 ⃗ φi ⟩∣2 V k ∣⟨φf ∣⃗ r ⋅ ˆλ (k)∣ (3.34) = ̵ e2 3 ̵ d Ω. 2 0 V (2π) hc h This expression is the differential rate for emission into the solid angle d2 Ω with definite photon polarization λ. In this derivation, we meticulously denote the degree of the differentials; the solid angle is thus denoted as d2 Ω = sin θ dθ dϕ, where θ is the polar, and ϕ is the azimuth angle. Note that we cannot simply integrate over dΩ → 4π because the photon polarization vectors ˆkλ ⃗ depend on the emission angle. In order to obtain the total rate, we have to sum over the polarizations of the emitted photon and integrate over the solid angle. Remembering in each step that ⃗ we obtain ω = ck = c ∣k∣, 2 ̵ω h 2π ⃗ φi ⟩∣2 V k ∣⟨φf ∣⃗ r ⋅ ˆλ (k)∣ Wf i = ̵ ∑ ∫ d2 Ω e2 ̵ h 2 0 V (2π)3 hc =

λ 2

2 2 e ω d2 Ω ⃗ φi ⟩∣2 (2π) (4π) k ∣⟨φf ∣⃗ r ⋅ ˆλ (k)∣ ∑∫ 3 ̵ 4π 0 hc 2 λ 4π 8π

= 2α ω k 2 ∑ ∫ λ

d2 Ω ⃗ φi ⟩∣2 , ∣⟨φf ∣⃗ r ⋅ ˆλ (k)∣ 4π

(3.35)

̵ is the fine-structure constant. The angular integral over the where α = e2 /(4π0 hc) polarization vectors is finally carried out with the help of Eq. (2.63), Wf i = 2α = 2α

ω3 d2 Ω kj k` j ` j` ⟨φ ∣r ∣ ⟩ ⟨φ ∣r ∣ ⟩ φ φ (δ − ) ∑ f i i f ∫ c2 ij 4π k⃗2 ω3 1 ⟨φf ∣ rj ∣ φi ⟩ ⟨φi ∣ rj ∣ φf ⟩ (1 − ) . 2 c 3

(3.36)

The result Wf i =

4α ω 3 2 ∣⟨φf ∣⃗ r∣ φi ⟩∣ , 3 c2

ω=

Ei − Ef , ̵ h

(3.37)

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is of fundamental importance for the description of atomic processes. In Eq. (3.37), we assume an average over the magnetic quantum numbers of the initial state, and a sum over the magnetic quantum numbers of the final state. One should still be aware that if any degeneracy persists in the final state (e.g., magnetic quantum numbers), then a sum over the degenerate final states needs to be performed in order to obtain the total decay rate. A potentially necessary averaging over initial-state configurations also is implied in Eq. (3.37). 3.3 3.3.1

Dynamic Stark Effect Way 1: Time-Dependent Perturbation Theory

The dynamic Stark effect (sometimes called ac Stark shift, where “ac” stands for “alternating current”) is an energy shift of an atomic reference state, caused by far off-resonant electromagnetic driving. Better stated, it is an energy shift of the atomic level, averaged over a period of the irradiating electromagnetic field, which is present if the frequency of the radiation cannot excite a transition to an excited state because it is far off resonance. The ac Stark shift, therefore, constitutes a perturbation and can be treated using time-dependent perturbation theory, but we have to go to second order (in contrast to Fermi’s golden rule, where a firstorder treatment was fully sufficient). Let us figure an analogy, in order to better understand the physical essence of the effect: If one drives a swing far off resonance, the child sitting on the swing will never get very high up, but will feel an average “quiver” energy due to an oscillatory motion of the swing. The position of the child, somewhat uncomfortably, will oscillate, and due to the average displacement from the equilibrium position of the swing, the child will acquire an average potential energy. By analogy, in a situation where an atom subject to far off-resonance electromagnetic radiation is unable to excite transitions to excited states, it feels a moderate “quiver” energy equal to the ac Stark shift. The most “classical” way of calculating this energy shift is given by timedependent perturbation theory, with a classical, oscillating, electric laser field. Thus, in this section, we rederive the classical expressions for the ac Stark shift, using a classical description of the laser field, inspired by Chap. 5 of Ref. [26]. The Hamiltonian is H = H0 + H1 (t) , H1 (t) = Hd cos (ωL t) ,

H0 = ∑ Ej ∣φj ⟩ ⟨φj ∣ , j

Hd = −e z EL ,

(3.38)

where Hd is the dipole interaction Hamiltonian for the laser field, polarized in the z direction. The atomic eigenstates have energies Ej , and the monochromatic laser light has an angular frequency ωL . We expect the wavelength of the laser radiation to be large compared to the spatial extent of the atomic wave function, so that the dipole approximation implicit in formulating Hd is justified (see also the pertinent discussion in Sec. 3.2.3).

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In contrast to our discussion of Fermi’s golden rule, we now use time-dependent perturbation theory in the interaction picture. Furthermore, because an energy shift is calculated, we have to employ adiabatic damping in the infinite past and in the infinite future. Similar considerations can be employed in the derivation of quantum electrodynamic energy shifts of atomic levels (for a review see, e.g., Chap. 2 of Ref. [116]). Thus, the interaction Hamiltonian is represented as ̵

̵

HI (, t) = e− ∣t∣ (ei H0 t/h H1 (t) e−i H0 t/h ) ,

(3.39)

where H0 is the unperturbed Hamiltonian of the atom. This interaction Hamiltonian is different from the one introduced in Eq. (2.108), in the sense that H1 (t) has an explicit time dependence (laser field oscillation). It also contains exponentials which transform it into the interaction picture, and an exponential prefactor which leads to adiabatic damping in the distant past (t → −∞) and the distant future (t → ∞), based on an infinitesimal positive damping parameter . In contrast to the derivation of Fermi’s golden rule, we do not consider a decay process, which starts at a definite time with a prepared state, but we consider the energy perturbation of a stationary state. For this reason, we need to introduce some adiabatic damping, so that the reference state can be used as an adiabatic eigenstate of the full Hamiltonian in the infinite future and infinite past, where the interaction is not present. Formally, the equation of motion of the wave function ∣φI (t)⟩ in the interaction picture, ̵ ih

∂ ∣φI (t)⟩ = HI (, t) ∣φI (t)⟩ , ∂t

(3.40)

is solved by the time-ordered exponential t i UI (, t) = T exp (− ̵ ∫ dt′ HI (, t)) , h −∞

(3.41)

where T denotes the time ordering, and UI (, t) is a unitary time-translation operator. Expanding up to second order in the interaction, we calculate UI (t) as follows, t i UI (, t) ≈ 1 − ̵ ∫ dt′ HI (, t′ ) + h −∞ t i = 1 − ̵ ∫ dt′ HI (, t′ ) − h −∞

2

t t (−i) ′ dt dt′′ T (HI (, t′ ) HI (, t′′ )) ∫ ∫ ̵ 2 −∞ 2! h −∞ ′

t t 1 ′ ′′ ′ ′′ ̵h2 ∫−∞ dt ∫−∞ dt HI (, t ) HI (, t ) .

(3.42)

We are now interested in the time evolution of the initial atomic reference state, which we denote as ∣φn ⟩, with H0 ∣φn ⟩ = En ∣φn ⟩. The time-dependent wave function in the interaction picture is ∣φI (t)⟩ = UI (, t) ∣φI (t = −∞)⟩ = ∑ cm (t) ∣φm ⟩ . m

(3.43)

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The initial boundary condition is cm (−∞) = δmn for the reference state ∣φn ⟩, with all other cm (−∞) vanishing in the infinite past. In the interaction picture, ∣φn ⟩ is stationary, because it is an eigenstate of the unperturbed Hamiltonian. We are interested in the projection cn (t) = ⟨φn ∣φI (t)⟩ = ⟨φn ∣UI (, t)∣φn ⟩ .

(3.44)

We now insert UI (, t) from Eq. (3.42) and calculate the matrix element ′

t t 1 cn (t) = ⟨φn ∣UI (, t)∣φn ⟩ = 1 − ̵ 2 ∫ dt′ ∫ dt′′ ⟨φn ∣HI (, t′ ) HI (, t′′ )∣φn ⟩ h −∞ −∞ ′ t t 1 (3.45) = 1 − ̵ 2 ∑ ∫ dt′ ∫ dt′′ ⟨φn ∣HI (, t′ )∣φm ⟩ ⟨φm ∣ HI (, t′′ )∣φn ⟩ , h m −∞ −∞

where m is a multi-index which counts all bound and continuum states of the unperturbed atom. Throughout this book, we will understand the summation symbol ∑ in a generalized sense, to include an integral over a continuous spectrum when appropriate. The first-order perturbation vanishes because the reference state ∣φn ⟩ cannot be coupled to itself in a dipole transition. Based on the result ̵

⟨φn ∣HI (, t)∣φm ⟩ = e− ∣t∣ ei (En −Em )t/h cos(ωL t) ⟨φn ∣Hd ∣φm ⟩ ,

(3.46)

the time integrals can be carried out without much difficulty. The final result for the matrix element cn (t) in second-order perturbation theory is cn (t) = 1 −

⟨φn ∣Hd ∣φm ⟩ ⟨φm ∣Hd ∣φn ⟩ ⎞ exp(2  t) ⎛ i ∑ ̵ ̵ ωL + ih ̵ ⎠, 2h ⎝ 4 m,± En − Em ± h

(3.47)

where we ignore terms that vanish in the limit  → 0. The easiest way to derive Eq. (3.47) is to assume that t < 0, which implies that t′ < 0, so that we do not have to split the integration intervals into domains with negative and positive integration variables. Furthermore, we note that differentiation with respect to t of the numerator exp(2  t) brings down another factor . The ± summation over the signs (±) in Eq. (3.47) covers two terms which differ in the sign of ωL in the denominator. The logarithmic derivative reads as ⎧ ⎫ ⎪ ⟨φn ∣Hd ∣φm ⟩ ⟨φm ∣Hd ∣φn ⟩ ⎪ ∂ ∆En ⎪ i ⎪ = −i ̵ , lim ( ln cn (t)) = lim ⎨− ̵ ∑ (3.48) ̵ ωL + ih ̵ ⎬ →0 ⎪ →0 ∂t ⎪ 4h m,± En − Em ± h h ⎪ ⎪ ⎩ ⎭ ̵ From now on, we shall assume that if we interpret cn (t) as cn (t) = exp(−i ∆En t/h). the limit  → 0 is taken at the end of a calculation, if an infinitesimal parameter  is introduced into an expression. A remark is in order. We had calculated our matrix element cn (t) = ⟨φn ∣UI (, t)∣φn ⟩. Had we projected onto a different final state, as in the matrix element ⟨φm ∣UI (, t)∣φn ⟩ with Em ≠ En , then we would not have obtained the infinitesimal  term in the denominator, as in Eq. (3.47). Then, the logarithmic derivative calculated below in Eq. (3.48) would be of order , and the result would vanish in the limit  → 0. This is a manifestation of a general principle that in adiabatic time-dependent perturbation theory, the final and initial states should be

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energetically degenerate in order to obtain nonvanishing final results for transition rates and energy shifts. We recall that energy conservation only holds if all emitted and absorbed quanta are counted in, so the degeneracy effectively amounts to an energy conservation condition. We have already encountered this principle in our derivation of Fermi’s golden rule. The dynamic Stark shift of the reference state ∣φn ⟩ is thus obtained as ∆En =

⟨φn ∣Hd ∣φm ⟩ ⟨φm ∣Hd ∣φn ⟩ 1 . ∑ ̵ L 4 m,± En − Em ± hω

(3.49)

In order to gain additional insight into the time evolution, we observe that the process is calculated in the interaction picture. Our ansatz for the wave function in the interaction picture has been given in Eq. (3.43), and we single out the component of the reference state ∣n⟩, ∣ψI (t)⟩ = UI (, t) ∣ψI (0)⟩ = ∑ cm (t) ∣φm ⟩ ≈ cn (t) ∣φn ⟩ .

(3.50)

m

This translates to a Schr¨ odinger-picture time evolution ̵

∣ψI (t)⟩ = e−i H0 t/h ∣ψI (t)⟩ ≈ e−i (En +∆En ) t ∣φn ⟩ .

(3.51)

This justifies, a posteriori, the identification cn (t) = exp(−i ∆En t) given in Eq. (3.48). Complementing the sum over the spectrum of virtual states in operator form in Eq. (3.49), we can express the dynamic Stark shift of the reference state ∣φn ⟩ as ∆En =

1 1 ∑ ⟨φn ∣Hd ̵ L Hd ∣ φ n ⟩ , 4 ± En − H0 ± hω

(3.52)

where Hd is time-independent [see Eq. (3.38)]. For a spherically symmetric reference state ∣φn ⟩, we define the dynamic dipole polarizability α(ωL ) as α(ωL ) = e2 ∑ ⟨φn ∣z ±

=

1 ̵ L z∣ φn ⟩ H0 − En ± hω

2

e 1 r ∑ ⟨φn ∣⃗ ̵ L r⃗∣ φn ⟩ , 3 ± H0 − En ± hω

(3.53)

which is not to be confused with the fine-structure constant α. The scalar product of the two coordinate vectors r⃗ is implied (for a complex atom, one has to sum over the coordinates r⃗a of all electrons labeled by the index a). The polarizability is normalized in such a way that it has the physical dimension of a dipole moment when multiplied by an electric field. This dynamically induced dipole is the dipole moment dynamically induced by the exciting, oscillatory electric field. In SI mksA units, the unit of polarizability is C2 m2 /J. The ac Stark shift of the reference state is finally expressed as ∆En = −

EL2 IL α(ωL ) = − α(ωL ) , 4 2 0 c

(3.54)

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where the laser field intensity of a plane electromagnetic wave in SI mksA units reads as IL =

1 0 c EL2 . 2

(3.55)

The real part of ∆En represents the shift of the energy eigenvalue, induced by the perturbation. If the frequency of the laser field is too low to excite a transition to an energetically upper state, then the imaginary part of the energy shift vanishes. ̵ L is located within a continuum of states in the However, if the energy En + hω spectrum of H0 , then the ac Stark shift develops an imaginary part. The imaginary part changes the amplitude modulus of ∣n⟩ with time, and represents a “width” of the state ∣φn ⟩ acquired in the laser field, due to ionization into the continuum, En = Re En −

i Γn . 2

(3.56)

This leads to the following evolution of the state vector with time, with an exponential factor exp(− h̵i En t), i ∣φn (t)⟩ = exp (− 21h̵ Γn t) exp [− ̵ (En + Re ∆En ) t] ∣φn (0)⟩ . h

(3.57)

Formally, we can establish a time-dependent norm (a nonvanishing decay rate) of the state ∣φn ⟩ in the laser field, because of depletion into the continuum by ionization, 2 according to the formula ∣∣φn (t)⟩∣ = exp (−Γn t). It is instructive to match this result with the transition rate, calculated using Fermi’s Golden Rule in first-order time-dependent perturbation theory. The result for the ionization cross section σioni reads [6, 117] σioni =

ωL ωL Γn = Im α(ωL ) . IL 0 c

(3.58)

This result has the right physical dimension: The angular frequency has units of radians per second, whereas Γn is (the imaginary part of) an energy. The intensity has units of energy per time per area; the ratio given in Eq. (3.58) therefore is a cross sectional area, i.e., it has a physical dimension of length squared. It is highly instructive to rewrite Eq. (3.58) as follows, IL Γn . γn = ̵ = σioni ̵ h hωL

(3.59)

Here, γn is the rate of transition from the bound to the continuum state, while ̵ IL /(hω) is the number of photons per cross sectional area per time. Multiplied by the ionization cross section σioni , the latter quantity gives the transition rate. For typical cases, ionization cross sections are of the order of the square of the Bohr radius, and they scale with 1/Z 4 for hydrogenlike ions (of nuclear charge number Z), because these ions become smaller with increasing nuclear charge.

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Way 2: Quantized Fields

It is highly instructive to repeat the above derivation using time-independent perturbation theory in the second-quantized formalism. To be specific, we adopt the electromagnetic field in the second-quantized formalism, whereas the atom is firstquantized. In Eqs. (2.85) and (2.96), we had encountered the Hamiltonian of the electromagnetic field as a sum over the available modes. For a single, highly populated laser mode, the Hamiltonian HL of the laser mode simply counts the laser photons on the basis of the number operator a+L aL . The angular laser frequency is denoted as ωL , and aL ≡ aλL (k⃗L ) is the annihilation operator of a laser mode photon (in the discretized representation (2.96), with [aL , a+L ] = 1). The Hamiltonian for the atom is kept in symbolic form, as in Eq. (3.38), and the unperturbed Hamiltonian of the atom and laser field (hence, with two zero indices) is given as ̵ ωL a + aL , H00 = ∑ Em ∣φm ⟩ ⟨φm ∣ + h L

(3.60)

m

where the sum over m covers the entire bound and continuous spectra of the atom. We reserve the notation H0 for the unperturbed Hamiltonian of the atom alone, in agreement with Eq. (3.38). Furthermore, in agreement with the discussion in Sec. 3.3.1, we assume that the polarization vector ˆL = ˆez of the laser field is aligned along the z axis. It is necessary to restrict the operator of the electric field in the discrete representation, given in Eq. (2.129b), to the laser mode, √ ̵ L hω ⃗ ⃗ ⃗ ˆL (aL ei k⋅⃗r − a+L e−i k⋅⃗r ) . (3.61) EL (⃗ r) = i 2 0 V In the dipole approximation, we have ⃗L (⃗ ⃗L (⃗ E r) ≈ E r = 0) = i

√

̵ L hω ˆL (aL − a+L ) . 2 0 V

(3.62)

On the level of second quantization, the electromagnetic field operator can thus be formulated as a time-independent quantity, whereas on the level of classical field theory, one cannot transform the time evolution of the electromagnetic fields away. The number states in the Fock space are the states ∣nL ⟩ with a+L aL ∣nL ⟩ = nL ∣nL ⟩. They fulfill the following relations, √ √ aL ∣nL ⟩ = nL ∣nL − 1⟩ , a+L ∣nL ⟩ = nL + 1 ∣nL + 1⟩ . (3.63) The quantized electric-dipole interaction of the electron enters the interaction Hamiltonian, which reads as follows, √ ̵ ωL h ⃗ H1 = −e r⃗ ⋅ EL = −i e z (aL − a+L ) . (3.64) 2 0 V In second quantization, the Hamiltonian for the coupled system consisting of atom and radiation field reads H = H00 + H1 ,

(3.65)

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where H00 is given in Eq. (3.60). Eigenstates of H00 are product states with the atom in state ∣φn ⟩ and nL photons in the laser mode. We may choose ∣φn , nL ⟩ as the reference state, where the subscript n in φn and nL have totally different physical meanings. The Hamiltonian multiplies the state by the (unperturbed) energy eigenvalue, ̵ nL ωL ) ∣φn , nL ⟩ . H00 ∣φn , nL ⟩ = (En + h (3.66) The first-order perturbation vanishes, ⟨φn , nL ∣H1 ∣ φn , nL ⟩ = 0. For second-order energy shifts, we need the reduced Green function which excludes the reference state, i.e., the product state of the atom being in the reference state and the photon field being in the state with nL laser photons. The spectral decomposition of the reduced Green function reads as ′ ∣φm , mL ⟩ ⟨φm , mL ∣ 1 ) = ( (3.67) ∑∑ ̵ ωL . z − H00 z − Em − mL h mL m ² ∣φm ,mL ⟩≠∣φn ,nL ⟩

Here, we exclude only the reference state ∣φn , nL ⟩ from the sum over virtual, intermediate states. For the state ∣φn , nL ⟩, both the atomic part as well as the field part of the product state are equal to the reference state. The second-order energy shift is ′ 1 ) ∆En = ⟨φn , nL ∣ H1 ( ̵ ωL − H00 H1 ∣ φn , nL ⟩ En + nL h ̵ ωL e2 h 1 = (⟨φn , nL ∣ z aL z a+L ∣ φn , nL ⟩ ̵ 2V En + nL h ωL − H00 1 (3.68) ̵ ωL − H00 z aL ∣ φn , nL ⟩) . En + nL h Because aL annihilates, while a+L generates, exactly one photon, the second-order energy shift involves a very restricted number of intermediate photon states, with one photon more, or one photon less, than the reference state, ̵ ωL ⟨φn , nL ∣ z aL ∣ φm , nL + 1⟩ ⟨φm , nL + 1 ∣ z a+L ∣ φn , nL ⟩ e2 h ∆En = ∑{ ̵ ωL − (Em + (nL + 1) h ̵ ωL ) 2 0 V m E n + nL h + ⟨φn , nL ∣ z a+L

⟨φn , nL ∣ z a+L ∣ φm , nL − 1⟩ ⟨φm , nL − 1 ∣ z aL ∣ φn , nL ⟩ } ̵ ωL − (Em + (nL − 1) h ̵ ωL ) En + nL h ̵ ωL 1 e2 h = ((nL + 1) ⟨φn ∣ z z ∣ φn ⟩ ̵ 2 0 V E n − h ω L − H0 +

1 (3.69) ̵ ωL − H0 z ∣ φn ⟩) . En + h Here, H0 refers to the atomic part of the unperturbed Hamiltonian. For large photon number nL and large normalization volume V , with a finite ratio nL /V , we can replace (nL + 1)/V → nL /V and obtain + nL ⟨φn ∣ z

∆En =

nL e2 ωL 1 1 ⟨φn ∣ z ̵ L − H0 z + z En + hω ̵ L − H0 z ∣ φ n ⟩ . 20 V En − hω

(3.70)

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With the dynamic polarizability α(ωL ) as defined in Eq. (3.53), we can write ̵ ωL nL h ∆En = − α(ωL ) . (3.71) 2 0 V It remains to clarify the role of the laser intensity in this result. The energy density w of the laser field and the intensity IL are related by ̵ ωL ̵ ωL c nL h nL h w= , IL = w c = . (3.72) V V For ∆En , we finally obtain the same expression as in Eq. (3.54), ∆En = −

IL α(ωL ) . 2 0 c

(3.73)

We thus conclude that the classical-field and the quantized-field approach give consistent results in the limit of a strong, macroscopic laser field. 3.3.3

Way 3: Gell-Mann–Low–Sucher Theorem

It is quite surprising that there actually is a third way to derive the energy shift of an atom in an oscillatory laser field. The third way is almost completely independent of the two derivations given above and exclusively relies on a rather abstract theorem, known as the Gell-Mann–Low–Sucher theorem, which is given in Eq. (2) of Ref. [118]. This theorem relates the auto-correlative matrix element ⟨φn ∣UI (, 0)∣φn ⟩ for a reference state ∣φn ⟩ to the energy shift experienced by the state under a perturbation (3.39), ̵

̵

HI (, t) = e− ∣t∣ HI (t) = e− ∣t∣ (ei H0 t/h H1 (t) e−i H0 t/h ) ,

(3.74)

where HI (t) is the interaction Hamiltonian in the interaction picture. From the matrix element ⟨φn ∣UI (, t = 0)∣φn ⟩ alone, which describes the time evolution from the infinite past, where the interaction is adiabatically switched off, to the time t = 0, one can derive the energy shift. Let us carry out the calculation for the ac Stark shift. We first recall the result given in Eq. (3.47), which reads cn (t) = ⟨φn ∣UI (, t)∣φn ⟩ = 1 −

⟨φn ∣Hd ∣φm ⟩⟨φm ∣Hd ∣φn ⟩ ⎞ exp(2t) ⎛ i ∑ ̵ ̵ L + i ⎠ . 2h ⎝ 4 m,± En − Em ± hω

(3.75)

The matrix element of the time evolution operator, at time zero, through secondorder time-dependent perturbation theory, therefore reads as 1 ⎛i ⟨φn ∣Hd ∣φm ⟩⟨φm ∣Hd ∣φn ⟩ ⎞ ⟨φn ∣UI (, 0)∣φn ⟩ = 1 − ̵ ∑ ̵ L + i ⎠ . 2h ⎝ 4 m,± En − Em ± hω

(3.76)

One first augments the interaction Hamiltonian HI (, t) defined in Eq. (3.39) by an artificial coupling parameter g, which serves to characterize the coupling strength and orders the perturbation series, HI (, t) → g HI (, t) .

(3.77)

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65

Then, according to the Gell-Mann–Low–Sucher theorem, we have ̵ g ∆En = lim lim (i h →0 g→1

∂ ln ⟨φ ∣UI (, 0)∣ φ⟩) ∂g

⎛ ̵ ⎛ ∂ g2 i ⟨φn ∣Hd ∣φm ⟩⟨φm ∣Hd ∣φn ⟩ ⎞⎞ = lim lim i h g ln 1 − ̵ ∑ ̵ L + i ⎠⎠ →0 g→1 ⎝ ∂g ⎝ 2h 4 m,± En − Em ± hω ≈ lim lim

→0 g→1

=

⟨φn ∣Hd ∣φm ⟩⟨φm ∣Hd ∣φn ⟩ g ∂ 2 g ∑ ̵ L + i 8 ∂g En − Em ± hω m,±

1 ⟨φn ∣Hd ∣φm ⟩⟨φm ∣Hd ∣φn ⟩ ∑ ̵ L + i . 4 m,± En − Em ± hω

(3.78)

The latter result for ∆En is in full agreement with the intermediate result given in Eq. (3.49) in Sec. 3.3.1. The next steps toward Eqs. (3.54) and (3.73) are completed as outlined in Sec. 3.3.1, demonstrating that the Gell-Mann–Low–Sucher theorem is in agreement with the alternative derivations presented in Secs. 3.3.1 and 3.3.2. In the transformation step denoted by “≈” in Eq. (3.78), an expansion of the logarithm in powers of g is implied, which is equivalent to a second-order expansion in the time-dependent perturbation HI . Higher-order terms in g would correspond to fourth-order interactions of the atom with the laser field, and would be proportional to so-called hyperpolarizabilities. 3.4 3.4.1

Static Stark Effect and Level Width Stark Shift and Large-Order Perturbation Theory

It is quite intriguing to observe that the above second-order perturbation theory treatment of the ac Stark effect has an interesting static limit that corresponds to a time-independent electric field. In this case, perturbation theory can be calculated to very high order, and a number of interesting phenomena can be observed, which will be described in the following. These observations are in fact connected with a set of deep questions regarding the asymptotic character of perturbation series in high orders. They are discussed here using the static Stark effect as a particularly simple, conceptually evident example, where a number of phenomena find an immediate physical interpretation. Let us therefore consider the static limit of the above treatment, i.e., the case ωL → 0. With Hd = −e z EL , the energy shift simplifies to ⟨φn ∣Hd ∣φm ⟩ ⟨φm ∣Hd ∣n⟩ e2 EL2 ⟨φn ∣z∣φm ⟩ ⟨φm ∣z∣φn ⟩ 1 = . (3.79) ∑ ∑ ̵ ωL →0 4 m,± En − Em ± hωL + i  2 m≠n En − Em

∆En = lim

The symmetric limit ±ωL → 0 eliminates the reference state ∣φn ⟩ from the sum over intermediate states ∣φj ⟩ and leads to an overall factor 2. However, in order to calculate the static limit properly, we have to remember that this result has been obtained for an oscillatory electric field of the form EL cos(ωL t). In this result, the

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term EL2 /2 represents the square of the electric field, averaged over a laser period. ⃗ = Ez ˆez , with an interaction Hamiltonian For a static field E ⃗ ⋅ r⃗ = −e Ez z , H1 = −e E

(3.80)

the second-order energy shift therefore reads as ′ ⟨φn ∣z∣φm ⟩ ⟨φj ∣z∣φn ⟩ 1 = −e2 Ez2 ⟨φn ∣z ( ) z∣ φn ⟩ . En − Em H0 − E0 m≠n

∆En = e2 Ez2 ∑

(3.81)

Here, H0 = ∑m Em ∣φm ⟩ ⟨φm ∣ is the unperturbed atomic Hamiltonian, and the prime denotes the (atomic) reduced Green function. In general, when an atom or ion is exposed to a static electric field, the energy levels are displaced from their field-free positions (Stark effect). For a nondegenerate reference state, the expectation value of the electric dipole operator in a particular quantum eigenstate (e.g., the ground state) of an atom vanishes due to parity symmetry, and the leading effect is of order Ez2 . For the ground state ∣φ1S ⟩ of the hydrogen atom, the second-order shift reads, in operator notation, ∆E0 = −e2 Ez2 ⟨φ1S ∣z (

′ 9 1 ) z∣ φ1S ⟩ = − Eh G2 < 0 , H0 − E0 4

(3.82)

where the numerical result refers to hydrogen. Here, Eh is the Hartree energy, as defined in Eq. (2.27), while G is a coupling parameter, ˆa.u. . G = Ez /E

(3.83)

ˆa.u. of the electric field strength is equal to the electric field felt The atomic unit E by a bound electron at a distance of one Bohr radius from a proton (see Sec. 2.3.3). ˆa.u. = 1.) In SI mksA units, the atomic field (In atomic units, we therefore have E strength is given by e α3 m2e c3 ˆa.u. = ∣ − ∂ (− e )∣ E ∣= = . 2 ̵ ∂r 4π0 r r=a0 4π0 a0 eh

(3.84)

Here, me is the electron mass (we calculate in the non-recoil limit, with an infinitely ˆa.u. = 5.1422 × 1011 V/m, illustrating heavy atomic nucleus). Numerically, we have E that the electric fields, which hold the atoms together, are typically much larger than macroscopically relevant electric fields, but smaller than the electric fields needed to “make the vacuum spark”, i.e., to generate electron-positron pairs from the vacuum. This latter effect will be discussed in greater detail in Chap. 18. The energy shift in Eq. (3.82) is negative, because all virtual states entering H0 have a higher energy than the reference state. An interesting phenomenon associated with the static Stark effect has given rise to some discussion. Namely, the perturbation series that describes the static Stark energy shift is nonalternating (the terms have the same sign), and it is factorially divergent in large orders in the coupling parameter G. For the ground state of

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Time-Ordered Perturbations

hydrogen, which we denote by the subscript ∣0⟩, the first perturbative terms read as follows, 1 9 3 555 4 2 512 779 6 13 012 777 803 8 E0 (G) = Eh (− − G2 − G − G − G + ⋯) . 2 4 64 512 16 384 (3.85) Here, E0 is the shifted ground-state energy of the hydrogen atom as a function of coupling parameter G. By inspection, one can infer that the magnitude of the perturbative coefficients grows rather drastically with the order of perturbation theory. One can formally write the perturbation series as [119–122] E0 (G) ∼ −Eh

∞

∑

K=0,2,4,...

aK GK ,

aK ∼

3 K! 3 K ( ) , π 2

K → ∞.

(3.86)

The series in Eq. (3.85) is divergent, no matter how small the electric field strength is. The ratio of two successive terms in the perturbation series is 9 aK+2 GK+2 9 ∼ K (K + 1) G2 ∼ K 2 G2 (3.87) aK GK 4 4 for large K. This implies that the perturbative terms grow with the order K when K ≥ 2 (3 G)−1 no matter how small G is. The minimal term in the perturbation series is reached at order K ∼ 2/(3 G), and it is at this order that the perturbation series can be “optimally truncated”. The physics behind the divergent perturbation series can be analyzed as follows: First, it is necessary to remark that the atom will not experience an arbitrarily large energy shift in a small electric field. Rather, the rigid interpretation of a perturbation series as a convergent rock-solid physical prediction should be reconsidered. A perturbation series, in general, is an asymptotic series which does not converge, but whose first terms still give us a good indication of the physically relevant answer, provided the coupling parameter G is not too large, i.e., for the case G ≪ 1. In this case, it is possible to obtain a good prediction by considering only the first few terms of the asymptotic series until the minimum term is reached. For the Stark effect, the nonalternating divergent perturbation series given in Eqs. (3.85) and (3.86) has a further physical interpretation. This additional point can be understood if we realize that non-alternating divergent series typically correspond to Borel-type integrals like those given in Eq. (3.93), which are evaluated “on the cut” in the complex plane, i.e., in a parameter region where a slight displacement of the coupling parameter leads to the emergence of a nonvanishing imaginary part for the resonance energy. In the current case, the physical interpretation for the emerging imaginary part is evident: The bound electron is tied to the nucleus by the Coulomb potential. However, in a uniform, infinitely extended electric field, it may eventually tunnel through the barrier into a region where the total electrostatic potential (sum of Coulomb field and external static field) is lower than the binding energy of the electron. Quantum mechanically, the tunneling probability is nonvanishing even for a small external electric field (small coupling parameter G),

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and it gives rise to a decay width Γ0 (G) which represents the imaginary part of the perturbed ground-state energy E0 (G), E0 (G) = Re E0 (G) −

i Γ0 (G) , 2

(3.88)

in full analogy to Eq. (3.56). The decay width Γ0 (G) can be approximated, for small coupling parameter G, as follows (see Refs. [119–122]), Γ0 (G) ∼

2 4 Eh exp (− ), G 3G

G → 0.

(3.89)

The leading term in the large-order asymptotics of the perturbation series (3.86) and the leading term in the decay width (3.89) are related by a dispersion relation. For the Stark effect, the dispersion relation is not completely trivial; it is given in Eq. (1) of Ref. [122]. One has to take into account cuts of the resonance energy E0 (G) for both positive as well as negative coupling parameter G. A detailed analysis of the implications of the complex structure of the resonance energies can be found in Ref. [123]. Under certain conditions, it is possible to calculate the decay width using Borel–Pad´e summation [124], using nontrivial complex contours in the evaluation of the Borel integral [125, 126]. The resonance energy is obtained as a complex quantity even if the perturbative coefficients are purely real. In these cases, it is possible to overcome the limit set by the smallest term in the asymptotic series. The numerical values for the real and imaginary parts of the resonance energies of the Schr¨ odinger–Stark problem, obtained using resummation, can be verified against directly obtained solutions of complex-scaled radial differential equations (see Refs. [127, 128]). One may ask if the divergent perturbation series (3.85) can be mapped onto a mathematical model problem. To this end, let us consider the integral A(g) = −

∞ 1 1 1 1 u 1 exp ( ) Ei (− ) = ∫ exp (− ) du . g g g g 0 1+u g

(3.90)

Here, the so-called exponential integral Ei is defined by a principal-value prescription, denoted as (P.V.), Ei(x) = −(P.V.) ∫

x −∞

t−1 et dt .

(3.91)

Upon expansion in powers of g, we obtain the perturbation series, ∞

A(g) ∼ ∑ (−1)K K! g K , K=0

g → 0,

(3.92)

which is divergent no matter how small g is. Note that the “∼” sign means that the series on the right is an asymptotic, alternating series. An even more interesting example is the integral N (g) =

1 u 1 exp (− ) du , ∫ g C 1−u g

(3.93)

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for g > 0. Here, C is a contour which starts at u = 0, ends at u = +∞, and encircles the pole at u = 1 as specified below. Upon expansion in powers of g, N (g) gives rise to a nonalternating series, ∞

N (g) = ∑ K! g K , K=0

g → 0.

(3.94)

From the integral representation in Eq. (3.93), it is clear that there is some sort of ambiguity for real, positive g in the final evaluation of N (g), depending on how the pole at u = 1 is encircled. Indeed, depending on how the pole term is encircled in Eq. (3.93), we have the result ⎧ 1 1 ⎪ ⎪ exp (− ) Ei(g −1 ) (principal value) ⎪ ⎪ ⎪ g g ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 iπ 1 ⎪1 exp (− ) (upper contour) . (3.95) N (g > 0) = ⎨ exp (− ) Ei(g −1 ) + ⎪ g g g g ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 iπ 1 ⎪ ⎪ ⎪ exp (− ) Ei(g −1 ) − exp (− ) (lower contour) ⎪ ⎪ g g g ⎩g

An interesting observation can be made: Although the original perturbation series (3.94) is purely real for real coupling g, the result of the “summation” of the series can be manifestly complex. The “summation” in this case consists in the evaluation of the integral (3.93). We recall that the series (3.94) was generated upon expansion of (3.93) in powers of g. A numerical evaluation of (3.93) along a contour infinitesimally above or below the pole at u = 1 therefore corresponds to an evaluation of the very expression from which the perturbation series (3.94) originated by a perturbative expansion. This is the meaning of the idea of resummation as first imagined by Euler3 who wrote in a letter to Goldbach4 in 1745: “Summa cuiusque seriei est valor expressionis illius finitae, ex cuius evolutione illa series oritur.” This is translated as follows: The sum of any (possibly divergent) series is equal to the numerical value of that particular finite expression, which gave rise to the (possibly divergent) series upon asymptotic expansion in the first place. There is some ambiguity in identifying the latter finite expression: Namely, as illustrated in Eq. (3.95), one may add or subtract a term whose perturbation series vanishes, i.e., a non-analytic term for g → 0. Indeed, the Taylor expansion of exp(−1/g) vanishes to all orders in g. Still, it is intriguing to observe that the imaginary parts of the expressions on the right-hand side of Eq. (3.95) come in discrete steps: Either one has no imaginary part at all, or one has a positive, or a negative, nonanalytic expression. For a hydrogen atom in an electrostatic field, the imaginary part of the energy shift has a physical interpretation: It describes the probability for the bound electron to escape to infinity, in any electric field, by tunneling through the potential barrier. This imaginary part of the energy shift should be negative, therefore describing a decaying state [see also Eq. (3.56)]. It is very intriguing to ask whether or not, given the 3 Leonhard 4 Christian

Euler (1707–1783). Goldbach (1690–1764).

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purely real character of the perturbation series (3.94), and of the series (3.85) which describes a hydrogen atom in an electrostatic field, one may in effect recalculate the decay width of the state against tunneling, i.e., the imaginary part of the energy shift. We have already mentioned the related investigations in Refs. [123, 125, 126]. For our mathematical model problem (3.94), the negative imaginary part of the final answer is obtained when using the “lower” contour, i.e., the contour which encircles the pole in the mathematically positive direction. In the power series (3.94), all coefficients are positive. By contrast, the coefficients are negative in the series (3.85). So, the “upper” contour should be used, in order to obtain a negative imaginary part of the energy, as it was employed in Refs. [123, 125]. According to Table I of Ref. [123], by a resummation of the perturbation series (3.85), one obtains a numerical result for the manifestly complex ground-state resonance energy E0 of i E0 (G = 0.04) = −0.503 771 591 013 654 2(5) − 3.892 699 990(1) × 10−6 . (3.96) Eh 2 The numerical accuracy, which is quoted in parentheses, deserves a comment. Namely, in the resummation methods described in Refs. [123, 125], one first has to perform an analytic continuation of the so-called Borel transform of the perturbation series (3.85) beyond its original circle of convergence, in order to be able to evaluate a numerical approximation to the integral representation (3.93) corresponding to the shifted energy (3.85) in the electrostatic field. This analytic continuation can be accomplished by Pad´e approximants, conformal mappings in the Borel plane, and combinations thereof. So-called order-dependent mappings represent a viable alternative [129–131]. A full optimization of these methods for the (dc) Stark effect, to go beyond the numerical accuracy of reference data such as the result given in Eq. (3.96), remains an open problem. 3.4.2

Quantum Electrodynamics and Large-Order Perturbations

Considerations regarding the asymptotic character of a perturbation series have important consequences for QED. As argued in Ref. [132], the vacuum of quantum electrodynamics becomes unstable against pair creation if one reverses the sign of the fine-structure constant α. This is because e2 = 4πα. So, in a hypothetical world in which e2 = −4πα, opposite charges would repel each other, and the vacuum would become unstable against the creation of electron-positron pairs. These would move away from each other, thereby accumulating positrons at one end of the Universe, and electrons at the other. Dyson [132] argued that this consideration conclusively shows that the QED perturbation series (which describe energy shifts, corrections to the photon propagator, or vacuum-to-vacuum transition amplitudes) must constitute a divergent series, because its radius of convergence about the origin must vanish. Otherwise, there would be no room for the decay processes in the case α < 0. From today’s point of view, this semi-quantitative consideration of Dyson is quite self-evident. This is because one has obtained a much more intuitive

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understanding of the mechanism that generates the divergence. E.g., Zinn-Justin and coworkers [129, 130] have investigated the perturbation series describing socalled renormalization-group function of various scalar theories, or theories describing self-interacting fields with an O(N ) symmetry group, with the notion of determining critical exponents, inspired by the numerical evaluations of higherorder Feynman diagrams performed by other groups [133]. In O(N )-symmetric field theories with self-interaction terms, the divergence of the perturbation series follows quite naturally from the path integral formulation. Let us consider a onedimensional analogy of a path integral, namely, an integral of the type J(g) = ∫

∞ −∞

1 dx exp (− x2 − g x4 ) , 2

(3.97)

where the factor exp (− 21 x2 ) models the Gaussian measure of the full field theory and the term exp (−g x4 ) describes the quartic self-interaction. Upon expansion in g, formula (3.97) generates a perturbation series whose coefficients entail integrals that evaluate to Euler Γ functions. The resulting perturbation series in g, ∞

J(g) ∼ ∑ CK g K ,

(3.98)

K=0

is divergent. In general, the structure of the coefficients CK of order K in a divergent perturbation series in scalar field theories with self-interactions is known [130] to be of the form CK ∼ AK K b Γ(K) g K ,

K → ∞,

(3.99)

with appropriate parameters A and b and coupling parameter g. For O(N ) symmetric φ4 field theories, all renormalization-group functions (β functions and the anomalous field dimensions) have been shown to exhibit a divergent character for large orders, and adapted summation methods (“order-dependent mappings”) have been applied to the resummation of the series [129, 130]. Indeed, the analysis of large orders in perturbation theory now is a well established field of research, and there is no doubt about its divergent character, for wide classes of quantum mechanical perturbations, and field theories. The analysis of quantum mechanical operators [134–137] has revealed that any perturbation series in quantum mechanics must be divergent unless the perturbation is Kato-bounded with regard to the terms in the unperturbed Hamiltonian. Paradigmatic examples for the divergence of perturbation theory (almost in the status of a “toy model”) are anharmonic oscillators. For even anharmonic oscillators, described by Hamiltonians of the form H = − 12 ∂x2 + 12 x2 + g xN ,

N = 4, 6, 8, . . . ,

(3.100)

the large-order asymptotics of the perturbative coefficients, for arbitrary even N and an arbitrary bound-state energy level, are given by the famous Bender–Wu

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formulas [138]. For the level of the Hamiltonian (3.100) with quantum number n, we have (N )

En(N ) (g) ∼ ∑ En,K g K ,

(3.101a)

K

(N )

En,K ∼

(−1)K+1 (N − 2) N −2 Γ( K + n + 12 ) 2 π 3/2 n! 2K+1−n 1

− 2 K−n− 2 N N , )] , (3.101b) N −2 N −2 for K → ∞. Here, B(x, y) = Γ(x) Γ(y)/Γ(x + y) is the Euler Beta Function. The formulas have recently been generalized to odd anharmonic oscillators with a Hamiltonian √ H = − 21 ∂x2 + 12 x2 + g xM , M = 3, 5, 7, . . . . (3.102) N −2

× [B (

In this case, the result for the nth resonance energy of an odd anharmonic oscillator (M ) of degree M , denoted as n (g), reads as (M )

) (M (g) ∼ ∑ n,K g K , n

(3.103)

K

(M )

n,K ∼

2−M Γ ((M − 2)K + n + 21 ) π 3/2 n! 22K+1−n 1

−(M −2)K−n− 2 M M , )] . (3.104) × [B ( M −2 M −2 This result is derived in Refs. [139, 140]. A deeper analysis of the problem has meanwhile led to the conclusion that so-called resurgent trans-series, which involve powers of the structures exp(−1/g), ln(−g), and g itself, adequately describe the resonance energies of rather wide classes of anharmonic oscillators [141, 142]. Furthermore, the utility of these resurgent transseries for field-theoretical problems has been investigated [143], and discussed at international meetings (see Ref. [144]). This includes the nonlinear σ model, as a model problem for quantum chromodynamics which, nevertheless, exceeds the complexity of a pure “toy” model (see Ref. [145]). For QED, the situation is complicated because one has both bosons (photons) as well as fermions (leptons) in the theory. The fermions are massive, so one can integrate them out, in order to obtain an estimate of the resulting, nonlinear interaction of the photons. The integration of the fermions leads to a so-called fermionic determinant. This has been analyzed in Refs. [146–150]. The result for the photonphoton interaction can be inserted in various perturbation series, in order to estimate the large-order growth of vacuum-polarization insertions, or loop diagrams which contribute to the anomalous magnetic moment of the electron, and other effects. In general, the structure of the perturbative coefficients CK of the series describing the QED effects is conjectured to be proportional to (see Chap. 39 of Ref. [130]),

CK ∝ (−1)K A−K Γ(K/2) ,

A = 4.886 ,

K → ∞,

(3.105)

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which is different from the estimate (3.99) in the sense that the Γ function is evaluated at argument K/2, not K. The perturbation coefficients in QED do not grow as drastically with the order of perturbation theory as in scalar and O(N ) symmetric field theories. The conceivable nonperturbative contributions are thus smaller than otherwise expected. One can then infer from Eq. (3.105) that the nonperturbative contributions in QED are proportional to A2 (3.106) exp (− 2 ) = 2.55 × 10−194 698 , α relative to the leading term. So, the original argument supporting the divergent character of the perturbation series, originally brought forward by Dyson [132], is correct, but the limitations to the predictive power of QED are not practically relevant. The situation is different for the analysis of critical exponents in field theory, where the zero of the so-called β function is obtained at a critical coupling g = g ∗ of order unity [129, 130]. The static Stark effect, which initiated this discussion in the context of the current treatise, remains a transparent example case for the realization of such a divergence, with a particularly transparent interpretation, namely, the tunneling autoionization width of a level in the external, static, electric, field. 3.5

Further Thoughts

Here are some suggestions for further thought. (1) Time Evolution. Using Eq. (3.4), derive Eq. (3.5). (2) Density of States. Reconsider Eq. (3.20). Show that the equation is consis̵ which pertains to a one-dimensional quantum tent with the density dn = L dp/h, system. Here, L is the overall length dimension of the one-dimensional system. (3) Coefficients and Expansions. Derive the time-dependent expression Eq. (3.47) based on Eqs. (3.45) and (3.46). (4) Laser Intensity. Verify Eq. (3.55) by calculating the contribution of the electric-field component of the electromagnetic wave, adding the magnetic-field component (same magnitude) and averaging over a laser period. (5) Ionization Cross-Section. Derive Eq. (3.58) on the basis of Fermi’s Golden Rule, as introduced in Sec. 3.2. Start from the overlap integral of a bound and a continuum wave function. Then, study the evaluation of the overlap integrals for hydrogen, on the basis of Chap. 71 of Ref. [6] and Chap. 4 which follows here. (6) Ionization Cross-Section. Verify Eq. (3.58) by matching the imaginary part of the ac Stark shift with a beam of incoming photons. (7) Large-Order Perturbation Theory. Do a literature search on Eq. (3.85) and try to find ways to obtain as many terms in the perturbation series in powers of G as possible. (8) Numerical Calculations. It is a good exercise to evaluate the integral A(g) numerically and accurately, for a few numerical example cases such as, for

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example, to pick a certain value, the case g = 0.07. For comparison, one may sum the first few terms of the asymptotic expansion for the same value of g, and compare. One will find that the first terms give an excellent approximation of the numerical value of the entire series even if the series eventually diverges. Also, one should investigate the integral (3.93), substituting u → exp(iθ) u with θ > 0 (for the “upper” contour) and u → exp(−iθ) u with θ > 0 (for the “lower” contour). The numerical result, obtained using Gauss–Laguerre quadrature, should be quite independent of the angle 0 < θ < π/2 because the integrand is suppressed for u → ∞. Automatic integration routines in modern computer algebra systems [151] may help in the endeavour.

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Chapter 4

Bound-Electron Self–Energy and Bethe Logarithm

4.1

Overview

At this point, we have prepared ourselves for the analysis of a number of physical processes which involve atoms and the quantized electromagnetic field. In the current chapter, concrete applications of the quantized field-theoretical formalism will be discussed, with particular emphasis on one-electron (hydrogenlike) ions. It will turn out to be advantageous that a number of quantities of interest, which can only be calculated numerically for many-electron ions, are amenable to an analytic treatment in the case of a hydrogenlike ion. Two examples are the bound-state and continuum wave functions of the eigenstates of the hydrogenlike ion, and the Green function, which can be calculated either as a double-sum over associated Laguerre polynomials (so-called Sturmian decomposition), or in terms of Whittaker functions. The final expressions of the Green functions, which we derive, may be used for calculations; yet, these expressions provide little insight without a deeper understanding of how these results can actually be derived. Analytic expressions for the Green functions will be developed based on the “concatenation approach”, which is based on the matching of solutions of the homogeneous radial differential equation at the cusp (see Chaps. 2 and 4 of Ref. [81]). A further, very elegant way of deriving the Schr¨odinger–Coulomb Green function, based on a momentum-space approach, profits from the SO(4) symmetry of the hydrogen atom, and leads to a very elegant splitting of the hydrogen Green function into a free term, a one-potential term, and a many-potential term. These can be combined toward the end of the calculation into a compact integral representation. We apply the results to the calculation of the dynamic polarizability of the hydrogen atom, and use the result to calculate the Bethe logarithm of the hydrogen atom, which is a sum over the virtual excitations of the bound system, mediated by interactions of the bound electron with its own radiation field. This calculation is presented in detail, and reference values of Bethe logarithms of unprecedented accuracy are given, as a benchmark for future studies in this and related areas. It should be noted that atomic physics calculations probably belong to the most 75

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sophisticated calculations carried out by mankind so far. The analytic techniques combine mathematical insight, physical inspiration and a great deal of calculational dedication. Nevertheless, a deeper look into these calculations is of utmost interest to the phenomenologically interested theoretical physicist because it combines two established areas of theory: atomic physics of bound states and field theory. Training in the solution of bound-state differential equations will find applications later, as we discuss solutions of the Dirac equation in Chaps. 7 and 8. The calculation of bound-state wave functions, and Green functions, involves finding solutions to bound-state differential equations with specific boundary conditions, and matching them appropriately at the cusp, i.e., at the point where the radial arguments are equal. 4.2 4.2.1

Schr¨ odinger–Coulomb Hamiltonian and Wave Functions Spectrum of the Hydrogen Atom and SO((4)) Symmetry

When discussing the hydrogen atom, it is often stated (see, e.g., Ref. [152]) that oneelectron ions possess an SO(4) symmetry. However, this statement is sometimes used as a tongue-in-the-cheek remark, to imply that whoever is not familiar with the SO(4) symmetry fails to understand what the hydrogen atom is all about. So, it is justified to ask in which sense the SO(4) symmetry needs to be understood. One might guess, at first, that spherically symmetric bound states of the hydrogen atom exhibit SO(3) (rotational) symmetry, and that the SO(4) symmetry is nothing but a typographical error, mixing up the dimensionality of the space on which the Schr¨ odinger Hamiltonian is acting. This is not the case; yet, it makes the problem only more intriguing. Our treatment below is inspired by Chap. 30 of Ref. [152]. We switch to natural ̵ = c = 0 = 1. After the transformation to center-of-mass coordinates, units with h the Schr¨ odinger–Coulomb Hamiltonian of the hydrogen atom can be written as HS =

p⃗ 2 Zα − , 2µ r

⃗, p⃗ = −i ∇

(4.1)

where µ = m1 m2 /(m1 + m2 ) is the reduced mass, Z is the nuclear charge number, and α is the fine-structure constant. The masses of the two constituent particles (electron and nucleus) are denoted as m1 and m2 , and r⃗ is the distance between the two constituent particles (“relative coordinate”). The first decisive observation is that, in addition to the angular momentum vector ⃗ = r⃗ × p⃗ = −i⃗ ⃗, L r×∇

(4.2)

the Runge–Lenz operator also is conserved, i.e., it commutes with HS . Indeed, the ⃗ is given as vector-valued Runge–Lenz operator M ⃗ −L ⃗ × p⃗) − Zα r⃗ , ⃗ = 1 (⃗ p×L M 2µ r

(4.3)

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⃗ ] = 0. The Runge–Lenz operator is and it commutes with the Hamiltonian, [HS , M perpendicular to the angular momentum. In operator notation, we have ⃗ ⋅L ⃗=L ⃗ ⋅M ⃗ = 0. M (4.4)

⃗ does not commute with M ⃗ . The The two statements above are not redundant; L square of the Runge–Lenz operator is ⃗ 2 + 1) + (Zα)2 . ⃗ 2 = 2HS (L (4.5) M µ Some important commutator relations are

[Li , Lj ] = i ijk Lk , [M i , Lj ] = i ijk M k , (4.6a) 2HS [M i , M j ] = i (− ) ijk Lk . (4.6b) µ The algebra involved in verifying Eq. (4.6) is rather lengthy and left as an exercise to the reader. It is helpful to notice that in verifying Eq. (4.6b), the Schr¨odinger Hamiltonian (4.1) appears on the right-hand side of the equation explicitly, i.e., one has to keep track of all momentum operators in the intermediate stages of the calculation. We now restrict the considerations to a bound-state subspace with (possibly degenerate) energy ⟨HS ⟩ = En < 0, and define √ √ ⃗ ≈ − µ M ⃗. ⃗′ = − µ M (4.7) M 2 En 2 HS ⃗ ′ has the same physical dimension as M ⃗ , because In natural units, the operator M of mass-energy equivalence. Then we get the following algebraic relations, [Li , Lj ] = i ijk Lk ,

[M ′ , Lj ] = i ijk M ′k , i

[M ′i , M ′j ] = i ijk Lk .

(4.8)

These relations can be mapped onto those that are fulfilled by the six generators of rotations in four-dimensional space. In three-dimensional space, we can rotate about all three axes, but in four-dimensional space, there are three more rotations possible, which correspond to rotations of the three “spatial” axes about the fourth axis. The SO(4) generators can thus be ordered into a six-dimensional vector structure and identified as follows, x ˜ 23 ⎞ ⎛ X2 P3 − X3 P2 ⎞ ⎛ L ⎞ ⎛L ˜ 31 ⎟ ⎜ X3 P1 − X1 P3 ⎟ ⎜ Ly ⎟ ⎜ L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Lz ⎟ ⎜ L ⎜X P − X P ⎟ ˜ 12 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 2 2 1 △ ˜=⎜ ⎟ ⎜ ˜ 14 ⎟ = ⎜ ⎟. L (4.9) ⎜ M ′x ⎟ = ⎜ L ⎟ ⎜ X1 P4 − X4 P1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ M ′y ⎟ ⎜ L ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ˜ 24 ⎟ ⎜ X2 P4 − X4 P2 ⎟ ˜ 34 ⎠ ⎝ X3 P4 − X4 P3 ⎠ ⎝ M ′z ⎠ ⎝ L ˜ ij = X i P j − X j P i They fulfill the same commutation relations as the components L of angular momentum in four-dimensional space, but with a Euclidean metric. We have [P i , X j ] = −i δ ij , and the commutator relations are ˜ ij , L ˜ k` ] = i (δ ik L ˜ j` + δ j` L ˜ ik − δ jk L ˜ i` − δ i` L ˜ jk ) . [L (4.10)

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In our discussion of the angular momentum in special relativity (see Chap. 8 of Ref. [81]), we had encountered the relation [pµ , xν ] = i g µν for the components pµ of the four-momentum and space-time coordinates xν , and thus [Lµν , Lκλ ] = i (g µλ Lνκ + g νκ Lµλ − g µκ Lνλ − g νλ Lµκ ) .

(4.11)

Here, L = x p − x p is the generalized angular momentum in four-dimensional space-time. We recall that the metric of space-time is g ρσ = diag(1, −1, −1, −1) which is the metric of SO(3, 1). The identification of Eqs. (4.10) and (4.11) proceeds via the replacement g µλ → −δij and an appropriate change in the indices. Furthermore, we can transform the generators as µν

µ

ν

ν

µ

1 ⃗ ⃗′ ⃗ = 1 (L ⃗ −M ⃗ ′) , +M ) , K (4.12) I⃗ = (L 2 2 with [Ii , Kj ] = 0. The algebra is thus separated into two separate SO(3) algebras, [Ii , Ij ] = i ijk Ik ,

[Ki , Kj ] = i ijk Kk ,

[Ii , Kj ] = 0 .

(4.13)

We recall that the angular momentum generates rotations, and fulfills the algebraic relations of the Lie group that generates length-preserving rotations of threedimensional space, which corresponds to the SO(3) group. Then, by virtue of the ⃗ are angular momentum algebra, the eigenvalues of I⃗ and K I⃗2 = i (i + 1) ,

⃗ 2 = k (k + 1) . K

(4.14)

Here, one should remember that the spinor representations of the rotation group allow i and k to be half integer, while in the usual multipole representations (see Chap. 2 of Ref. [81]), i and k assume non-negative integer values. So, the possible values are i, k = 0, 21 , 1, 23 , 2, . . . . Because the Runge–Lentz vector is perpendicular to the angular momentum [see Eq. (4.4)], we have 1 ⃗ 2 ⃗ ′2 ⃗ 2 = 1 (L ⃗2 + M ⃗ ′2 ) . +M ) , K (4.15) I⃗2 = (L 4 4 The following two operators, commonly referred to as Casimir operators, thus com⃗ mute with all components of I⃗ and K, 1 ⃗ 2 ⃗ ′2 (L + M ) , 2 ⃗ 2 = i (i + 1) − k (k + 1) = 0 . C2 = I⃗2 − K

⃗ 2 = i (i + 1) + k (k + 1) = C1 = I⃗2 + K

(4.16a) (4.16b)

The latter relation follows by direct inspection of Eq. (4.15) and implies that we are only dealing with the subspace of SO(4) for which i = k. We now project Eq. (4.5) onto the eigenspace of the hydrogen atom with ⟨HS ⟩ = En , ⃗ 2 + 1) + (Zα)2 . ⃗ 2 = 2En (L M µ

(4.17)

⃗ ′, Upon multiplication by [−µ/(2En )], we obtain the corresponding relation of M ⃗ ′2 = −L ⃗ 2 − 1 − µ (Zα)2 . (4.18) M 2En

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We now combine Eqs. (4.14), (4.15) and (4.18) to obtain the following relation in the space with ⟨HS ⟩ = En , ⃗2 + M ⃗ ′2 ) ⃗ 2 = 1 (L 2k(k + 1) = 2K 2 1 ⃗2 ⃗2 µ 1 µ = (L −L −1− (Zα)2 ) = (−1 − (Zα)2 ) . (4.19) 2 2En 2 2En Solving for En , we find En = −

(Zα)2 µ µ (Zα)2 =− , 2 2(2k + 1) 2n2

n ≡ (2k + 1) .

(4.20)

We have obtained the Schr¨ odinger spectrum on the basis of purely algebraic manipulations, because k ≥ 0 may be integer or half integer. 4.2.2

Differential Equations and the Hydrogen Atom

We have just determined the Schr¨odinger spectrum but not the wave functions of the hydrogen atom. We shall now find the hydrogen wave functions, using natural ̵ = c = 0 = 1. The three-dimensional Schr¨odinger equation for a twounits with h body system of reduced mass µ in a scalar, time-independent potential V (⃗ r) is given by ⃗2 ∇ HS ψE`m (⃗ r) = (− + V (⃗ r)) ψE`m (⃗ r) = E ψE`m (⃗ r) , (4.21) 2µ where the subscript of the wave function ψ denotes the energy E, the angular momentum `, and the magnetic quantum number m. For an electron of charge e in the Coulomb potential of a nucleus of charge −Ze, the potential is spherically symmetric and given by V (⃗ r) = −

Ze2 , 4π0 r

(4.22)

where e2 = 4πα. In spherical coordinates, the Laplace operator is given by ⃗2 ∂2 2 ∂ L ⃗ = −i r⃗ × ∇ ⃗2 = ⃗, ∇ + − 2 , L (4.23) 2 ∂r r ∂r r ⃗ is the angular momentum operator. The full equation in spherical coordiwhere L nates then reads as ⃗2 2 ∂ L Zα 1 ∂2 )+ − ] ψE`m (r, θ, ϕ) = E ψE`m (r, θ, ϕ) . (4.24) [− ( 2 + 2µ ∂r r ∂r 2 µ r2 r It is solved by the separation ansatz ψE`m (r, θ, ϕ) = RE` (r) Y`m (θ, ϕ) , where Y`m (θ, ϕ) is the spherical harmonic defined by ¿ Á 2` + 1 (` − m)! m À Y`m (θ, ϕ) = Á P (cos(θ)) eimϕ , 4π (` + m)! `

(4.25)

(4.26)

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with the associated Legendre polynomials P`m (cos(θ)). For an introduction to these quantities, see Chap. 2 of Ref. [81]. The spherical harmonic has the properties ⃗ 2 Y`m (θ, ϕ) = `(` + 1) Y`m (θ, ϕ) , L

Lz Y`m (θ, ϕ) = m Y`m (θ, ϕ) .

(4.27)

After the separation, we have to solve the radial equation, (−

2 d 1 d2 `(` + 1) Zα − ( 2+ )+ ) RE` (r) = E RE` (r) . 2µ dr r dr 2 µ r2 r

(4.28)

One uses the substitution uE` (r) = rRE` (r) and arrives at the differential equation (−

`(` + 1) Zα 1 d2 + − − E) uE` (r) = 0 . 2 2µ dr 2µ r2 r

(4.29)

It is helpful to rescale the differential equation, and to express it in terms of dimensionless quantities, 1/2

ρ = (−8 µ E)

r,

ν = Zα (−

µ 1/2 ) , 2E

̃E` (ρ) = uE` (r) . u

(4.30)

Here, ν is a generalized energy variable, which can be real or imaginary. It generalizes the concept of the principal quantum number, E=−

(Zα)2 µ . 2 ν2

(4.31)

For bound states with negative energy E, the generalized quantum number ν = n is real. For continuum states with E > 0, ν is purely imaginary. We choose the branch cut of the square root in such a way as to give ρ a positive imaginary part. Let us assume that the positive energy E obtains a small negative imaginary part (in the sense of a nonvanishing decay width Γ, so that E → Re E − iΓ/2). Then, 1/2 the argument −8 µ E of the square root in the expression ρ = (−8 µ E) r obtains a small positive imaginary part, while the real part of the expression −8 µ E is negative. With the standard choice of the branch cut of the complex square root, the variable ρ becomes purely imaginary, with a positive imaginary part, while ν becomes negative imaginary [see also Eq. (4.70)]. The dimensionless equation then reads as (

`(` + 1) ν 1 d2 ̃E` (ρ) = 0 . − + − )u 2 dρ ρ2 ρ 4

(4.32)

We can trivially rewrite the equation as (

1 ν d2 − + + 2 dρ 4 ρ

1 4

− (` + 12 )2 ̃E` (ρ) = 0 , )u ρ2

(4.33)

and match it with the definition of Whittaker’s differential equation [153], d2 w 1 κ + (− + + dρ2 4 ρ

1 4

− µ2 ) w = 0, ρ2

(4.34)

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where κ = ν and µ = ` + 12 . The linearly independent solutions of this differential equation are called the Whittaker M and W functions, where 1

w = Mκ,µ (ρ) = e− 2 ρ ρµ+1/2 1 F1 (µ + w = Wκ,µ (ρ) =

1 e− 2 ρ

ρµ+1/2 U (µ +

1 2

1 2

− κ, 2µ + 1, ρ) ,

− κ, 2µ + 1, ρ) .

(4.35a) (4.35b)

Here, the confluent hypergeometric function is denoted as 1 F1 , and we also encounter the hypergeometric U function [153]. A priori, the equation (4.33) defines a valid differential equation for any values of `, ν, and ρ. However, we shall see below that a special case is encountered for ν = n where n is a positive integer. We first discuss the M function, ̃E` (ρ) = M u =

1 (ρ) ν,`+ 2 1 − ρ `+1 e 2 ρ

1

= e− 2 ρ ρ`+1 1 F1 (` + 1 − ν, 2` + 2, ρ)

1 Γ(2` + 2) u ρ `−ν `+ν ∫ e u (1 − u) du . Γ(` + 1 − ν)Γ(` + 1 + ν) 0

(4.36)

Let us dwell a little on the definition of the so-called confluent (Kummer’s) hypergeometric function. In order to understand this function, we first remember that the exponential function is given by a convergent series as ∞

zn = exp(z) . n=0 n! ∑

(4.37)

When we decorate the terms in the infinite sum over n by appropriate prefactors, which contain Gamma functions, we get 1 F1 (a, c; z)

∞

Γ(a + n) Γ(c) z n , Γ(a) Γ(c + n) n! n=0

=∑

(4.38)

where the definition of the Gamma function for z > 0 is [154, 155] Γ(z) = ∫

∞ 0

tz−1 exp(−t) dt ,

Γ(n) = (n − 1)! .

(4.39)

In particular, we have as a special case of the confluent hypergeometric function, 1 F1 (1, 1; z)

∞ zn Γ(1 + n) Γ(1) z n =∑ = exp(z) . Γ(1) Γ(1 + n) n! n=0 n! n=0 ∞

=∑

(4.40)

From the functional form of Mν,`+1/2 (ρ) in Eq. (4.36), we can immediately conclude that Mν,`+1/2 (ρ) is regular at ρ = 0 because of the fundamental properties of the confluent hypergeometric 1 F1 function. In many cases, it is useful to introduce the Pochhammer symbol (a)n =

Γ(a + n) , Γ(a)

(4.41)

while emphasizing that a thorough discussion of properties of hypergeometric, and other special functions, can be found in Refs. [154, 156, 157]. Let us consider the case of a so-called terminating hypergeometric function, which occurs for a nonpositive integer first argument a = −m ≤ 0. We then have the

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following situation: When n ≤ m, we have a Γ function of negative argument in both the numerator as well as the denominator of the expression Γ(a + n)/Γ(a) = Γ(−m + n)/Γ(−m), and the whole term therefore is finite. In order to see this, one displaces the arguments of the Gamma function according to Γ(−m + n + )/Γ(−m + ) and invokes the recursion relation Γ(x) = (x−1) Γ(x−1) until the Gamma function in the numerator and denominator cancel. On the other hand, when n > m, the expression Γ(a + n)/Γ(a) = Γ(−m + n)/Γ(−m) = 0, because the denominator is infinite, whereas the numerator is finite. The series thus terminates at the summation index n = m. Furthermore, a remark is in order. One may write the general solution of Eq. (4.33) either in terms of Mν,`+1/2 (ρ) and Wν,`+1/2 (ρ), or in terms of Mν,`+1/2 (ρ) and Mν,−`−1/2 (ρ). One might thus think that the second required solution to Eq. (4.33) is given by Mν,−`−1/2 (ρ). However, one runs into trouble because upon substituting ` → −` − 1 in Eq. (4.36), the argument of the Gamma function transforms as Γ(2` + 2) → Γ(−2 `). Here, the quantity Γ(−2 `) becomes ill-defined for integer `. This is problematic because the physically interesting case corresponds to an integer angular momentum quantum number `. The complementary solution to Eq. (4.36) is found by replacing in Eq. (4.36) the hypergeometric 1 F1 function by the hypergeometric U function, ̃E` (ρ) = W u =

1 (ρ) 2 1 − ρ ν 2

ν,`+

1

= e− 2 ρ ρ`+1 U (` + 1 − ν, 2` + 2, ρ)

∞ e ρ t `+ν e−t t`−ν (1 + ) dt . ∫ Γ(` + 1 − ν) 0 ρ

(4.42)

For the integral representation to be valid, we must exclude the case of ` + 1 − ν ̃E` (ρ) = W being a negative integer. The function u 1 (ρ) has a cut along the ν,`+

2

negative ρ axis. A very useful relation connects the Whittaker W function with the hypergeometric 2 F0 function, namely [153] W

ν,`+

1 (ρ) 2

1

= e− 2 ρ ρν 2 F0 (` + 1 − ν, −` − ν, −ρ−1 ) .

(4.43)

The W function is regular at infinity due to the asymptotic expansion [153] W

1 (ρ) ν,`+ 2

1

∞

(` + 1 − ν)ν (−` − ν)ν , n! (−ρ)n n=0

∼ e − 2 ρ ρν ∑

ρ → ∞.

(4.44)

The solution Mν,`+1/2 (ρ) is regular at the origin, whereas Wν,`+1/2 (ρ) is not. The situation is inverted at ρ = ∞. In order to fix ideas, let us summarize the unnormalized eigenstates of the hydrogen atom that we have found so far. Specifically, we have encountered solutions of the differential equation (4.29) for arbitrary real energy parameter E. The solutions

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of the eigenvalue problem HS ψE`m (⃗ r) = E ψE`m (⃗ r) read as follows [see Eq. (4.30)], 1 (ρ) 2

=M

1 (ρ) ν,`+ 2

=W

uE` (r) = M uE` (r) = W

ν,`+

ν,`+

1 ν,`+ 2

1/2

r) ,

1/2

r) ,

((−8 µ E)

1 2

((−8 µ E)

(4.45a) ν = Zα (−

µ 1/2 ) , 2E

(4.45b)

uE` (r) , ψE`m (⃗ r) = RE ` (r) Y`m (θ, ϕ) . (4.45c) r The Schr¨ odinger–Coulomb Hamiltonian HS is given in Eq. (4.1). Because we would like to describe bound and continuum wave functions, the remaining task is to find appropriate normalizations for the wave functions. For bound states, we must choose the solution regular at the origin, with the exception of a few cases where the two solutions actually coincide. Up to now, we have solved the radial Schr¨odinger equation for all possible values of the energy. This is not sufficient, though, for a particular value of the energy to be an eigenvalue. At the bound-state eigenvalues, the solution must the regular at both the origin as well as at infinity, and for a continuum eigenstate, the solution must be normalized to a Dirac-δ in the energy variable. The remaining task in finding the eigenstates therefore implies the calculation of appropriate normalization factors. RE` (r) =

4.2.3

Schr¨ odinger–Coulomb Bound States

In Eq. (4.31), we have defined the generalized energy variable ν. For ν = n, where n is a positive integer, we have (Zα)2 µ , (4.46) En = − 2 n2 which is a bound-state energy of the Schr¨odinger Hamiltonian. In this case, the Whittaker M function given in Eq. (4.36) assumes the following form, M

1 (ρ) n,`+ 2

1

= e− 2 ρ ρ`+1 1 F1 (` + 1 − n, 2` + 2; ρ) .

(4.47)

For n ≥ ` + 1, the hypergeometric series is terminating, the state is normalizable, and we can write the M function as follows, M

1 (ρ) n,`+ 2

n−`−1

1

= e− 2 ρ ρ`+1 ∑

j=0

Γ(` + 1 − n + j) Γ(2` + 2) ρj . Γ(` + 1 − n) Γ(2` + 2 + j) j!

(4.48)

This function is finite at both the origin ρ = 0 as well as for ρ → ∞. With the recursion relation of the Γ function, one may simplify the ratio Γ(` + 1 − n + j)/Γ(` + 1 − n), which leads to the expression M

1 (ρ) n,`+ 2

1

n−`−1

= e− 2 ρ ρ`+1 ∑ (−1)j j=0

(n − ` − 1)! (2` + 1)! ρj . (n − ` − 1 − j)! (2` + 1 + j)! j!

(4.49)

This is (almost) the canonical form of the radial bound-state wave function. We now define the associated Laguerre polynomials, ν (ν + m)! ρp (m + ν)! p = (4.50) Lm 1 F1 (−ν, m + 1, ρ) . ν (ρ) = ∑ (−1) (ν − p)!(m + p)! p! m! ν! p=0

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For illustration, it is useful to indicate the first few polynomials explicitly, L00 (ρ) = 1 ,

L01 (ρ) = 1 − ρ ,

L02 (ρ) = 1 − 2ρ +

ρ2 , 2

L11 (ρ) = 2 − ρ ,

L12 (ρ) = 3 − 3ρ +

ρ2 , 2

L22 (ρ) = 6 − 4ρ +

ρ2 . 2

(4.51)

In terms of the associated Laguerre polynomials, we can finally write Eq. (4.49) as follows, 1

Mn,`+1/2 (ρ) = e− 2 ρ ρ`+1

(n − ` − 1)! (2` + 1)! 2`+1 Ln−`−1 (ρ) , (n + `)!

(4.52)

̃n` (ρ) = provided n and ` are positive integers with n ≥ ` + 1. The solution u Mn,`+1/2 (ρ) fulfills the boundary conditions for a bound state in quantum mechanics, namely according to Ref. [158], ̃n` (ρ) = lim u ̃n` (ρ) = 0 , lim u

ρ→0

̃n` (ρ) ∈ L2 (R) . u

̃n` (ρ) = ρ R` (ρ) , u

ρ→∞

(4.53) According to Eq. (4.30), we have the following identification for the ρ variable, for a bound state of quantum number n, ρ = (−8 µ En )

1/2

r=

2Zαµr 2Zr = , n a0 n

a0 =

1 , αµ

(4.54)

where we use the convenient notation a0 for the Bohr radius of the bound system with reduced mass µ. The radial Whittaker function becomes Mn,`+1/2 (

2Zr Zr 2Zr `+1 (n − ` − 1)! (2` + 1)! 2`+1 2Zr ) = exp (− )( ) Ln−`−1 ( ). a0 n a0 n a0 n (n + `)! a0 n (4.55)

At the bound-state energy, the solutions regular at the origin, and regular at infinity, actually have to coincide up to a constant prefactor, because the eigenstate is normalizable, and the M and W function span the space of solutions. Indeed, we have for ν = n, Mn,`+1/2 (

(2` + 1)! 2Zr 2Zr ) = (−1)n+`+1 Wn,`+1/2 ( ). a0 n (n + `)! a0 n

(4.56)

We remember that Mn,`+1/2 (ρ) was the solution for un` (ρ), so in order to obtain the radial wave function Rn` (r) of the bound state, we have to calculate Rn` (r) = un` (r)/r, within the separation ansatz given in Eq. (4.25). Furthermore, we apply a normalization prefactor √ (n + `)! Z 1/2 √ N= ( ) . (4.57) n (2` + 1)! (n − ` − 1)! a0 The normalized radial wave function Rn` (r) = N RE` (r) =

N N 2Zr uE` (r) = Mn,`+1/2 ( ) r r a0 n

(4.58)

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can be written as √ (n − ` − 1)! 2`+1 Z 3/2 Zr ` Zr 2Zr Rn` (r) = √ ( ) ( ) exp (− ) L2`+1 ) . (4.59) n−`−1 ( 2 n a a n a n a0 n 0 0 0 (n + `)! Combining the radial with the angular part, we find for the discrete bound-state eigenfunctions of the Schr¨ odinger–Coulomb Hamiltonian, ψn`m (r, θ, ϕ) = Rn` (r) Y`m (θ, ϕ) , HS ψn`m (r, θ, ϕ) = En ψn`m (r, θ, ϕ) = (−

(4.60a) (Zα)2 µ ) ψn`m (r, θ, ϕ) , 2n2

(4.60b)

where HS is given in Eq. (4.1). The normalization prefactor N ensures that the wave functions ψn`m (r, θ, ϕ) form an orthonormal set of basis functions, 3 ∗ ∫ d r ψn`m (r, θ, ϕ) ψn′ `′ m′ (r, θ, ϕ) = δnn′ δ``′ δmm′ .

(4.61)

Further useful properties of the associated Laguerre polynomials are summarized in Ref. [159]. 4.2.4

Schr¨ odinger–Coulomb Virial Theorem

The virial theorem connects the expectation values of the kinetic operator and the potential in an eigenstate of the Schr¨odinger–Coulomb Hamiltonian. It is useful to afford a small digression on this topic as the virial immediately leads to analytic formulas for the expectation values of the kinetic and potential energy operators. We recall the explicit form of the Schr¨odinger–Coulomb Hamiltonian, Zα p⃗ 2 +V , V =− , p⃗ = i m [HS , r⃗] . (4.62) 2m r Let us write the expectation value of an operator A in an eigenstate of the Schr¨ odinger–Coulomb Hamiltonian as ⟨A⟩ ≡ ⟨φ∣A∣φ⟩. For any operator A, in view of the eigenvalue property (HS − ES )∣φ⟩ = 0, we have HS =

[HS − ES , A] ∣φ⟩ = (HS − ES )A∣φ⟩ .

(4.63)

The expectation value of the kinetic operator in a Schr¨odinger–Coulomb energy eigenstate is given as ⟨

p⃗ 2 1 1 ⟩ = ⟨i [HS − ES , r⃗] ⋅ p⃗⟩ = − ⟨i r⃗ ⋅ (HS − ES ) p⃗⟩ (4.64) 2m 2 2 1 1 1 ⃗ )⟩ . = − ⟨i r⃗ ⋅ [HS − ES , p⃗ ]⟩ = ⟨i r⃗ ⋅ [⃗ p, HS − ES ]⟩ = ⟨⃗ r ⋅ ∇(V 2 2 2

Here, we use the shorthand notation ⟨A⟩ for the expectation value ⟨φ∣A∣φ⟩ of an operator A in the quantum state ∣φ⟩. If V is the Coulomb potential, then 1 1 Zα 1 Zα⃗ r 1 Zα 1 ⃗ )⟩ = ⟨⃗ ⃗ (− ⟨⃗ r ⋅ ∇(V r⋅ ∇ )⟩ = ⟨⃗ r⋅ 3 ⟩= ⟨ ⟩ = − ⟨V ⟩ , 2 2 r 2 r 2 r 2

(4.65)

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and the virial theorem follows as ⟨

1 p⃗ 2 ⟩ = − ⟨V ⟩ . 2m 2

(4.66)

Furthermore, for a bound-state ∣φ⟩ with ⟨φ∣HS ∣φ⟩ = ⟨φ ∣

(Zα)2 µ p⃗ 2 + V ∣ φ⟩ = − , 2m 2n2

(4.67)

one has (Zα)2 µ2 , n2 as a consequence of the virial theorem. ⟨φ∣⃗ p 2 ∣φ⟩ =

4.2.5

⟨φ∣V ∣φ⟩ = −

(Zα)2 µ , n2

(4.68)

Schr¨ odinger–Coulomb Continuum States

The continuum wave functions cover the energy spectrum from E = 0 to E = ∞, ⃗ of and it is customary to parameterize the energy according to the modulus k = ∣k∣ the wave vector, writing E=

k2 , 2µ

E > 0.

(4.69)

We recall the essential formulas for the radial part of the wave function regular at the origin, and convert these for continuum wave functions, ρ = (−8 µ E) ν = Zα (−

1/2

r = 2i k r ,

(4.70a)

µ 1/2 Zαµ ) = −i . 2E k

(4.70b)

According to Eqs. (4.30) and (4.36), the radial (Whittaker) wave function is a 1 F1 hypergeometric function ̃E` (ρ) = uE` (r) = M u

1 (ρ) n,`+ 2

1

= e− 2 ρ ρ`+1 1 F1 (` + 1 − ν, 2` + 2, ρ) ,

(4.70c)

which evaluates as follows, for a continuum state, Zαµ , 2` + 2, 2ikr) . (4.70d) k It is extremely important to get the analytic continuation into the complex plane right for all quantities mentioned above; it helps to assign an infinitesimal imaginary part to the energy E and to lay the branch cut of the square root along the negative real axis. The exponential factor exp(−i kr) is oscillatory for the continuum functions, and this behavior is typical for the continuum wave functions, which are normalized to a Dirac-δ. We write the continuum Schr¨odinger wave function ψk`m (⃗ r) as follows, uE` (r) = (2 i k r)`+1 e−i kr 1 F1 (` + 1 + i

ψk`m (⃗ r) = Rk` (r) Y`m (θ, ϕ) ,

Rk` (r) =

N′ uE` (r) , r

E=

k2 . 2µ

(4.71)

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These functions should now be normalized to a Dirac-δ in the momentum scale, and the normalization factor N ′ needs to be determined. Solutions for manifestly different wave vectors k and k ′ are orthogonal to each other, and the result for the overlap integral in the limit k → k ′ can therefore only be proportional to the Dirac-δ, which in turn implies that the result should be obtained by integration over a wide integration region. The only option for this is to integrate over the region of large radial variable r. We thus need to calculate the behavior of the confluent hypergeometric function for large argument. It is given by 1 F1 (a, c, z)

∼ Γ(c) (

(−z)−a ez z a−c + ), Γ(a) Γ(c − a)

∣z∣ → ∞ .

(4.72)

Substituting this asymptotic form into Eq. (4.70), one can obtain the following result after a somewhat tedious calculation [both terms in Eq. (4.72) contribute], Rk∗′ ` (r) Rk` (r) ∼

∣N ′ ∣2 i (k′ −k) r πZαµ Γ2 (2` + 2) exp (− e ) 2 r2 k ∣Γ (` + 1 + i Zαµ )∣ k +

∣N ′ ∣2 −i (k′ −k) r πZαµ Γ2 (2` + 2) , e exp (− ) 2 r2 k )∣ ∣Γ (` + 1 + i Zαµ k

r → ∞. (4.73)

i (k′ +k) r

On the right-hand side, we may ignore terms proportional to e and e which do not contribute to the Dirac-δ. The normalization integral becomes ∞ πZαµ Γ2 (2` + 2) dr r2 Rk∗′ ` (r) Rk` (r) = ∣N ′ ∣2 exp (− ) 2π δ(k ′ − k) . ∫ 2 k 0 ∣Γ (` + 1 + i Zαµ )∣ −i (k′ +k) r

k

(4.74)

The normalization constant may thus be chosen as follows, Zαµ πZαµ ∣Γ (` + 1 + i k )∣ ) . (4.75) 2k Γ(2` + 2) This leads to the following form for the radial part of the continuum wave functions,

N ′ = i−(`+1) exp (

2k (2 k r)` πZαµ Zαµ exp ( ) ∣Γ (` + 1 + i )∣ (2` + 1)! 2k k Zαµ , 2` + 2, 2ikr) . × e−i kr 1 F1 (` + 1 + i k The eigenvalue properties and normalization conditions are as follows, Rk` (r) =

ψk`m (⃗ r) = Rk` (r) Y`m (θ, ϕ) ,

HS ψk`m (⃗ r) =

k2 ψk`m (⃗ r) , 2µ

3 ∗ r) ψk′ `′ m′ (⃗ r) = 2πδ(k ′ − k) δ``′ δmm′ . ∫ d r ψk`m (⃗

(4.76a)

(4.76b) (4.76c)

In order to renormalize to the energy scale in the Eq. (4.76a), we observe that δ (E ′ − E) = δ (

k ′2 k 2 µ − ) = δ(k ′ − k) , 2µ 2µ k

∂ k2 k ( )= . ∂k 2µ µ

(4.77)

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Therefore, the continuum wave functions, normalized to the energy scale, are obtained from those normalized to the 2π δ(k − k ′ ) scale via a multiplicative factor √ µ/(2πk) and read as √ 2k µ (2 k r)` πZαµ Zαµ RE` (r) = exp ( ) ∣Γ (` + 1 + i )∣ π (2` + 1)! 2k k √ Zαµ × e−i kr 1 F1 (` + 1 + i , 2` + 2, 2ikr) , k = 2µ E . (4.78a) k The eigenvalue properties and normalization conditions change into ψE`m (⃗ r) = RE` (r) Y`m (θ, ϕ) , 3

∫ d

∗ r ψE`m (⃗ r) ψE ′ `′ m′ (⃗ r)

HS ψE`m (⃗ r) = E ψk`m (⃗ r) , ′

= δ(E − E) δ``′ δmm′ .

(4.78b) (4.78c)

A few remarks regarding the momentum-normalized, and energy-normalized, solutions are in order. For momentum normalization, our formulas are in agreement with the ones given in Eqs. (36.18) and (36.22) of Chap. 36 of Ref. [160]. The normalization integral is given in Eq. (33.4) of Chap. 33 of Ref. [160]. Our results also are in agreement with the results given in Eqs. (4.2), (4.8) and (4.20) of Ref. [6]. However, one has to take notice of the fact that the wave functions in Ref. [6] are normalized to the δ(k ′ − k)-scale (without the 2π prefactor). Of course, wave functions normalized to the √ scale δ(k ′ − k) can be obtained from Eq.√(4.76a) via multiplication by a factor 1/ 2π. This amounts to a replacement 2 → 2/π in the prefactor in Eq. (4.76a). A remark is in order concerning Chap. 4 of Ref. [6]. Namely, one has to read Chap. 4 of Ref. [6] carefully. The formulas given in Eqs. (4.22) and (4.23) of Ref. [6], in contrast to the normalization factor ck given in Eq. (4.20) of Ref. [6], are normalized to the energy scale, not to the wave number scale. This is implicitly stated in Ref. [6] but may give rise to confusion, including confusion in rather recent publications (see, e.g., the Erratum published for Ref. [161]). One may observe that in Refs. [6, 160], formulas for the continuum wave functions and normalization factors are scattered over a range of pages. Natural units are used in Eq. (4.76a). In atomic units, one would use the relation k = α me ka.u. ≈ α µ ka.u. , where me is the electron mass and ka.u. is the momentum expressed in atomic units. This is because αme c is the unit of momentum in atomic units. In natural units, we have c = 1. Furthermore, in atomic units, we have µ ≈ me = 1, because the atomic mass scale is the electron mass scale, which is approximately equal to the reduced mass µ. The natural unit system used in Eq. (4.76a) appears to be a good compromise between compact notation on one hand, and the retained possibility of keeping track of physical mass and length scales on the other hand, for the calculation of ionization cross sections. Our k in the prefactor in Eq. (4.76a) is dimensionful; all other factors in the wave function are dimensionless. Therefore, the physical dimension of Rk` equals that of a wave number.

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For the calculation of transition rates and cross sections, with the help of Fermi’s golden rule as discussed in Chap. 3, the energy-normalized wave functions have to be used. This is immediately evident if one takes into account that the final result for Fermi’s golden rule, given in Eq. (3.16), assumes that the transition is into a single, well-defined final state, which lies within an “allowed” energy band, over which one integrates in Eq. (3.15). In order for this integration to be well-defined, one has to discretize the energy continuum of the final states, say, in equal (small) intervals ∆E, with one state occupying an energy interval (E − ∆E/2, E + ∆E/2), in much the same way that one keeps a finite, but large normalization volume V in the derivation of Fermi’s golden rule. If one picks a representative energy E ∗ within the interval (E − ∆E/2, E + ∆E/2) and postulates that exactly one state should be found within the allowed interval ∆E, then the uniqueness of the final state in Eq. (3.16) is ensured. In the limit ∆E → 0, this condition is exactly fulfilled by wave functions normalized to a Dirac-δ in the energy variable.

4.2.6

Continuum States for a Repulsive Potential

For a repulsive Coulomb potential, the Hamiltonian (4.1) changes to HS =

p⃗ 2 Zα + , 2µ r

⃗. p⃗ = −i ∇

(4.79)

This Hamiltonian is obtained from Eq. (4.1) by the formal replacement r → −r in the potential term. A decomposition of the Laplacian operator in spherical coordinates reveals that the kinetic term in (4.79) is invariant under the substitution r → −r. Therefore, the eigenstates of the Hamiltonian (4.79) are obtained by the corresponding replacement r → −r in the Whittaker functions. As a consequence, there are no bound states for a repulsive Coulomb interaction. The exponential suppression of the bound-state wave function for large r is replaced by exponential growth under the replacement r → −r; the solutions are not normalizable. By contrast, continuum wave functions proportional to exp(−i k r) remain oscillatory under the replacement r → −r. A careful analysis shows that the argument of the exponential prefactor in Eq. (4.76a) also changes sign as one normalizes the continuum radial wave functions. When normalized to the energy scale, the continuum eigenstates of the repulsive potential can be written as ψE`m (⃗ r) = RE` (r) Y`m (θ, ϕ) , √ 2k µ (2 k r)` πZαµ Zαµ exp (− ) ∣Γ (` + 1 + i )∣ RE` (r) = π (2` + 1)! 2k k √ Zαµ × ei kr 1 F1 (` + 1 + i , 2` + 2, −2ikr) , k = 2µE . k

(4.80a)

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The eigenvalue properties are given as follows, ψE`m (⃗ r) = RE` (r) Y`m (θ, ϕ) ,

HS ψE`m (⃗ r) = E ψE`m (⃗ r) ,

3 ∗ r) ψE ′ `′ m′ (⃗ r) = δ(E ′ − E) δ``′ δmm′ . ∫ d r ψE`m (⃗

(4.80b) (4.80c)

The latter equation, again, implies that the wave functions are normalized to the energy scale. 4.3 4.3.1

Schr¨ odinger–Coulomb Green Function in Coordinate Space Green Function and Radial Equation

The Schr¨ odinger–Coulomb Green function G(⃗ r1 , r⃗2 , E) is the Green function of the Hamiltonian (4.1), which we recall for convenience, p⃗ 2 Zα − , 2µ r which fulfills the defining differential equation HS =

⃗, p⃗ = −i ∇

(HS − E) G(⃗ r1 , r⃗2 , E) = δ (3) (⃗ r1 − r⃗2 ) ,

(4.81)

(4.82)

where HS acts on the variable r⃗1 . We shall use techniques familiar from classical electrodynamics, namely, an algorithm which we would like to refer to as the concatenation approach for the calculation of the Green function (see Chaps. 2 and 4 of Ref. [81]), after the separation into radial and angular variables. For the Schr¨ odinger–Coulomb Hamiltonian, the angular structure of the Hamiltonian (4.81) helps in reducing Eq. (4.82) to a radial equation. The quantity E in Eq. (4.82) is not necessarily real, and in some problems of practical interest, it is helpful to promote the energy variable E to a general complex variable z. One sometimes sets z = En − ω where En is a bound-state energy, and ω is the (complex) energy of a virtual photon. We use the generalized principal quantum number ν in order to parameterize the energy argument E [see also Eq. (4.30)], Z 2 α2 µ . (4.83) 2E Our separation ansatz for the Green function is familiar from Chaps. 2 and 4 of Ref. [81], ν2 = −

∞

`

G(⃗ r1 , r⃗2 , ν) = ∑ ∑ f`m (r1 , r2 , θ2 , ϕ2 ) Y`m (θ1 , ϕ1 ) .

(4.84)

`=0 m=−`

Inserting the ansatz into the defining equation (4.82), we find ⃗2 1 ∂2 2 ∂ L Zα Z 2 α2 µ ∞ ` ( 2+ )+ 1 2 − + ] ∑ ∑ f`m (r1 , r2 , θ2 , ϕ2 ) Y`m (θ1 , ϕ1 ) 2µ ∂r1 r1 ∂r1 2µ r1 r1 2ν 2 `=0 m=−` 1 = δ(r1 − r2 ) δ(cos θ1 − cos θ2 ) δ(ϕ1 − ϕ2 ) , (4.85) r1 r2

[−

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where we have used (r1 r2 )−1 as a symmetric prefactor. One multiplies the equation ̵ 2 and obtains by −2µ/h ⃗ 2 2Zαµ (Zαµ)2 ∞ ` ∂2 2 ∂ L + − 21 + − ) ∑ ∑ f`m (r1 , r2 , θ2 , ϕ2 ) Y`m (θ1 , ϕ1 ) 2 ∂r1 r1 ∂r1 r1 r1 ν2 `=0 m=−` 2µ δ(r1 − r2 ) δ(cos θ1 − cos θ2 ) δ(ϕ1 − ϕ2 ) . (4.86) =− r1 r2

(

We multiply both sides by Y`∗′ m′ (θ1 , ϕ1 ) and integrate over the solid angle of r⃗1 , ∫ dΩ1 = ∫

1 −1

d(cos θ1 ) ∫

2π 0

dϕ1 .

(4.87)

The orthonormality of the spherical harmonics dictates that ∞

`

∑ ∑ [(

`=0 m=−`

∂2 2 ∂ `(` + 1) 2Zαµ (Zαµ)2 ) f`m (r1 , r2 , θ2 , ϕ2 )] δ``′ δmm′ + − + − ∂r12 r1 ∂r1 r12 r1 ν2

∂2 2 ∂ `(` + 1) 2Zαµ (Zαµ)2 + − + − ) f`′ m′ (r1 , r2 , θ2 , ϕ2 ) ∂r12 r1 ∂r1 r12 r1 ν2 2µ =− δ(r1 − r2 ) Y`∗′ m′ (θ2 , ϕ2 ) . r1 r2 =(

(4.88)

As expected, we find that the angular dependence of f`′ m′ (r1 , r2 , θ2 , ϕ2 ) is given by Y`∗′ m′ (θ2 , ϕ2 ). One now redefines `′ → ` and m′ → m and separates f`m into a radial and angular part, writing ∗ f`m (r1 , r2 , θ2 , ϕ2 ) = g` (r1 , r2 , ν) Y`m (θ2 , ϕ2 ) ,

(4.89)

which amounts to the following representation of the Green function, ∞

`

∗ G(⃗ r1 , r⃗2 , ν) = ∑ ∑ g` (r1 , r2 , ν) Y`m (θ1 , ϕ1 ) Y`m (θ2 , ϕ2 ) .

(4.90)

`=0 m=−`

We thus have to find a purely radial function g` (r1 , r2 , ν) satisfying the equation ∂2 2 ∂ `(` + 1) 2Zαµ (Zαµ)2 2µ + − + − ) g` (r1 , r2 , ν) = − δ(r1 − r2 ) . 2 ∂r1 r1 ∂r1 r12 r1 ν2 r1 r2 (4.91) In analogy to our analysis of bound-state wave functions, where we wrote the radial part as RE` (r) = uE` (r)/r, we now define the rescaled radial function G, (

G` (r1 , r2 , ν) = r1 r2 g` (r1 , r2 , ν) ,

(4.92)

which fulfills ∂2 `(` + 1) 2Zαµ (Zαµ)2 2µ 1 ( 2− + − ) G` (r1 , r2 , ν) = − δ(r1 − r2 ) . (4.93) r1 r2 ∂r1 r12 r1 ν2 r1 r2 Our experience with the Schr¨ odinger–Coulomb Hamiltonian suggests a rescaling to dimensionless quantities, ρ1 =

2Zα µ r1 , ν

ρ2 =

2Zα µ r2 , ν

(4.94)

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to find the dimensionless equation ⎛ ∂2 1 ν − + + 2 ∂ρ 4 ρ1 ⎝ 1

1 4

2

− (` + 12 ) ⎞ ν G` (ρ1 , ρ2 , ν) = − δ(ρ1 − ρ2 ) . ρ21 Zα ⎠

(4.95)

We will solve this equation in two ways, (i) in the same way as for the Poisson equation (Chap. 2 of Ref. [81]), by an appropriate construction using Whittaker functions, and (ii) using a separation ansatz based on the associated Laguerre polynomials. 4.3.2

Solution using Whittaker Functions

We recall from Sec. 4.2.2 that the general eigenfunction solution of the Schr¨odinger– Coulomb Hamiltonian is given by the linear combination of Whittaker functions M and W , where the M function is regular at the origin while the W function is not. We thus have two solutions to the homogeneous equation for ρ1 < ρ2 and ρ1 > ρ2 . Inspired by our treatment of Green functions in classical electrodynamics (Chaps. 2 and 4 of Ref. [81]), we concatenate the radial Green function G` (ρ1 , ρ2 , ν) at ρ1 = ρ2 by integrating Eq. (4.95) from ρ1 = ρ2 −  to ρ1 = ρ2 + . A comparison of Eqs. (4.33) and (4.95) reveals that the general solution to the homogeneous equation corresponding to Eq. (4.95) (i.e., the equation obtained by replacing the Dirac-δ function by zero) reads as follows, G` (ρ1 , ρ2 , ν) = α` (ρ2 ) M

ν,`+

1 (ρ1 ) . 1 (ρ1 ) + β` (ρ2 ) W ν,`+ 2 2

(4.96)

For ρ1 < ρ2 , we assume that G` (ρ1 , ρ2 , ν) = α` (ρ2 ) Mν,`+1/2 (ρ1 ) because the solution needs to be regular at the origin. For ρ1 > ρ2 on the other hand, we have G` (ρ1 , ρ2 , ν) = β` (ρ2 ) Wν,`+1/2 (ρ1 ) because the solution needs to be regular at infinity. Continuity at ρ1 = ρ2 requires that (we assume that a is a constant) G` (ρ1 , ρ2 , ν) = a M

ν,`+

1 (ρ> ) , 1 (ρ< ) W ν,`+ 2 2

ρ< = min(ρ1 , ρ2 ) ,

ρ> = max(ρ1 , ρ2 ) . (4.97)

We now integrate Eq. (4.95) from ρ1 = ρ2 −  to ρ1 = ρ2 +  to obtain ∫

ρ2 + ρ2 −

⎛ ∂2 1 ν dρ1 − + + ⎝ ∂ρ21 4 ρ1

1 4

2

− (` + 12 ) ⎞ G` (ρ1 , ρ2 , ν) ρ21 ⎠

ρ2 + ν dρ1 δ(ρ1 − ρ2 ) . (4.98) ∫ Zα ρ2 − The terms without derivative operators on the left-hand side drop out because G` (ρ1 , ρ2 , ν) is assumed to be continuous at ρ1 = ρ2 ,

=−

ρ2 +

ν ∂2 G` (ρ1 , ρ2 , ν)] = − . 2 ∂ρ1 Zα ρ2 − An integration immediately leads to ⎡ ⎤ ⎢ ∂G` (ρ1 , ρ2 , ν) ⎥ ∂G` (ρ1 , ρ2 , ν) ⎢ ⎥=− ν . lim ⎢( ) −( ) ⎥ →0 ⎢ ∂ρ1 ∂ρ Zα 1 ρ1 =ρ2 + ρ1 =ρ2 − ⎥ ⎣ ⎦ ∫

dρ1 [

(4.99)

(4.100)

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We now use the specific forms of G` (ρ1 , ρ2 , ν) for the two terms in Eq. (4.97), observing that in each case, only one of the Whittaker functions depends on ρ1 . Specifically, ρ1 = ρ2 +  implies that ρ1 = ρ> , while ρ1 = ρ2 −  implies that ρ2 = ρ> . We can thus pull out the respective other Whittaker function to write a [M

∂

∂

W M 1 (ρ1 ) − W 1 (ρ1 ) 1 (ρ1 )] 1 (ρ1 ) ν,`+ ∂ρ1 ν,`+ 2 ∂ρ1 ν,`+ 2 2 2

ν,`+

=−

ν . Zα

(4.101)

The Wronskian of the two solutions (M and W functions) is independent of ρ and can therefore be calculated for any ρ. We take notice of the following asymptotic forms, M

ν,`+

1 (ρ) 2

→ ρ`+1 ,

W

ν,`+

1 (ρ) 2

→

Γ(2` + 1) , Γ(1 + ` − ν) ρ`

ρ → 0.

(4.102)

The Wronskian thus evaluates to M

ν,`+

1 (ρ) 2

(

∂ ∂ Γ(2` + 2) W M . (4.103) 1 (ρ)) − W 1 (ρ) ( 1 (ρ)) = − ν,`+ ν,`+ ν,`+ ∂ρ ∂ρ Γ(1 + ` − ν) 2 2 2

Solving Eq. (4.101) for a, we find a=

ν Γ(1 + ` − ν) . Zα (2` + 1)!

(4.104)

We immediately find the G` function as G` (ρ1 , ρ2 , ν) =

ν Γ(1 + ` − ν) M 1 (ρ< ) W 1 (ρ> ) , ν,`+ ν,`+ Zα (2` + 1)! 2 2

(4.105)

and g` is obtained by dividing G` by r1 r2 , g` (ρ1 , ρ2 , ν) =

Γ(1 + ` − ν) ν M 1 (ρ< ) W 1 (ρ> ) . ν,`+ (2` + 1)! Zα r1 r2 ν,`+ 2 2

(4.106)

Furthermore, ∞

`

∗ G(⃗ r1 , r⃗2 , ν) = ∑ ∑ g` (r1 , r2 , ν) Y`m (θ1 , ϕ1 ) Y`m (θ2 , ϕ2 ) ,

(4.107)

`=0 m=−`

so that g` (r1 , r2 , ν) =

ν Γ(1 + ` − ν) 2Zαµ 2Zαµ M r< ) W r> ) . (4.108) 1 ( 1 ( ν,`+ ν,`+ (2` + 1)! Zα r1 r2 ν ν 2 2

A final word on the integration ranges for r< and r> is in order. Within classical electrodynamics (Chap. 5 of Ref. [81]), we can generally associate r< with the point at which radiation is generated, whereas r> is the point at which radiation is observed. Within typical atomic physics calculations, one cannot avoid integrations over the entire range of r ∈ (0, ∞), and the integration over the full range of r is suitable for numerical evaluation in many cases.

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4.3.3

Solution using Laguerre Polynomials

For analytic calculations, a representation in terms of so-called Sturmian functions is sometimes preferred over the representation (4.108). The so-called Sturmians are special forms of Laguerre polynomials, and the Sturmian representation reminds us of an eigenfunction representation (see Sec. 2.5 of Ref. [81]), because it is formulated in terms of the variables r1 and r2 , not r< and r> , and therefore facilitates integrations over r1 and r2 separately. We incur a further explicit sum over the Sturmians, though, in exchange for discarding the need to use the variables r< and r> , in lieu of the more convenient r1 and r2 . We use the ansatz ∞

G` (ρ1 , ρ2 , ν) = ∑ Rk` (ρ1 ) fk` (ρ2 , ν) .

(4.109)

k=0

We recall Eq. (4.95), ⎛ ∂2 1 ν − + + 2 ⎝ ∂ρ1 4 ρ1

1 4

2

− (` + 12 ) ⎞ ν G` (ρ1 , ρ2 , ν) = − δ(ρ1 − ρ2 ) . 2 ρ1 Zα ⎠

(4.110)

This leads to ∞

⎛ ∂2 1 ν − + + ∑ fk` (ρ2 , ν) 2 ⎝ ∂ρ1 4 ρ1 k=0

1 4

2

− (` + 12 ) ⎞ ν Rk` (ρ1 ) = − δ(ρ1 − ρ2 ) . 2 ρ1 Zα ⎠

(4.111)

We know that Rk` (ρ1 ) = Mν,`+1/2 (ρ1 ) and Rk` (ρ1 ) = Wν,`+1/2 (ρ) are solutions of the homogeneous equation. Let us pretend, for a moment, that ν = n is an integer. Then, the solution to the homogeneous equation is [see Eq. (4.52)] M

1 (ρ) n,`+ 2

1

= e− 2 ρ ρ`+1

(n − ` − 1)! (2` + 1)! 2`+1 Ln−`−1 (ρ) . (n + `)!

(4.112)

Identifying k = n − ` − 1, we enter into Eq. (4.109) with the ansatz Rk` (ρ) =

k! (2` + 1)! `+1 2`+1 ρ Lk (ρ) e−ρ/2 , (k + 2` + 1)!

(4.113)

which, for ν = n, would just be the radial eigenfunction. For non-integer ν, however, a residual term remains which can be used in order to fulfill the inhomogeneous equation. Performing elementary differentiations, we obtain the following result, ( =

∂2 1 ν − + + ∂ρ21 4 ρ1

1 4

− (` + 12 )2 ) Rk` (ρ1 ) ρ21

k! (2` + 1)! −ρ1 /2 ` ∂2 ∂ e ρ1 (ρ1 2 + (2` + 2 − ρ1 ) + (ν − ` − 1)) L2`+1 (ρ1 ) . k (k + 2` + 1)! ∂ρ1 ∂ρ1 (4.114)

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The last part of Eq. (4.114) is close to the defining differential equation of the Laguerre polynomials [159]. We can easily rearrange the differential equation in Eq. (4.114) to the following form: (ρ1

∂ ∂2 + (2` + 2 − ρ1 ) + (ν − ` − 1)) L2`+1 (ρ1 ) k 2 ∂ρ1 ∂ρ1

= (ρ1

∂2 ∂ + (2` + 2 − ρ1 ) + k) L2`+1 (ρ1 ) + (ν − ` − 1 − k) L2`+1 (ρ1 ) k k ∂ρ21 ∂ρ1

= (ν − ` − 1 − k) L2`+1 (ρ1 ) . k

(4.115)

We have used the differential equation solved by the associated Laguerre polynomials Lξk (ρ1 ), which reads as follows [see Eq. (22.6.15) of Ref. [159]] (ρ1

∂ ∂2 + (ξ + 1 − ρ1 ) + k) Lξk (ρ1 ) = 0 , 2 ∂ρ1 ∂ρ1

(4.116)

where in our case ξ = 2` + 1. Combining Eqs. (4.114) and (4.115), we find ∂2 `(` + 1) ν 1 k! (2` + 1)! −ρ1 /2 ` − + − ) Rk` (ρ1 ) = e ρ1 (ν − ` − 1 − k) L2`+1 (ρ1 ) . k ∂ρ21 ρ21 ρ1 4 (k + 2` + 1)! (4.117) Plugging the result back into Eq. (4.111) yields (

∞

ν k! (2` + 1)! −ρ1 /2 ` e ρ1 (ν − ` − 1 − k) L2`+1 (ρ1 ) = − δ(ρ1 − ρ2 ) . k (k + 2` + 1)! Zα k=0 (4.118) 2`+1 (ρ ) and integrate over the remaining We multiply both sides with e−ρ1 /2 ρ`+1 L ′ 1 1 k radial part of the volume integral ∫ dρ1 ρ21 , leading to the expression ∑ fk` (ρ2 , ν)

∞

∑ (ν − ` − 1 − k)

k=0

∞ k! (2` + 1)! L2`+1 (ρ1 )L2`+1 fk` (ρ2 , ν) ∫ dρ1 e−ρ1 ρ2`+1 1 k k′ (ρ1 ) (k + 2` + 1)! 0

∞ ν −ρ1 /2 2`+1 dρ1 ρ`+1 Lk′ (ρ1 ) δ(ρ1 − ρ2 ) . (4.119) ∫ 1 e Zα 0 The orthogonality relation of the associated Laguerre polynomials is given by [see Eq. (22.2.12) of Ref. [159]]

=−

(k + 2` + 1)! δkk′ . k! 0 Consequently, if we carry out the integration, we obtain ∫

∞

dρ1 e−ρ1 ρ2`+1 L2`+1 (ρ1 )L2`+1 1 k k′ (ρ1 ) = ∞

∑ (ν − ` − 1 − k)

k=0

k! (2` + 1)! (k + 2` + 1)! fk` (ρ2 , ν) δkk′ (k + 2` + 1)! k!

(4.120)

ν `+1 −ρ2 /2 2`+1 ρ e Lk′ (ρ2 ) . (4.121) Zα 2 Collapsing the sum using the Kronecker-δ and redefining k ′ → k, one finds that ν `+1 −ρ2 /2 2`+1 (ν − ` − 1 − k) (2` + 1)! fk` (ρ2 , ν) = − ρ e Lk (ρ2 ) , (4.122) Zα 2 =−

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which immediately gives a result for fk` (ρ2 , ν). We plug the result for fk` (ρ2 , ν) back into Eq. (4.109), and use Eq. (4.113), G` (ρ1 , ρ2 , ν) = =

e−ρ2 /2 L2`+1 (ρ2 ) ν ∞ `+1 k ∑ Rk` (ρ1 ) ρ2 Zα k=0 (2` + 1)! (k + ` + 1 − ν) k! L2`+1 (ρ1 ) L2`+1 (ρ2 ) ν ∞ − 1 (ρ1 +ρ2 ) k k (ρ1 ρ2 )`+1 . ∑e 2 Zα k=0 (k + 2` + 1)! (k + ` + 1 − ν)

(4.123)

Recalling the definitions from Eq. (4.94), one arrives at 2Z 2`+1 r1 + r2 ) exp (−Z ) (r1 r2 )`+1 a0 ν a0 ν 2Zr2 2`+1 2Zr1 ) L2`+1 ( ) ( ∞ k! Lk k a0 ν a0 ν ×∑ . (k + 2` + 1)! (k + ` + 1 − ν) k=0

G` (ρ1 , ρ2 , ν) = 2µ (

(4.124)

Using Eq. (4.92), we obtain the radial Green function g` (r1 , r2 , ν) by dividing G` by the product r1 r2 , 2Zr2 2`+1 2Zr1 ( ) L2`+1 ) ( 2`+1 ∞ k! Lk k Z(r +r ) 1 2 2Z a0 ν a0 ν − ) e a0 ν (r1 r2 )` ∑ , g` (r1 , r2 , ν) = 2µ ( a0 ν (k + 2` + 1)! (k + ` + 1 − ν) k=0 (4.125) where a0 = 1/(αµ) is the Bohr radius including the reduced-mass dependence. Our result is in agreement with the one derived in Ref. [158] by a Laplace transform of the differential equation for the radial Green function. For convenience, we recall that the total Green function is given as ∞

`

∗ G(⃗ r1 , r⃗2 , ν) = ∑ ∑ g` (r1 , r2 , ν) Y`m (θ1 , ϕ1 ) Y`m (θ2 , ϕ2 ) .

(4.126)

`=0 m=−`

The representation (4.125) separates the two variables r1 and r2 and is especially suited for analytic calculations, because the radial integrations can be carried out independently, before the sum over k is taken. 4.3.4

Green Function and Dynamic Polarizability

The Green function as derived in Eq. (4.125) is to be applied to the evaluation of matrix elements of the Schr¨ odinger–Coulomb propagator. Two important matrix elements are as follows, 1 1 P (φn , ω) = ⟨φn ∣pi pi ∣ φn ⟩ , (4.127a) 3µ HS − E n + ω (Zα)4 µ3 1 Q(φn , ω) = ⟨φn ∣xi x i ∣ φn ⟩ , (4.127b) 3 HS − En + ω where the sum over i = 1, 2, 3 is implicit in view of the summation convention. We use superscripts in order to denote the Cartesian components, and we have to include prefactors of µ in order to render the matrix elements dimensionless. In

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Eqs. (4.127a) and (4.127b), φn is the bound reference state with energy En < 0. We assume that the subscript n in φn is a multi-index that specifies the principal quantum number and the orbital angular momentum quantum number `. The matrix elements P and Q are independent of the magnetic projection quantum number m. They are related to the polarizability α(ω) defined in Eq. (3.53). While implicit formulas for these matrix elements were derived around 1970 in Ref. [162], it was not until 1993 that convenient expressions were derived for the matrix elements of the propagator, in Ref. [163]. The calculation has to be well organized, and one has to be familiar with a number of rather peculiar identities fulfilled by hypergeometric functions (so-called contiguous relations). We shall describe the evaluation in some detail, for the reference state being the 1S ground state of the hydrogen atom, and first concentrate on the matrix element 1 1 1 1 ⟨1S ∣pi pi ∣ 1S⟩ = ⟨1S ∣pi pi ∣ 1S⟩ , z = En − ω , P (1S, ω) = 3µ HS − En + ω 3µ HS − z (4.128) where the Schr¨ odinger–Coulomb Hamiltonian HS [see also Eq. (4.1)] and the 1S wave function are given as follows, p⃗ 2 Zα − , 2µ r

Z 3/2 −Zr/a0 ) e . a0 (4.129) We had previously parameterized the energy argument z of the Green function in terms of the generalized principal quantum number ν as follows [see Eq. (4.83)], √ n2 En ν (Zα)2 µ = En 2 , t= = . (4.130) z = En − ω = − 2 2ν ν z n For the ground state, we have ν = t because n = 1. The t variable has certain advantages. Namely, if z = En − ω, then ω = 0 corresponds to t = 1 and ω = ∞ corresponds to t = 0. Any conceivable integral over ω, which we may have to do in the following, therefore is mapped onto an integral over t in the range t ∈ (0, 1). In terms of the photon energy ω, we write √ −1/2 En 2n2 ω (Zα)2 µ 1 − t2 = (1 + ) , ω= . (4.131) t= 2 En − ω (Zα) µ 2n2 t2 HS =

ψ1S (⃗ r) = R1S (r) Y00 (θ, ϕ) ,

R1S (r) = 2 (

An integral over a range of ω values can be transformed into a t integral as follows, ∫

ω1 ω0

dω ω f (ω) =

t0 (Zα)4 µ2 1 − t2 ˜ dt f (t) , ∫ 2n4 t5 t1

(4.132)

where f˜(t) ≡ f (ω(t)) is the original function, expressed in terms of t. All of these identities will become useful in the following. We now return to the evaluation of the matrix element of the propagator and use the coordinate-space representation of the Schr¨ odinger–Coulomb Green function, ∞

`

∗ G(⃗ r1 , r⃗2 , ν) = ∑ ∑ g` (r1 , r2 , ν) Y`m (θ1 , ϕ1 ) Y`m (θ2 , ϕ2 ) . `=0 m=−`

(4.133)

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For the radial Green function, we use the Sturmian decomposition given in Eq. (4.125), ∞ 2Z 2`+1 −Z ra1 +rν2 0 ) e g` (r1 , r2 , ν) = 2µ ( (r1 r2 )` ∑ a0 ν k=0

2Zr2 2Zr1 ) L2`+1 ( ) k a0 ν a0 ν . (k + 2` + 1)! (k + ` + 1 − ν) (4.134)

k! L2`+1 ( k

The (` = 1)-component reads as follows, 2Zr2 2Zr1 ) L3k ( ) a0 ν a0 ν . (k + 3)! (2 + k − ν) (4.135) After the angular integration, which is easy to perform using the explicit representation of the spherical harmonics, we reduce P (1S, ω) to the following radial integral, ∞ r1 + r2 Z 3 g`=1 (r1 , r2 , ν) = 16µ ( ) exp (−Z ) r1 r2 ∑ a0 ν a0 ν k=0

k! L3k (

∞ ∞ ∂ ∂ 1 dr1 r12 ∫ dr2 r22 ( R1S (r1 )) g`=1 (r1 , r2 , t) ( R1S (r2 )) . ∫ 3µ 0 ∂r1 ∂r2 0 (4.136) Here and elsewhere, partial derivatives are indicated in order to denote the differentiation with respect to the radial variable only (strictly speaking, the wave function also depends on other parameters like the Bohr radius). We use the rescaling of variables as in Eq. (4.94),

P (1S, ω) =

2Zα µ 2Zα µ r1 , ρ2 = r2 , (4.137) t t where we take into account that ν = t for the ground state, to bring the integral into the form ρ1 =

∞

t5 k! k=0 12 (k + 3)! (2 + k − t)

P (1S, t) = ∑

×∫

∞ 0

dρ1 ρ21 ∫

∞ 0

1

1

dρ2 ρ22 e− 2 (1+t) ρ1 − 2 (1+t) ρ2 L3k (ρ1 ) L3k (ρ2 ) , (4.138)

changing the argument of the function from ω to t, which parameterizes the energy according to Eq. (4.131). The integrals over ρ1 and ρ2 may be evaluated using the formula (10.12.33) of Ref. [156], ∫

∞ 0

dρ e−sρ ργ−1 Lm k (ρ) =

1 Γ(γ) Γ(k + m + 1) −γ s 2 F1 (−k, γ, 1 + m, ) . k! Γ(m + 1) s

(4.139)

The hypergeometric function resulting from the integration is terminating in view of the negative first argument −k < 0. In typical cases of practical interest, after the use of contiguous relations, the hypergeometrics can be expressed in closed analytic form, using formulas such as 2 F1 (−k, a, a, z)

= (1−z)k ,

2 F1 (−k, a+1, a, z)

= (1−z)k

(z − 1) a + k z . a (z − 1)

(4.140)

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The matrix element P (1S, t) can thus be written as a rather compact sum, P (1S, t) =

∞ 64 t5 (k + 1) (k + 2) (k + 3) (−ζ)2k , ∑ 8 3 (1 + t) k=0 (2 + k − t)

ζ=

1−t . (4.141) 1+t

The summation over k can be carried out using various formulas. We here give a selection of useful sums, which are relevant to matrix elements which involve the Schr¨ odinger–Coulomb propagator, and refer to standard encyclopediae for larger collections of formulas [164–169]. Three relatively easy formulas are as follows, ∞

k ∑ ks =

k=0

∞

s , (1 − s)2

2 k ∑k s =

k=0

s + s2 , (1 − s)3

∞

3 k ∑k s =

k=0

s + 4 s2 + s3 . (4.142) (1 − s)4

A slightly more involved formula is ∞

sk 2 F1 (1, a, 1 + a, s) = Φ(s, 1, a) = . a + k a k=0 ∑

(4.143)

Here, Φ is the Lerch transcendent, whose general defining relation is ∞

sk . γ k=0 (a + k)

Φ(s, γ, a) = ∑

(4.144)

A formula valid in a larger domain of the complex plane is Φ(s, γ, a) =

∞ 1 −s k k (−1)j k ) ∑ ( ). ∑( 1 − s k=0 1 − s j=0 (a + j)γ j

(4.145)

The Lerch Φ transcendent can be thought of as a generalization of the Riemann zeta function, because Φ(1, s, 1) = ζ(s). Sometimes, more complex mathematical formulae have to be used, like the following one, found in Ref. [154], ∞

sz Γ(k + λ) k s 2 F1 (−k, b, c, z) = Γ(λ) (1 − s)−λ 2 F1 (λ, b, c, − ). k! 1−s k=0 ∑

(4.146)

For our expression given in Eq. (4.141), the summation over k leads to the following result, P (1S, t) =

2t2 (1 + 5t + 7t2 + 11t3 ) 64 (1 − t) t6 1−t 2 − F (1, 2 − t, 3 − t, ( ) ). 2 1 3 (1 + t)5 3 (2 − t) (1 + t)7 1+t (4.147)

This expression already is quite compact, but it is possible to bring it into some more standard form, using the so-called contiguous relations for hypergeometric functions. These are given in Sec. 2.8 of Ref. [154]. Two of them, conveniently written to allow for a lowering of the second and third arguments, read as follows, c−1 1 [2 F1 (a, b − 1, c − 1, z) + (z − 1) 2 F1 (a, b, c − 1, z)] , 2 F1 (a, b, c, z) = z (c − a − 1) (4.148a) 1 [(c − 1) 2 F1 (a, b − 1, c − 1, z) + (b − c) 2 F1 (a, b − 1, c, z)] . 2 F1 (a, b, c, z) = b−1 (4.148b)

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Judicious repeated application of the contiguous relations, lowering the second and third arguments of the hypergeometric functions in turn, then leads to the result P (1S, t) =

1−t 2 64 t5 2t2 (1 − t − 5t2 − 11t3 ) F (1, −t, 1 − t, ( + ) ) . (4.149) 2 1 3 (1 + t)2 (1 − t)3 3 (1 − t2 )3 1+t

For an excited reference state with principal quantum number n ≥ 2, the hypergeometric function in the result takes a slightly more general form, 2 F1 (1, −n t, 1 − n t, (

1−t 2 ) ), 1+t

(4.150)

as exemplified in the results P (2S, t) =

2t2 (1 − 2 t − 8 t2 − 46 t3 + 151 t4 ) 4

2

3 (1 − t) (1 + t) +

256 t5 (1 − 4 t2 ) 4

3 (1 − t) (1 + t)

F (1, −2t, 1 − 2t, ( 4 2 1

1−t 2 ) ), 1+t

(4.151)

and P (2P, t) =

2t2 (3 − 6 t − 3 t2 + 12 t3 + 29 t4 + 122 t5 − 413 t6 ) 5

9 (1 − t) (1 + t) +

3

256 t7 (−3 + 11 t2 ) 5

9 (1 − t) (1 + t)

F (1, −2t, 1 − 2t, ( 5 2 1

1−t 2 ) ). 1+t

(4.152)

In the results (4.149), (4.151), and (4.152), t parameterizes the photon energy ω according to Eq. (4.131); it is important that t in this case carries a dependence on ̵ = c = 0 = 1, the principal quantum number n. As we are using natural units with h it may be helpful to convert Eq. (4.131) to SI mksA units, t = (1 +

̵ 2n2 hω ) 2 (Zα) µ c2

−1/2

.

(4.153)

The result (4.152) for the dynamic polarizability, expressed in terms of the energy variable, has been derived in Ref. [170]. We recall the conventions implied by Eq. (4.130). For completeness, we also indicate explicit results for Q matrix elements, for the states with principal quantum numbers n = 1 and n = 2. For the ground state, we have Q(1S, t) =

2t2 (3 − 3t − 12t2 + 12t3 + 19t4 − 19t5 − 26t6 − 38t7 ) 3 (1 − t)5 (1 + t)4 +

256 t9 1−t 2 F (1, −t, 1 − t, ( ) ). 2 1 3 (1 + t)5 (1 − t)5 1+t

(4.154)

The static limit of the polarizability is proportional to the limit as t → 1 of the matrix element (4.154).

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101

An expansion of the hypergeometric 2 F1 function in powers of its argument [(1 − t)/(1 + t)]2 , amounts to an expansion for t → 1, or ω → 0. With the resulting expression, one easily obtains the well-known result limω→0 Q(1S, ω) = 9/4. The 2S state has the following matrix element, Q(2S, t) =

16t2 (21 − 42 t − 48 t2 + 138 t3 + 14 t4 − 166 t5 − 16 t6 − 314 t7 + 1181 t8 ) 3 (1 − t)6 (1 + t)4 +

1−t 2 16384 t9 (1 − 4t2 ) ) ). 2 F1 (1, −2t, 1 − 2t, ( 6 6 3 (1 − t) (1 + t) 1+t

(4.155)

This result is divergent in the limit ω → 0, because of the virtual 2P state, which is energetically degenerate with respect to the 2S reference state. This behavior has been studied in detail in Ref. [171] and, in the context of black-body energy shifts, in Ref. [172]. For t → 1, one has 36 9(Zα)2 µ + 60 + O(t − 1) = + 60 + O(ω) . (4.156) 1−t ω Finally, for the 2P reference state, we have a more complex result, which reads Q(2S, t) =

Q(2P, t) =

16t2 (45 − 90t − 153t2 + 396t3 + 150t4 − 696t5 9 (1 − t)7 (1 + t)5 + 66t6 + 564t7 + 61t8 + 850t9 − 3241t10 ) −

16384 t11 (3 − 11 t2 ) 1−t 2 ) ). 2 F1 (1, −2t, 1 − 2t, ( 7 7 9 (1 + t) (1 − t) 1+t

(4.157)

For t → 1, one has 3(Zα)2 µ 12 + 70 + O(t − 1) = + 88 + O(ω) . (4.158) 1−t ω The results is Eqs. (4.152) and (4.157) require the use of the (` = 0)- and (` = 2)components in the Schr¨ odinger–Coulomb propagator (4.134). Let us conclude the discussion of dipole matrix elements of the Schr¨odinger– Coulomb Hamiltonian, by establishing the connection to the dynamic polarizability defined in Eq. (3.53), which we recall as Q(2P, t) =

e2 1 1 (⟨φn ∣xi xi ∣ φn ⟩ + ⟨φn ∣xi xi ∣ φn ⟩) , 3 HS − E n + ω HS − En − ω (4.159) where the two terms with the different sign of ω are explicitly written out. The matrix element involving the denominator HS − En + ω is proportional to Q(φn , t), whereas the one with the denominator H0 − En − ω is proportional to Q(φn , s) with α(φn , ω) =

s = (1 −

̵ 2n2 hω ) 2 (Zα) µ c2

−1/2

.

(4.160)

̵ exceeds the bound-state energy (Zα)2 µ c2 /(2n2 ), the argument If the energy hω s acquires an imaginary part which eventually gives rise to the imaginary part of

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the dipole polarizability [which is proportional to the one-photon ionization crosssection, see Eq. (3.58)]. The results for the Q matrix elements in Eqs. (4.154) and (4.155) are dimensionless. In SI mksA units, the polarizability has a dimension of the square of the dipole moment, divided by the energy. Applying this scaling consistently, we have in SI mksA units a0 2 1 ) (Q(φn , t) + Q(φn , s)) Z (Zα)2 µc2 ̵2 e2 h = (Q(φn , t) + Q(φn , s)) . (Zα)4 µ3 c4

α(φn , ω) = e2 (

(4.161)

The static limits for 1S, 2S, and 2P states are ̵2 9 e2 h , 2 (Zα)4 µ3 c4 ̵2 e2 h , α(2S, ω → 0) = 120 4 (Zα) µ3 c4 ̵2 e2 h α(2P, ω → 0) = 176 . (Zα)4 µ3 c4 α(1S, ω → 0) =

(4.162a) (4.162b) (4.162c)

For the 2S and 2P states, the divergence of the Q matrix elements as ω → 0 cancels out in the final result because the limit is approached symmetrically in the two terms involving the two signs ±ω in Eq. (4.159). From the atomic physics point of view, we now have all necessary analytic techniques at our disposal, which are required in order to calculate the Bethe logarithm, which is a quantity obtained as an integral over matrix elements of the P type, i.e., over (a part of) the dynamic polarizability of the atom. However, before we can finally proceed to the calculation of this quantity, and be able to interpret the result in a suitable context, we still need to discuss two further ideas: the renormalization and the overlapping parameter. 4.4 4.4.1

Schr¨ odinger–Coulomb Green Function in Momentum Space Free Green Function in Momentum Space

We have just finished our discussion of the Schr¨odinger–Coulomb Green function in coordinate space, and evaluated a number of phenomenologically interesting matrix elements. For a number of calculations, though, which include the quadrupole radiation correction to the Lamb shift [163,173], a different formulation is useful, namely, in momentum space. In addition, the derivation of the Schr¨odinger–Coulomb Green function in momentum space offers the opportunity for an ideal illustration of the algebraic beauty of the SO(4) symmetry of the hydrogen atom, based on ideas originally formulated in Ref. [174]. However, a close inspection of Ref. [174] reveals that a number of details of the calculation are being “swept under the rug”, and the connection of the Schr¨ odinger–Coulomb wave functions in momentum space to the Green function is not very clearly displayed.

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103

We shall attempt to fill the missing pieces in the derivation presented in Ref. [174] and start with the clarification of certain normalization factors relevant to the formulation of the Green function in momentum space. First, one recalls that the Green function of the free Schr¨ odinger equation fulfills (H0 − E) G(⃗ r, r⃗′ ) = (−

⃗2 ∇ − E) G(⃗ r, r⃗′ ) = δ (3) (⃗ r − r⃗′ ) , 2m

(4.163)

or ⃗ 2 + 2mE) G(⃗ (∇ r, r⃗′ ) = −2mδ (3) (⃗ r − r⃗′ ) .

(4.164)

So, we have, with reference to the well-known Green function of the Helmholtz equation, √

(2m) e− 2mE ∣⃗r−⃗r ∣ G(⃗ r, r⃗ ) = . 4π∣⃗ r − r⃗′ ∣ ′

′

(4.165)

In calculating the Fourier transform, the second argument gets transformed with the complex conjugate exp(+i⃗ p ⋅ r⃗) of the Fourier factor exp(−i⃗ p ⋅ r⃗). This is because we associate G(⃗ r, r⃗′ ), in an eigenfunction decomposition, with G(⃗ r, r⃗′ ) = ∑ n

ψn (⃗ r) ψn∗ (⃗ r′ ) , λn − E

−

⃗2 ∇ ψn (⃗ r) = λn ψn (⃗ r) , 2m

(4.166)

and the completeness relation for the wave functions is ∗

′

r) ψn (⃗ r)=δ ∑ ψn (⃗ n

(3)

(⃗ r − r⃗′ ) ,

(4.167)

where n is a multi-index that enumerates the eigenstates in a particular basis, say, the plane-wave basis or the spherical basis. The latter involves the spherical Bessel functions in the radial variable (see Chap. 2.5 of Ref. [81]). In Eq. (4.166), the sum over states may include a continuum. Let us investigate the Fourier transform of Eq. (4.163), with the proviso that the second argument is transformed according to the complex conjugate: 3 3 ′ ∫ d r ∫ d r (−

⃗2 ∇ ⃗ r +ip⃗′ ⋅⃗ r′ − E) G(⃗ r − r⃗′ )e−ip⋅⃗ 2m

⃗ r +ip⃗ ⋅⃗ r = ∫ d3 r ∫ d3 r′ δ (3) (⃗ r − r⃗′ )e−ip⋅⃗ . ′

′

(4.168)

A double partial integration of the term on the left-hand side leads to (

p⃗ 2 − E) G(⃗ p, p⃗′ ) = (2π)3 δ (3) (⃗ p − p⃗′ ) , 2m

(4.169)

where ⃗ r +ip⃗ ⋅⃗ r G(⃗ p, p⃗′ ) = ∫ d3 r ∫ d3 r′ e−ip⋅⃗ G(⃗ r, r⃗′ ) . ′

′

(4.170)

Within the conventions usually adopted in the physics literature for the Fourier amplitudes, one thus incurs a factor of (2π)3 in momentum space, in the defining equation of the Green function (4.169).

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4.4.2

Toward the SO((4)) Symmetry

The defining equation of the Schr¨odinger–Coulomb Green function is (

p⃗ 2 + V − E) G(⃗ r, r⃗′ ) = δ (3) (⃗ r − r⃗′ ) . 2m

(4.171)

The Coulomb potential, in momentum space, is V (⃗ p − p⃗′ ) = −

4πZα . (⃗ p − p⃗′ )2

(4.172)

The defining equation for the Green function thus becomes, in momentum space, (

p⃗ 2 d3 p′′ 4πZα − E) G(⃗ p, p⃗′ ) − ∫ G(⃗ p′′ , p⃗′ ) = (2π)3 δ (3) (⃗ p − p⃗′ ) . 2m (2π)3 (⃗ p − p⃗′′ )2

(4.173)

One may go into (Euclidean) momentum space, setting X2 , (4.174) 2m where X = p0 has the interpretation of the zeroth component of the Euclidean fourmomentum. For a free particle, the parameter X has the interpretation of being equal to E=−

√

2 p⃗free . (4.175) 2m The Schr¨ odinger–Coulomb Green function describes an energy-conserving system, and hence one needs only one energy argument, X = X ′ . One can map the momentum space onto the unit sphere, via the conformal transformation

X = p0 =

ξ0 =

X 2 − p⃗ 2 , X 2 + p⃗ 2

−2mE = i ∣⃗ pfree ∣ ,

ξ⃗ =

2X p⃗ , X 2 + p⃗ 2

E=

ξ02 + ξ⃗2 = 1 .

(4.176)

It is useful to derive the equation d3 Ω =

d3 ξ ∣ξ0 ∣

(4.177)

for the area element on the three-dimensional unit sphere, embedded in fourdimensional space. Here, one should remember that the three-dimensional com⃗ may have varying magnitude and we ponents of the four-dimensional vector (ξ0 , ξ) only have the restriction ξ02 + ξ⃗2 = 1. The derivation of Eq. (4.177) is less trivial than one might think. One first ⃗=x ⃗(t1 , t2 ), remembers that the infinitesimal element of a two-dimensional surface x parameterized by the variables t1 and t2 , reads as: RRR ∂x RRR⎛ ∂t1 R⎜ ∂y d2 S = RRRR⎜ ∂t RRR⎜ 1 ∂z RRRR⎝ ∂t 1

∂x

⎞ ⎛ ∂t2 ⎟ ⎜ ∂y ⎟ × ⎜ ∂t2 ⎟ ⎜ ⎠ ⎝ ∂z ∂t2

R R RRR ⎞RRRR ⎛ ˆex ˆey ˆez ⎞RRRR RRR R R ⎟RRR ⎜ ∂x ∂y ∂z ⎟RRRR ⎟RR dt1 dt2 = RRRRdet ⎜ ∂t ⎟ dt dt . ⎟RR ⎜ 1 ∂t1 ∂t1 ⎟RRRR 1 2 RRR R RRR ⎠RRR ⎝ ∂x ∂y ∂z ⎠RRRR ∂t2 ∂t2 ∂t2 R R R

(4.178)

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105

The appropriate generalization to the three-dimensional “surface” of a manifold embedded into four-dimensional space is, with ξ0 = ξ0 (t1 , t2 , t3 ), ξx = ξx (t1 , t2 , t3 ), ξy = ξy (t1 , t2 , t3 ), ξz = ξz (t1 , t2 , t3 ), where ξ0 is the fourth coordinate, RRR R ⎛ ˆe0 ˆex ˆey ˆez ⎞RRRR RRR RRR ⎜ ∂ξ0 ∂ξx ∂ξy ∂ξz ⎟RRRR ⎜ ∂t ∂t ∂t ∂t ⎟RR R R ⎜ ⎟ d3 Ω = RRRRdet ⎜ ∂ξ1 ∂ξ1 ∂ξ1y ∂ξ1 ⎟RRRR dt1 dt2 dt3 . ⎜ 0 z z ⎟R RRR ⎟R ⎜ RRR ⎜ ∂t2 ∂t2 ∂t2 ∂t2 ⎟RRRR RRR ∂ξ ∂ξ ∂ξ ∂ξ ⎝ ∂t 0 ∂tx ∂ty ∂tz ⎠RRRR RR R 3 3 3 3

(4.179)

First, one calculates the four-dimensional vector described by the determinant, and then calculates its vector modulus. The three-dimensional unit sphere, embedded in four-dimensional space, can be interpreted as a three-dimensional manifold, param√ eterized by the coordinates ξx = t1 , ξy = t2 , and ξz = t3 , while ξ0 = 1 − ξx2 − ξy2 − ξz2 . One finds that d3 Ω = √

dξx dξy dξz 1 − ξx2 − ξy2 − ξz2

=

d3 ξ . ∣ξ0 ∣

(4.180)

One should now transform d3 ξ into d3 p. To this end, one interprets the second member of the relation (4.176), keeping X constant, in the sense that ∂ξx RRR ⎛ ∂p RRR x ⎜ ∂ξx R d3 ξ = RRRRRdet ⎜ ⎜ ∂py ⎜ RRRR ∂ξx RRR ⎝ ∂p z

∂ξy ∂px ∂ξy ∂py ∂ξy ∂pz

∂ξz ∂px ∂ξz ∂py ∂ξz ∂pz

⎞RRRR 3 2 2 ⎟RRRR ⎟RR dpx dpy dpz = 8X (X − p⃗ ) d3 p . ⎟RRR (X 2 + p⃗ 2 )4 ⎟RR ⎠RRRR

(4.181)

We remember that according to Eq. (4.176), we have ξ0 = (X 2 − p⃗ 2 )/(X 2 + p⃗ 2 ), and so one obtains d3 Ω =

3 2X d3 ξ =( 2 ) d3 p , 2 ∣ξ0 ∣ X + p⃗

(4.182)

and, as a trivial consequence, δ

(3)

3

2 2 ⃗ −Ω ⃗ ′ ) = ( X + p⃗ ) δ (3) (⃗ (Ω p − p⃗′ ) . 2X

(4.183)

By an elementary calculation, one can show that (ξ − ξ ′ )2 = (ξ0 − ξ0′ )2 + (ξ⃗ − ξ⃗′ )2 =

4 X2 (⃗ p − p⃗′ )2 . (X 2 + p⃗ 2 ) (X 2 + p⃗ ′ 2 )

(4.184)

⃗ on the three-dimensional unit sphere can be One observes that every point Ω mapped uniquely onto a vector p⃗. Hence, suppressing in the notation the depen⃗ Ω ⃗ ′ ) as follows, dence on X or E, one may define the generalized Green function Γ(Ω, ⃗ Ω ⃗ ′) = − Γ(Ω,

′ 1 (X 2 + p⃗ 2 )2 G(⃗ p, p⃗′ ) (X 2 + p⃗ 2 )2 . 3 16mX

(4.185)

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It is advantageous to define D(ξ − ξ ′ ) = 1/[4π 2 (ξ − ξ ′ )2 ], which according to Eq. (4.184) can be simplified as follows for ∣ξ∣ = ∣ξ ′ ∣ = 1 (in the sense of a modulus of a four-dimensional unit vector), D(ξ − ξ ′ ) =

1 1 1 1 = 2 2 ′ 2 ′ 4π (ξ − ξ ) 4π (ξ0 − ξ0 )2 + (ξ⃗ − ξ⃗′ )2 1 (X 2 + p⃗ 2 ) (X 2 + p⃗ 2 ) . 2 2 16 π X (⃗ p − p⃗′ )2 ′

=

(4.186)

⃗ ′′ , Ω ⃗ ′ ) as Let us reformulate the expression H = ∫ d3 Ω′′ D(ξ − ξ ′′ )Γ(Ω H = ∫ d3 p′′ ( × {−

3 (X 2 + p⃗ 2 ) (X 2 + p⃗′′ 2 ) 2X 1 ) { } X 2 + p⃗′′ 2 16 π 2 X 2 (⃗ p − p⃗′′ )2

′ 1 (X 2 + p⃗′′ 2 )2 G(⃗ p′′ , p⃗′ ) (X 2 + p⃗ 2 )2 } 16mX 3

1 1 (X 2 + p⃗ 2 )(X 2 + p⃗ 2 )2 } {− G(⃗ p′′ , p⃗′ )} = ∫ d p 8X { 16 π 2 X 2 (⃗ p − p⃗′′ )2 16mX 3 ′

3 ′′

3

=−

′ 1 1 d3 p′′ π {(X 2 + p⃗ 2 ) (X 2 + p⃗ 2 )2 } G(⃗ p′′ , p⃗′ ) ∫ 2 3 4mX 8π (⃗ p − p⃗′′ )2

=−

′ d3 p′′ 4π 1 2 2 2 2 2 ⃗ ⃗ (X + p ) (X + p ) G(⃗ p′′ , p⃗′ ) . ∫ 16 m X 2 (2π)3 (⃗ p − p⃗′′ )2

(4.187)

Therefore, 3 ′′ ′′ ⃗ ′′ , Ω ⃗ ′) ∫ d Ω D(ξ − ξ ) Γ(Ω 3

2

X X 2 + p⃗ 2 X 2 + p⃗ 2 d3 p′′ 4π =− ( ) ( 2 ) G(⃗ p′′ , p⃗′ ) . ∫ 2m 2X X + p⃗ 2 (2π)3 (⃗ p − p⃗′′ )2 ′

(4.188)

We define the energy variable [see Eq. (4.83)], ν=

Zαm Zαm =√ , X −2mE

(4.189)

which is some sort of generalized principal quantum number, as it attains an integer value at a bound-state energy E = −(Zα)2 m/(2n2 ), ν=√

Zαm −2m [−(Zα)2 m/(2n2 )]

=√

Zαm (Zα)2 m2 /n2

= n.

(4.190)

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Bound-Electron Self–Energy and Bethe Logarithm

It is useful to investigate the combination 2 2 2 p, p⃗′ ) (X 2 + p⃗ 2 )2 ⃗ Ω ⃗ ′ ) − 2ν ∫ d3 Ω′′ D(ξ − ξ ′′ )Γ(Ω ⃗ ′′ , Ω ⃗ ′ ) = − (X + p⃗ ) G(⃗ Γ(Ω, 16mX 3 2 ′ 3 ⎫ ⎧ ⎪ d3 p′′ Zαm ⎪ X 2 + p⃗ 2 4π ⎪ X X 2 + p⃗ 2 ′′ ′ ⎪ ⃗ ) G(⃗ p , p ) ⎬ −2 ⎨− ( ) ( 2 ∫ 2 3 ′′ 2 ⎪ ⎪ 2m X ⎪ 2X X + p⃗ (2π) (⃗ p − p⃗ ) ⎪ ⎭ ⎩ ′

2

3

2

3

=(

X 2 + p⃗ 2 X 2 + p⃗ 2 X 2 + p⃗ 2 d3 p′′ 4πZα ) ( ) [− G(⃗ p, p⃗′ ) + ∫ G(⃗ p′′ , p⃗′ )] 2 2 X + p⃗ 2X 2m 8π 3 (⃗ p − p⃗′′ )2

=(

X 2 + p⃗ 2 X 2 + p⃗ 2 ⃗ −Ω ⃗ ′) , p − p⃗′ )] = −(2π)3 δ (3) (Ω ) ( ) [−(2π)3 δ (3) (⃗ 2 2 X + p⃗ 2X

′

′

(4.191)

where we have used Eq. (4.183). So, we have derived the compact formula ⃗ Ω ⃗ ′ ) − 2ν ∫ d3 Ω′′ D(ξ − ξ ′′ ) Γ(Ω ⃗ ′′ , Ω ⃗ ′ ) = −(2π)3 δ (3) (Ω ⃗ −Ω ⃗ ′) . Γ(Ω,

(4.192)

The Fourier transformation of D(ξ − ξ ′ ) is D(k) = ∫ d4 ξ e−ik⋅(ξ−ξ ) ′

1 1 1 = , 4π 2 (ξ − ξ ′ )2 k 2

(4.193)

where k 2 = ∑4µ=1 kµ kµ is the Euclidean vector square, and the four-vectors ξ and ξ ′ are, a priori, not restricted to be unit vectors. Let us suppose now that we had a decomposition of the form ∞

ρn< 1 ∗ Y (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) , n+2 n`m n=0 `=0 m=−` 2(n + 1) ρ> n

`

D(ξ − ξ ′ ) = ∑ ∑ ∑

(4.194)

where the Yn`m (χ, θ, ϕ) are four-dimensional spherical harmonics. (See Eq. (4.229) for a proof.) These depend on the angular variables χ, θ and ϕ which refer to the angular components of the ξ vector, ξx = ∣ξ∣ cos ϕ sin θ sin χ ,

ξy = ∣ξ∣ sin ϕ sin θ sin χ ,

(4.195a)

ξz = ∣ξ∣ cos θ sin χ ,

ξ0 = ∣ξ∣ cos χ .

(4.195b)

For completeness, we note that d4 ξ = ξ 3 dξ d3 Ω ,

d3 Ω = sin2 χ sin θ dχ dθ dϕ .

(4.196)

For ∣ξ∣ = 1, the angular variables alone suffice to define the unit vector. In Eq. (4.194), one has ρ = ∣ξ∣ and ρ′ = ∣ξ ′ ∣. So, for two points on the unit sphere, we would set ρ = ρ′ = 1. We assume that the four-dimensional spherical harmonics Yn`m have the properties 3 ∗ ∫ d Ω Yn`m (χ, θ, ϕ) Yn′ `′ m′ (χ, θ, ϕ) = δnn′ δ``′ δmm′ , ∗ ′ ′ ′ (3) ⃗ ⃗ ′) . −Ω ∑ Yn`m (χ, θ, ϕ) Yn`m (χ , θ , ϕ ) = δ (Ω

n`m

(4.197) (4.198)

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Let us guess that the Green function Γ can be expressed as ∗ ′ ′ ′ ⃗ Ω ⃗ ′ ) = −(2π)3 ∑ Yn`m (χ, θ, ϕ) Yn`m (χ , θ , ϕ ) , Γ(Ω, 1 − ν/(n + 1) n`m

(4.199)

and verify this guess with the help of Eq. (4.192). Using Eq. (4.194) for ρ = ∣ξ∣ = ρ′ = ⃗ Ω ⃗ ′ )−2ν ∫ d3 Ω′′ D(ξ−ξ ′′ ) Γ(Ω ⃗ ′′ , Ω ⃗ ′) ∣ξ ′ ∣ = 1, one can derive for the expression Ξ = Γ(Ω, the representation ∗ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) Ξ = − (2π)3 ∑ 1 − ν/(n + 1) n`m 1 ∗ + (2π)3 2ν ∑ ∑ ∫ d3 Ω′′ Yn`m (χ, θ, ϕ) Yn`m (χ′′ , θ′′ , ϕ′′ ) 2(n + 1) ′ ′ ′ n`m n ` m ×

∗ Yn′ `′ m′ (χ′′ , θ′′ , ϕ′′ ) Yn`m (χ′ , θ′ , ϕ′ ) 1 − ν/(n + 1)

= − (2π)3 ∑ ( n`m

1 ν 1 − ) 1 − ν/(n + 1) n + 1 1 − ν/(n + 1)

∗ ⃗ −Ω ⃗ ′) . × Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) = −(2π)3 δ (3) (Ω

(4.200)

We have thus shown Eq. (4.199) as a remarkable consequence of the SO(4) symmetry of the hydrogen atom. The bound-state spectrum is obtained in view of the poles at ν = n + 1 = 1, 2, 3, . . . .

(4.201)

This consideration suggests that n + 1 takes the role of the principal quantum number within the hydrogen spectrum; hence, the four-dimensional spherical harmonic with n = 0 is associated with the (innermost) K shell of hydrogen, while n = 1 is associated with the (next in order) L shell, and so on. We will see, in a moment, that Yn`m is proportional to the Schr¨odinger–Coulomb momentum-space eigenfunction of principal quantum number n + 1. Around the pole, one may expand, using (Zα)2 m Zαm (n + 1) + , X= −  + O(2 ) , 2(n + 1)2 n+1 Zα with n = 0, 1, 2, 3, . . . , and obtain, on one hand, E=−

(4.202)

2

⃗ Ω ⃗ ′ ) ≈ (2π)3 X Yn`m (χ, θ, ϕ) Y ∗ (χ′ , θ′ , ϕ′ ) Γ(Ω, n`m m 1 ∗ [(2π)3/2 X Yn`m (χ, θ, ϕ)] [(2π)3/2 X Yn`m = (χ′ , θ′ , ϕ′ )] . m On the other hand, we have from Eq. (4.185), ′ 1 ⃗ Ω ⃗ ′) = − Γ(Ω, (X 2 + p⃗ 2 )2 G(⃗ p, p⃗′ ) (X 2 + p⃗ 2 )2 3 16mX p) ψ(n+1)`m (⃗ p′ ) ′ 1 2 2 2 ψ(n+1)`m (⃗ ⃗ (X + p ) = − (X 2 + p⃗ 2 )2 3 16mX (−) 1 (X 2 + p⃗ 2 )2 (X 2 + p⃗ 2 )2 [ ψ(n+1)`m (⃗ p)] [ ψ(n+1)`m (⃗ p′ )] . 3/2 m 4X 4X 3/2

(4.203)

′

=

(4.204)

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A comparison of terms in square brackets in Eqs. (4.203) and (4.204) yields the wave functions (we replace n → n − 1) ψn`m (⃗ p) = (2π)3/2

4X 5/2 Y(n−1) `m (χ, θ, ϕ) . (X 2 + p⃗ 2 )2

(4.205)

The four-dimensional spherical harmonics are normalized according to Eq. (4.197), whereas the wave functions in momentum space are normalized according to d3 p 2 ∣ψn`m (⃗ p)∣ = 1 . ∫ (2π)3

(4.206)

The integration measure explains the prefactor (2π)3/2 in Eq. (4.205). There is something to be said about the comparison to Eq. (4.182), which we recall for convenience, 3

d3 p = (

X 2 + p⃗ 2 ) d3 Ω . 2X

(4.207)

Namely, in view of the orthogonality properties (4.197), one may define the functions 3/2 2X ) Yn `m (χ, θ, ϕ) , X 2 + p⃗ 2 X 2 − p⃗ 2 2X ξ0 = cos χ = 2 , ξ⃗ = 2 p⃗ , 2 X + p⃗ X + p⃗ 2

χn`m (⃗ p) = (2π)3/2 (

θ = θξ⃗ = θp⃗ ,

ϕ = ϕξ⃗ = ϕp⃗ .

(4.208a) (4.208b) (4.208c)

For amusement, one notes the identity d3 Ω = sin2 χ sin θ dχ dθ dϕ = (1 − cos2 χ) =(

∂χ d∣⃗ p∣ sin θ dθ dϕ ∂∣⃗ p∣

3 3 2X 2X 2 ⃗ ) p d∣⃗ p ∣ sin θ dθ dϕ = ( ) d3 p , X 2 + p⃗ 2 X 2 + p⃗ 2

(4.209)

verifying Eq. (4.182) once more. The functions defined in Eq. (4.208) have the property d3 p ∗ ∗ χ (⃗ p) χn′ `′ m′ (⃗ p) = ∫ d3 Ω Yn`m (⃗ p) Yn′ `′ m′ (⃗ p) = δnn′ δ``′ δmm′ . (4.210) ∫ (2π)3 n`m The prefactor in Eq. (4.205) is manifestly different from the one in Eq. (4.208); still, both ψn`m (⃗ p) and χn`m (⃗ p) are normalized to unity when integrated over d3 p/(2π)3 . 4.4.3

Four-Dimensional Spherical Harmonics

We now need to define the four-dimensional spherical harmonics used in Eq. (4.194), (1) and it will turn out that the Gegenbauer polynomials Cn (x) are instrumental in our investigations. Indeed, the Gegenbauer polynomials are generated by the relation ∞ 1 = ∑ Cn(1) (x) tn . (4.211) 2 1 − 2xt + t n=0

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Here, we shall elect to define associated Gegenbauer polynomials in a somewhat non-standard way, which is designed to highlight the connection of the associated Gegenbauer polynomials with the associated Legendre polynomials. The associated Legendre polynomials are usually denoted as Pn` . One may thus define the Gegenbauer polynomials as Q, Qn (x) = Cn(1) (x) .

(4.212)

∂ ∂ 1 sin2 χ Qn (cos χ) = −n (n + 2) Qn (cos χ) . 2 ∂χ sin χ ∂χ

(4.213)

They have the property

In full analogy with the associated Legendre polynomials, one may define the associated Gegenbauer-type polynomials as ∂` Qn (x) , ` ≤ n. (4.214) ∂x` For ` > n, the associated Gegenbauer polynomials vanish because Qn (x) is of order n. The associated Gegenbauer polynomials have the property `/2

Q`n (x) = (−1)` (1 − x2 )

(

∂ 1 ∂ ` (` + 1) sin2 χ − ) Q`n (cos χ) = −n (n + 2) Q`n (cos χ) . ∂χ sin2 χ ∂χ sin2 χ

(4.215)

For the normalization, one finds π

π , 2 π 2 π (n + ` + 1)! 2 ` . ∫ dχ sin χ [Qn (cos χ)] = 2 (n + 1) (n − `)! 0 ∫

0

2

dχ sin2 χ [Qn (cos χ)] =

One may thus define the four-dimensional spherical harmonics as √ ¿ 2Á Á À (n + 1) (n − `)! Q` (cos χ) Y`m (θ, ϕ) , Yn`m (χ, θ, ϕ) = n π (n + ` + 1)!

(4.216a) (4.216b)

(4.217)

where the Y`m ’s are the usual spherical harmonics, and verify that they fulfill Eq. (4.197), ∫

0

2π

dϕ ∫

0

π

dθ ∫

π 0

dχ sin2 χ sin θ Yn`m (χ, θ, ϕ) Yn∗′ `′ m′ (χ, θ, ϕ) = δnn′ δ``′ δmm′ . (4.218)

The addition theorem is ∗ ′ ′ ′ ∑ Yn`m (χ, θ, ϕ) Yn`m (χ , θ , ϕ ) =

n`m

n+1 Qn (ˆ x⋅x ˆ′ ) . 2π 2

(4.219)

Here, x ˆ is a four-dimensional unit vector, associated with the angles χ, θ and ϕ. Analogously, x ˆ′ is a four-dimensional unit vector, associated with the angles χ′ , θ′ ′ and ϕ . The four-dimensional Laplacian is ⃗2 = ∇

⃗2 ∂2 3 ∂ 1 1 ∂ ∂ L ). + + 2( 2 sin2 χ − 2 ∂ρ ρ ∂ρ ρ sin χ ∂χ ∂χ sin2 χ

(4.220)

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111

Let us return to the explicit form of the D function, given in Eq. (4.193), D(ξ − ξ ′ ) =

1 1 , 2 4π (ξ − ξ ′ )2

D(k) = ∫ d4 ξ e−ik⋅(ξ−ξ ) ′

1 1 1 = 2. 2 ′ 2 4π (ξ − ξ ) k (4.221)

The Fourier transform dictates that ∇2 D(ξ − ξ ′ ) = −δ (4) (ξ − ξ ′ ) .

(4.222)

Here, ∇2 is the four-dimensional Laplacian. Let us also determine A, in coordinate space, from an ansatz borrowed from Eq. (4.194), ∞

n

`

D(ξ − ξ ′ ) = ∑ ∑ ∑ A n=0 `=0 m=−`

ρn< ∗ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) . ρn+2 >

(4.223)

Of course, δ (4) (ξ − ξ ′ ) =

1 δ(ρ − ρ′ ) δ(χ − χ′ ) δ(θ − θ′ ) δ(ϕ − ϕ′ ) . ρ3 sin2 χ sin θ

(4.224)

We already know that ∗ ′ ′ ′ ∑ Yn`m (χ, θ, ϕ) Yn`m (χ , θ , ϕ ) =

n`m

1 δ(χ − χ′ ) δ(θ − θ′ ) δ(ϕ − ϕ′ ) . (4.225) sin χ sin θ 2

So, we need to solve the radial equation A(

∂2 1 3 ∂ ρn< = − 3 δ(ρ − ρ′ ) . + − n(n + 2)) n+2 2 ∂ρ ρ ∂ρ ρ> ρ

(4.226)

Integrating this equation from ρ = ρ′ −  to ρ = ρ′ +  using techniques discussed in Chap. 2 of Ref. [81], we are led to A

⎛ ∂ ρn< ⎞ ∂ ρn< 1 ∣ − ∣ = − ′3 , n+2 n+2 ρ ⎝ ∂ρ ρ> ρ=ρ′ + ∂ρ ρ> ρ=ρ′ − ⎠

(4.227)

and so A (

∂ ρ′n ∂ ρn 1 − )∣ = − ′3 , n+2 ′n+2 ∂ρ ρ ∂ρ ρ ρ ρ=ρ′

A=

1 . 2(n + 1)

(4.228)

We thus confirm Eq. (4.194), D(ξ − ξ ′ ) =

1 1 4π 2 (ξ − ξ ′ )2 ∞

1 ρn< ∗ Y (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) . n+2 n`m n=0 `=0 m=−` 2(n + 1) ρ> n

`

= ∑∑ ∑

(4.229)

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Wave Functions in Momentum Space

According to Eq. (4.205), one has the wave functions ψn`m (⃗ p) = (2π)3/2

4X 5/2 Y(n−1) `m (χ, θ, ϕ) , (X 2 + p⃗ 2 )2

where we recall from Eq. (4.217), √ ¿ 2Á Á À (n + 1) (n − `)! Q` (cos χ) Y`m (θ, ϕ) . Yn`m (χ, θ, ϕ) = n π (n + ` + 1)!

(4.230)

(4.231)

We also recall that according to Eq. (4.176), 2X X 2 − p⃗ 2 , ξ⃗ = 2 p⃗ , ξˆ = pˆ . (4.232) ξ0 = cos χ = 2 X + p⃗ 2 X + p⃗ 2 Then, 4X 5/2 ψn`m (⃗ p) = (2π)3/2 Y(n−1) `m (χ, θ, ϕ) (X 2 + p⃗ 2 )2 √ ¿ 4X 5/2 2Á 3/2 Á À n (n − 1 − `)! Q` (cos χ) Y`m (θ, ϕ) = (2π) n−1 2 2 2 (X + p⃗ ) π (n + `)! ¿ Á n (n − 1 − `)! X 5/2 X 2 − p⃗ 2 ` À Q ( ) Y`m (θ, ϕ) , (4.233) = 16 π Á (n + `)! (X 2 + p⃗ 2 )2 n−1 X 2 + p⃗ 2 where θ = θp⃗ and ϕ = ϕp⃗. A comparison of Eq. (4.174) with the bound-state energies, X2 (Zαm)2 Zαm =− , X= , (4.234) 2 2m 2mn n allows us to bring the bound-state wave function into the form ¿ 2 2 2 3/2 ⎛ 1 − n a02 p⃗ ⎞ 16 π a0 n2 Á (n − 1 − `)! 1 ` Z Á À ⎟ Y`m (θ, ϕ) , ψn`m (⃗ p) = Qn−1 ⎜ n2 a20 p⃗ 2 n2 a20 p⃗ 2 2 (n + `)! Z 3/2 ⎝ 1 + Z2 ⎠ (1 + Z 2 ) (4.235) where a0 = 1/(αm) is the Bohr radius. The identity E=−

`+1 Q`n−1 (x) = (−1)n−1 2` `! (1 − x2 )`/2 Cn−`−1 (−x) ,

(4.236)

`+1 Cn−`−1 (x)

where the are the Gegenbauer polynomials in the conventional definition (see Ref. [159]), can be used in order to bring the wave function, in momentum space, into the familiar form [see page 39 of Ref. [6]], ¿ −(`+2) Á (n − 1 − `)! n2 a20 p⃗ 2 À ) ψn`m (⃗ p) = 22`+4 π `! Á (1 + (n + `)! Z2 ×n

`+2

(a0 /Z)

`+3/2

∣⃗ p∣

`

2 2 2 ⎛ n a02 p⃗ (`+1) Cn−`−1 ⎜ n2 Za2 p⃗ 2 ⎝ Z02

− 1⎞ ⎟ Y`m (θ, ϕ) . + 1⎠

(4.237)

Factors of Z are suppressed in Ref. [6] because of the atomic unit system used. We absorb the overall prefactor (−1)n−1 incurred in the conversion from Eq. (4.235) to (4.237) into the global phase of the wave function.

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Integral Representation

The formula in Eq. (4.199) leads to a convenient representation of the wave functions in momentum space. One may obtain a convenient integral representation of the Green function by exploring the connection of the four-dimensional multipole decomposition (4.194) with the explicit representation (4.199) of the Green function. However, in order to do so, it is necessary to use an additional, not very obvious trick and specialize (4.194) to the cases ξ = r1 , and ξ ′ = r2 , as follows, r1 = ρ ξ ,

∣ξ∣ = 1 ,

∣r1 ∣ = ρ ,

rˆ1 = ξˆ,

r2 = ξ ′ ,

∣ξ ′ ∣ = 1 ,

∣r2 ∣ = 1 ,

rˆ2 = ξˆ′ ,

(4.238)

with 0 < ρ < 1, so that ∣r1 ∣ = ρ = r< , and ∣r2 ∣ = 1 = r> . Then, ∣r1 − r2 ∣2 = ∣r1 ∣2 + ∣r2 ∣2 − 2ρ (ξ ⋅ ξ ′ ) = 1 + ρ2 − 2ρ (ξ ⋅ ξ ′ ) = (1 + ρ2 − 2ρ) + 2ρ − 2ρ (ξ ⋅ ξ ′ ) = (1 − ρ)2 + ρ(ξ 2 + ξ ′2 ) − 2ρ (ξ ⋅ ξ ′ ) = (1 − ρ)2 + ρ(ξ − ξ ′ )2 .

(4.239)

The decomposition of Eq. (4.194) becomes D(r1 − r2 ) =

1 1 4π 2 (1 − ρ)2 + ρ(ξ − ξ ′ )2 ∞

ρn ∗ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) . n=0 `=0 m=−` 2(n + 1) n

`

= ∑∑ ∑

(4.240)

Incidentally, one can verify the multiplicity of the hydrogen spectrum in this way. One simply sets ξ = ξ ′ , but keeps ρ variable. To this end, one observes that the four-dimensional total solid angle is ∫ dΩ = ∫

2π 0

dϕ ∫

π 0

dθ ∫

π 0

dχ sin2 χ sin θ = 2π 2 .

(4.241)

Now, setting ξ = ξ ′ in Eq. (4.240), and integrating over dΩ, one finds that ` ∞ n 2π 2 1 ρn = 1. ∑ ∑ ∑ 4π 2 (1 − ρ)2 n=0 2(n + 1) `=0 m=−`

(4.242)

In view of the result n

`

2 ∑ ∑ 1 = (n + 1) ,

`=0 m=−`

∞ ∞ 1 ρn 2 = (n + 1) = ρn (n + 1) , (4.243) ∑ ∑ (1 − ρ)2 n=0 (n + 1) n=0

one confirms the multiplicity of the eigenvalue for the principal quantum number n+1, as (n+1)2 , where n is a nonnegative integer. Let us trivially rewrite Eq. (4.240) in the form ∞ n ` 1 1 ρn ∗ = Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) . (4.244) ∑ ∑ ∑ 2π 2 (1 − ρ)2 + ρ(ξ − ξ ′ )2 n=0 `=0 m=−` n + 1

It is a priori not obvious how the latter expression could be brought into correspondence with Eq. (4.199), in view of the presence of the factor 1/[1 − ν/(n + 1)] in the

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latter case. However, one may use the algebraic identity ν 1 1 =1+ + ν2 1 − ν/(n + 1) n+1 (n + 1)(n + 1 − ν) 1 ν ν2 n−ν =1+ + , (4.245) ∫ dρ ρ n + 1 (n + 1) 0 ⃗ Ω ⃗ ′ ) as follows, starting from to reformulate the expression S = −(2π)−3 Γ(Ω, Eq. (4.199), ∗ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) S= ∑ 1 − ν/(n + 1) n`m ∗ = ∑ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) (1 + n`m

1 ν ν2 n−ν + ∫ dρ ρ ) n+1 n+1 0

∗ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) n+1 n`m

∗ = ∑ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) + ν ∑ n`m

+ ν2 ∫

1 0

ρn ∗ Yn`m (χ, θ, ϕ) Yn`m (χ′ , θ′ , ϕ′ ) n`m n + 1

dρ ρ−ν ∑

1 1 ν ν2 1 + 2 ∫ dρ ρ−ν . 2 ′ 2 2π (ξ − ξ ) 2π (1 − ρ)2 + ρ(ξ − ξ ′ )2 0 With one partial integration, one can show that 1 1 ρ ⃗ Ω ⃗ ′ ) = δ (3) (Ω ⃗ −Ω ⃗ ′ ) + ν ∫ dρ ρ−ν ∂ − Γ(Ω, . 3 2 (2π) 2π ∂ρ (1 − ρ)2 + ρ(ξ − ξ ′ )2 0 In view of the identity ∂ ρ 1 − ρ2 = 2 ′ 2 2 ∂ρ (1 − ρ) + ρ(ξ − ξ ) [(1 − ρ) + ρ(ξ − ξ ′ )2 ]2 and the relation 1 − ρ2 ⃗ −Ω ⃗ ′ ) = lim 1 δ (3) (Ω ρ→1 2π 2 [(1 − ρ)2 + ρ(ξ − ξ ′ )2 ]2 one can even write the very compact formula 1 ρ(1 − ρ2 ) 1 ⃗ Ω ⃗ ′ ) = 1 ∫ dρ ρ−ν ∂ − Γ( Ω, , (2π)3 2π 2 0 ∂ρ [(1 − ρ)2 + ρ(ξ − ξ ′ )2 ]2 because the latter expression is equivalent to 1 1 ρ1−ν (1 − ρ2 ) ⃗ Ω ⃗ ′ ) = 1 ∫ dρ ∂ − Γ(Ω, 3 2 (2π) 2π ∂ρ [(1 − ρ)2 + ρ(ξ − ξ ′ )2 ]2 0 1 ρ(1 − ρ2 ) 1 − 2 ∫ dρ (−ν)ρ−ν−1 2π [(1 − ρ)2 + ρ(ξ − ξ ′ )2 ]2 0 1 ρ1−ν (1 − ρ2 ) = 2 lim 2π ρ→1 [(1 − ρ)2 + ρ(ξ − ξ ′ )2 ]2 1 ν ρ(1 − ρ2 ) + 2 ∫ dρ ρ−ν 2π [(1 − ρ)2 + ρ(ξ − ξ ′ )2 ]2 0 1 ρ ⃗ −Ω ⃗ ′ ) + ν ∫ dρ ρ−ν ∂ = δ (3) (Ω . 2 2π ∂ρ (1 − ρ)2 + ρ(ξ − ξ ′ )2 0

⃗ −Ω ⃗ ′) + = δ (3) (Ω

(4.246)

(4.247)

(4.248)

(4.249)

(4.250)

(4.251)

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Bound-Electron Self–Energy and Bethe Logarithm

Using Eq. (4.250) and recalling Eqs. (4.184) and (4.185), 1 1 ⃗ Ω ⃗ ′) G(⃗ p, p⃗′ ) = − 16mX 3 Γ(Ω, , 2 2 2 2 (X + p⃗ ) (X + p⃗ ′ 2 )2

(4.252a)

4 X 2 (⃗ p − p⃗′ )2 . (4.252b) 2 + p⃗ ) (X 2 + p⃗ ′ 2 ) One may write G = −(2π)−3 G(⃗ p, p⃗′ ) as 1 16mX 3 ∂ ρ(1 − ρ2 ) 1 1 G= − dρ ρ−ν ∫ 2 2 2 2 2π (X + p⃗ ) ∂ρ [(1 − ρ)2 + ρ(ξ − ξ ′ )2 ]2 (X 2 + p⃗ ′ 2 )2 0 (ξ − ξ ′ )2 =

(X 2

= −

1 16mX 3 ρ(1 − ρ2 ) −ν ∂ dρ ρ ∫ 2 2π 2 ∂ρ [(1 − ρ)2 (X 2 + p⃗ 2 ) (X 2 + p⃗ ′ 2 ) + 4 ρ X 2 (⃗ 0 p − p⃗′ )2 ]

= −

mX 3 ieiπν −ν ∂ ∫ dρρ 2π 2 2 sin(πν) C ∂ρ [X 2 (⃗ p − p⃗′ )2 +

(1 − ρ2 )/ρ (1−ρ)2 4ρ

(X 2 + p⃗ 2 ) (X 2 + p⃗ ′ 2 )]

2

.

(4.253) The latter formula demands an explanation; we have replaced the simple integral over dρ over the interval ρ ∈ (0, 1) by a contour integral. This is done in order to lift the restriction ν < 1. The precise definition of the contour C is illustrated in the following. With the standard definition of the logarithm, the function ρ−ν = exp[−ν ln(ρ)] (4.254) has a branch cut along the negative ρ axis. So, with the standard definition of the branch cut of the logarithm, one has ln(−∣ρ∣ + i) = ln(∣ρ∣) + i π , ln(−∣ρ∣ − i) = ln(∣ρ∣) − i π . (4.255) Otherwise, with the branch cut of the logarithm along the positive real axis, one has ln(∣ρ∣ + i) = ln(∣ρ∣) , ln(∣ρ∣ − i) = ln(∣ρ∣) + 2πi , (4.256) because the imaginary part has to vary continuously as the phase of the complex argument is being ramped up from zero to 2π. Let us define an integration contour C that starts at ρ = 1 + i, follows infinitesimally above the real axis, encircles the origin in the mathematically positive sense, and finishes at ρ = 1 − i. One can replace the contour C equivalently by an expression which is minus the contribution from the “upper” contour from ρ = 0 to ρ = 1 + i, plus the contribution from the contour from ρ = 0 to ρ = 1 − i. Then, for any test function F (ρ), with the contour C as just defined, one has ieiπν ieiπν −ν ∫ dρ ρ F (ρ) = 2 sin(πν) C 2[exp(iπν) − exp(−iπν)]/(2i) × (− ∫ =

1 0

dρ ρ−ν + ∫

1 0

dρ ρ−ν e−2πiν ) F (ρ)

1 1 (1 − e−2πiν ) ∫ dρ ρ−ν F (ρ) 1 − exp(−2iπν) 0

=∫

1 0

dρ ρ−ν F (ρ) .

(4.257)

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We have derived the operator identity 1 ieiπν −ν −ν ∫ dρ ρ F (ρ) = ∫ dρ ρ F (ρ) . 2 sin(πν) C 0

(4.258)

Our investigations have culminated in the compact integral representation G(⃗ p, p⃗′ ) = 4πmX 3 (

ieiπν ) 2 sin(πν)

× ∫ dρ ρ−ν C

∂ ∂ρ [X 2 (⃗ p − p⃗′ )2 +

(4.259) (1 − ρ2 )/ρ (1−ρ)2 4ρ

(X 2 + p⃗ 2 ) (X 2 + p⃗ ′ 2 )]

2

.

We also recall that the energy argument of the Green function is parameterized as Zαm Zαm ν= . (4.260) =√ X −2mE For Lamb shift calculations, one typically replaces E → E −ω,

(4.261)

where E is the reference state energy, and therefore [see also Eq. (4.130)] ν=√

Zαm

Zαm 1 =√ =n√ = nt. 2m 2n2 ω −2m(E − ω) 1 + 2 −2m (− (Zα) − ω) (Zα) m 2n2

(4.262)

The integral representation (4.259) has been used with good effect in Lamb shift treatments [163, 173]. 4.5

Modern Ideas and Bound-State Self–Energy

4.5.1

Essence of Renormalization

The renormalization program of quantum electrodynamics has been outlined at various places in the literature, e.g., in Sec. 8.4 of Ref. [81], based on the Casimir effect (attractive force between two perfectly conducting plates). The calculation involves the zero-point energy of the electromagnetic field and its modification in the presence of nontrivial boundary conditions. We remember the general paradigmatic equation [see Eq. (8.144) of Ref. [81]] Physical Observable = Bare Observable∣reg. − Counter Term∣reg. =

lim

regulator→0

∫ (Bare Observable∣reg.−int. − Counter Term∣reg.−int. ) ,

(4.263)

where we recall that both the bare observable as well as the counter term need to be regularized. Here, we understand that “reg.” denotes the regularized physical quantity, while “reg-int.” denotes the regularized integrand. Integrals are typically involved in the calculation. In our derivation of the Bethe logarithm, which is an energy shift to be discussed in detail below, we have to regularize a logarithmic divergence. On the

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level of classical field theory, we may actually encounter a divergence of this type in the calculation of an electrostatic potential, in quite a different setting, and we would like to illustrate the concept of renormalization once more, in a toy-model calculation [see also Sec. 8.5 of Ref. [81]]. We consider an infinitely long, charged linear structure (long rod) that extends along the x axis, and evaluate the electrostatic potential at a point r⃗ = (0, 0, z), elevated from the x axis by a distance z. Although the resulting integrals are divergent (at face value), one can do this calculation using the standard formalism for the evaluation of an electrostatic potential, but only if one subtracts a counterterm. We start from the familiar equation Φ(⃗ r) =

r′ ) 1 3 ′ ρ(⃗ ∫ d r 4π0 ∣⃗ r − r⃗′ ∣

(4.264)

and use a charge density ρ of a long charged rod along the x-axis, ρ(⃗ r′ ) = λ δ(y ′ ) δ(z ′ ) .

(4.265)

The potential is to be evaluated at the point r⃗ = (0, 0, z), Φ(z) =

∞ ∞ λ dx′ λ dx′ √ √ = . ∫ ∫ 4π0 −∞ z 2 + (x′ )2 2π0 0 z 2 + (x′ )2

(4.266)

This expression is formally infinite, but the divergence at the upper limit of the integration domain is only logarithmic. First, we regularize this integral by introducing an upper cutoff Λ for the x′ -integration, Φ(z)∣reg. =

Λ dx′ λ λ (ln(2) − ln(z) − ln(Λ) + O(Λ−1 )) . (4.267) = ∫ √ 2 ′ 2 2π0 0 2π0 z + (x )

Because the integral diverges, we have to subtract a counterterm. Fixing the potential to be zero at z = z0 , we can devise the following regularized counterterm, C∣reg. =

Λ λ dx′ λ (ln(2) − ln(z0 ) − ln(Λ) + O(Λ−1 )) , (4.268) = ∫ √ ′ 2 2π0 0 2π 0 z0 + (x )

which is infinite in the limit Λ → ∞ and independent of z. The renormalized expression reads Φ(z) = lim (Φ(z)∣reg. − C∣reg. ) = − Λ→∞

λ z ln ( ) , 2π0 z0

(4.269)

where the precise value of z0 is physically irrelevant as it vanishes in calculating any potential differences; the term with z0 can otherwise be thought of as fixing a zero of the potential. This limit Λ → ∞ has been taken after evaluating the difference Φ(z)∣reg. − C∣reg. . Our model example illustrates that after regularization, one has to evaluate both the original integral as well as the counterterm as a function of the regulator, while keeping the regulator Λ finite up to the last step in the calculation.

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Overlapping Parameter

The Bethe logarithm term, to be discussed below, is only part of the bound-state self-energy of an electron. The calculation of the self-energy necessitates a global renormalization and requires the introduction of a scale-separation parameter, in order to distinguish the self-interactions due to high-energy, and low-energy photons. The scale-separation parameter is alternatively called an “overlapping parameter”. Indeed, the introduction of such a parameter is nothing unnatural, but, as we shall see below, can be necessary even on a very basic level, namely, in the evaluation of innocent-looking integrals such as ∞ √ dω e−ω (Zα)2 + ω 2 . (4.270) F (Zα) = ∫ 0

Here, Zα is a coupling parameter which we might as well have called g. However, in quantum electrodynamic bound-state calculations, the expansion parameter is Zα, where Z is the nuclear charge number and α is the fine-structure constant. For any value of Zα, the integral (4.270) can easily be evaluated by numerical quadrature. One surprisingly encounters problems when trying to evaluate the asymptotic expansion of the integral as Zα → 0. If we naively try to expand the integrand in powers of Zα, in order to obtain an expansion of F (Zα), then we run into a problem, √

ω 2 + (Zα)2 = ω +

(Zα)2 (Zα)4 − + O((Zα)6 ) , 2ω 8 ω3

(4.271)

because the expansion generates inverse powers of ω, and these lead to divergences at the lower limit of integration in Eq. (4.270). If we think of ω as a dimensionless photon energy variable, then the physics of the problem is completely different, depending on the energy of the photon ω, which necessitates a different treatment depending on whether ω is commensurate with the electron-mass scale ω ∼ 1 or commensurate with the atomic binding-energy scale ω ∼ (Zα)2 (we set the electron mass scale equal to unity in our model problem). Equally surprisingly, the problem generated by the inverse powers of ω in Eq. (4.271) can be solved if one introduces a so-called overlapping parameter, or scale-separation parameter  which fulfills the inequality (Zα)2 ≪  ≪ 1. We cut off the integral (4.270) at a lower limit ω = , and expand first in Zα, then in , ∞ √ H(Zα, ) = ∫ dω e−ω (Zα)2 + ω 2 

= 1 + (Zα)2 (− + (Zα)4 (−

γE ln() − ) 2 2

3 γE 1 1 ln() + − + + ) + O(Zα)5 , 2 32 16 16  8 16

(4.272)

where we neglect all terms of order O(Zα)5 and all terms of order  and higher. The Euler–Mascheroni constant is γE = 0.57721 56649 . . . . Only the divergent terms

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in  are kept, including the logarithmic ones. Because ω >  in Eq. (4.272), we can intuitively identify H(Zα, ) as the high-energy part of the photon integral (4.270). The original integral (4.270) is independent of , so, there must be a compensating term which eliminates the dependence on . It can only come from the low-energy region ω < , i.e., from the integral  √ (4.273) L(Zα, ) = ∫ dω e−ω (Zα)2 + ω 2 . 0

In this region, we cannot apply the expansion (4.271), but ω is “genuinely” small in this region, and so we can expand the exponential as follows, ω2 e−ω = 1 − ω + + ⋯, (4.274) 2! √ while keeping the square root (Zα)2 + ω 2 in unexpanded form. We should point out that each and every term in the ω-expansion of the exponential actually gives rise to terms of arbitrarily high order in the Zα-expansion. However, if we are interested in the expansion of L(Zα, ) in powers of Zα, then we are still justified in applying the expansion, because higher-order terms in the ω-expansion of the exponential give rise to higher orders in the Zα-expansion (for the low-energy part). The low-energy part finally evaluates to 2 (Zα)3 1 1 )) + L(Zα, ) = (Zα)2 ( + ln ( 4 2 Zα 3 1 1 1 1 2 4 + (Zα) (+ + − − ln ( )) + O(Zα)5 . (4.275) 2 64 16  8  16 Zα Again, we neglect all terms of order  and higher, and we group the terms in powers of Zα. This is achieved by first letting Zα → 0, then  → 0. The sum of H(Zα, ) and L(Zα, ) is finite and free of any dependence on , F (Zα) = H(Zα, ) + L(Zα, ) 1 γE 1 2 (Zα)3 = 1 + (Zα)2 ( − + ln ( )) + 4 2 2 Zα 3 5 γE 1 2 4 + (Zα) (− + − ln ( )) + O(Zα)5 . (4.276) 64 16 16 Zα The result is correct up to terms of order O(Zα)5 . One may still have doubts about the necessity of including logarithmic terms in the semi-analytic expansion. However, for Zα = 1/1000, a numerical evaluation of the integral (4.270), in agreement with the first few terms in powers of Zα, evaluates to 1.00000 37621 76214. Indeed, the overlapping parameter is an utmost versatile instrument for handling logarithmic terms in physical calculations where a simple power series is not sufficient to represent a semi-analytic expansion. In Lamb shift calculations, it is customary to label the coefficients in the semi-analytic expansion systematically, indicating both the power of Zα as well as the power of the logarithm. Formally, we can write the result for F (Zα) as ∞

1

F (Zα) = ∑ ∑ Amn (Zα)m lnn ( m=0 n=0

2 ). Zα

(4.277)

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By comparison of coefficients, we read off the following results from Eq. (4.276), 1 γE 1 1 A00 = 1 , A20 = − , A21 = , A30 = , (4.278) 4 2 2 3 1 as well as A41 = − 16 . For Zα → 0, the numerical contribution of the logarithmic terms is enhanced. Let us try to analyze the -dependence somewhat more deeply, by writing H(Zα, ) = ∑(Zα)m Hm (Zα, ) , m

L(Zα, ) = ∑(Zα)m Lm (Zα, ) ,

(4.279)

m

for the high- and low-energy parts, respectively. Here, the coefficient functions Hm (Zα, ) and Lm (Zα, ) typically contain logarithmic terms of the form ln() as well as powers of . Because the original function F (Zα) is independent of , the dependence on  separately cancels for each order in Zα expansion, i.e., we have for any m = 0, 1, 2, . . . , 1

n ∑ Amn ln (Zα) = Hm (Zα, ) + Lm (Zα, ) ,

(4.280)

n=0

independent of the numerical value of  (m is not repeated and not summed over). We have obtained our functions Hm and Lm by first expanding in Zα → 0, then in  → 0. This procedure is justified provided we assume that Zα ≪  ≪ 1. Under this assumption, we are free to choose the value of . We first choose a value of  larger than Zα, so that the expansion first in Zα, then in  is indicated. We should also stress that we are free to choose the degree to which the functions Hm and Lm should be expanded in , provided all logarithmic and constant terms are kept. Namely, in the sum Hm (Zα, ) + Lm (Zα, ), the dependence on  necessarily has to cancel at the end of the calculation. We can choose to expand the functions Hm (Zα, ) and Lm (Zα, ) only up to the constant term in powers of  for  → 0. All the dependence on , both in negative as well as positive powers of , has to cancel between the low- and high-energy parts. Furthermore, the positive powers of  vanish for both L and H, separately, in the limit  → 0, and, therefore, do not have to be kept. Keeping these terms in the calculation for illustrative purposes, it is an instructive exercise to show the additional cancelation of the positive powers of  for the first few terms as given in Eqs. (4.272) and (4.273). This scale separation parameter  therefore acts as an “overlapping parameter” for the two energy scales present in the self-energy of the bound electron. The method is necessary for the separation of the two different energy scales for the virtual photons: the nonrelativistic domain, in which the virtual photon assumes values of the order of the atomic binding energy, and the relativistic domain, in which the virtual photon assumes values of the order of the electron rest mass. The √ analogy to self-energy calculations is approximately as follows: The quantity ω 2 + (Zα)2 in the numerator is approximately equivalent to the electron propagator, where Zα is the coupling strength to the nucleus. We can directly expand in Zα for the high-energy part, but we have to keep the entire bound-state “prop√ 2 agator” expression ω + (Zα)2 with all powers of the Coulomb field strength for the low-energy part.

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Bound-State Self–Energy: Low-Energy Part Length Gauge

The Bethe logarithm arises naturally from the low-energy part of the self-energy of a bound electron. In its evaluation, we shall need the general formalism of the quantized electromagnetic field derived in Chap. 3, the concept of second-order perturbation theory in both atomic and field variables, as derived in Sec. 3.3.2, the radial Schr¨ odinger–Coulomb Green function as derived in Sec. 4.3.3, and the concept of renormalization explained in Sec. 4.5.1 as well as the concept of an overlapping parameter developed in Sec. 4.5.2. Indeed, in the first derivation of the Lamb shift [175], an upper cutoff ω ∼  was used for the energy of the virtual photon, in order to cut off a logarithmic ultraviolet divergence of the bound-electron selfenergy; later, the parameter  was identified as a scale-separation parameter that separates the scale of the binding energy from that of the electron mass in much the same way as in our toy model in Sec. 4.5.1. The Bethe logarithm was thus identified as the low-energy part of the energy shift. We shall evaluate the starting expression for the evaluation of the Bethe logarithm in two ways, namely, in the so-called length gauge, where the atom-field interaction Hamiltonian is ⃗, H1 = −e r⃗ ⋅ E

(4.281)

and in the so-called velocity gauge, where one uses a p⃗ ⋅ A⃗ coupling. Here, e is the charge of the orbiting particle (electron), r⃗ is the position coordinate of the electron, ⃗ is the electric-dipole field operator. and E There is an important subtlety with regard to the length gauge in fieldtheoretical calculations. Namely, the self-energy of a bound electron is calculated with regard to the interaction of the bound electron with the radiation field; the interaction of the compound two-body system with the radiation field would have to be formulated differently. The Schr¨odinger–Coulomb Hamiltonian (4.1) is expressed in terms of the relative coordinate of electron and proton, not in terms of the electron coordinate. In a system with an infinitely heavy nucleus, where the reduced mass µ is equal to the electron mass m, the electron position operator r⃗ is equal to the relative coordinate. We thus assume that µ=m

(length gauge)

(4.282)

for the rest of the current section on the length gauge. The interpretation of the renormalization of the self-energy relies on the comparison of the bound-electron self-energy with the free self-energy of a free electron, and is facilitated by the assumption (4.282). In our treatment of the length-gauge self-energy, we shall therefore assume that the nucleus is infinitely heavy. The correct reduced-mass dependence is easier to evaluate in the velocity gauge, while the length gauge has the advantage that is formulated in terms of the physically observable electric field strength (operator), rather than the gauge-dependent vector potential.

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In the length gauge, the interaction Hamiltonian is the “classic” dipole interaction (4.281). The task is to evaluate the energy shift of an atom-field state ∣φn , 0⟩ with the atom in state ∣φn ⟩ (and no photons) in second order of stationary perturbation theory. The electric-dipole field operator, in the discrete representation, is given by Eq. (2.129b) and recalled here for convenience, in the dipole approximation ̵ = c = 0 = 1, and in units with h ⃗ ≈ E(⃗ ⃗ r = 0) = ∑ E ⃗ kλ

√

ωk⃗ ⃗ (i aλ (k) ⃗ − i a+ (k)) ⃗ . ˆλ (k) λ 2V

(4.283)

The sum extends over all modes of the vacuum, in contrast to Eq. (3.64) where only the laser mode is treated. The atom-field interaction thus reads √ ωL ⃗ ⋅ r⃗) (i aλ (k) ⃗ − i a+ (k)) ⃗ . ⃗ = −e ∑ (ˆ λ (k) (4.284) H1 = −e r⃗ ⋅ E λ 2 V ⃗ kλ The unperturbed atom+field system is described by the Hamiltonian (3.31), which we recall for convenience, H0 = ∑ Em ∣φm ⟩ ⟨φm ∣ + ∑ ωk⃗ m

⃗ kλ

⃗ aλ (k) ⃗ . a+λ (k)

(4.285)

The sum over m contains the bound and the continuous spectrum (m is a multiindex that contains all necessary quantum numbers to specify the state). We here assume that the energies Em and eigenstates ∣φm ⟩ correspond to those of the Schr¨ odinger–Coulomb problem. As in Sec. 3.3.2, we assume that the unperturbed state of the system contains the atom in the state ∣φn ⟩ and the electromagnetic field in the vacuum state ∣0⟩, without any photons present. The unpertured energy eigenvalue with respect to H0 of this state simply is En , where En is the energy of the atomic bound state. Virtual states with the atom in the state ∣φm ⟩ and one photon in the radiation field are denoted as ∣φm , 1kλ ⃗ ⟩ and have the energy H0 ∣φm , 1kλ ⃗ ⟩ = (Em + ωk ⃗ ) ∣φm , 1kλ ⃗ ⟩.

(4.286)

We denote the wave vector and the polarization state of the single virtual photon, in the usual sense, by k⃗ and λ, respectively, and use the shorthand notation ω in the following derivation because there will only be one photon involved. The secondorder perturbation (operators are used in the Schr¨odinger picture) can be written as ∆E = ⟨φn , 0 ∣H1

1 H1 ∣ φn , 0⟩ . En − H0

(4.287)

It involves the coupling of unperturbed states ∣φn , 0⟩ to virtual states ∣φj , 1kλ ⃗ ⟩. The

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calculation, which we give in compact but complete form, proceeds as follows, ∆E = ∑ ∑

⃗ ⟨φa , 1λ (k) ⃗ ∣HI ∣ φn , 0⟩ ⟨φn , 0 ∣HI ∣ φa , 1λ (k)⟩

En − Ea − ωk⃗ √ RRR RRR ωk⃗′ 1 ⃗ =∑∑ ⟨φn , 0 RRRRR(−i e) ∑ ˆλ′ (k⃗′ ) ⋅ r⃗ aλ′ (k⃗′ )RRRRR φa , 1λ (k)⟩ 2V RRR RRR ⃗ a kλ ⃗ En − Ea − ωk ⃗ ′ λ′ k √ RR RR ωk⃗′′ ′′ + ⃗ ′′ RRR ⃗ RRRRi e ∑ ⃗ ′′ ⃗  ˆ × ⟨φa , 1λ (k) ( k ) ⋅ r a ( k ) RRR φn , 0⟩ λ λ′′ RRR 2V RRR RR k⃗′′ λ′′ ⃗ ⋅ r⃗∣ φa ⟩ ⟨φa ∣ˆ ⃗ ⋅ r⃗∣ φn ⟩ λ (k) λ (k) e2 ωk⃗ ⟨φn ∣ˆ =∑∑ 2V En − Ea − ωk⃗ a kλ ⃗ a kλ ⃗

=∑∑ a

⃗ k

i j e2 ωk⃗ k i k j ⟨φn ∣x ∣ φa ⟩ ⟨φa ∣x ∣ φn ⟩ ) (δ ij − . 2V En − Ea − ωk⃗ k⃗2

(4.288)

Using the matching relation (2.81) for the summation over continuum modes, we now obtain ∆E = e2 ∑ ∫ a

j i d3 k k ij k i k j ⟨φn ∣x ∣ φa ⟩ ⟨φa ∣x ∣ φn ⟩ (δ − ) (2π)3 2 En − Ea − k k⃗2

dΩk⃗ k ij k i k j ⟨φn ∣xi ∣ φa ⟩ ⟨φa ∣xj ∣ φn ⟩ dk k 2 (δ − ) ∫ 2 π2 4π 2 En − Ea − k k⃗2 a 2α 1 3 i = x i ∣ φn ⟩ (4.289) ∫ dk k ⟨φn ∣x 3π En − HS − k 1 pi 2α 3 pi = − k ⟨φn ∣xi xi ∣ φn ⟩ + ⟨φn ∣ ∣ φn ⟩) . ∫ dk k ( 3π 2m m En − HS − k m

= e2 ∑ ∫

A few explanatory remarks are now in order. In the derivation (4.289), the Einstein summation convention is used for the spatial indices i and j with i, j = 1, . . . , 3. In ̵ = c = 0 = 1), we have e2 = 4πα, where α is the fine-structure our unit system (h ⃗ and ω⃗ constant. The infinitesimal solid angle is dΩk⃗ . Also, k is the magnitude of k, k ⃗ is the frequency of the vacuum mode with wave vector k so we have ωk⃗ = c k = k. The sum over the photon polarizations is carried out using Eq. (2.63). In Eq. (4.289), the integration limits for k have not been specified, but it is clear that, without a renormalization, the photon modes available range from k = 0 to k = ∞. Let us anticipate the introduction of an upper cutoff kmax = . The last line of Eq. (4.289) shows that the self-energy is strongly divergent for large . For the first two terms in round brackets in the last line of Eq. (4.289), this is obvious, and for the third term, one observes that ⟨φn ∣

pi 1 pi 1 p⃗ 2 ∣ φn ⟩ → − ⟨φn ∣ 2 ∣ φn ⟩ , m En − HS − k m k m

k → ∞.

(4.290)

The term En − HS can be neglected in comparison to k for k → ∞ even if En − HS is operator-valued. In intermediate steps of the derivation (4.289), the following two

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results are useful,

√ RRR RRR ωk⃗′ ⃗ (ˆ ⟨φn , 0 RRRRR(−i e) ∑ λ′ (k⃗′ ) ⋅ r⃗) aλ′ (k⃗′ )RRRRR φa , 1λ (k)⟩ 2V RRR R ′ ′ ⃗ R kλ R √ ωk⃗ ⃗ ⋅ r⃗ ∣ φa ⟩ , ⟨φn ∣ˆ λ (k) (4.291a) = −i e 2V √ RRR RRR ωk⃗′′ ⃗ RRRi e ∑ (ˆ ⟨φn , 1λ (k) λ′′ (k⃗′′ ) ⋅ r⃗) a+λ′′ (k⃗′′ )RRRRR φa , 0⟩ RRR 2V RRR RR k⃗′′ λ′′ √ ωk⃗ ⃗ ⋅ r⃗ ∣ φa ⟩ . ⟨φn ∣ˆ λ (k) (4.291b) = ie 2V In going from the last-but-one to the last line of Eq. (4.289), the following identity has been used, pi 1 pi p i xi ⟨φn ∣ ∣ φn ⟩ = − i ⟨φn ∣ ∣ φn ⟩ m E − HS − k m m 1 + k ⟨φn ∣⃗ r 2 ∣ φn ⟩ + k 2 ⟨φn ∣xi xi ∣ φn ⟩ . (4.292) E − HS − k The identity can be shown by applying the operator relations p⃗ 2 Zα p⃗ = i [HS , r⃗] , HS = − , µ = m, (4.293) m 2m r repeatedly to the left-hand side of (4.292). In reformulating the first term on the right-hand side of Eq. (4.292), use is made of the identities pi xi 3 i ⟨φn ∣ ∣ φn ⟩ = , ⟨φn ∣{pi , xi }∣φn ⟩ = 0 . (4.294) m 2m This latter identity can be proven by observing that i m (En − Ea ) ⟨φn ∣xi xj ∣φa ⟩ = i m ⟨φn ∣[HS , xi xj ]∣φa ⟩ = ⟨φn ∣xi pj + xj pi ∣φa ⟩ . (4.295) For ∣φn ⟩ = ∣φa ⟩, the left-hand side of (4.295) trivially vanishes. Now we turn to the evaluation of the mass counter term. The physically observable self-energy actually is a residual effect, equal to the self-energy of a bound electron, minus the self-energy of a free electron. However, the reasoning is a little subtle in the nonrelativistic domain. The first guess would be to consider the free self-energy operator, where the Schr¨odinger–Coulomb Hamiltonian is replaced by the free Schr¨ odinger Hamiltonian in the propagator denominator. One would then calculate the expectation value of this free self-energy operator, sandwiched for an electron in a bound state. However, it is better to proceed in two steps. Namely, one first calculates the full free self-energy operator, which is equivalent to calculating all its matrix elements in a basis of eigenstates of the free Hamiltonian. In the second step, one projects this operator onto a basis of eigenstates of the free Hamiltonian. Let us therefore calculate with the eigenstates of the free Schr¨odinger equation, p⃗ q⃗2 p⃗ 2 , = i [H0 , r⃗] , ψq⃗(⃗ r) = ei⃗q⋅⃗r , H0 ψq⃗(⃗ r) = Eq⃗ ψq⃗(⃗ r) = ψq⃗(⃗ r) , H0 = 2m m 2m (4.296)

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normalized so that ⟨ψq⃗∣ψq⃗′ ⟩ = (2π)3 δ (3) (⃗ q − q⃗′ ). We should recall that pi here is the momentum operator, which acts as follows on a momentum-space wave function, ⃗ i⃗q⋅⃗r = q⃗ ei⃗q⋅⃗r . p⃗ ψq⃗(⃗ r) = −i∇e

(4.297)

The self-energy operator in the length gauge is not diagonal in the momentum eigenstate basis because the position operator xi does not commute with the free Schr¨ odinger Hamiltonian H0 . Let us denote an off-diagonal element of the secondorder perturbation ⟨ψq⃗∣H1 [1/(Eq⃗′ − H0 − k)]H1 ∣ψq⃗′ ⟩ as ⟨ψq⃗∣δm∣ψq⃗′ ⟩, reinterpreting the free self-energy operator as a mass-counter-term operator δm. We can write this matrix element as ⟨ψq⃗∣δm∣ψq⃗′ ⟩ = =

1 2α 3 i xi ∣ ψq⃗′ ⟩ ∫ dk k ⟨ψq⃗ ∣x ′ 3π Eq⃗ − H0 − k pi xi 2α ∣ ψq⃗′ ⟩ − k ⟨ψq⃗ ∣xi xi ∣ ψq⃗′ ⟩ ∫ dk k {i ⟨ψq⃗ ∣ 3π m + ⟨ψq⃗ ∣

pi pi 1 ∣ ψq⃗′ ⟩ m Eq⃗′ − H0 + k m

−k (⟨ψq′⃗ ∣xi

1 xi H0 ∣ ψq⃗′ ⟩ − h.c.)} . Eq⃗′ − H0 − k

(4.298)

The integrand has been transformed using identities akin to those used in deriving Eq. (4.292). Because the free Hamiltonian H0 commutes with the momentum operator, we have ⟨ψq⃗ ∣

pi pi q⃗ ⋅ q⃗′ q⃗2 1 1 1 ∣ ψq⃗′ ⟩ = − ⟨ψq⃗ ∣ 2 ∣ ψq⃗′ ⟩ = − ⟨ψq⃗ ∣ 2 ∣ ψq⃗′ ⟩ . m Eq⃗′ − H0 − k m k m k m

(4.299)

The first three terms in curly brackets in Eq. (4.298) are diagonal in the momentum basis, i.e., proportional to δ (3) (⃗ q − q⃗′ ). Now, the expectation value of the mass counter term, in a bound state ∣φn ⟩, can be easily calculated by inserting, twice, a complete basis of momentum eigenstates of the free Hamiltonian, ⟨φn ∣δm∣φn ⟩ =

pi xi 2α 1 p⃗ 2 − k r⃗ 2 − ∣ φn ⟩ ∫ dk k ⟨φn ∣i 3π m k m2 1 2α i −∑ ∑ xi H0 ∣ ψq⃗′ ⟩ ⟨ψq⃗′ ∣φn ⟩ ∫ dk k ⟨φn ∣ψq⃗⟩ ⟨ψq⃗ ∣x 3 π E − H − k ′ 0 q⃗ q⃗ +∑ ∑ q⃗

=

q⃗′

2α 1 i xi ∣ ψq⃗′ ⟩ ⟨ψq⃗′ ∣φn ⟩ ∫ dk k ⟨φn ∣ψq⃗⟩ ⟨ψq⃗ ∣H0 x 3π E − H0 − k

2α 3 1 p⃗ 2 − k ⟨φn ∣⃗ r 2 ∣ φn ⟩ − ⟨φn ∣ 2 ∣ φn ⟩) . ∫ dk k ( 3π 2m k m

(4.300)

Here, the closure relation for the spectrum of the free Hamiltonian, ′

r∣ψq⃗⟩ ⟨ψq⃗∣⃗ r⟩=∫ ∑⟨⃗ q⃗

d3 q i⃗q⋅(⃗r′ −⃗r) e = δ (3) (⃗ r − r⃗′ ) , (2π)3

(4.301)

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has been used repeatedly. The renormalized low-energy part ∆ELEP of the selfenergy can be obtained by forming the difference ∆E − ⟨φn ∣δm∣φn ⟩, where ∆E is given in Eq. (4.289) and ⟨φn ∣δm∣φn ⟩ is given in Eq. (4.300), ∆ELEP = ∆E − ⟨φn ∣δm∣φn ⟩ =

 pi 1 1 pi 2α + ) ∣ φn ⟩ , ∫ dk k ⟨φn ∣ ( 3π 0 m En − HS − k k m

µ = m. (4.302)

This length-gauge energy shift is valid in the non-recoil limit and still divergent at the upper limit of integration , but only logarithmically. The integral over k diverges at infinity and this constitutes an ultraviolet divergence. However, this divergence is of a different nature as compared to the ultraviolet divergences in quantum field theory. Rather, it is an intermediate divergence which is later canceled against the infrared divergence or “infrared catastrophe” of the vertex correction to the forward scattering amplitude for an electron in a Coulomb field, which can be expressed as a the so-called Dirac F1 form factor correction. The cancelation of the  parameter between the low- and high-energy parts has been discussed for a toy model in Sec. 4.5.2, and we shall see later that the calculation described here models the low-energy part in our toy model of Sec. 4.5.2 rather well. 4.6.2

Velocity Gauge

In Sec. 3.3.2, we had formulated the dipole interaction of the atom with the quantized electromagnetic field in terms of the dipole interaction with the electric field ⃗ which is referred to as the length gauge. However, it is also possible to (−e⃗ r ⋅ E), formulate the interaction in the velocity gauge, with the interaction Hamiltonian 2 ⃗ 2 p⃗ 2 e (⃗ p − e A) ⃗ p⃗} + e A⃗2 , {A, − =− (4.303) H1 = 2m 2m 2m 2m ⃗ where {⋅, ⋅} is the anticommutator. The term quadratic in A, e2 ⃗2 A , (4.304) 2m is commonly referred to as the seagull term, because in a diagrammatic language, it corresponds to a graph with two photons emerging from the same interaction vertex. In the velocity gauge, the coupling of the bound electron to the radiation field is exactly given by Eq. (4.303), with p⃗ being the relative momentum, i.e., the conjugate variable of the relative distance r⃗. There actually is a second term with the electron mass m replaced by the mass M of the nucleus, which leads to the self-energy of the nucleus. In the absence of the vector potential, the two kinetic-energy terms, for the electron and the nucleus, consistently add up to HS =

p⃗ 2 p⃗ 2 p⃗ 2 + = , 2m 2M 2µ

(4.305)

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where µ is the reduced mass. Here we ignore the kinetic-energy term of the nucleus. Indeed, Eq. (4.303) exclusively describes the interaction of the bound electron with the radiation field, not that of the nucleus or the combined two-body system. Incidentally, it is helpful to note that Power and Zienau [176] have shown that the two interaction Hamiltonians in length and velocity gauge are related by a gauge transformation, which, strictly speaking, also involves a gauge transformation of the wave function. For the moment, we shall only briefly mention questions regarding the general gauge invariance of quantum processes with regard to the transition from velocity to length gauge (work on related questions continues up to this day, see Ref. [177]). One decisive observation is that in length gauge, the interaction is formulated in terms of the gauge-independent electric field (which is a physical observable, or, in the quantized formalism, an operator corresponding to a physical observable), while in the velocity gauge, we have the gauge-dependent vector potential. In the presence of a nonvanishing vector potential, the kinetic momentum of the particle is no longer described by the canonical momentum p⃗, but by the ⃗ The latter is the conjugate variable of the position canonical momentum p⃗ − e A. operator in the presence of a nonvanishing vector potential. According to pertinent remarks given by Lamb on page 268 of Ref. [178], the wave function φn (⃗ r) of a bound state therefore preserves its probabilistic interpretation only in the length gauge, and it is this gauge which should be used for off-resonance excitation processes [179]. Otherwise, one also has to gauge-transform the wave function (which would introduce more calculational difficulties, because the gauge-transformed wave functions differ from the Schr¨ odinger wave functions discussed in Secs. 4.2.3, 4.2.5 and 4.2.6). Recently, limitations on the gauge invariance due to changed physical interpretations have been discussed in Refs. [177, 180]. Let us briefly examine the connection of gauge transformations, minimal cou⃗ pling, and canonical momenta. Under a gauge transformation A⃗ → A⃗ + ∇Λ, the canonical momentum transforms according to ⃗ . ⃗ = p⃗ − e A⃗ → π ⃗ ′ = p⃗ − e A⃗ − e ∇Λ π (4.306) The gauge transformation leaves the canonical commutation relations invariant, i.e., ∂ ∂ i j (4.307) [π i , π j ] = [π ′ , π ′ ] = −i ( i Aj (x) − j Ai (x)) , ∂x ∂x because partial derivatives interchange, ∂i ∂j Λ = ∂j ∂i Λ. Also, a quick calculation shows that ∂ j [x′i , π ′ ] = [xi , π j − e j Λ] = [xi , π j ] = i δ ij . (4.308) ∂x We conclude that canonical commutation relations are gauge invariant, but not the conjugate momenta which receive a modification according to (4.306). The conjugate momentum is not a physical observable. This could otherwise suggest that the length gauge is far preferable over the velocity gauge. For energy perturbations, however, both gauges can be used while ignoring the gauge transform of the wave function. This may seem quite surprising. The reason

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is that all quantum electrodynamic energy shifts, which involve interactions with quantized fields, can actually be formulated in terms of adiabatically damped interactions that vanish in the infinite future and the infinite past, like those described in the classical-field derivation of the dynamic Stark shift given in Sec. 3.3.1. Because the state φn (⃗ r) (without any photons in the radiation field) is an eigenstate of the unperturbed Hamiltonian (without interaction), and because the interaction is adiabatically switched off for the in- and out-states (past and future), both gauges are permissible. A deeper analysis of related problems will be performed in Sec. 11.2.2. The Power–Zienau transformation which connects both gauges is discussed in Sec. 12.6.1. For the time being, we contend ourselves with a brief, but instructive derivation of the gauge-independence of the bound-state self-energy. The interaction Hamiltonian in Eq. (4.303) is a sum of the A⃗ ⋅ p⃗ coupling and the seagull term which is proportional to A⃗2 . In the discrete representation, the vector-potential operator (2.129a) in the dipole approximation is given by ⃗ (aλ (k) ⃗ + a+ (k)) ⃗ . ⃗ r) ≈ A(⃗ ⃗ r = 0) ⃗ =∑√ 1 ˆλ (k) A(⃗ λ V 2ω ⃗ ⃗ k kλ

(4.309)

In the dipole approximation, it is independent of position and commutes with the ⃗ p⃗} = 2A⋅ ⃗ p⃗. We start from the same unperturbed state momentum operator, i.e., {A, ∣φn , 0⟩ as in Sec. 4.6.2 and calculate all perturbative corrections to the energy in the order e2 = 4πα due to the dipole interaction Hamiltonian given in Eq. (4.303). Again, the subscript n in φn is a multi-index containing all relevant quantum numbers of the reference state. The first-order perturbation due to the seagull term ⟨φn , 0 ∣

e2 ⃗2 A ∣ φn , 0⟩ 2m

(4.310)

shifts all atomic levels by the same amount and can be discarded. With the unperturbed atom+field Hamiltonian given in Eq. (3.31), the second-order perturbation due to the A⃗ ⋅ p⃗ coupling is easily evaluated in analogy to Eq. (4.289), ∆E = ∑ ∑ a kλ ⃗

= → =

⟨φn , 0 ∣(−

e ⃗ e ⃗ A ⋅ p⃗)∣ φa , 1kλ A ⋅ p⃗)∣ φn , 0⟩ ⃗ ∣(− ⃗ ⟩ ⟨φa , 1kλ m m En − Ea − ωk⃗

⃗ ⋅ p⃗ ∣ φa ⟩ ⟨φa ∣ˆ ⃗ ⋅ p⃗ ∣ φn ⟩ λ (k) λ (k) 1 ⟨φn ∣ˆ e2 ∑∑ 2 m a kλ En − Ea + ωk⃗ ⃗V ⃗ 2 ωk i j e2 d3 k 1 k i k j ⟨φn ∣p ∣ φa ⟩ ⟨φa ∣p ∣ φn ⟩ ij (δ − ) ∑ ∫ m2 a (2π)3 2 k En − Ea − k k⃗2

2α pi 1 pi ∣ φn ⟩ . ∫ dk k ⟨φn ∣ 3π m En − HS − k m

(4.311)

The mass counter term is easy to calculate in the velocity gauge. We take a general matrix element of the self-energy, for a free electron, sandwiched between normalized eigenstates ∣ψq⃗⟩ and ∣ψq⃗′ ⟩ of free Hamiltonian H0 = p⃗ 2 /(2µ). Because the free

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Hamiltonian commutes with the momentum operator, we have in the same conventions as in Eq. (4.296), ⟨ψq⃗∣δm∣ψq⃗′ ⟩ =

pi 1 pi 2α ∣ ψq⃗′ ⟩ ∫ dk k ⟨ψq⃗ ∣ 3π m Eq⃗ − H0 − k m

= −

q⃗2 2α ∫ dk ⟨ψq⃗ ∣ 2 ∣ ψq⃗′ ⟩ . 3π m

(4.312)

The mass counter term for a free electron is diagonal in the momentum basis. In analogy to Eq. (4.300), we now easily calculate the expectation value of the mass counter term in a bound state ∣φn ⟩, with the result ⟨φn ∣δm∣φn ⟩ = −

2α p⃗ 2 ∫ dk ⟨φn ∣ 2 ∣ φn ⟩ . 3π m

(4.313)

The renormalized bound-state self-energy is obtained as the difference of the velocity-gauge shift (4.311) and the mass counter term (4.313), ∆ELEP = ∆E − ⟨φn ∣δm∣φn ⟩ =

 pi 1 1 pi 2α + ) ∣ φn ⟩ , ∫ dk k ⟨φn ∣ ( 3π 0 m En − HS − k k m

(4.314)

in full agreement with Eq. (4.289), and with the same Schr¨odinger wave function (without gauge transformation). However, the velocity-gauge result is valid even if µ ≠ m and clarifies the reduced-mass dependence of the bound-electron self-energy in the two-body system. In retrospect, the nonrelativistic renormalization turns out to be a little more problematic in the length gauge as opposed to the velocity gauge, because it involves fewer terms in the velocity gauge. However, the length gauge has the advantage that we do not have to worry about the seagull term in the Hamiltonian. Gauge invariance is fully recovered after renormalization. 4.6.3

Calculation of the Bethe Logarithm

After renormalization, as we have shown, length and velocity gauges yield identical results for the low-energy part (LEP) of the renormalized self-energy ∆ELEP , as given in Eqs. (4.289) and (4.311). There are two ways to evaluate the expression given in Eqs. (4.289) and (4.311). The first way is merely symbolic and relies on an analytic evaluation of the photon energy integral, and the identification of the logarithmically divergent terms. This calculation exhibits in a particularly clear manner the origin of the name “Bethe logarithm”. The second way, which is discussed here, relies on an analytic subtraction of the logarithmically divergent terms, and on a formulation of the finite remainder term, which contains the Bethe logarithm, in terms of the dynamic polarizability matrix elements calculated in Sec. 4.3.4. We start from Eq. (4.314), use the expression for the Schr¨odinger–Coulomb Hamiltonian given in Eq. (4.1), and keep HS in operator form until the end of the

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calculation, pi 2α  1 1 pi − ) ∣ φn ⟩ ∫ dk k ⟨φn ∣ ( 3π 0 m HS − E n + k k m

∆ELEP = −

pi (k + HS − En ) − (HS − En ) 2α  pi − 1) ∣ φn ⟩ ∫ dk ⟨φn ∣ ( 3π 0 m HS − En + k m

= − =

 2α µ 2 pi HS − En pi ( ) ∫ dk ⟨φn ∣ ∣ φn ⟩ . 3π m µ HS − E n + k µ 0

(4.315)

We now take the principal value of this integral. Note that the imaginary part of the energy shift otherwise incurred (for an excited reference state) finds a natural interpretation in terms of the radiative decay rate of the reference state against dipole decay (see Refs. [181–183], and also Sec. 3.2.3). Here, we are only interested in the real part of the energy shift and find ∆ELEP =

pi  pi 2α µ 2 ( ) ⟨φn ∣ (HS − En ) ln ( ) ∣ φn ⟩ 3π m µ ∣HS − En ∣ µ

=

2α µ 2 pi pi 2 ( ) (⟨φn ∣ (HS − En ) ∣ φn ⟩ ln ( ) 3π m µ µ (Zα)2 µ − ⟨φn ∣

=

α µ 2 pi pi 2 ( ) ⟨φn ∣[ , [(HS − En ), ]] ∣ φn ⟩ ln ( ) 3π m µ µ (Zα)2 µ −

=

pi 2 ∣HS − En ∣ pi (HS − En ) ln ( ) ∣ φn ⟩) µ (Zα)2 µ µ

pi 2 ∣HS − En ∣ pi 2α µ 2 ( ) ⟨φn ∣ (HS − En ) ln ( ) ∣ φn ⟩ 3π m µ (Zα)2 µ µ

4 α (Zα)4 m µ 3 2 ( ) [ln ( ) δ`0 − ln k0 (n`)] . 3π n3 m (Zα)2 µ

(4.316)

In the first step, we use the fact that  must be large against ∣HS − En ∣, and ignore terms of order O(α (Zα)6 µ). The Bethe logarithm ln k0 (n`) is identified as ln k0 (n`) =

pi HS − En 2 ∣HS − En ∣ pi n3 ⟨φ ∣ ln ( ) ∣ φn ⟩ , n 2(Zα)4 µ µ (Zα)2 µ µ

(4.317)

where the sum over i = 1, 2, 3 is implied by the Einstein summation convention. The Bethe logarithm depends on the principal quantum number n, and on the orbital angular momentum ` of the reference state ∣φn ⟩, but the reduced mass µ scales out of the problem. Written as a sum over intermediate, virtual states, we have ln k0 (n`) =

2 n3 p⃗ Ea − En 2 ∣Ea − En ∣ ∣⟨φ ∣ ∣ φ ⟩∣ ln ( ). ∑ n a 2(Zα)4 a µ µ (Zα)2 µ

(4.318)

The absolute value is very important in the argument of the logarithm. For excited reference states, the argument of the logarithm Ea − En might otherwise become negative, which would imply a nonvanishing imaginary part of the Bethe logarithm. While the imaginary part actually has a physical interpretation, as discussed in

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Sec. 4.7.1 below, we stress that the Bethe logarithm is defined in terms of the principal-value integral given in Eq. (4.316), which implies that it is purely real. The argument of the logarithm in Eq. (4.318) contains energy differences ∣Ea − En ∣ of atomic states scaled to the reference quantity (Zα)2 µ , (4.319) 2 where we generally assume that the reduced mass µ of the system is close to the electron mass m. The formulas (4.317) and (4.318) are valid for all atomic states, including S states. Several steps in the derivation (4.316) are in need of further explanation. We have already explained the principal-value prescription used in the step labeled (P.V.). The right-hand side of the same transformation step also contains a hint that higher-order terms O(α(Zα)6 ) have been neglected in the evaluation. These terms are convergent as  → ∞, for example terms proportional to ⟨φn ∣[(Zα)2 /r4 ]∣φn ⟩/, and will be discussed below in Sec. 4.7.3. A further remark is in order. Namely, the general commutator relation Z 2 R∞ ≈

ABA =

1 1 ([A, [B, A]] + A2 B + BA2 ) = ([A, [B, A]] + {A2 , B}) 2 2

(4.320)

implies that ⟨φn ∣

pi pi 1 pi pi (HS − En ) ∣ φn ⟩ = ⟨φn ∣[ , [(HS − En ), ]]∣ φn ⟩ , µ µ 2 µ µ

(4.321)

while, furthermore, one has [

1 pi pi 1 1 2 ⃗ (V ) = , [(HS − En ), ]] = − 2 p⃗ 2 (V ) = 2 ∇ 4πZα δ (3) (⃗ r) . µ µ µ µ µ2

(4.322)

Only S states have a nonvanishing probability density at the origin, (Zα)3 µ3 δ`0 . (4.323) πn3 Here, as always, the subscript n in φn needs to be understood as a multi-index denoting the nonrelativistic quantum numbers n, ` and m, where n is the principal quantum number, ` is the orbital angular momentum quantum number, and m is the magnetic projection quantum number. A further, very important remark is in order. The low-energy part of the boundelectron self-energy, after renormalization, is of order α (Zα)4 µ, as evident from Eq. (4.316). This order-of-magnitude result could have been obtained from the first line of Eq. (4.316) independently. Bound systems follow rather strict hierarchies in terms of the momentum and energy scales associated to specific operators. Specifically, bound-state energies are of order (Zα)2 µ, as evident from the Schr¨odinger– Coulomb spectrum. The same order-of-magnitude, (Zα)2 µ, can be associated with the Schr¨ odinger–Coulomb Hamiltonian as an operator, because it contains a sum over virtual bound and continuum states, and the inverse 1/((Zα)2 µ) should thus ⟨φn ∣δ (3) (⃗ r)∣φn ⟩ =

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be associated with the Green function. The same order-of-magnitude applies to the Coulomb potential alone. A bound-state momentum is of order Zαµ, in accordance with the nonrelativistic identification E ∼ p2 /µ, where p is a characteristic momentum scale. The exchanged photon energy k in the low-energy part also is of order (Zα)2 µ. We thus have pi ∼ Zαµ ,

ri ∼

1 , Zα µ

HS ∼ En ∼ k ∼ (Zα)2 µ ,

 pi 1 2α − ∫ dk k ⟨φn ∣ ( 3π 0 m HS − E n + k Zαm 1 ∼ α ((Zα)2 m) ((Zα)2 m) m (Zα)2 m

∆ELEP = −

µ ≈ m,

1 pi ) ∣ φn ⟩ k m Zαm = α (Zα)4 m . m

(4.324)

The Zα-scaling sometimes has to be adopted to a more general method of (integration) regions (see Chaps. 16 and 17). In the current context, it provides, in a very straightforward fashion, for a useful basis in the study of bound systems. The scaling of the atomic coordinate ri is given for reference; it is useful analyzing the Zα-scaling of intermediate expressions in Eq. (4.289). One can easily convince oneself that the Bethe logarithm (4.317) is independent of the values of the reduced mass µ of the bound system. In accordance with our toy model calculation discussed at length in Sec. 4.5.2, we recall that the result obtained in Eq. (4.316) has to be interpreted as the lowenergy part of the bound-electron self-energy, which is obtained after first expanding in Zα, then in . This procedure implies, first of all, that  ≫ Zα, so that, in the integrals over the photon energy ω, one integrates over the range ω ∈ (0, ∞) (unless divergences are encountered), because  is much larger than the atomic energy scale. However, based on the general paradigm discussed in Sec. 4.5.2, one later neglects all terms proportional to  and higher in the final result for the low-energy part, ∆ELEP =

2 4 α (Zα)4 m µ 3 ( ) [ln ( ) δ`0 − ln k0 (n`)] . 3π n3 m (Zα)2 µ

(4.325)

This is based on the paradigm, also discussed in Sec. 4.5.2, which states that the -dependent terms have to cancel in each order of the Zα-expansion separately, whether they be divergent or not in the limit  → 0. The physically relevant regime for the  parameter in this case is the regime where (Zα)2 µ ≪  ≪ µ ,

(4.326)

i.e.,  is large on the scale of the atomic binding energy, which is why it is permissible to extend all convergent integrals over ω to infinity, but small on the scale of the electron mass, so that  acts as a cut-off parameter for the high-energy contributions that involve infrared divergences which are naturally cut off at the atomic binding energy scale. In Sec. 4.5.2, this subtle interplay is illustrated using a compact, analytic example calculation.

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Bound-Electron Self–Energy and Bethe Logarithm

Table 4.1 Bethe Logarithms for all excited states in hydrogenlike systems with ` ≤ 4 and n ≤ 12. For ` = 2, 3, 4, the given numerical results represent the quantity ln k0 (n`) × 102 , so the Bethe logarithm for n = 12 and ` = 2 reads as ln k0 (n = 12, ` = 2) = −0.009 342 953 99. Bethe Logarithms ln k0 ( n`)) for n ≤ 12 and ` ≤ 4 `=0

`=1

`=2

`=3

`=4

ln k0 × 10−1

ln k0 × 101

ln k0 × 102

ln k0 × 102

ln k0 × 102

n=1

0.298 412 856

—

—

—

—

n=2

0.281 176 989

-0.300 167 086

—

—

—

n=3

0.276 766 361

-0.381 902 294

-0.523 214 814

—

—

n=4

0.274 981 184

-0.419 548 946

-0.674 093 888

-0.173 366 148

—

n=5

0.274 082 373

-0.440 346 956

-0.760 075 126

-0.220 216 838

-0.077 209 890

n=6

0.273 566 421

-0.453 121 977

-0.814 720 396

-0.250 217 976

-0.096 279 743

n=7

0.273 242 913

-0.461 551 773

-0.851 922 329

-0.270 909 573

-0.109 447 274

n=8

0.273 026 726

-0.467 413 520

-0.878 504 298

-0.285 911 456

-0.119 043 204

n=9

0.272 875 117

-0.471 657 000

-0.898 203 229

-0.297 190 149

-0.126 309 451

n = 10

0.272 764 694

-0.474 828 934

-0.913 227 225

-0.305 909 428

-0.131 971 806

n = 11

0.272 681 778

-0.477 262 681

-0.924 957 082

-0.312 802 113

-0.136 484 485

n = 12

0.272 617 934

-0.479 171 116

-0.934 295 399

-0.318 351 910

-0.140 146 873

4.6.4

Numerical Values of Bethe Logarithms

For the numerical evaluation of the Bethe logarithm, which is important for the comparison of theoretical predictions to experimental results, a representation in terms of transition elements and energy differences to intermediate, virtual states is not of great help. One may in fact use such a representation to good effect if one considers a discretized representation of the Schr¨odinger–Coulomb propagator on a carefully adjusted lattice of points, which takes into account the importance of the region near the nucleus and uses a highly concentrated array of lattice points in this area [184, 185]. Another representation relies on explicit results for transition matrix elements for bound-bound, and bound-continuum transitions [186, 187]. Indeed, explicit matrix elements of discrete-discrete transitions have been given rather early in the history of quantum mechanics [188], and discrete-continuum transitions

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have also been analyzed [161,189]. The results given in the literature are not free of typographical errors: In Eq. (6) of Ref. [161], the argument of the inverse cotangent (arccot) function in the exponential should be replaced according to n′ n/n → n′ /n. Certain normalization problems of the continuum wave functions used in Ref. [161] motivate a very careful discussion, as already mentioned in the discussion following Eq. (4.76a) in Sec. 4.2.5 above. For selected low-lying states coupled to the continuum, formulas have been given as early as 1930 (see Ref. [190]), but one should be aware of multiplicative correction factors given below Eq. (11) of Ref. [191]. The numerical results obtained using either lattice calculations or explicit transition matrix elements are somewhat “hanging in the air” because they are obtained as “pure numbers”, without any control over the numerics (or analytic limits) in intermediate steps of the calculation. For low-lying states of hydrogenlike systems, a transparent method in evaluating the Bethe logarithm relies on the analytic results for the dynamic polarizability matrix elements given in Sec. 4.3.4. Let us recall from Sec. 4.3.4 the definition of the P matrix elements, P (φn , ω) =

1 1 ⟨φn ∣pi pi ∣ φn ⟩ , 3µ HS − z

z = En − ω .

The definition of the t variable according to Eq. (4.131), √ −1/2 En 2n2 ω (Zα)2 µ 1 − t2 = (1 + ) , ω = , t= En − ω (Zα)2 µ 2n2 t2

(4.327)

(4.328)

implies that ultraviolet divergences from the integration region ω ∼  ≫ (Zα)2 µ, i.e., ω → ∞, are mapped onto the integration region t → 0. We also recall the analytic result for the dynamic polarizability of the ground state, given originally in Eq. (4.149), P (1S, t) =

2t2 (1 − t − 5t2 − 11t3 ) 64 t5 1−t 2 + ) ) . (4.329) 2 F1 (1, −t, 1 − t, ( 2 3 2 3 3 (1 + t) (1 − t) 3 (1 − t ) 1+t

The integration region ω ∈ (0, ∞) is mapped onto the integration region t ∈ (0, 1). Following the calculation reported in Eq. (4.316) in Sec. 4.6.3, the ultraviolet divergences can easily be calculated analytically and subtracted. Inspired by Eq. (4.325), we define a scaled function FLEP = FLEP (φn , , Zα) as follows, ∆ELEP =

α (Zα)4 m µ 3 ( ) FLEP , π n3 m

(4.330)

where FLEP is given by FLEP = ∫

1 0

dt t2 − 1 2 8 2 4 2 n2 ( P (φ , t) + − t δ ) + δ ln ( ). n `,0 `0 t3 n t2 3n 3 3 (Zα)2 µ

(4.331)

A comparison of Eqs. (4.330), (4.317) and (4.325) reveals that FLEP =

4 2 4 δ`0 ln ( ) − ln k0 (n`) . 3 (Zα)2 µ 3

(4.332)

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Bound-Electron Self–Energy and Bethe Logarithm

Combining (4.331) and (4.332), we immediately read off the integral representation 1 3 1 t2 − 1 2 8 (P.V.) ∫ dt 3 { P (φn , t) + − t2 δ`0 } − 2 ln(n) δ`0 . 4 t n t2 3n 3 0 (4.333) The remaining t integral can be done semi-analytically, using the series representation of the hypergeometric function in Eq. (4.329) as a guide, and the remaining terms can be expressed in terms of a generalized zeta function, which is known as the Lerch Φ transcendent. We should briefly discuss these functions. The Lerch Φ transcendent is defined as

ln k0 (n`) = −

∞

zn . s n=0 (n + a)

Φ(z, s, a) = ∑

(4.334)

A formula valid in a larger domain of the complex plane is Φ(z, s, a) =

∞ −z n n (−1)k n 1 ) ∑ ( ). ∑( 1 − z n=0 1 − z k=0 (a + k)s k

(4.335)

The Lerch Φ transcendent can be thought of as a generalization of the zeta function, because Φ(1, s, 1) = ζ(s). The connection of the Lerch transcendent to the hypergeometric function is Φ(z, 1, a) =

2 F1 (1, a, 1 + a, z)

a

.

(4.336)

After the t integration, hydrogenic Bethe logarithms of 1S, 2S and 2P states can be written as rather compact sums over Lerch Φ functions, and evaluated to essentially arbitrary accuracy. We have for the ground state, ∞

1+k 16 k Φ( , 1, 2 k) 2 (k + 1)2 (k − 1) 1 −k k=2

ln k0 (1S) = 10 ln(2) − 2ζ(2) − 1 + ∑

= 2.98412 85557 65497 61075 97770 90013 79796 99751 80566 17002 00048 15926 13924 06576 62306 75532 86860 62013 30404 72249 . (4.337) Here ζ denotes the Riemann zeta function. The logarithmic sum, for the 2S state, is given by 545 22 + ln(2) − 14 ζ(2) + 24 ζ(3) 36 3 ∞ 1024 k (k − 1) (k + 1) 2+k +∑ Φ( , 1, 2 k) 3 3 (k − 2) (k + 1) 2−k k=3

ln k0 (2S) = −

(4.338)

= 2.81176 98931 20563 51521 97427 85941 63611 28935 51470 29732 41909 18696 96453 24020 20118 89106 87017 48612 02831 24031 ,

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whereas for the 2P state, it reads 3437 3280 14 136 64 ln k0 (2P ) = − + ln(2) − ζ(2) + ζ(3) − ζ(4) 2916 2187 3 9 3 ∞ 2+k 256 k 3 (11 k 2 − 12) Φ( , 1, 2 k) +∑ 4 4 2−k k=3 3 (k − 2) (k + 2)

(4.339)

= − 0.03001 67086 30212 90244 36757 10951 14406 39409 33044 23103 04668 98525 32719 44796 89622 57183 26244 10312 70799 73828 . While these formulas do not provide direct, analytic results for Bethe logarithms, they still illustrate that one can think of the Bethe logarithm as a sum of logarithms, zeta functions, and generalized zeta functions. For other high-precision evaluations of Bethe logarithms, see Refs. [192, 193]. While the 100-figure numerical accuracy of the results (4.337), (4.338) and (4.339) far exceeds the requirements of overall theoretical predictions for the Lamb shift of the atomic states under investigation, the reference values given here may serve as a guide for conceivable future test calculations of accurate Green functions of the hydrogen atom, obtained using alternative lattice and other methods. An extensive list of 9-figure numerical results for all bound states with quantum numbers n ≤ 200 has been given in Refs. [186, 187]. A convenient list for n ≤ 12 and ` ≤ 4 is given in Table 4.1. 4.6.5

Outline of the High-Energy Part

The low-energy part of the self-energy has been evaluated in Eq. (4.325), and the Bethe logarithm has been given in Eq. (4.317). Furthermore, we have discussed the numerical evaluation of the Bethe logarithm in Sec. 4.6.4. We shall now briefly discuss the high-energy part of the self-energy, which is spin-dependent and requires fully relativistic loop calculations. We formulate the high-energy part in terms of ¨ dinger1 –Pauli2 wave effective potentials, which are evaluated on so-called Schro functions, i.e., wave functions whose radial part is equal to the wave functions discussed in Sec. 4.2.2, but whose angular part allows for the presence of an internal degree of freedom of the electron, which is the spin. The Schr¨odinger–Pauli wave functions are defined in Eq. (6.99); they combine a nonrelativistic radial part with a spin-angular component which describes the coupling of the orbital angular momentum ` and the spin angular momentum s = 1/2, to result in the total angular momentum j. We can only give the essential steps here; some parts have to be relegated to a more thorough discussion of the radiatively corrected Dirac equation (see Chap. 10). The first high-energy potential we consider is ∆V1 = α (Zα) [ 1 Erwin

m 11 δ (3) (⃗ r) 4 , ln ( ) + ] 3 2 18 m2

Rudolf Josef Alexander Schr¨ odinger (1887–1961). Ernst Pauli (1900–1958).

2 Wolfgang

(4.340)

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which is proportional to the slope of the Dirac form factor of the bound electron at zero momentum transfer. Hence, it is manifestly the electron mass m that enters the denominator, not the reduced mass µ. The corresponding energy shift is ∆E1 = ⟨φn ∣∆V1 ∣ φn ⟩ =

m 11 µ 3 α (Zα)4 m 4 [ ln ( ) + ] ( ) δ`0 . 3 π n 3 2 18 m

(4.341)

The correction ∆E1 is spin independent and nonvanishing only for S states with ` = 0. The second potential comes from the anomalous magnetic moment of the ⃗ electron. Relativistically, it is obtained as ∆V2 = F2 (0) i e γ⃗ ⋅ E/(2m), where F2 (0) = α/(2π) is the anomalous magnetic moment (Schwinger factor). The nonrelativistic limit of the second perturbative, “radiative” potential is ∆V2 =

⃗ ⃗⋅L r) α (Zα) σ α 1 π (Zα) δ (3) (⃗ + , 2 3 π 2 m π 4mµr

(4.342)

⃗ is the orbital angular momentum, and σ ⃗ is the vector of Pauli spin matrices where L [see Eq. (8.66) of Ref. [81]], which we recall for convenience, σx = σ1 = (

01 ), 10

σy = σ2 = (

0 −i ), i 0

σz = σ3 = (

1 0 ). 0 −1

(4.343)

The reduced-mass dependence of the second term (spin-orbit term) follows from the two-body Hamiltonian of the system (Breit Hamiltonian) and is a nontrivial result that takes into account the manifestly two-body nature of the bound system. In writing Eq. (4.342), we have explicitly displayed the expression ∆Vstd =

π (Zα) δ (3) (⃗ r) , m2

⟨φn ∣∆Vstd ∣ φn ⟩ =

(Zα)4 m µ 3 ( ) δ`0 , n3 m

(4.344)

which is a “standard” Dirac-δ potential that is sometimes useful in atomic physics calculations. We note that the evaluation of the spin-dependent term given in Eq. (4.342) actually entails the use of the Schr¨odinger–Pauli wave function discussed in Sec. 6.4.1 below. The second term in Eq. (4.342) necessitates the inclusion of the electron spin into the formalism, which can easily be accomplished by the use of the Schr¨odinger– Pauli wave function [see Eq. (6.99) below]. After the tracing of the spin degrees of freedom, what remains is the evaluation of a matrix element of r−3 on a hydrogenic bound state, and we are now interested in its numerical value for a general reference state with quantum numbers n, ` and m. Modern computer algebra systems enable us to map all momentum operators and integrations onto a well-defined subset of integrals, once the explicit form of a reference-state wave function is known. The explicit form of the reference-state wave function can thus be calculated from the formulas given in Sec. 4.2.3, or from the general formulas given below. Alternatively, the following recursion relation, due to Kramers and Pasternack [194, 195], can be used in order to establish relations among matrix elements

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of different powers of the radial variable r. It reads as 2k + 1 k+1 ⟨φn ∣rk ∣ φn ⟩ − ⟨φn ∣rk−1 ∣ φn ⟩ n2 Zαµ k [(2` + 1)2 − k 2 ] ⟨φn ∣rk−2 ∣ φn ⟩ = 0 , (4.345) + 4 (Zαµ)2 being valid for both positive as well as negative k. Closer inspection reveals that the result for the matrix element of 1/r can be derived by setting k = 0 in Eq. (4.345), whereas the result for 1/r3 can be derived by setting k = −2 in Eq. (4.345), and having a result for 1/r2 handy. Furthermore, this relation can be used to derive matrix elements of higher inverse powers of r. One obtains the following results [see Eq. (3.24) ff. of Ref. [6]] 1 1 ⟨φn ∣ ∣ φn ⟩ = Zαµ 2 , (4.346a) r n 1 2 ⟨φn ∣ 2 ∣ φn ⟩ = (Zαµ)2 3 , (4.346b) r n (2 ` + 1) 1 2 ⟨φn ∣ 3 ∣ φn ⟩ = (Zαµ)3 3 , (4.346c) r n ` (` + 1) (2` + 1) 1 4 [3 n2 − ` (` + 1)] ⟨φn ∣ 4 ∣ φn ⟩ = (Zαµ)4 5 . (4.346d) r n ` (` + 1) (2` − 1) (2` + 1) (2` + 3) Higher inverse power of the radial variable lead to matrix elements divergent on P states, 4 [5 n2 − 3` (` + 1) + 1] 1 ∣ φn ⟩ = (Zαµ)5 5 , (4.346e) 5 r n (` − 1) ` (` + 1) (` + 2) (2` − 1) (2` + 1) (2` + 3) 1 64 [35 n4 − 5n2 (6` (` + 1) − 5) + 3(` − 1)`(` + 1)(` + 2)] . ⟨φn ∣ 6 ∣ φn ⟩ = (Zαµ)6 r n7 ∏5j=−3 (2` + j) (4.346f) ⟨φn ∣

Matrix elements of 1/r3 and 1/r4 are already divergent for S states with ` = 0. This is reflected in the denominators of the formulas. The expectation value of ∆V2 for a reference bound state ∣φn ⟩ is ∆E2 = ⟨φn ∣∆V2 ∣ φn ⟩ =

=

α (Zα)4 m µ 3 δ`0 ( ) π n3 m 2 α (Zα)4 m j(j + 1) − `(` + 1) − s(s + 1) µ 2 + ( ) (1 − δ`0 ) π n3 2` (` + 1) (2` + 1) m α (Zα)4 m µ 3 δ`0 1 µ 2 [( ) + (− ) ( ) (1 − δ`0 )] . π n3 m 2 2κ (2 ` + 1) m

(4.347)

Here, the Dirac quantum number κ is obtained from the orbital angular momentum ` and the total momentum j of the bound electron as 1 κ = (−1)j+`+1/2 (j + ) . (4.348) 2

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E.g., for S states, we have κ = −1, and ` = 0, so that the term −1/[2κ (2 ` + 1)] in Eq. (4.347) evaluates to 1/2 for S states and nominally becomes equal to the first term, in the non-recoil limit µ → m. The high-energy part ∆EHEP = ∆E1 + ∆E2 is obtained as ∆EHEP =

α (Zα)4 m 4 m 10 µ 3 1 − δ`0 µ 2 {[ ln ( ) + ] ( ) δ`0 − ( ) }, 3 π n 3 2 9 m 2κ (2 ` + 1) m

(4.349)

which includes the full reduced-mass dependence. In the limit µ → m, we can add the low-energy part (4.325) to obtain the full self-energy shift ∆ESE = ∆EHEP + ∆ELEP as α (Zα)4 m 4 11 1 4 [( ln [(Zα)−2 ] + ) δ`0 − − ln k0 (n`)] , π n3 3 18 2κ (2 ` + 1) 3 (4.350) which is a nice general formula for µ = m, but it does not include the full reducedmass dependence of the result. If we wish to include the reduced-mass dependence, then it is easier to separate the cases of S versus non-S states. For S states, the high-energy part evaluates to µ=m

∆ESE =

∆EHEP (nS) =

m 10 α (Zα)4 m µ 3 4 ( ) ( ln ( ) + ) . π n3 m 3 2 9

(4.351)

Adding the low-energy part (4.325), the auxiliary overlapping parameter  cancels, and we obtain the following total self-energy shift for S states, ∆E(nS) =

α (Zα)4 m µ 3 4 m 1 10 4 ( ) [( ln [ ] + ) − ln k0 (nS)] , 3 2 π n m 3 µ (Zα) 9 3

(4.352)

which includes the full reduced-mass dependence, including the argument of the logarithm. By contrast, for non-S states with ` ≠ 0, the first term in square brackets in Eq. (4.349) vanishes. Adding the low-energy part (4.316), we thus obtain the following total self-energy shift for non-S states, ∆E(n`j )∣`≠0 =

α (Zα)4 m µ 3 1 m 4 ( ) (− − ln k0 (n`)) , 3 π n m 2 κ (2` + 1) µ 3

(4.353)

which depends on the total angular momentum j of the bound electron i.e., it is fine-structure dependent. We have denoted the reference state in spectroscopic notation as n`j , i.e., for a 2P3/2 state we would have n = 2, ` = 1 and j = 3/2. The reduced-mass dependence of the anomalous-magnetic-moment correction given in Eq. (4.353) finds an explanation in Eq. (12.118). The calculation of the bound selfenergy for a general state, for arbitrary masses of orbiting particle and nucleus, is a nontrivial achievement. The results (4.352) and (4.353) can directly be applied to other bound systems such as muonic hydrogen, under an appropriate replacement of m for the muon mass and µ for the reduced mass of muonic hydrogen. Our only additional assumption concerns the spin-1/2 character of the orbiting particle.

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Applications of the Developed Formalism

4.7.1

Electric Dipole Decay and Imaginary Part of Self–Energy

The calculation of the (real part of the) self-energy shift of a bound atomic state, with final results reported in Eqs. (4.350), (4.352) and (4.353), represents the most significant result of the current chapter, together with the numerical results for Bethe logarithms listed in Table 4.1 and Eq. (4.333) (for a general integral representation of the Bethe logarithm), as well as the numerical result for n = 1 and n = 2 states in Eqs. (4.337) and (4.338). However, the potential applications of the developed formalism are much more general, and we shall indicate several of them in the following. First, it is actually quite interesting to note that there is a connection of the imaginary part of the self-energy shift to the decay rate of an atomic state. We recall that in going from Eq. (4.315) to (4.316), we have deliberately ignored the imaginary part of the energy shift, focusing only on the real part, which we extracted using a principal-value prescription. In general, an energy shift can be written as [see also Eq. (3.56)] ∆E = Re ∆E −

i Γ, 2

Γ = −2Im ∆E .

(4.354)

In specifying the imaginary part of the self-energy shift (4.315), we should first find a valid prescription concerning the poles along the contour used for the integration over k. One might ask which energy in the propagator denominator should get an imaginary part: the reference state energy En or the virtual-state energy described by the Hamiltonian HS ? The answer is that in our formalism, the reference state is thought of as a perfect, asymptotic state, and the interaction is switched off in the infinite past and future. So, the virtual states, not the reference state, obtain an imaginary part (infinitesimal damping term), and we have ∆ELEP = =

pi 1 pi 2α µ 2 ( ) ∫ dk k ⟨φn ∣ ∣ φn ⟩ + . . . 3π m µ En − (HS − i ) − k µ 2 2α µ 2 p⃗ 1 ( ) ∑ ∫ dk k ∣⟨φn ∣ ∣ φm ⟩∣ . 3π m µ En − Em − k + i  m

(4.355)

The omitted terms in the first line correspond to the renormalization counter-term, which we can ignore in calculating the imaginary part of the self-energy shift, because it does not contain any virtual-state energies in the denominator and cannot lead to an imaginary energy shift. Because we are eventually interested only in the imaginary part of the energy, we can use the following formula, which follows from a theorem proven by Sochocki3 and Plemelj4 (for a derivation, see page 66 of 3 Julian 4 Josip

Sochocki (1842–1927). Plemelj (1873–1967).

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Ref. [26]) 1 1 = En − (Em − i ) − k En − Em − k + i  1 = (P.V.) − i π δ(En − Em − k) , (4.356) En − Em − k which essentially describes how the poles along the k-integration contour are to be encircled. If En − Em > 0, i.e., if the virtual state has a lower energy than the reference state, then the Dirac-δ “fires” and a nonvanishing imaginary part is found. Extracting the imaginary part, we have 2 ∞ 2α µ 2 p⃗ Im ∆ELEP = − ( ) ∑∫ dk k ∣⟨φn ∣ ∣ φm ⟩∣ π δ(En − Em − k) 3 m µ m 0 =−

2α µ 2 ( ) 3 m

2 p⃗ ∑ (En − Em ) ∣⟨φn ∣ ∣ φm ⟩∣ µ Em a0 , atomic interaction is being calculated for a finite, nonvanishing separation ∣R∣ where a0 is the Bohr radius. Thus, e4 Ri Rk Rj R` δ ij δ k` ∆E (4) ≈ − (δ ik − 3 ) (δ j` − 3 ) 2 2 6 2 16π 0 R R R2 3 3 × ∑∑ ρ

σ

m n n ⟨1S, A∣xm 1 ∣ρ⟩ ⟨ρ∣x1 ∣1S, A⟩ ⟨1S, B∣x2 ∣σ⟩ ⟨σ∣x2 ∣1S, B⟩ , Eρ − E1S,A + Eσ − E1S,B

(5.90)

where the property (5.72) has been used, the superscripts m and n are summed over, and the sign of the terms in the propagator has been reversed. Although initially not obvious, it is advantageous to write the propagator denominator as an integral, ∞ (Eρ − E1S,A ) (Eσ − E1S,B ) 1 1 = ∫ du , Eρ − E1S,A + Eσ − E1S,B π −∞ [(Eρ − E1S,A )2 + u2 ] [(Eσ − E1S,B )2 + u2 ] (5.91) and we finally obtain ∆E (4) ≈ − ×

e4

∞

du ∑ (Eρ − E1S,A ) (Eσ − E1S,B ) ∫ 24π 3 20 R6 −∞ ρ,σ m ⟨1S, A∣xm ∣ρ⟩ ⟨ρ∣x ∣1S, A⟩ ⟨1S, B∣xn2 ∣σ⟩ ⟨σ∣xn2 ∣1S, B⟩ 1 1 [(Eρ − E1S,A )2 + u2 ] [(Eσ − E1S,B )2 + u2 ]

.

(5.92)

̵ this expression can be rewritten as Using the substitution u → hω, 2 ∞ ̵ HA − E1S,A 2h 1 i ∆E (4) = − ( ) dω {e2 ⟨1S, A ∣xi1 ∫ ̵ 2 x1 ∣ 1S, A⟩} 3πR6 4π0 (HA − E1S,A )2 + (hω) −∞ HB − E1S,B j (5.93) ̵ 2 x2 ∣ 1S, B⟩} , (HB − E1S,B )2 + (hω) where the atomic Hamiltonians HA and HB have been defined in Eq. (5.55). Using the decomposition HA − E1S,A 1 1 1 = ( + (5.94) 2 2 ̵ ̵ ̵ ), (HA − E1S,A ) + (hω) 2 HA − E1S,A + ihω HA − E1S,A − ihω and an analogous decomposition for atom B, one may identify the integrand as a sum over dynamic polarizabilities of complex argument. Since we are dealing with ground-state atoms, we can write α(A, ω) ≡ α(1S, A, ω), suppressing the reference state in the notation, which results in e2 1 i i α(A, i ω) = ∑ ⟨1S, A ∣x1 ̵ x1 ∣ 1S, A⟩ 3 ± HA − E1S,A ± ihω × {e2 ⟨1S, B ∣xj2

=

2e2 HA − E1SA i ⟨1S, A ∣xi1 ̵ 2 x1 ∣ 1S, A⟩ . 3 (H − E1S,A )2 + (hω)

(5.95)

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When applied to Eq. (5.93), the result reads ̵ 3h 1 2 ∞ ( dω α(A, iω) α(B, iω) ) ∫ 2πR6 4π0 −∞ ̵ 1 2 ∞ 3h dω α(A, iω) α(B, iω) , ( ) ∫ = − πR6 4π0 0 ̵ ̵ hc hc a0 . a0 ≪ R ≪ , ∼ ∣E1S,A − Eρ ∣ ∣E1S,B − Eσ ∣ α

∆E (4) = −

(5.96)

We observe the change in the integration domain in the first and second lines of Eq. (5.96) and find the well-known 1/R6 van der Waals interaction to be expressible in terms of dynamic polarizabilities of the two atoms involved in the interaction. The formula is valid only in a distance range bounded from below by the Bohr radius, and from above by a typical atomic transition wavelength. 5.4.5

Interpolating Formula

Summing all contributions to ∆E (4) given in Eqs. (5.66) and (5.67), one would hope to obtain a formula which interpolates between the limiting cases (5.82) and (5.96), still under the assumption of non-overlapping wave functions (a0 ≪ R). The −1 integrand necessarily involves the sum ∑12 i=1 [fi (k1 , k2 )] . In the calculation, one 3 has the advantage that one integrates over both d k1 and d3 k2 , over the entire space R3 . The integrand involves a dependence on both k1 and k2 , and it is possible to reorder the expressions in terms of symmetric or asymmetric combinations of the two photon momenta. In the following, we consider a calculational scheme which is based on an asymmetric combination of the two momenta, in order to facilitate the k2 -integration. The basic rationale is that after the reordering and replacement, one should generate a factor k23 k23 (−k2 )3 k23 − = + , k1 + k2 k1 − k2 k1 + k2 k1 + (−k2 )

(5.97)

which eventually allows one to carry out the k2 integration by principal value, after extending the k2 integration interval from (0, ∞) to (−∞, ∞), due to the symmetry of the integrand. After some algebra, one can show the identity 1 1 1 1 1 1 + + + + + f1 (k1 , k2 ) f3 (k1 , k2 ) f4 (k1 , k2 ) f7 (k1 , k2 ) f9 (k1 , k2 ) f10 (k1 , k2 ) = T1 + T2 ,

(5.98)

where T1 = −

1 1 1 1 1 1 ( + ) ̵ ( − ), ̵ ̵ EρA + EσB EρA + hck1 EσB + hck1 hc k1 + k2 k1 − k2

(5.99)

T2 = −

1 1 1 1 1 1 ( ̵ 2 + EσB + hck ̵ 2 ) hc ̵ ( k1 + k2 + k1 − k2 ) , EρA + EσB EρA + hck

(5.100)

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where we have introduced the notation EρA = Eρ − E1S,A ,

EσB = Eσ − E1S,B .

(5.101)

The second term T2 is equal to the first term T1 , upon the interchange of the photon momenta k1 and k2 . Because one integrates over d3 k1 and d3 k2 , the two terms are equivalent, and one can replace 1 1 1 1 1 1 + + + + + → 2 T1 f1 (k1 , k2 ) f3 (k1 , k2 ) f4 (k1 , k2 ) f7 (k1 , k2 ) f9 (k1 , k2 ) f10 (k1 , k2 ) (5.102) for the purpose of the later k1 and k2 integrations. One can also show the identities 1 1 1 + + = G1 + G2 , f2 (k1 , k2 ) f5 (k1 , k2 ) f6 (k1 , k2 ) 1 1 1 1 1 ( + ), G1 = − ̵ ̵ ̵ EρA + hck2 EσB + hck2 hc k1 + k2 k1 − k2 1 1 1 1 . G2 = ̵ ̵ ̵ EρA + hck2 EσB + hck1 hc k1 − k2 Another combination is 1 1 1 + + = H1 + H2 , f8 (k1 , k2 ) f11 (k1 , k2 ) f12 (k1 , k2 ) 1 1 1 1 1 H1 = − ( + ), ̵ ̵ ̵ EρA + hck2 EσB + hck2 hc k1 + k2 k1 − k2 1 1 1 1 H2 = ̵ 1 EσB + hck ̵ 2 hc ̵ k1 − k2 . EρA + hck

(5.103a) (5.103b) (5.103c)

(5.104a) (5.104b) (5.104c)

If we exchange in H2 the variables k1 and k2 , then we end up with −G2 . Hence, we can ignore the sum of G2 and H2 for later integrations. In the sum G1 + H1 , we can exchange k1 and k2 and replace G1 + H1 → T3 ≡ −

1 1 1 2 1 − ). ( ̵ ̵ ̵ EρA + hck1 EσB + hck1 hc k1 + k2 k1 − k2

(5.105)

We can then replace the entire sum of relevant terms by 12

1 1 2 1 + → 2T1 + T3 = − ( ̵ ̵ 1) EρA + EσB EρA + hck1 EσB + hck i=1 fi (k1 , k2 ) 1 1 1 1 1 1 1 1 ×̵ ( ( − )− − ) ̵ ̵ ̵ hc k1 + k2 k1 − k2 EρA + hck1 EσB + hck1 hc k1 + k2 k1 − k2 ̵ 1) 4(EρA + EσB + hck 1 1 1 =− ( − ). (5.106) ̵ ̵ ̵ (EρA + EσB ) (EρA + hck1 ) (EσB + hck1 ) hc k1 + k2 k1 − k2 ∑

This is the most economic way of writing the integrand, in terms of a k1 -dependent (pre-)factor, and the term k23 /(k1 + k2 ) − k23 /(k1 − k2 ), which is generated upon multiplication by d3 k2 k2 . This combination generates the factor given in Eq. (5.97), which eventually allows us to symmetrize the integrand for the k2 -integration.

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eik2 r

e−ik2 r

Fig. 5.2 In the evaluation of the integral (5.108), one defines the variable x = k2 r and writes the trigonometric factors in the integrand in terms of the expressions exp(i k2 r) and exp(−i k2 r). One then closes the integration contour either above or below the real axis, such as to ensure the convergence of the integral. The principal value of the integral is obtained as the arithmetic mean of the integrals infinitesimally above, and below, the real axis, i.e., as the arithmetic mean of the dashed and solid contours. By inspection of the figure, one realizes that this principal-value calculation involves taking the half pole contribution at k1 + k2 = 0.

Another important intermediate result is the following,

∫

dΩk ij k i k j ik⋅ ⃗ ⃗ ˆiR ˆ j )(cos(kR) − sin(kR)) ˆiR ˆ j ) sin(kR) +(δ ij − 3R (δ − )e R =(δ ij − R 2 ⃗ 4π kR (kR)2 (kR)3 k ij ⃗ ≡ P (k, R). (5.107)

In the integration, one may treat the factor 1/k⃗2 as a constant factor, then perform the dΩk integration, and represent the k i operators by differentiations under the integral sign, k i → −i ∂/∂Ri . We also need the principal-value (P.V.) integral (P.V.) ∫

∞

−∞

dk2

k23 ⃗ = π k 3 {(δ ij − R ˆi R ˆ j ) cos(k1 R) P ij (k2 R) 1 k1 + k2 k1 R

⃗ . ˆi R ˆ j ) ( sin(k1 R) + cos(k1 R) )} ≡ π k13 Qij (k1 , R) − (δ ij − 3R (k1 R)2 (k1 R)3

(5.108)

In the evaluation of this integral, the principal value (P.V.) is to be understood as the arithmetic mean of results obtained by encircling the pole at k2 = −k1 either infinitesimally above or infinitesimally below the real axis (see Fig. 5.2). The derivation starts from Eq. (5.66), which we recall for convenience, ∆E (4) = ∫

̵ 1 hck ̵ 2 k2j k2` i (k⃗1 +k⃗2 )⋅R⃗ d3 k1 d3 k2 hck k1i k1k ik j` (δ − ) (δ − )e ∫ (2π)3 (2π)3 2 0 2 0 k12 k22 12

× ∑ ⟨1S, A∣xi1 ∣ρ⟩⟨ρ∣xj1 ∣1S, A⟩⟨1S, B∣xk2 ∣σ⟩⟨σ∣x`2 ∣1S, B⟩ ∑ fi−1 (k1 , k2 ) . ρ,σ

i=1

(5.109)

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After ordering prefactors, one can write this as ∞ ∞ ̵ hc 4e4 dk2 k23 dk1 k13 ∫ ∆E (4) = − ∫ 2 2 (4π0 ) 9π 0 0 × (∫

j dΩk1 ik k1i k1k i k⃗1 ⋅R⃗ dΩk2 j` k2 k2` ⃗ ⃗ (δ − 2 ) e ) (∫ (δ − 2 ) ei k2 ⋅R ) 4π k1 4π k2

m n n × δ ij δ k` ∑ ⟨1S, A∣xm 1 ∣ρ⟩⟨ρ∣x1 ∣1S, A⟩⟨1S, B∣x2 ∣σ⟩⟨σ∣x2 ∣1S, B⟩ ρ,σ

̵ 1 EρA + EσB + hck 1 1 ̵ 1 ) (EσB + hck ̵ 1 ) ( k1 + k2 − k1 − k2 ) . (5.110) (EρA + EσB ) (EρA + hck Using the intermediate results (5.107) and (5.108), we can express the fourth-order energy shift as ∞ ̵ hc 4e4 ⃗ Qj` (k1 , R) ⃗ δ ij δ k` dk1 k16 Pik (k1 , R) ∆E (4) = − ∫ 2 (4π0 ) 9π 0 m n n ̵ ⟨1S, A∣xm 1 ∣ρ⟩⟨ρ∣x1 ∣1S, A⟩⟨1S, B∣x2 ∣σ⟩⟨σ∣x2 ∣1S, B⟩(EρA + EσB + hck1 ) ×∑ . ̵ ̵ (EρA + EσB ) (EρA + hck1 ) (EσB + hck1 ) ρ,σ (5.111) Finally, ∞ ̵ 4e4 hc dk1 ∆E (4) = − ∫ 2 6 (4π0 ) 9π R 0 ×

× (2k1 R [(k1 R)2 − 3] cos(2k1 R) + [(k1 R)4 − 5(k1 R)2 + 3] sin(2k1 R)) m n n ̵ ⟨1S, A∣xm 1 ∣ρ⟩⟨ρ∣x1 ∣1S, A⟩⟨1S, B∣x2 ∣σ⟩⟨σ∣x2 ∣1S, B⟩(EρA + EσB + hck1 ) . ×∑ ̵ 1 ) (EσB + hck ̵ 1) (EρA + EσB ) (EρA + hck ρ,σ (5.112) One now writes the trigonometric factors in terms of exponentials, e2ik1 R + e−2ik1 R e2ik1 R − e−2ik1 R cos(2k1 R) = , sin(2k1 R) = , (5.113) 2 2i to obtain the result 2k1 R [(k1 R)2 − 3] cos(2k1 R) + [(k1 R)4 − 5(k1 R)2 + 3] sin(2k1 R) 1 = e2ik1 R (−3i − 6(k1 R) + 5i(k1 R)2 + 2(k1 R)3 − i(k1 R)4 ) + c.c., (5.114) 2 where c.c. denotes the complex conjugate. A Wick rotation changes the integration contour into ∞ ∞ dω iω dk F (k) → i ∫ F( ), (5.115) ∫ c c 0 0 where for F we take the first term on the right-hand side of in Eq. (5.114). Finally, after the addition of the complex conjugate, one obtains ∞ ̵ hc ω 4 e−2ωR/c 2c 5c2 6c3 3c4 ∆E (4) = − dω (1 + + + + ) ∫ (4π0 )2 0 π R2 ω R (ω R)2 (ω R)3 (ω R)4 ⎡ 2e2 ⎤ EρA ⎢ ⎥ m m ⎥ ×⎢ ⟨1S, A∣x ∣ρ⟩⟨ρ∣x ∣1S, A⟩ ∑ 1 1 2 ⎢ 3 ρ E + ω2 ⎥ ρA ⎣ ⎦ 2 EσB 2e ⟨1S, B∣xn2 ∣σ⟩⟨σ∣xn2 ∣1S, B⟩] . (5.116) ×[ ∑ 2 3 σ EσB + ω2

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Using Eq. (5.95), one may write the complete result, valid for any distance between the atoms, as follows, ∞ ̵ h dω α(A, iω) α(B, iω) ∆E (4) = − ∫ 4 2 π c (4π0 ) 0 ×

ω 4 e−2ωR/c 2c 5c2 6c3 3c4 (1 + + + + ). R2 ωR (ωR)2 (ωR)3 (ωR)4

(5.117)

We now proceed to the study of the two limiting cases already discussed in Secs. 5.4.3 and 5.4.4. For very large R, where the condition (5.69) is valid, the exponential e−2ωR/c in Eq. (5.117) leads to a rapid damping of the contribution of finite ω, which overcompensates the large value of the speed of light in the denominator. All terms in the retardation expansion for c → ∞ are thus relevant. To a very good approximation, we may therefore approximate α(A, iω) → α(A, ω = 0) ≡ α(A) and analogously for atom B. Then, ∫

∞ 0

dω

2c 5c2 6c3 3c4 23 c5 ω 4 e−2ωR/c (1 + + + + ) = . R2 ωR (ωR)2 (ωR)3 (ωR)4 4R7

Thus, we may verify that for large R, ̵ 23 hc ∆E (4) ≈ − α(A) α(B) , 4πR7 (4π0 )2

R≫

a0 , α

(5.118)

(5.119)

in exact agreement with Eq. (5.82). For intermediate distances, governed by the condition (5.84), one may ignore all but the last term in brackets in the integrand in Eq. (5.117), ω 4 e−2ωR/c 2c 5c2 6c3 3c4 3c4 (1 + + + + ) → , R2 ωR (ωR)2 (ωR)3 (ωR)4 R6

(5.120)

which is precisely equal to the limit of the entire expression as c → ∞ (the “nonretardation limit”). Notice that the approximation e−2ωR/c ≈ 1 uniformly holds over the entire range of ω values relevant to the integration, provided R is sufficiently small. In the short-range limit, one thus obtains ∞ ̵ 1 3h a0 dω α(A, iω) α(B, iω) , a0 ≪ R ≪ . (5.121) ∆E (4) ≈ − 6 ∫ 2 πR (4π0 ) α 0 We take careful note of the integration limits and recover Eq. (5.93) in the shortrange limit. The result (5.117) interpolates between the two limiting forms. The 1/R7 form is valid for large R. The long-range 1/R7 involves all the terms in the retardation expansion, i.e., the expansion in ωR/c. One often says that the 1/R6 van der Waals result is changed to the 1/R7 Casimir–Polder interaction “due to retardation”. This is slightly misleading, because it could otherwise suggest that a single “retardation correction term” is responsible for the change in the functional form, whereas from our discussion above, it is clear that the difference is due to the complete failure of the retardation expansion in the long-range limit, i.e., we cannot expand the expression exp(−2ωR/c) in terms of its argument for very large R even if c has a large numerical value in SI mksA units.

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Our consideration offers a smooth path toward an understanding of the concept of retardation. Namely, when the phase of the atomic oscillation during a virtual transition changes appreciably on the time scale it takes light to travel the interatomic separation distance R, then it is intuitively clear that some form of “retarded” form of the van der Waals interaction would have to be used. The criterion for the validity of the nonretardation approximation thus is ̵ h a0 R ≪ = , (5.122) c Eh α c or, more precisely, ̵ ̵ h h a0 a0 = ≪R≪ 2 = , (5.123) αmc α mc α if we take into account that substantial overlap of the electronic wavefunctions is to be avoided. This picture is consistent with our calculations. One important remark should follow our discussion: The entire treatment of the van der Waals and Casimir–Polder interaction energies has been limited to the case of ground-state interactions. For excited states, one decisive feature is the placement of the poles in the propagator denominators of the polarizability, in Eq. (5.67). A careful analysis, carried out very recently by a number of authors [208–211], reveals that in this case, the 1/R7 long-range asymptotics actually is changed, toward a 1/R2 term, though, with a numerically small coefficient. The functional form of the short-range 1/R6 term is not affected for excited states. Here, we were able to carry out the Wick rotation ω → iω in Eq. (5.115) without worrying about the infinitesimal displacement from the real axis, because both atoms are assumed to be in their respective ground states. This treatment remains approximately valid for the interaction of metastable hydrogen 2S atoms with ground-state atoms, because there are no lower states accessible from the 2S state via dipole transitions. However, one still needs to take the quasi-degenerate 2P states into account very carefully [212]. For highly excited states, the energetically lower states lead to the characteristic pole terms which give rise to a 1/R2 oscillatory tail [208–211]. 5.5 5.5.1

Atom-Surface Interactions Perfectly Conducting Wall, Long Range

We have already seen that the asymptotic behavior of the interaction potential for two polarizable atoms changes from 1/R6 for small distances to 1/R7 for large distances, due to retardation. This result has originally been derived in Ref. [213]. We now discuss a somewhat related effect, which is the interaction potential of an atom close to an idealized, perfectly conducting surface, or to a realistic, dielectric surface. Here, the surface takes over the role of the “other” atom. It is not charged, but the change in the boundary conditions in the vicinity of the surface provide for enough of a disturbance of the “vacuum” structure in the vicinity of the surface to induce a nontrivial interaction potential. The surface is canonically placed in the

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xy plane, and the atom is supposed to be displaced from the surface by a distance z. We shall find that the behavior of the interaction potential changes from 1/z 3 for small distances to 1/z 4 for larger distances, where “large” and “small” distances are understood as compared to a characteristic wavelength of radiation used for inducing a virtual transition of the atom. In general, the Casimir–Polder interactions are important for micro- and nanostructure devices, where the dimensions of the involved objects become comparable to the atomic transition wavelengths of typical virtual atomic transitions. The derivation of the Casimir–Polder interaction for arbitrary distances, as reported in Ref. [213], was very important for the understanding of QED interactions of atoms and surfaces. Nevertheless, an ideal conductor is a poor approximation for most realistic materials. For a material with an arbitrary (frequency-dependent) dielectric constant, the result has been derived in Ref. [214]. For a number of recent theoretical investigations on multipole corrections to the Casimir–Polder atom-surface interaction, we refer to the literature [215–223] and to complementary experiments [224–228].

5.5.2

Dipole Interaction

We assume a perfectly conducting wall in the xy plane (z = 0). In order to calculate the interaction energy, we assume that the surface located at z = 0 is made out of perfectly conducting material for all relevant incident light frequencies. The task then is to determine the allowed modes of the electromagnetic field, and calculate the second-order shift due to the virtual excitations of the modes, while subtracting the contribution of those modes that would otherwise contribute if there were no surface present (renormalization). We shall use a trick in doing the calculation: Namely, we shall assume that there is a perfectly conducting surface at z = 0, but we shall also assume perfectly conducting surfaces at z = Lz , and at x = 0, x = L, and y = 0, y = L. The atom is placed at x = 12 L, y = 12 L, and at a specific z coordinate, so that the artificially introduced boundary conditions at x = 0 and x = L, at y = 0 and y = L, and the one at z = Lz move away to infinity as we let L, Lz → ∞ at the end of the calculation. So, the “virtual” boundary conditions can have no effect on the final result for the interaction energy, but they help in writing down the mode functions that are essential in calculating the interaction energy in the first place. The details of the calculation will be described in the following. We work in Cartesian coordinates, with mode functions (see Sec. 7.3.2 of Ref. [81]) A⃗k,λ r, t) = Ak,λ,x (⃗ r, t) ˆex + Ak,λ,y (⃗ r, t) ˆey + Ak,λ,z (⃗ r, t) ˆez . ⃗ (⃗ ⃗ ⃗ ⃗

(5.124)

The vector potential of the fundamental modes appropriate to the interior of a rectangular parallelepiped of side lengths Lx = Ly = L and Lz , with all surfaces

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being perfectly conducting, reads as follows (Coulomb gauge), √ 8 ⃗ cos(kx x) sin(ky y) sin(kz z) e−i ω t , (5.125a) Ak,λ,x = A0 ⃗ V k,λ,x √ 8 Ak,λ,y = A0 ⃗ sin(kx x) cos(ky y) sin(kz z) e−i ω t , (5.125b) ⃗ V k,λ,y √ 8 Ak,λ,z ⃗ sin(kx x) sin(ky y) cos(kz z) e−i ω t , (5.125c) = A0 ⃗ V k,λ,z where V = L2 Lz , and A0 is a global amplitude. The energy of the electromagnetic mode is ω = ωk⃗ = c k. The polarization vector for the mode with wave vector k⃗ and polarization λ is ˆk,λ , k,λ,y , and k,λ,z (λ = 1, 2). ⃗ , with Cartesian components k,λ,x ⃗ ⃗ ⃗ The components of the wave vectors for the discrete modes are given as follows, mπ nπ `π kx = , ky = , kz = , (5.126) L L Lz where `, m , and n are integer. For the vector potential given in Eq. (5.125) not to vanish, two of these need to be nonzero. Furthermore, for Lz ≪ L, the fundamental mode has quantum numbers n = 0, and ` = m = 1, and therefore is z-polarized (only one polarization is available). For all modes other than the fundamental, one has two polarizations available. In the limit of a large quantization volume, the contribution of the fundamental mode to the sum over virtual excitations in Eq. (5.140) vanishes, and we can safely sum over both available polarizations (see also Sec. 7.3.2 of Ref. [81]). The transversality (Coulomb gauge) condition for the vector potential is verified as follows, √ 8 ⃗ ⃗ ⋅ A⃗k,λ (k ⋅ ⃗k,λ (5.127) r) = − A0 ∇ ⃗ ) sin(kx x) sin(ky y) sin(kz z) = 0 , ⃗ (⃗ V which is fulfilled if k⃗ ⋅ ⃗k,λ (5.128) ⃗ = 0, which we shall assume in the following discussion. Alternatively, we can write the vector k⃗ as follows, kx = k sin θ cos ϕ , ky = k sin θ sin ϕ , kz = k cos θ , (5.129a) k⃗ = kx ˆex + ky ˆey + kz ˆez . (5.129b) Here, kx , ky and kz must be matched against the quantum numbers `, m, and n, as ⃗ not given in Eq. (5.126). The angles θ and ϕ in Eq. (5.129a) of course belong to k, to the coordinate vector r⃗. The two polarization vectors (corresponding to λ = 1, 2) for TE (transverse electric) and TM (transverse magnetic) modes can be written as follows, ˆk,TE = sin ϕ ˆex − cos ϕ ˆey , (5.130a) ⃗ ˆk,TM = − cos θ cos ϕ ˆex − cos θ sin ϕ ˆey + sin θ ˆez , ⃗ k⃗ ⋅ ˆk,TE = k⃗ ⋅ ˆk,TM =0 ⃗ ⃗

⃗ k∣ ⃗. ˆk,TM × ˆk,TE = kˆ = k/∣ ⃗ ⃗

(5.130b) (5.130c)

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Summing over the polarizations λ = 1, 2 where 1 stands for TE and 2 for TM, one obtains kx2 + ky2 ky2 + kz2 kx2 + kz2 2 2 2 ) ) ) = (ˆ (ˆ = = ,  ,  . (5.131) k,λ,x ∑ ∑ ∑ (ˆ ⃗ ⃗ ⃗ k,λ,y k,λ,z k2 k2 k2 λ λ λ The mode functions in Eq. (5.125) are defined so that there is a cosine function if the Cartesian component of the wave vector in the argument of the trigonometric function is the same Cartesian component as the vector potential component itself. The presence of the sine functions implies that the tangential component of the vector potential (the component lying inside the boundary planes) vanishes for all three boundaries, Ak,λ,x (x, y, z = 0) = Ak,λ,x (x, y, z = Lz ) ⃗ ⃗ = Ak,λ,x (x, y = 0, z) = Ak,λ,x (x, y = L, z) = 0 , ⃗ ⃗

(5.132a)

Ak,λ,y (x, y, z = 0) = Ak,λ,y (x, y, z = Lz ) ⃗ ⃗ = Ak,λ,y (x = 0, y, z) = Ak,λ,y (x = L, y, z) = 0 , ⃗ ⃗

(5.132b)

Ak,λ,z (x = 0, y, z) = Ak,λ,z (x = L, y, z) ⃗ ⃗ = Ak,λ,z (x, y = 0, z) = Ak,λ,z (x, y = L, z) = 0 . ⃗ ⃗

(5.132c)

The electric field is given by differentiation of the mode functions with respect to the time, where we assume a factor exp(−iωt) for the time dependence [see Eq. (5.125)], ⃗ = i ω A. ⃗ So, E ⃗ is parallel to A. ⃗ As we can assume a perfectly conducting so that E plate, the tangential component of the electric field has to vanish on the boundaries. The transversal (normal) component of the electric field does not need to vanish at the boundaries, and indeed we have, e.g., for the planes with z = 0 and z = Lz , Ek,λ,z (x, y, z = 0) = (−1)n Ek,λ,z (x, y, z = Lz ) ⃗ ⃗ √ 8 ⃗ sin(kx x) sin(ky y) . = iω A0 ⃗ (k) V k,λ,z

(5.133)

An explicit calculation of the curl of the vector potentials given in Eq. (5.125) also shows that the normal component of the magnetic field (the component in the direction of the normal to the boundary surface) vanishes on the boundaries, as it should by virtue of Faraday’s law. Let us now assume that one of the indices `, m, n is zero, which implies that one Cartesian component of the wave vector k⃗ and two Cartesian components of the vector potential given in Eq. (5.125) vanish, because A⃗k,λ r, t) is proportional ⃗ (⃗ to ⃗k,λ . This implies that two Cartesian components of the polarization vector ⃗ ⃗k,λ also have to vanish, and indeed, ⃗k,λ has to point into the same direction as ⃗ ⃗ ⃗ As an example, if n = 0, then kz = 0, the vanishing component of the vector k. and in view of A⃗k,λ,x = A⃗k,λ,y = 0, the polarization vector ⃗k,λ has to point into ⃗ ⃗ ⃗ the z direction in order to have any nonvanishing vector potential at all. Then, √ 8 ⃗⃗ sin(kx x) sin(ky y) exp(−iωt), so that ˆk,λ = ˆez , and we have Ak,λ,z = ⃗ ⃗ V k,λ,z

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only one polarization, namely, z polarization. Also, we may consider, for kz = 0, the curl of the vector potential, i.e., the magnetic field. It has nonvanishing components in the x and y directions. Therefore, at the “lower” and “upper” boundary planes ⃗ = ˆez ⋅ B ⃗ = 0, where n (z = 0 and z = Lz ), we have n ˆ⋅B ˆ = ˆez is the normal to the boundary surface. This is necessary in order to show consistency with Faraday’s law. For the normalization, we can use the well-known result that ∫L dx cos2 (x) = L/2 which holds whenever cos2 (x) is integrated over a whole period. Then, it is easy to show that 2 2 3 2 ∫ d r cos (kx x) sin (ky y) sin (kz z) = V

V , 8

(5.134)

and the same result holds for the integrals with sine functions interchanged with cosines. So, 2 3 r)∣ = ∣A0 ∣2 . ⃗ (⃗ ∫ d r ∣A⃗k,λ V

(5.135)

Here and in the following, we identify A⃗k,λ r) ≡ A⃗k,λ r, t = 0). ⃗ (⃗ ⃗ (⃗ We must now analyze the behavior of the vector potential operator in the quantized formulation. In the presence of the boundary conditions, we can write a generalization of Eq. (2.129a), for the discretized representation of the vector potential operator in the vacuum, with normalization volume V , + ⃗ ⃗∗ ⃗ A⃗⃗ (⃗ ⃗ r) = ∑ [aλ (k) r)] . A(⃗ ⃗ (⃗ k,λ r ) + aλ (k) Ak,λ

(5.136)

⃗ kλ

In analogy with Eq. (2.93), we have ⃗ a+′ (k⃗′ )] = δ⃗ ⃗′ δλλ′ . [aλ (k), λ kk

(5.137)

In vacuum, the “mode functions” are given as follows [see Eq. (2.129a)], √ 1/2 ̵ ̵ ̵ 2 h h h ⃗r ik⋅⃗ 3 ⃗ ⃗ ∣ Ak,λ r) = ( r)∣ = ) ˆk,λ , ∫ d r Ak,λ , A0 = . ⃗ (⃗ ⃗ e ⃗ (⃗ 20 ω V 20 ω 2 0 ω V (5.138) The formula for A0 follows from a comparison to Eq. (5.135). We now consider an atom located at x = 12 L, y = 12 L, i.e., r⃗ = r⃗z =

L L ˆex + ˆey + z ˆez , 2 2

(5.139)

while the atom’s z coordinate denotes the distance from the perfectly conducting ⃗ ⃗ = −∂ A⃗⃗ /∂t = iω A⃗⃗ . According to plate located at z = 0. We use the result E kλ kλ kλ Eq. (8.4) of Ref. [28], one can use the following ansatz for the energy shift E`=1 (z) due to the dipole interaction with the vacuum modes (the subscript refers to the

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dipole interaction), 2 ̵ A⃗kλ rz ) 2 h 1 1 ⃗ (⃗ 2 ⃗ ∣ E1 (z) = − ∑ α(ω) Ekλ rz )∣ = − ∑ α(ω) ω ( )∣ ∣ ⃗ (⃗ 2 kλ 2 kλ 20 ω A0 ⃗ ⃗ ̵ ky L 4 hω kx L 2 ) cos2 ( =− ) [(ˆ k,λ,x ) sin2 ( ) sin2 (kz z) ∑ α(ω) ( ⃗ V kλ 2 2 2 0 ⃗

ky L kx L ) cos2 ( ) sin2 (kz z) 2 2 ky L kx L 2 ) sin2 ( + (ˆ k,λ,z ) sin2 ( ) cos2 (kz z)] . ⃗ 2 2 2

) sin2 ( + (ˆ k,λ,y ⃗

(5.140)

Here, the fundamental modes A⃗kλ ⃗ are given in Eq. (5.125). We write the dipole polarizability [see Eq. (3.53)] as α(ω), where we assume the reference state to be the atomic ground state. The justification of the ansatz given in Eq. (5.140) is not as easy as one might assume. Namely, we recall from Secs. 4.6.1 and 4.6.2 that the calculation of energy shifts involving a reference state with the atom in the ground state, and the photon field in the vacuum state, implies that the intermediate (virtual) state has to be a one-photon state. In turn, this implies that we have only one propagator denominator, as is evident from Eq. (4.302), not two, as we would otherwise need for the polarizability [see Eqs. (3.68)–(3.70)]. Indeed, the ansatz given in Eq. (5.140) can be justified based on a Wick rotation, as explained in Eqs. (7.116)–(7.120) of Ref. [28]. In the limit L → ∞, the energy shift has to be independent of the precise location of the atom in the x and the y directions, and we can thus average over x and y, which is equivalent to an average over L. Thus, we replace every square of a trigonometric function whose argument depends only on x and y, by 21 , and obtain the result E1 (z) = − ∑ ⃗ kλ

̵ hα(ω) ω 2 2 2 ) } sin2 (kz z) + (ˆ ) cos2 (kz z)] . ) + (ˆ [{(ˆ k,λ,x k,λ,y k,λ,z ⃗ ⃗ ⃗ 20 V (5.141)

We recall that ω = c k, so that it is impossible to pull the ω’s out of the sum. We can renormalize this expression by considering the limit z → ∞. For very large distance from the perfect conductor, the energy shift has to vanish. Also, in this limit, one can replace the squares of the trigonometric functions by their average values, sin2 (kz z) → 1/2 and cos2 (kz z) → 1/2, arguing that deviations from the average values could be damped away by a convergent factor exp(−η z) for infinitesimal η. We can thus implement the renormalization by the replacements sin2 (kz z) → sin2 (kz z) − 21 , cos2 (kz z) → cos2 (kz z) −

1 2

=

(5.142a) 1 2

1 − sin2 (kz z) = − (sin2 (kz z) − ) . 2

(5.142b)

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So, after renormalization and the transition to the continuum limit of k vectors, one obtains with the help of Eq. (5.131), ̵ h ⃗ 2 + (ˆ ⃗ 2 − (ˆ ⃗ 2] λ x (k)) E1 (z) = λ y (k)) λ z (k)) ∑ α(ω) ω cos (2kz z) [(ˆ 40 V kλ ⃗ = =

̵ ky2 + kz2 kx2 + kz2 kx2 + ky2 h + − ] ∑ α(ω) ω cos (2kz z) [ 40 V k⃗ k2 k2 k2

̵ ̵ h kz2 h V kz2 3 ∑ α(ω) ω cos (2kz z) 2 = ∫ d k α(ω) ω cos (2kz z) 2 . 3 20 V k⃗ k 20 V 8π k

(5.143) In the last step, we have used Eq. (2.81) one more time. If we now use the relation ω = c k and transform to spherical coordinates, then the z-dependent Casimir energy is obtained as π ∞ ̵ 1 h 2πc ∫ dθ sin θ cos2 θ ∫ dk α(c k) k 3 cos (2kz cos θ) E1 (z) = 3 20 8π 0 0 ∞ π ̵ hc dk α(c k) k 3 ∫ dθ sin θ cos2 θ cos (2kz cos θ) = 2 ∫ 8π 0 0 0 ∞ ̵ hc 2iωz i ω iω 2 = lim 2 ∫ dωα(ω) exp ( )( + − ) e−η ω/c + c.c. η→0 8π 0 0 c 4cz 3 2c2 z 2 2c3 z ∞ ̵ h ωz ωz 2 2ωz =− ) [1 + 2 + 2 ( ) ] dω α(iω) exp (− ∫ 16π 2 0 z 3 0 c c c ∞ ̵ h ⎧ ⎪ dω α(iω) (z → 0) − ⎪ ∫ ⎪ 2 3 ⎪ ⎪ 16π 0 z 0 . (5.144) =⎨ ̵ ⎪ 3hc α(ω = 0) ⎪ ⎪ ⎪ , (z → ∞) − ⎪ ⎩ 32π 2 0 z 4 In the last two lines in this equation, we assume that the reference state is the ground state, so that the Wick rotation can be performed without problems. Considerable additional complexity is encountered in the analysis of excited-state interactions [229, 230]. We have used a convergence generating factor exp(−η ω/c). In the calculation of the short- and long-distance limits in Eq. (5.144), the meaning of the retardation expansion is very obvious. Namely, for c → ∞, we can simply replace 2ωz ωz ωz 2 exp (− ) [1 + 2 + 2( ) ] → 1, c → ∞, (5.145) c c c which is the appropriate expansion for the short-distance limit. In the nonretardation limit, the speed of light can be approximated as being infinite, and one can let 1/c → 0. By contrast, for z → ∞, the quantity ω z/c is not small, and thus all the terms in the retardation expansion [i.e., the expansion of the factor (5.145)] in terms of its argument ω z/c are relevant. In that limit, the magnitude of the quantity c/z relevant to the integration region over ω in Eq. (5.144) is so tiny that it can be neglected when compared to an atomic excitation frequency, and thus, we can approximate α(ω) → α(ω = 0).

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5.5.3

Multipole Interactions

It is very instructive to generalize the result (5.144) to higher multipole orders. The higher multipoles correspond to corrections to the excitation dynamics due to the small changes in the phases of the virtual electromagnetic field quanta on length scales commensurate with the Bohr radius (radius of the atom). The 2` -pole dynamic polarizability α` is the generalization of the dipole (` = 1) expression in Eq. (3.53) to higher multipole orders; these are called dipole (2`=1 = 2), quadrupole (2`=2 = 4), octupole (2`=3 = 8), and hexadecupole (2`=4 = 16) corrections. In order to approach the problem of a consistent definition of the multipole oscillator strength, let us make contact with the literature and consult Ref. [231], which contains a number of useful formulas for quantities of universal interest in atomic physics. We start from Eq. (27) of Ref. [231], where the Einstein A coefficient is defined for spontaneous emission from an upper level, denoted as level 2, to a lower level 1, 3 e2 ω21 r∣2 m2 ⟩∣2 . (5.146) A21 = ∑ ∣⟨1 m1 ∣⃗ ̵ 3 m 3π0 hc 1 A comparison with Eq. (3.37), which was written in natural units, reveals that A21 = Wf i = γ21 ,

(5.147)

where γ21 = 1/τ21 is the contribution to the decay rate of the upper level 2 due to electric-dipole (E1) decay, to the lower level 1, due to spontaneous emission of a resonant photon. By τ21 , we denote the lifetime of level 2 against spontaneous electric-dipole decay to level 1. If there is more than one decay channel for the spontaneous transition 2 → 1, then the corresponding contributions to the decay rate have to be added to the electric-dipole decay rate γ21 = A21 . Additional decay channels are common. For example, in the case of the nS → 1S decay in hydrogenlike ions, the two most important contributions to the total decay rate are the sum of two-photon decay (2E1, see Refs. [232,233]) and one-photon magnetic-dipole (M1) decay. Two-photon decay nS → 1S decay proceeds via a virtual intermediate n′ P state, where n′ can be smaller or larger than n. For nS → 1S decay, with n ≥ 3, it is an interesting problem to separate the coherent contribution due to two-photon decay from the cascade process nS → n′ P → 1S, where, for the cascade process, one has n > n′ > 1 (see Refs. [183, 196, 197, 234]). For P → S transition, the (dominant) electric-dipole decay rate receives corrections due to more complex processes like two-photon E1M1 and E1M2 decay (see Refs. [235, 236]). For clarity and ease of understanding, we remark that in Eq. (27) of Ref. [231], ̵ is used. The (dipole) oscillator strength, the trivial relationship 2/(3h) = 1/(3π h) according to Eq. (22) of Ref. [231], is defined as 1 −1 f21 = − A21 γcl , (5.148) 3 where the classical decay rate for a single-electron oscillator at frequency ω21 is 2 e2 ω21 γcl = . (5.149) 6π0 mc3

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Here, ω21 has the interpretation of the (angular) transition frequency, in atomic units. Note that the oscillator strength defined according to Eq. (5.148) is a manifestly dimensionless quantity, obtained as the ratio of the spontaneous (atomic) dipole decay rate to the classical decay rate. According to Eq. (23) of Ref. [231], the so-called gf value for a transition is given as g1 f12 = −g2 f21 ≡ gf .

(5.150)

Furthermore, according to Eq. (24) of Ref. [231], the oscillator strength f12 (as opposed to f21 ) for the transition 1 → 2 (lower to upper level) is given as f12 = −

g2 g2 2π0 mc3 2 g2 m r∣2 m2 ⟩∣2 . f21 = A21 2 2 = ∑ ∣⟨1 m1 ∣⃗ ̵ ω21 m g1 g1 e ω21 3 g1 h 1

(5.151)

Here, g1 and g2 are the multiplicities of levels 1 and 2 (due to magnetic sublevels). The sum over m1 ensures that f12 is independent of m2 . In Eq. (5.151), the sum over m1 and the division by g1 correspond to an averaging over the magnetic projections of level 1. If ∣1m1 ⟩ is the 1S ground state of a hydrogenlike system, then one can alternatively write f12 =

2m r∣2 m2 ⟩∣2 , ∑ ∣⟨1S∣⃗ ̵ ω21 m 3 h 2

(5.152)

where the factor 1/g1 cancels the sum over m1 , and the factor g2 is alternatively written as a sum over m2 . The above considerations will now be applied to the dynamic polarizability of an atom, which we assume has a spherically symmetric ground state. We recall the definition of the ground-state dynamic polarizability α(ω) from Eq. (3.53), α(ω) =

1 e2 r ∑ ⟨1S ∣⃗ ̵ r⃗∣ 1S⟩ , 3 ± HS − E1S ± hω

(5.153)

where HS is the Schr¨ odinger–Coulomb Hamiltonian. (For a many-electron atom, one needs to sum over the coordinates of all electrons, as described by the substitution r⃗ → ∑i r⃗i .) Inserting a complete set of virtual states ∣k⟩, one easily shows that α(ω) = ∑

2 k Ek1

F1k ̵ 2, − (hω)

(5.154)

where the sum over k comprises the continuous spectrum, and F1k is a dimension̵ k1 is the ful oscillator strength defined in Eq. (5.155). Here, Ek1 = Ek − E1 = hω transition energy for the transition ∣1⟩ → ∣k⟩, where ∣1⟩ is the reference 1S ground state and ∣k⟩ ≡ ∣k mk ⟩ is the virtual state which is connected to the ground state by an allowed electric-dipole transition and has magnetic projection mk . We should add that we here do not consider the infinitesimal displacement of the poles (as a function of ω) in Eq. (5.154). For a comprehensive discussion of this question, see

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Sec. 12.2.2. Furthermore, F1k is the dimensionful (dipole) oscillator strength for the transition, F1k =

2 2 2 e Ek1 ∑ ∣⟨1S ∣⃗ r∣ k mk ⟩∣ = e2 a20 Eh f1k , 3 mk

(5.155)

where f1k is the dimensionless oscillator strength for the transition ∣1⟩ → ∣k⟩, defined according to Eq. (5.152). The prefactors e2 a20 Eh imply that the dimensionful oscillator strength is equal to its value expressed in atomic units (we recall that Eh is the Hartree energy). The question now is how to generalize the concept of the oscillator strength to multipole transition. To this end, we first write the dimensionful dipole oscillator strength as 2 2 2 e Ekn ∑ ∣⟨1S∣⃗ r∣ k mk ⟩∣ 3 mk √ RRR RRR2 4π R 2 1 R = 2 e Ekn ∑ ∑ RRR⟨1S∣ r Y1m (ˆ r)∣ k mk ⟩RRRRR . (5.156) 3 3 RRR m mk RRR For the argument of the spherical harmonics, we have used a more compact notation, denoting the arguments of the spherical harmonic as F1k =

Y`m (ˆ r) ≡ Y`m (θ, ϕ) ,

(5.157)

observing that the polar and azimuth angles θ and ϕ define rˆ uniquely, and vice versa. The 2` -pole polarizability can alternatively be written as α` (ω) =

1 1 1 ∗ + ∑ ⟨1S ∣Q`m ( ̵ ̵ ) Q`m ∣ 1S⟩ , 2` + 1 m H − E1S − hω H − E1S + hω

where the Q`m tensor is given as

√

Q`m = e

4π r Y`m (ˆ ri ) . 2` + 1

(5.158)

(5.159)

The physical dimension of the 2` -pole polarizability is charge squared, times energy multiplied by length to the power of 2`. Therefore, the multipole corrections to the dipole-induced energy shift E1 (z) given in Eq. (5.144) have more inverse powers of z in order to compensate for the larger number of atomic coordinates in the multipole polarizabilities. The generalization of E1 (z) given in Eq. (5.140), which we recall 2 ⃗ ⃗ (⃗ for convenience as E1 (z) = − 21 ∑kλ rz )∣ , to the quadrupole and octupole ⃗ α(ω) ∣Ekλ interactions, is as follows. For the quadrupole term (` = 2), one finds E`=2 = −

1 2 j i r)}∣`=2 ∣ , ∑ ∑ α2 (ω) ∣ {∇ Ekλ ⃗ (⃗ 12 kλ ⃗ ij

(5.160)

where we reserve the index i for the component of the electric field. Here, T ij ∣`=2 =

1 1 (T ij + T ji ) − δ ij Tr(T ) 2 3

(5.161)

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is the traceless quadrupole component of a general tensor T ij (see Sec. 6.2.1). The octupole energy shift (` = 3) is E3 = −

1 2 j k i r)}∣`=3 ∣ . ∑ ∑ α3 (ω) ∣ {∇ ∇ Ekλ ⃗ (⃗ 180 kλ ⃗ ijk

(5.162)

The ` = 3 component can be extracted from a rank-3 tensor T ijk as follows [237]. One first decomposes T ijk into a totally symmetric part S ijk , a totally skewsymmetric part Aijk , and a remainder term N ijk , 1 ijk (T + T jki + T kij + T jik + T kji + T ikj ) , 6 1 = T [ijk] = (T ijk + T jki + T kij − T jik − T kji − T ikj ) , 6 = T [ijk] − S ijk − Aijk .

S ijk = T (ijk) =

(5.163a)

Aijk

(5.163b)

N ijk

(5.163c)

The matrix S ijk contains both ` = 1 as well as ` = 3 components. One then forms a vector αk which one promotes to a matrix K ijk , αk = δ ij S ijk = K ijk =

1 ij ijk (δ T + δ ij T ikj + δ ij T kij ) , 9

1 (αi δ jk + αj δ ik + αk δ ij ) . 5

(5.164)

Finally, the ` = 3 components is extracted as follows, T ijk ∣`=3 = S ijk − K ijk .

(5.165)

From Eqs. (5.160) and (5.162), one derives the following results for the energy shifts in the case of a perfectly conducting surface. For ` = 2 (quadrupole case), the energy shift is found as follows, ∞ ̵ h 2ωz dω α2 (i ω) exp (− ) ∫ 16π 2 0 z 5 0 c 3 3 ωz 4 ωz 2 2 ωz 3 1 ωz 4 ×[ + + ( ) + ( ) + ( ) ]. 4 2 c 3 c 3 c 6 c

E2 (z) = −

(5.166)

For the octupole term, we have E3 (z) = − +

∞ ̵ 69 ωz 2863 ωz 2 h 69 −2ωz/c + + ( ) dω α (i ω)e [ 3 ∫ 16π 2 0 z 7 0 320 320 c 7200 c

793 ωz 3 71 ωz 4 47 ωz 5 1 ωz 6 ( ) + ( ) + ( ) + ( ) ]. 3600 c 900 c 2700 c 675 c

(5.167)

From the above formulas, it becomes clear that, for large distances, the multipole polarizabilities are suppressed by higher powers of z.

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Interactions with a Dielectric Surface

It is very instructive and useful to generalize the above results, which were obtained for atom-surface interactions with a perfect conductor, to a nontrivial frequencydependent dielectric function (ω). In this case, the reflection and transmission at the boundary (plane z = 0) of the upper half space (z > 0, outside of the dielectric) and the lower half space z < 0 (inside the dielectric) need to be considered. In principle, one needs to enlarge the reference volume V = L2 Lz used in the previous considerations to include a region filled by the dielectric, and to calculate the field modes in the enlarged reference volume. In practice, it is possible to simplify the calculation by considering the field modes only in the region above the surface, z > 0, as a superposition of incident and reflected waves. If the incident wave has a wave vector k⃗ = (kx , ky , kz ), then the wave vector of the reflected wave reads ⃗ = ∣k⃗R ∣. as k⃗R = (kx , ky , −kz ). There is no change in the wavelength; we have ∣k∣ However, the amplitude of the reflected wave is less than that of the incident wave, due to a nonvanishing transmission coefficient. One then has to consider the socalled TE and TM modes separately, where TE in this case stands for incident waves whose electric field is transverse to the plane of incidence, whereas TM stands for waves whose electric field is in the plane of incidence. The calculation is described in some detail in Ref. [238]. For given wave vector ⃗ one has only one polarization vector characterizing the TE mode, and one for k, the TM mode. Alternatively, one can average over the polarizations, which corresponds to an average over a conical region of incident waves. The weight factor in √ formulating the superposition of incident and reflected waves is 1/ 2, which preserves the normalization of the modes given in Eq. (5.135) in the limit of a perfect conductor. With these ideas in mind, it is relatively easy to rederive the following result, originally derived in Ref. [214] for a dipole polarizable particle in contact with a dielectric surface, ∞ ∞ ̵ h 2ξ ωz E1 (z) = − 2 3 ∫ dω ω 3 α1 (iω) ∫ dξ exp (− ) H(ξ, (iω)) , 8π 0 c 0 c 1 (5.168a) √ √ ξ2 +  − 1 −  ξ ξ2 +  − 1 − ξ H(ξ, ) = (1 − 2 ξ 2 ) √ +√ . (5.168b) ξ2 +  − 1 +  ξ ξ2 +  − 1 + ξ It is useful to investigate the function ∞

2ξ ωz ) H(ξ, ) . c

(5.169)

1 c 3 zω zω 2 ( ) [1 + 2 +2 ( ) ] , 2 zω c c

 → ∞ , (5.170)

K(, z) = ∫

1

dξ exp (−

For  → ∞, we have H(ξ, ) ∼ 2 ξ 2 ,

K(, z) ∼

which shows that the formula (5.168a) is consistent with the result given in Eq. (5.144) for the dipole interaction with a perfect conductor. For z → 0, we

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have, on the other hand, K(, z) =

1 c 3 −1 1 c ( − 1) (3  + 1) ( ) − ( ) + O(z 0 ) , 2 zω +1 2 zω ( + 1)2

Inserting (5.171) into (5.168a), we have ∞ ̵ (iω) − 1 h dω α1 (iω) , E1 (z) = − ∫ 2 3 16π 0 z 0 (iω) + 1

z → 0,

z → 0 . (5.171)

 → ∞.

(5.172)

The generalization to the quadrupole polarizability reads as follows, ̵ h

∞

∞

ξ2 1 − ) H(ξ, (iω)) . 2 3 1 0 0 (5.173) In the perfect conductor limit ( → ∞), this result is in agreement with the previously derived result given in Eq. (5.166). The octupole energy shift reads ∞ ∞ ̵ h E3 (z) = − dω ω 7 α3 (iω) ∫ dξ e−2ξωz/c ∫ 2 7 16π 0 c 0 1 23 4 121 2 17 ×( ξ − ξ + ) H(ξ, (iω)) . (5.174) 1200 5400 3600 E2 (z) = −

16π 2 

c5

∫

dω ω 5 α2 (iω) ∫

dξ e−2ξωz/c (

In the perfect conductor limit ( → ∞), the result given in Eq. (5.167) is recovered. For completeness, we should mention that the calculation of the dielectric response function for imaginary input frequencies is a nontrivial problem. For a typical solid, it involves the application of the Kramers3 –Kronig4 relation to tabulated data for the frequency-dependent dielectric function of a particular material. A comprehensive compilation of pertinent data can be found in Ref. [239]. The reader should be aware, though, that some of the data compiled in Ref. [239] needs to be seen as outdated from today’s point of view. 5.6

Further Thoughts

Here are some suggestions for further thought. (1) Ion–Ion and Ion–Atom Interactions. In Sec. 5.4.2, we had observed that, if the two nuclei of the hydrogenlike systems have charge numbers other than unity, then the two-photon exchange energy is dominated by either the electrostatic interaction among the ions, or the ion–atom interaction. Calculate the leading functional form of both interactions. (2) Expressions for the Propagator Denominators. Derive the propagator denominators listed in Eq. (5.67), one by one. (3) Interpolating Formula. Derive Eq. (5.117) for the interpolating formula, in between the Van-der-Waals and Casimir–Polder regimes, on the basis of collecting all terms listed in Eq. (5.67). 3 Hendrik 4 Ralph

Anthony Kramers (1894–1952). de Laer Kronig (1904–1995).

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(4) Symmetrized Integrand. The asymmetric (in k1 and k2 ) form of the integrand given in Eq. (5.106) allows us to carry out the k2 -integration as a principal-value integral [see Eqs. (5.97), (5.106) and (5.108)]. Convince yourself, conceivably with the help of a computer algebra system, that the sym−1 metrized form of the integrand ∑12 is equal to the symmetrized i=1 [fi (k1 , k2 )] form of the integrand given in Eq. (5.106), and try to complete the calculation of the long-range energy shift, with the aim of verifying Eq. (5.117). (5) Long-Range Tail for Excited States. Derive the oscillating tail of the long-range interaction for excited reference states, by matching the energy shift with the corresponding S-matrix element (see Sec. 12.2.2). Contrast with the treatment in Sec. 5.4.5. The matching of the S-matrix element with the energy shift must be used for excited states, in order to correctly assign the position of poles in the propagator denominators, in the complex plane. Consult Refs. [208, 209, 240] for additional inspiration. (6) Atom-Surface Interactions. Calculate the curl of the vector potentials given in Eq. (5.125) and show that the normal component of the magnetic field (the component in the direction of the normal to the boundary surface) vanishes on the boundaries. Furthermore, show that this is necessary, in order to fulfill Faraday’s law. ⃗ r) and the (7) Atom-Surface Interactions. Generalize the vector potentials A(⃗ vector field operator given in Eq. (5.136) given in Eq. (5.125) to an interaction⃗ r, t). Write the electric-field and magnetic-field picture vector-field operator A(⃗ ⃗ ⃗ operators E(⃗ r, t) and B(⃗ r, t) and show that these are Hermitian. Give a physical argument why these operators should be Hermitian even under the presence of boundary conditions corresponding to a box. (8) Two-Photon Decays. Study the separation of the coherent two-photon decay rate correction from cascades (see Refs. [183, 196, 197, 234]). Generalize the treatment to forbidden transitions of current interest for high-precision spectroscopy. (9) Oscillator Strengths. Calculate the dimensionful oscillator strength, defined according to Eq. (5.155), in the nonrelativistic approximation for a hydrogen atom in the limit of infinite nuclear mass, for a 1S → nP transition. Hint: You might obtain a result similar to F1n =

n − 1 2n 2 2 256 n5 ( ) e a0 Eh . 3 (n2 − 1)4 n + 1

(5.175)

Compare your result to Ref. [241]. (10) Incident and Reflected Waves. Try to derive Eq. (5.168a), based on the incident and reflected waves off of a dielectric surface. Hint: Consult Ref. [238]. (11) Excited States. How would the formalism outlined in Sec. 5.5 have to be altered for excited states? Hint: Consult Refs. [229, 230]. Would you still be able to use the ansatz given in Eq. (5.140), for excited states?

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Chapter 6

Racah–Wigner Algebra

6.1

Overview

The most important global symmetry of an atom is the rotational symmetry of the Coulomb potential, which is generated by the nucleus. Therefore, atomic physics is intimately connected with the group of rotations in three dimensions, which is the special orthogonal group SO(3). This symmetry can be augmented, in the case of the hydrogen atom, to SO(4), as described in detail in Chap. 4. The theory of the rotation group has been perfectioned over the years by generations of physicists, starting from the work of Racah1 and Wigner;2 it is our goal here to present an intuitively accessible, concise chapter on angular integration methods in atomic physics which involve the Clebsch–Gordan coefficients. Emphasis is laid on an intuitive understanding, so that more specialized literature can be digested by the interested reader with ease. We refer to the treatises [14, 15, 242, 243] for general understanding, as well as to the monumental handbook [16] for use in research, and we also mention Ref. [23], which is very useful in the analysis of more complex systems. All of these books are self-contained and voluminous in nature. However, a student interested in gaining a working knowledge of the field, or a working scientist interested in refreshing one’s knowledge on related topics, sometimes does not have the time to work the way through an entire book in order to familiarize oneself, or refamiliarize oneself with the topic. In general, the Racah–Wigner algebra associated with a group (in this case, the group of rotations) is generally understood as the set of algebraic manipulations concerning the coupling and recoupling coefficients that pertain to the irreducible representations of the group. 6.2

Clebsch–Gordan Coefficients

6.2.1

Expansions and Clebsch–Gordan Coefficients

The motivation for the definition of Clebsch–Gordan coefficients is twofold: first, to find the expansion coefficients of a tensor of higher rank as it is composed out of 1 Giulio

(Yoel) Racah (1909–1965). Paul Wigner (1902–1995).

2 Eugene

191

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elements of tensors of lower rank, and second, to formalize a certain proportionality phenomenon that affects matrix elements of tensors of specific rank, all of which can be written in terms of a so-called reduced matrix element and an expansion factor, which is a Clebsch–Gordan coefficient. We start with the first motivation. As outlined in Chaps. 5 and 8 of Ref. [81], the matrix-valued elements of the Lie algebra of the rotation group can be chosen as (Sk )ij = −iijk ,

123 = 1 ,

(6.1)

ijk

where  is the totally antisymmetric Levi-Civit`a tensor. Here, k denotes the Cartesian component of the vector of S matrices. The rotation matrices fulfill the same algebraic relations as the components of the angular momentum operator, [ Si , Sj ] = iijk Sk ,

[Li , Lj ] = i ijk Lk ,

(6.2) ⃗ where the angular momentum operator L has been defined in Eq. (4.2) and the Einstein3 summation convention is used. The angular momentum operator can be written as ∂ ⃗ = −i r⃗ × ∇ ⃗, . (6.3) L Li = −iijk xj ∂xk ̵ = c = 0 = 1, which simplifies the notation conHere, we use natural units with h siderably. For all calculations reported here, SI mksA units can be restored easily. The three-dimensional rotation matrix R is obtained as the matrix exponential 3

R = exp (i ∑ ϕk Sk ) = exp (i ϕ⃗ ⋅ S⃗ ) ,

r⃗′ = R ⋅ r⃗ .

(6.4)

k=1

Here, ϕ⃗ is a vector of rotation angles about the three axes. For small ϕ⃗ = n ˆ δϕ, one can expand ⃗ )] ≈ [1 + iϕ⃗ ⋅ S ⃗ ] ≈ δ ij + ϕk kij . [exp (iϕ⃗ ⋅ S ij

ij

(6.5)

The components of a vector v⃗ thus transform as follows, ⃗ )] [exp (iϕ⃗ ⋅ S

ij

i

v j ≈ (δ ij + ϕk kij ) v j = v i − (ϕ⃗ × v⃗) .

(6.6)

⃗ in the case This is just the representation of an infinitesimal rotation by an angle ϕ, of a passive interpretation of the rotation. That means that the coordinate system ⃗ and the coordinate vector r⃗ → r⃗′ needs to turn by an angle −ϕ⃗ turns by an angle ϕ, with respect to the coordinate system, with the net result that r⃗′ and r⃗ represent the same vector in real space. Under a passive rotation, a scalar function transforms as f ′ (⃗ r′ ) = f (⃗ r)

(6.7)

and therefore f ′ (⃗ r) ≈ f (R−1 ⋅ r⃗) = f (xi − ϕk kij xj ) ≈ f (⃗ r) − ϕk kij xj = f (⃗ r) + i ϕk (−i kij xi 3 Albert

Einstein (1879–1955).

∂ f (⃗ r) ∂xi

∂ ⃗ f (⃗ ) f (⃗ r) = f (⃗ r) + i ϕ⃗ ⋅ L r) . ∂xj

(6.8)

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For finite rotations, we may thus summarize the transformation law of a vector r⃗ and of a scalar function as ⃗ ⋅ r⃗ , r⃗′ = exp(iϕ⃗ ⋅ S)

⃗ f (⃗ f ′ (⃗ r) = exp(i ϕ⃗ ⋅ L) r) .

(6.9)

We see that the angular momentum operator generates rotations. To fix ideas, let us study the case ϕ = 90○ = π/2, with ⎛ 0 1 0⎞

Rπ/2 = ⎜ −1 0 0 ⎟ ,

⎛ ux ⎞

⎛ uy ⎞

⎝ uz ⎠

⎝ uz ⎠

Rπ/2 ⋅ ⎜ uy ⎟ = ⎜ −ux ⎟ .

⎝ 0 0 1⎠

(6.10)

⃗′ = Rπ/2 ⋅ u ⃗, where u ⃗ has the Cartesian components as The rotated vector is u indicated. It is easy to show that the modulus of a vector or the scalar product of ⃗ and v⃗ is invariant under rotations, two vectors u ⃗) ⋅ (Rπ/2 ⋅ v⃗) = ux vx + uy vy + uz vz . ⃗′ ⋅ v⃗′ = (Rπ/2 ⋅ u u

(6.11)

We shall now attempt to convince ourselves that scalar functions, which depend only on the radial variable, vectors, and the components of the quadrupole tensor, transform separately under the rotation group, i.e., they do not mix. We consider a second-rank tensor ⎛ ux vx ux vy ux vz ⎞

T = u⃗ ⊗ v⃗ = ⎜ uy vx uy vy uy vz ⎟ ,

(6.12)

⎝ uz vx uz vy uz vz ⎠

which can be decomposed as follows,

T = T∣`=0 + T∣`=1 + T∣`=2 , T∣`=0 =

Tr(T) 13×3 , 3

(6.13)

Tr(T) = ux vx + uy vy + uz vz ,

ux vy − uy vx uz vx − ux vz ⎛ − 0 2 2 ⎜ ⎜ ⎜ u v −u v uy vz − uz vy 1 0 T∣`=1 = (T − TT ) = ⎜⎜ − x y y x 2 2 2 ⎜ ⎜ ⎜ uz vx − ux vz uy vz − uz vy − 0 ⎝ 2 2

(6.14) ⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠

(6.15)

A comparison of Eq. (6.14) with Eq. (6.11) shows that the (` = 0)-component is invariant under rotations. The entries of the matrix T∣`=1 can be identified as the component of the vec⃗ and v⃗, which will be discussed in the following. Finally, the tor product of u

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(` = 2)-component reads as trc(T) 1 13×3 (6.16) T∣`=2 = (T + TT ) − 2 3 ux vy + uy vx 2u v − uy vy − uz vz ux vz + uz vx ⎛ x x ⎞ 3 2 2 ⎜ ⎟ ⎜ ⎟ 2uy vy − ux vx − uz vz uy vz + uz vy ux vy + uy vx ⎜ ⎟ ⎟. =⎜ ⎜ ⎟ 2 3 2 ⎜ ⎟ ⎜ uy vz + uz vy 2uz vz − ux vx − uy vy ⎟ ux vz + uz vx ⎝ ⎠ 2 2 3 As already anticipated, we can order the components of T∣`=1 into a vector, which ⃗ and v⃗, is known as the vector product of u ⎛ uy vz − uz vy ⎞ i ⃗ × v⃗ = ⎜ uz vx − ux vz ⎟ , u (⃗ u × v⃗) = ijk uj v k . (6.17) ⎝ ux vy − uy vx ⎠ ⃗ ×⃗ On the example of our rotation by 90○ about the z axis, the fact that u v transforms as a (pseudo-)vector can be verified as follows, ⎛ uz vx − ux vz ⎞ ′ ′ ⃗ × v⃗ = (Rπ/2 ⋅ u ⃗) × (Rπ/2 ⋅ v⃗) = ⎜ −(uy vz − uz vy ) ⎟ = Rπ/2 ⋅ (⃗ u u × v⃗) . (6.18) ⎝ ux vy − uy vx ⎠ With the help of Eq. (6.5), we can generalize this statement to arbitrary rotations, ⃗′ × v⃗′ = (⃗ ⃗) × (⃗ u u + ϕ⃗ × u v + ϕ⃗ × v⃗) ⃗ × v⃗ + (ϕ⃗ × u ⃗) × v⃗ + u ⃗ × (ϕ⃗ × v⃗) = u ⃗ × v⃗ + ϕ⃗ × (⃗ ≈u u × v⃗) , (6.19) where we expand to first order in ϕ. It is an instructive exercise to show the identity ⃗) × v⃗ + u ⃗ × (ϕ⃗ × v⃗) = ϕ⃗ × (⃗ (ϕ⃗ × u u × v⃗), which can be done component-wise, using the  tensor. Indeed, T∣`=1 transforms as a (pseudo-)vector; it is possible to construct a (pseudo-)vector from products of vector components. In Cartesian components, the “coupling coefficients” can directly be read off from Eq. (6.17). However, the canonical formalism employs the spherical basis. The z component of the angular momentum operator has a particularly simple form in spherical coordinates, ∂ ∂ ∂ −y ) = −i . (6.20) Lz = −i (x ∂y ∂x ∂ϕ The spherical basis of vector components is chosen to generate an explicit ϕdependence of the form exp(imϕ), i.e., it consists of eigenfunctions of Lz . The components are denoted as x+1 , x0 , and x−1 in the spherical basis and read as √ 1 1 4π iφ r∣ sin θ e = ∣⃗ r∣ Y11 (θ, ϕ) , (6.21a) x+1 = − √ (x + i y) = − √ ∣⃗ 3 2 2 √ 4π x0 = z = ∣⃗ r∣ cos θ = ∣⃗ r∣ Y10 (θ, ϕ) , (6.21b) 3 √ 1 1 4π x−1 = √ (x − i y) = √ ∣⃗ r∣ sin θ e−iφ = ∣⃗ r∣ Y1−1 (θ, ϕ) , (6.21c) 3 2 2

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√ where we have the luxury to write r = ∣⃗ r∣ = x2 + y 2 + z 2 explicitly. The emergence of phase factors of the form exp(i m ϕ) is evident. The spherical components are complemented by spherical basis vectors as follows, 1 e⃗+1 = − √ (ˆex + i ˆey ) , 2

1 e⃗−1 = √ (ˆex − i ˆey ) . 2

e⃗0 = ˆez ,

(6.22)

The coordinate vector can easily be expanded into the spherical basis, r⃗ = ∑(−1)q x−q e⃗q , q

e⃗q ⋅ e⃗q′ = (−1)q δq −q′ ,

(6.23)

but the spherical basis vectors are not normalized to unity. Components can be extracted by calculating the scalar product of the coordinate vector r⃗ with a spherical basis vector, r⃗ ⋅ e⃗q = ∑(−1)q x−q′ e⃗q′ ⋅ e⃗q = ∑(−1)2q x−q′ δq −q′ = xq , ′

′

(6.24)

q′

q′

where the sum over q runs over the indices q = −1, 0, 1. The paradigm of the vector addition, or Clebsch–Gordan coefficients, is that one obtains a tensor component of magnetic quantum number m for a tensor of rank j by coupling two tensors of ranks j1 and j2 as follows, u(j1 m1 ) v(j2 m2 ) . w(jm) = ∑ Cjjm 1 m1 j2 m2

(6.25)

m1 m2

Here, u(j1 m1 ) and v(j2 m2 ) are two distinct tensors. In our example, we couple ⃗ and v⃗ with spherical components u+1 , u0 two tensors of rank one (the vectors u and u−1 , as well as v+1 , v0 and v−1 ), to a tensor of rank one, which is the vector ⃗=u ⃗ × v⃗. So, we have j1 = j2 = j = 1 for our example. The Clebsch–Gordan product w coefficients are tabulated and nowadays implemented in most modern computer algebra systems [151]. Using tabulated values, one finds 1q wq = ∑ C1q ′ 1q ′′ uq ′ vq ′′ ,

(6.26a)

q ′ q ′′

u+1 v0 − u0 v+1 √ = 2 u+1 v−1 − u−1 v+1 √ w0 = 2 u0 v−1 − u−1 v0 √ w−1 = = 2 i ⃗ = √ (⃗ u × v⃗) , w 2 w+1 =

i 1 √ {− √ [(⃗ u × v⃗)x + i (⃗ u × v⃗)y ]} , 2 2 i = √ (⃗ u × v⃗)z , 2 i 1 √ { √ [(⃗ u × v⃗)x − i (⃗ u × v⃗)y ]} , 2 2

(6.26b) (6.26c) (6.26d) (6.26e)

where the subscripts x, y, and z denote the Cartesian components of the vector ⃗ are √equal to those product, i.e., (u × v)z = ux vy − uy vx . The components of w ⃗ × v⃗ in the spherical basis up to a prefactor i/ 2. obtained by writing u

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One can form two further linear combinations of the spherical basis vectors e⃗q which are of interest, √ √ 1λ 00 ⃗λ , ⃗q′ . r⃗ = − 3 ∑ C1q1q e⃗q × e⃗q′ = i 2 ∑ C1q1q (6.27) ′ e ′ xq e qq ′

λ

The first of these illustrates that the vector product of two basis vectors in the spherical basis again is a vector; the second clarifies that the coordinate vector r⃗ actually is a scalar under rotations, obtained as the scalar combination (tensor of rank zero, component number zero) composed out of the spherical coordinates xq and the spherical basis vectors e⃗q . Of course, the vector of the xq is not a scalar. Here, however, we consider the scalar product of the vector composed of the xq with the vector composed of the e⃗q . Upon rotation, this scalar product is equal to the coordinate vector obtained using the new coordinates x′q , but multiplied with the rotated basis vectors e⃗′q , thus corresponding to a passive interpretation of the rotation which leaves the physical coordinate vector r⃗ = r⃗′ invariant. Let us now investigate the (` = 2)-component given in Eq. (6.16). It is symmetric and traceless and thus has five independent components. This leaves three offdiagonal and two diagonal components to be determined; the third component on the diagonal is fixed by the condition trc ( T∣`=2 ) = 0. A scalar (` = 0) has one component, a vector (` = 1) has three magnetic components which depend on ϕ as exp(imϕ) with m = −1, 0, 1, and a quadrupole tensor has five magnetic components [which depend on ϕ as exp(imϕ) with m = −2, −1, 0, 1, 2]. The generalization calls for 2` + 1 magnetic components for a tensor of rank `. Again, using tabulated 2q ′ ′′ values for Clebsch–Gordan coefficients of the form C1q = −1, 0, 1 and ′ 1q ′′ , with q , q q = −2, −1, 0, 1, 2, we have 2q (6.28a) tq = ∑ C1q ′ 1q ′′ uq ′ vq ′′ , q ′ q ′′

t+2 = u+1 v+1 ,

t+1 =

u+1 v0 + u0 v+1 √ , 2

(6.28b)

⃗ ⋅ v⃗ 3 u0 v0 − u √ , (6.28c) 6 u−1 v0 + u0 v−1 √ t−1 = , t−2 = u−1 v−1 , (6.28d) 2 where the spherical components u−1 , u0 , and u+1 are defined in Eq. (6.21). If the ⃗ and v⃗ “live” in the coordinate spaces of different particles 1 and 2, then vectors u ⃗ 2 = (L ⃗1 + L ⃗ 2 )2 with eigenvalue the functions tq are eigenfunctions of the operator L ⃗ 2 → `(`+1) = 6 and of the z component Lz = L1z +L2z with eigenvalue Lz → m = q. L t0 =

This can be seen easily by expressing the components of tq in terms of spherical harmonics, in the variables θ1 and ϕ1 (for the vector u), and in the variables θ2 and ϕ2 (for the vector v). 6.2.2

Matrix Elements and Clebsch–Gordan Coefficients

A separate foundation for the interest in Clebsch–Gordan coefficients lies in the fact that matrix elements, which have the same radial symmetry but different angular

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symmetry, can be unified in terms of reduced matrix elements. This is known as the Wigner4 –Eckart5 theorem and will be discussed below, by way of example. Let us consider the Cartesian components of the momentum operator, which are pi = −i ∂/∂xi . We have the following spherical components, ∂ i ∂ i ∂ ∂ +i ) , p0 = −i ∂z , p−1 = √ ( −i ) , (6.29) p+1 = − √ ( ∂x ∂y ∂x ∂y 2 2 while the corresponding components of the position vector are given in Eq. (6.21). Let us investigate both matrix elements of the position as well as momentum operator “sandwiched” in between the ground state and the nonrelativistic 2P state of hydrogen (Z = 1) with angular momentum projection quantum number m = 0, ⟨1S∣ x+1 ∣2P (m = 0)⟩ = ⟨1S∣ p+1 ∣2P (m = 0)⟩ = 0 , ⟨1S∣ x−1 ∣2P (m = 0)⟩ = ⟨1S∣ p−1 ∣2P (m = 0)⟩ = 0 , 8ia20

(6.30a) √

(6.30b)

128 2 a0 . (6.30c) 3 243 ̵ = c = 0 = 1. Here, a0 is the Bohr radius, and we remember that we use units with h The angular part of the 2P (m = 0) state, given by the spherical harmonic Y10 (θ, ϕ), is oriented along the z axis (see Fig. 2.2 of Ref. [81]). Physically, the fact that the only nonvanishing transition matrix element is the one with x0 or p0 means that the dipole radiation emitted during the 2P (m = 0) → 1S transition has a dipole pattern aligned with the z axis. If the “incoming” 2P state has magnetic projection m = 1, then we have ⟨1S∣ x0 ∣2P (m = 0)⟩ =

⟨1S∣ p0 ∣2P (m = 0)⟩ =

⟨1S∣ x+1 ∣2P (m = 1)⟩ = ⟨1S∣ p+1 ∣2P (m = 1)⟩ = 0 , ⟨1S∣ x−1 ∣2P (m = 1)⟩ =

8ia20

⟨1S∣ p−1 ∣2P (m = 1)⟩ = −

3 ⟨1S∣ x0 ∣2P (m = 1)⟩ = ⟨1S∣ p0 ∣2P (m = 1)⟩ = 0 .

√

128 2 a0 , 243

(6.31a) (6.31b) (6.31c)

Of course, the 1S state always has m = 0. The matrix elements vanish unless the magnetic quantum number of the rightmost state (2P state) and the magnetic quantum number of the transition operator (position or momentum operator) add up to zero. Physically, this means that the light emitted during the transition 2P (m = 1) → 1S has to be circularly polarized. Finally, for an incoming state 2P with magnetic projection m = −1, we have the following matrix elements: √ 8ia20 128 2 ⟨1S∣ x+1 ∣2P (m = −1)⟩ = ⟨1S∣ p+1 ∣2P (m = −1)⟩ = − a0 , (6.32a) 3 243 ⟨1S∣ x−1 ∣2P (m = −1)⟩ = ⟨1S∣ p−1 ∣2P (m = −1)⟩ = 0 , (6.32b) ⟨1S∣ x0 ∣2P (m = −1)⟩ = ⟨1S∣ p0 ∣2P (m = −1)⟩ = 0 .

(6.32c)

An inspection of Eqs. (6.30), (6.31), and Eq. (6.32) reveals a certain proportionality phenomenon among the matrix elements. Namely, either the matrix elements 4 Eugene 5 Carl

Paul Wigner (1902–1995). Henry Eckart (1902–1973).

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⟨1S∣ xq ∣2P (m = mz )⟩ and ⟨1S∣ pq ∣2P (m = mz )⟩ with q = −1, 0, 1 and mz = −1, 0, 1 both vanish, or they are proportional to each other with proportionality constant 8ia20 /3. The same phenomenon can be observed in matrix elements of other vectors “sandwiched” between states of the same symmetry. This observation can be formalized in terms of the Wigner–Eckart theorem. Let us consider states with quantum numbers γ, j and m, and γ ′ , j ′ and m′ , where j and j ′ are angular momenta and m and m′ are angular momentum projections, whereas γ and γ ′ are supplementary quantum numbers. When these are sandwiched with an operator of the form Tkq , which represents a spherical tensor of rank k with magnetic projection q, we have Cjjm ′ m′ k q ⟨γ j∣∣ T⃗(k) ∣∣γ ′ j ′ ⟩ , ⟨γ j m∣Tkq ∣γ ′ j ′ m′ ⟩ = √ 2j + 1

(6.33)

where ⟨γ j∣∣ T⃗(k) ∣∣γ ′ j ′ ⟩ is the so-called reduced matrix element, and the Clebsch– Gordan coefficient Cjjm The ′ m′ k q is the same as the one that enters Eq. (6.25). ⃗ vector arrow over Tk in the reduced matrix element indicates that the reduced elements are a genuine property of the tensor of rank k, not of its components. The functional dependence given in Eq. (6.33) is commonly referred to as the Wigner– Eckart theorem. Using tabulated values of Clebsch–Gordan coefficients, we can then reduce Eqs. (6.30), (6.31), and (6.32) to two matrix elements, appropriately called “reduced matrix elements”, which read as follows, √ √ 128 2 16 2 i ⟨1S∣∣ r⃗ ∣∣2P ⟩ = − a0 , ⟨1S∣∣ p⃗ ∣∣2P ⟩ = . (6.34) 81 3 27 3 a0 Using the Wigner–Eckart theorem, the nine equations given in the sub-equations of (6.30), (6.31), and Eq. (6.32) have been reduced to just two equations given in Eq. (6.34). The supplementary quantum numbers γ and γ ′ are often suppressed in writing down the Wigner–Eckart theorem (6.33). 6.3 6.3.1

Coefficients and Rotations Vector Addition or Clebsch–Gordan Coefficients

The Clebsch–Gordan coefficients are commonly referred to as vector addition coefficients, as evident from the discussion in Sec. 6.2.1. In the following, for general angular momentum eigenvectors, we use the convention u(j m) or u(γ j m), where j is the angular momentum and m is the angular momentum projection (supplementary quantum numbers are summarized into the multi-index γ). It is customary not to set a comma between the arguments to avoid an increased workload when writing down long expressions, except when necessary for clarity. In order to make contact with the literature, we should recall that some authors (e.g., Ref. [244]) use the alternative notation Cjjm = (j1 m1 j2 m2 ∣j1 j2 j m) 1 m 1 j2 m 2

(6.35)

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for the Clebsch–Gordan coefficients. The notation (j1 m1 j2 m2 ∣j1 j2 jm) is inspired by the Dirac bra-ket notation for scalar products and by the fact that the resulting vector v(j1 j2 jm) = ∣j1 j2 jm⟩ carries a “memory” of how it was constructed. In full generality, the definition of the Clebsch–Gordan coefficients is given by the following equation, u(γ1 j1 m1 ) v(γ2 j2 m2 ) , w(γ j1 j2 j m) = ∑ Cjjm 1 m 1 j2 m 2

(6.36)

m1 m2

where γ stands for the combined multi-index γ1 γ2 . Upon interchanging indices, Clebsch–Gordan coefficients typically acquire a phase factor, as in the relation cγ cγ Caαbβ = (−1)a+b−c Cbβaα .

(6.37)

According to Eq. (5.4.1) of Ref. [14], Clebsch–Gordan coefficients fulfill a number of symmetry relations, m3 m3 −m3 Cjj13m = (−1)j1 +j2 −j3 Cjj23m = (−1)j1 +j2 −j3 Cjj13 −m (6.38) 1 j2 m2 2 j1 m 1 1 j2 −m2 √ √ 2j3 + 1 j2 m2 2j3 + 1 j1 m1 = (−1)j1 −m1 C = (−1)j2 +m2 C . 2j2 + 1 j3 m3 j1 −m1 2j1 + 1 j2 −m2 j3 m3 In general, Clebsch–Gordan coefficients vanish unless j1 + j2 + j3 is integer. Two of the ji ’s (i = 1, 2, 3) can be half-integer valued, but then the last remaining j has to be integer rather than half-integer. Variations of the Wigner–Eckart theorem include various forms,

Ckjm Cjjm ′ m′ k q q j ′ m′ ′ k−j+j ′ ⃗ √ ⟨j∣∣T (k)∣∣j ⟩ = (−1) ⟨j∣∣T⃗(k)∣∣j ′ ⟩ ⟨j m∣Tkq ∣j m ⟩ = √ 2j + 1 2j + 1 ′

′

= (−1)

k−j+j ′

Cjj′−m C kq −m′ k −q ′ j ′ −m′ j m j ′ −m′ ⃗ √ √ ⟨j∣∣T⃗(k)∣∣j ′ ⟩ ⟨j∣∣T (k)∣∣j ⟩ = (−1) 2j + 1 2k + 1 ′

′

m Ckj −q jm √ ′ = (−1) ⟨j∣∣T⃗(k)∣∣j ′ ⟩ . (6.39) 2j + 1 The first representation on the right-hand side has the advantage over the other representations that no phase factors or minus signs have to be introduced in the projections m, q and m′ . The normalization of the reduced matrix element is chosen so that ′ ′ 2 ′ 2 (6.40) ∑ ∣⟨j m∣Tkq ∣j m ⟩∣ = ∣⟨j∣∣T⃗(k)∣∣j ⟩∣ . k+q

m q m′

Some example values for some reduced matrix elements have already been given in ⃗ of Pauli spin matrices, Eq. (6.34). Other examples are as follows. For the vector σ we have √ ⟨ 12 ∣∣⃗ σ ∣∣ 12 ⟩ = 6 . (6.41) This formula is relevant to the evaluation of matrix elements of the Pauli matrices in the basis of the states 1 0 χm = ∣ 12 m⟩ , ∣ 12 12 ⟩ = ∣ ↑⟩ = ( ) , ∣ 21 − 12 ⟩ = ∣ ↓⟩ = ( ) , (6.42) 0 1

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which are the “spin up” and “spin down” states. The unit vector operator has the reduced matrix element √ 0 . (6.43) ⟨`1 ∣∣ˆ r∣∣ `2 ⟩ = 2`2 + 1 C``21010 The matrix element is supposed to be evaluated among spherical harmonics, i.e., m1 C``21m 2 1q ∗ ⟨`1 ∣∣ˆ r∣∣ `2 ⟩ , ∫ dΩ Y`1 m1 (θ, ϕ) rˆq Y`2 m2 (θ, ϕ) = √ 2`1 + 1

(6.44)

where rˆq = xq /r [see Eq. (6.21)]. 6.3.2

Wigner 3j, 6j and 9j Symbols

The Wigner 3j symbol is optimized for maximum symmetry and is related to the Clebsch–Gordan coefficients, (

(−1)j1 −j2 −m3 j3 −m3 j1 j2 j3 Cj1 m1 j2 m2 . )= √ m1 m2 m3 2j3 + 1

(6.45)

The 3j symbol is invariant under an even permutation of columns, whereas an odd permutation of the columns leads to a multiplicative factor (−1)j1 +j2 +j3 . The Clebsch–Gordan coefficient is converted into a 3j symbol as follows, m3 = (−1)j2 −j1 −m3 Cjj13m 1 j2 m2

√

2j3 + 1 (

j1 j2 j3 ). m1 m2 −m3

(6.46)

In terms of the Wigner 3j symbol, the Wigner–Eckart theorem can be given in the form ⟨j m∣Tkq ∣j ′ m′ ⟩ = (−1)j −k+m ( ′

j′ k j ) ⟨j∣∣ T⃗(k) ∣∣j ′ ⟩ . m′ q −m

(6.47)

Upon three interchanges of columns, we can rewrite this as ⟨j m∣Tkq ∣j ′ m′ ⟩ = (−1)j−m+2k (

j k j′ ) ⟨j∣∣ T⃗(k) ∣∣j ′ ⟩ . −m q m′

(6.48)

If k is integer, then we can simplify the prefactor to read (−1)j−m . In general, one has to be extremely careful with the phase factors in Clebsch–Gordan coefficients and Wigner 3j symbols if one or two of the j’s are half-integer. For the 3j symbol not to vanish, the sum J = j + k + j ′ has to be integer-valued (half-integer would ′ ′ not be acceptable), in which case (−1)j+k+j = (−1)−j−k−j . An important special formula concerns the 3j symbol with three zero magnetic quantum numbers reads (see Appendix 2 on page 125 of Ref. [14]) (

(J − 2j1 )!(J − 2j2 )!(J − 2j3 )! j1 j2 j3 ) ) = (−1)J/2 ( 0 0 0 (J + 1)! ×

1/2

(J/2)! δmod(J,2),0 , (J/2 − j1 )!(J/2 − j2 )!(J/2 − j3 )!

(6.49)

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where J = j1 + j2 + j3 and mod(J, 2) is defined to be the remainder of the division of J by two, i.e., mod(J, 2) is zero if J is even, otherwise it is one. Note that the expression vanishes for odd J, because of the factor δmod(J,2),0 . The reduced matrix element of the unit vector operator, in terms of the Wigner 3j symbol, reads as ⎛ `1 1 `2 ⎞ . (6.50) ⎝0 0 0⎠ Here, an important shorthand notation is used for a mathematical expression that often occurs in angular momentum algebra (see Chap. 12 of Ref. [16]), √ (6.51) Πja jb jc ...jf = (2ja + 1) (2jb + 1) (2jc + 1)⋯(2jf + 1) . ⟨`1 ∣∣ˆ r∣∣ `2 ⟩ = (−1)`1 Π`1 `2

Because of the complex nature of angular momentum algebra, the notation is not always unique. E.g., the Π symbol with three arguments (rather than indices) is commonly used to denote the test function for an even sum of the arguments, 1 if j1 + j2 + j3 even . (6.52) 0 otherwise This is not to be confused with the ∆ symbol of three arguments which is defined as a test function for the triangularity condition, Π(j1 , j2 , j3 ) = {

1 if ∣ja − jb ∣ ≤ jc ≤ ∣ja + jb ∣ . (6.53) 0 otherwise Vector coupling coefficients among irreducible representations whose j values do not fulfill the triangularity condition, vanish. Note that in Sec. 9.8 of Ref. [16], the test function for the triangularity condition is denoted differently, namely as {ja jb jc } ≡ ∆(ja , jb , jc ), but this notation is not used here. The Wigner 6j symbols are the vector recoupling coefficients that switch between different j1 ’s and j2 ’s which can be used to construct the same j and m. This ominous definition can be illustrated as follows. Let us consider the re-coupling of angular momenta. Let us consider three angular momenta j1 , j2 , and j3 which are coupled to some resulting J. We may either couple j1 and j2 to j12 , before coupling j12 and j3 to J, or we may couple j2 and j3 to j23 , before coupling j23 and j1 to J. The resulting vectors are different, ∆(ja , jb , jc ) = {

w((j1 j2 )j12 , j3 , JM ) = ∑ CjJM v((j1 j2 )j12 m12 ) u(j3 m3 ) 12 m12 j3 m3 m12 m3

=

∑

m12 m2 m3

CjJM Cjj112mm1 j122 m2 u(j1 m1 ) u(j2 m2 ) u(j3 m3 ) 12 m12 j3 m3 (6.54)

whereas w(j1 , (j2 j3 )j23 , JM ) = ∑ CjJM u(j1 m1 ) v((j2 j3 )j23 m23 ) 1 m1 j23 m23 m1 m23

=

∑

m12 m2 m3

CjJM Cjj223mm2 j233 m3 u(j1 m1 ) u(j2 m2 ) u(j3 m3 ) . 1 m1 j23 m23 (6.55)

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The recoupling coefficients, which are independent of M , are proportional to the overlap integral of the two w vectors given above, and can be written according Eqs. (6.1.4) and (6.1.5) of Ref. [14], ⟨(j1 j2 )j12 , j3 , J∣j1 , (j2 j3 )j23 , J⟩ =

=

∑

∑

m1 m2 m3 m12 m23

√

Cjj223mm2 j233 m3 CjJM Cjj112mm1 j122 m2 CjJM 1 m1 j23 m23 12 m12 j3 m3

(2j12 + 1)(2j23 + 1) (−1)j1 +j2 +j3 +J {

j1 j2 j12 }. j3 J j23

(6.56)

The latter equality merely constitutes a definition of the Wigner 6j symbol. The 6j symbols are invariant under any permutation of the columns, whether even or odd. We have, for instance, {

j1 j2 j3 j j j }={ 2 1 3} . j4 j5 j6 j5 j4 j6

(6.57)

They are also invariant under a simultaneous exchange of two entries in each of the rows, {

j1 j2 j3 j j j }={ 4 5 3} . j4 j5 j6 j1 j2 j6

(6.58)

The 6j symbols have many applications which go far beyond their original definition, one of which concerns decoupled angular momenta. A particularly useful result can be found in Eq. (7.1.7) of Ref. [14], and is relevant for the case of a reduced matrix element of a coupled state with angular momenta γj1 j2 J, evaluated on an operator T⃗k which acts only on j1 , not j2 . The formula reads ′ j ′ J ′ j2 ⟨γj1′ j2 J ′ ∣∣T⃗(k)∣∣γj1 j2 J⟩ =(−1)j1 +j2 +J+k ΠJJ ′ { 1 } ⟨γj1′ ∣∣T⃗(k)∣∣γj1 ⟩ , J j1 k

(6.59)

where the Π symbol has been defined in Eq. (6.51). Let us also consider the case of two decoupled angular momenta. If a tensor X of rank k with spherical basis components q is composed of tensors T and U , as in X = T ⊗ U,

Xkq = ∑ Ckkq1 q1 k2 q2 Tk1 q1 Uk2 q2 ,

(6.60)

q1 q2

where T acts on j1 and U acts on j2 only, then a 9j symbol is incurred [see Eq. (7.1.7) of Ref. [14]] , ⎧ j1 j1′ k1 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ′ ′ ′ ′ ⃗ ⃗ (k2 )∣∣γ ′ j ′ ⟩ , ⟨γj1 j2 j∣∣X(k)∣∣γ j1 j2 j ⟩ = Πjj ′ k ⎨ j2 j2′ k2 ⎬ ∑⟨γj1 ∣∣T⃗(k1 )∣∣γ ′′ j1′ ⟩ ⟨γ ′′ j2 ∣∣U 2 ⎪ ⎪ ′′ ⎪ ⎪ ′ γ ⎪ ⎪ j j k ⎩ ⎭ (6.61) where the sum over γ ′′ extends over all virtual states necessary to complete the spectrum. The entity in curly brackets in Eq. (6.61) is known as a Wigner 9j symbol, and Πjj ′ k is defined in Eq. (6.51).

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Gaunt Coefficients

Finally, it is perhaps not surprising that different conventions for vector coupling coefficients have been proposed in the literature; it is unclear up to this point if the most appropriate notation is the one commonly adopted today. Still, it represents the consensus of the community. For example, the so-called Racah V and W coefficients are not used very often today; still, it is useful to know that they exist and are related by conversion formulas (which we do not discuss in any further detail) to the Clebsch–Gordan coefficients. One example of an alternative convention which we briefly mention, concerns the so-called Gaunt6 coefficients. Let us recall the definition of the spherical harmonic from Ref. [81], 1/2

Y`m (θ, ϕ) = (−1)m [

(2` + 1) (` − m)! ] 4π(` + m)!

P`m (cos θ) ei m ϕ ,

∗ Y`m (θ, ϕ) = (−1)m Y` −m (θ, ϕ) .

(6.62) (6.63)

In the notation Y` −m , there is no comma to separate the two indices. The Gaunt coefficient is defined as an integral over three spherical harmonics, `1 `2 `3 m1 ∗ ∫ dΩ Yl1 m1 (θ, ϕ) Yl2 m2 (θ, ϕ) Yl3 m3 (θ, ϕ) = (−1) Y−m1 m2 m3 ,

(6.64)

where √ 2 `3 Ym`11`m 2 m3

=

(2`1 + 1)(2`2 + 1)(2`3 + 1) `1 `2 `3 ` ` ` ( )( 1 2 3 ). 0 0 0 m1 m2 m3 4π

(6.65)

This formula involves a 3j symbol with three zero magnetic projections which has been indicated in Eq. (6.49). The literature is not always consistent in terms of the definitions used; e.g., in Eq. (107.14) of Chap. 107 of Ref. [245], the formula is a little different because different phase conventions for the spherical harmonics are used. According to Eq. (6.45), one may reformulate the 3j symbols in terms of Clebsch–Gordan coefficients. Conversely, it is very instructive to reformulate the Gaunt coefficient in terms of Clebsch–Gordan coefficients, in order to illustrate the general formalism of Racah–Wigner algebra. Using the fact that an even permutation of columns leaves a 3j symbol invariant, one may apply Eq. (6.45) to Eq. (6.64) in the following manner, (

(−1)`2 −`3 +m1 `1 m1 `1 `2 `3 ` ` `1 C`2 m2 `3 m3 . )=( 2 3 )= √ −m1 m2 m3 m2 m3 −m1 2`1 + 1

(6.66)

In terms of Clebsch–Gordan coefficients, the integral (6.64) may thus be 6 John

Arthur Gaunt (1904–1944).

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expressed as ∗ ∫ dΩ Y`1 m1 (θ, ϕ) Y`2 m2 (θ, ϕ) Y`3 m3 (θ, ϕ) √ (2`2 + 1) (2`3 + 1) `1 `2 `3 m1 `2 −`3 ( ) C``21 m = (−1) 2 `3 m3 0 0 0 4π ¿ Á (2`2 + 1) (2`3 + 1) ` 0 m1 À =Á C`21 0 `3 0 C``21 m . 2 `3 m 3 (2`1 + 1) 4π

(6.67)

This representation illustrates that the integral (6.64) is related to the Clebsch– Gordan coefficient for the coupling of ∣j2 m2 ⟩ and ∣j3 m3 ⟩ to ∣j1 m1 ⟩, but not exactly equal (due to nontrivial prefactors). On the other hand, one may also interpret the integral (6.67) using the Wigner–Eckart theorem given in Eq. (6.39) as ⟨`1 m1 ∣Y`2 m2 ∣`3 m3 ⟩ = ∫ dΩ Y`∗1 m1 (θ, ϕ) Y`2 m2 (θ, ϕ) Y`3 m3 (θ, ϕ) m1 C``1 m ` m = (−1)`2 −`3 +`1 √2 2 3 3 ⟨`1 ∣∣Y⃗ (`2 )∣∣`3 ⟩ . 2`1 + 1

(6.68)

A comparison of (6.68) to the second line of Eq. (6.67) leads to the highly symmetric and beautiful expression, √ (2`1 + 1) (2`2 + 1) (2`3 + 1) `1 `2 `3 ` ⟨`1 ∣∣Y⃗ (`2 )∣∣`3 ⟩ = (−1) 1 ( ), (6.69) 0 0 0 4π for the reduced matrix element of the spherical harmonic tensor of rank `2 . Let us set `2 = 1 and use √ 3 xq , (6.70) Y1q (θ, ϕ) = 4π where xq is the qth spherical basis component √ of the unit vector rˆ [see Eq. (6.21)]. Multiplying both sides of Eq. (6.69) by (4π)/3, we see that formula (6.69) is consistent with Eq. (6.50) for the case `2 = 1, where `2 in this case refers to the angular momentum `2 as given in Eq. (6.69). 6.3.4

Representation of Finite Rotations

We have shown that the Clebsch–Gordan coefficients enable us to construct higherrank tensors out of lower-rank ones, and to decompose tensors into individual components that transform separately under rotations, i.e., they do not mix. This is called the reduction of the rotation group into irreducible representations. Within the irreducible representations, the effect of a rotation can be represented in terms of so-called Wigner D matrices, to be discussed below; these do not mix tensors of different `. Suppose we wish to rotate a tensor of rank j by a finite rotation defined by the Euler angles α, β and γ. Here, the Euler angles assume their usual definition, as defined in Ref. [246] or in Eq. (1.3.1) of Ref. [14]. According to the

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fundamental paradigm underlying irreducible representations, a rotation transforms tensor components of rank j into components of the same rank. However, one may ask how this is done in practice, precisely. To answer the question, one may conjecture a linear dependence of the rotated components on the original components of the tensor, and then, find the matrix, within each irreducible representation of the rotation group, which describes this linear mapping. We have according to Eq. (4.1.11) of Ref. [14], for the action of a rotation by the Euler angles α, β and γ onto a spherical tensor of rank j and magnetic projection m, (j)

D(α β γ) ∣jm⟩ = ∑ ∣jm′ ⟩ ⟨jm′ ∣D(α β γ) ∣jm⟩ = ∑ ∣jm′ ⟩ Dm′ m (α β γ) , m′

(6.71)

m′

(j)

where Dm′ m (α β γ) is the element with indices m′ and m of the representation of the rotation D(α β γ) in the space of tensors of rank j. The relation (4.270) serves (j) as a definition of the matrix Dm′ m (α β γ). An analogous relation holds for general tensors [see Eq. (5.2.1) of Ref. [14]], k

(k)

D(α β γ)T (k q)D−1 (α β γ) = ∑ T (k q ′ ) Dq′ q (α β γ) .

(6.72)

q ′ =−k

Because the rotation about the Euler angles involves two rotations about (mutually tilted) z axes, which only add a phase, we have [see Eq. (4.1.12) of Ref. [14]] (j)

(j)

Dm′ m (α β γ) = exp(i m′ γ) dm′ m (β) exp(i m α) .

(6.73)

(j)

The rotation dm′ m (β) can finally be represented as [see Eq. (4.1.23) of Ref. [14]] ¿ ′ ′ Á (j + m′ )!(j − m′ )! β m +m β m −m (m′ −m,m′ +m) (j) Á À (cos β) . (cos ) (sin ) Pj−m′ dm′ m (β) = (j + m)!(j − m)! 2 2 (6.74) Here, P is a Jacobi polynomial, which carries three indices and one argument, not to be confused with the Legendre polynomial or associated Legendre polynomial, which is also denoted as P but carries a lesser number of indices. If desired, the integration over the Euler angles can be performed using ∫

2π 0

dα ∫

π 0

dβ sin β ∫

2π 0

(j )

(j )∗

dγ Dm11 m′ (α, β, γ) Dm22 m′ (α, β, γ) 1

2

8π 2 = δj j δm1 m2 δm′1 m′2 . (6.75) 2j1 + 1 1 2 Products of rotation matrices can be decomposed using Clebsch–Gordan coefficients, (j )

(j )

Dm11 m′ (α, β, γ) Dm22 m′ (α, β, γ) = 1

2

∣j1 +j2 ∣

∑

j=∣j1 −j2 ∣

(j)

′

jm jm ∑ Cj1 m1 j2 m2 Cj1 m′ j2 m′ Dmm′ (α, β, γ) , 1

mm′

2

(6.76) where the sum over j encompasses all angular momenta j that fulfill the triangularity condition. They have a symmetry property, (j)∗

(j)

Dµµ′ (α, β, γ) = (−1)µ −µ D−µ−µ′ (α, β, γ) , ′

(6.77)

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which can be applied as follows (example calculation), (j )

(j )∗

(j )

(j )

Dm11 λ Dm22 λ = (−1)λ−m2 Dm11 λ D−m2 2 −λ (j)

′

= (−1)λ−m2 ∑ Cjjm Cjjm Dmm′ (α, β, γ) 1 m1 j2 −m2 1 λj2 −λ jmm′

(j)

= (−1)λ−m2 ∑ Cjjm Cjj0 Dm0 (α, β, γ) . 1 m1 j2 −m2 1 λj2 −λ

(6.78)

jm

Here, the sum over m′ could be carried out because the magnetic projections λ and −λ have to add up to zero. Finally, let us point out that a large collection of formulas, symmetries, and useful relations can be found in a number of works, notably, Refs. [14] and [16]. 6.4 6.4.1

Composed Tensors of Higher Order Construction of the Spin-Angular Function

A spinor amplitude (with two components) is obtained by angular coupling of an orbital angular momentum to a spin. The two-component spinor wave functions are used to characterize eigenfunctions of the total angular momentum of an electron, composed of the orbital angular momentum and the spin. An electron carries spin1/2, and orbital angular momentum `. One may construct the spinor amplitude ∣j`m⟩ by coupling the orbital angular momentum to the spin, m u(` m1 ) v( 21 m2 ) . w(j m) = ∑ C`j m 1 1/2 m2

(6.79)

m1 m2

In coordinate space, one has u(` m1 ) = Y`1 m1 (θ, ϕ) and v( 21 m2 ) = χm2 , where χm2 is given in Eq. (6.42). The two-component spin-angular function χκ µ (ˆ r) is obtained by assembling the total angular momentum j of the electron from the orbital angular momentum ` and the spin-1/2, with the help of vector-coupling (Clebsch–Gordan) coefficients, µ χκ µ (ˆ r) = ∑ C`j µ−m Y (ˆ r) χm , 1/2 m `µ−m

(6.80)

m

where j = j(κ) = ∣κ∣ − 1/2 and ` = `(κ) = ∣κ + 1/2∣ − 1/2. Here and in the following, on occasion, we use the compact notation Y`m (ˆ r) ≡ Y`m (θ, ϕ), observing that the polar and azimuth angles θ and ϕ define rˆ uniquely, and vice versa [see also Eq. (5.157)]. The Dirac angular momentum operator K and its eigenvalue −κ (by convention) are defined as follows, ⃗ + 1, ⃗⋅L K=σ

K χκ µ (ˆ r) = −κ χκ µ (ˆ r) .

(6.81)

We recall from Eq. (4.348) that the Dirac angular quantum number κ fulfills −(j + 1/2) 1 κ = (−1)j+`+1/2 (j + ) = { 2 (j + 1/2)

for j = ` + 1/2 for j = ` − 1/2

.

(6.82)

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As such, the Dirac quantum number κ encodes information about both ` and j. Specifically, the Dirac quantum number κ is negative if the spin is added to the orbital angular momentum, and positive if the spin is subtracted from the orbital angular momentum. For ` = 0, 1, 2, . . . and j = 1/2, 3/2, . . . , the Dirac quantum number κ attains all possible positive and negative integer values except for zero. The orbital angular momentum ` = `(κ) and the total electron angular momentum j can be written in terms of κ, as follows, 1 1 1 j = ∣κ∣ − . (6.83) `(κ) = ∣κ + ∣ − , 2 2 2 The first cases of special interest are as follows. For S1/2 states, we have ` = 0 and j = 1/2, so κ = −1. For ` = 1 and j = 1/2 (P1/2 state), we have κ = 1. For a P3/2 state, we have ` = 1 and j = 3/2, so κ = −2. For D3/2 symmetry, one has ` = 2 and j = 3/2, so κ = 2. Upon using explicit formulas for the Clebsch–Gordan coefficients, one finds the exact form for the two-component spin-angular function χκ µ (ˆ r), 1/2

⎞ ⎛ κ κ + 1/2 − µ r) ⎟ ⎜ − ∣κ∣ ( 2κ + 1 ) Y` µ−1/2 (ˆ ⎜ ⎟ ⎟, χκ µ (ˆ r) = ⎜ ⎜ ⎟ 1/2 ⎜ ⎟ κ + 1/2 + µ ⎜ ⎟ ( ) Y (ˆ r ) ` µ+1/2 ⎝ ⎠ 2κ + 1

(6.84)

where ` = `(κ) is given in Eq. (6.83). The following identities, found in Eqs. (7.2.5.23) and (7.2.5.26) of Ref. [16], are quite important, (⃗ σ ⋅ rˆ) χκ µ (ˆ r) = −χ−κ µ (ˆ r) ,

⃗ χκ µ (ˆ r(⃗ σ ⋅ ∇) r) = −(1 + κ) χ−κ µ (ˆ r) .

(6.85)

Note that a change κ → −κ preserves j = ∣κ∣ − 1/2 but entails a change ` → ` = 2j − ` for `. If ` = j ± 1/2, then ` = j ∓ 1/2. It is somewhat hard to find a pertinent proof of the formula (6.85) in the lit⃗ ⋅ rˆ χκµ (ˆ erature. Let us try to indicate a possible proof for the first identity, σ r) = ⃗ ⋅ rˆ commutes with the total angular −χ−κµ (ˆ r). First, one shows that the operator σ ⃗ momentum J, ⃗σ ⃗ ⋅ rˆ] = 0 , ⃗ ⋅ rˆ] = 0 . [J, [J⃗2 , σ (6.86) Because ∣κ∣ = j + 1/2, we can conclude that the function [⃗ σ ⋅ rˆ χκµ (ˆ r)] must belong 2 ⃗ to the same J = j(j + 1) manifold as the original function χκµ (ˆ r). ⃗σ ⃗ ⋅ rˆ] = 0, one has a relation of the form Furthermore, because [J, ⃗ ⋅ rˆ χκµ (ˆ σ r) = c1 χκµ (ˆ r) + c2 χ−κµ (ˆ r) ,

(6.87)

where the coefficients c1 and c2 remain to be determined. On both sides of Eq. (6.87), we have the same magnetic projection µ and the same total angular momentum j in the spin-angular functions. The orbital angular momenta, ` = ∣κ+ 12 ∣− 12 , are different, though, under the replacement κ → −κ. Form Eq. (6.84), we can immediately infer that ⟨χκµ ∣⃗ σ ⋅ rˆ∣χκµ ⟩ = 0 and thus c1 = 0. In order to determine c2 , we observe that ∣∣⃗ σ ⋅ rˆ χκµ (ˆ r)∣∣2 = ⟨χκµ ∣(⃗ σ ⋅ rˆ)2 ∣χκµ ⟩ = ⟨χκµ ∣χκµ ⟩ = 1 = ∣c2 ∣2 .

(6.88)

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This means that c2 can only be a phase factor, c2 = eiδ , where δ remains to be determined. It can be determined as follows. Let us investigate the value of the spin-angular function at θ = 0, or, equivalently, rˆ = ˆez . We have Y`m (ˆez ) = (

2` + 1 1/2 ) δm0 . 4π

(6.89)

We recall that [see Eq. (6.80)] χκµ (ˆ r) = ∑ C`jµµ−m 1/2 m Y` µ−m (ˆ r) χm ,

(6.90)

χ−κµ (ˆ r) = ∑ C jµ Y (ˆ r) χm , ` µ−m 1/2 m ` µ−m

(6.91)

m

m

` = ∣κ + 12 ∣ − 12 ,

` = ∣ − κ + 12 ∣ − 12 .

(6.92)

Specialized to the case θ = 0, which is equivalent to rˆ = ˆez , one has χκµ (ˆez ) = ∑ C`jµµ−m 1/2 m ( m

2` + 1 1/2 2` + 1 1/2 ) δ(µ−m)0 χm = C`jµ0 1/2 µ ( ) χµ . (6.93) 4π 4π

⃗ ⋅ rˆ χκµ (ˆ We have considered the equation σ r) = c2 χ−κµ (ˆ r) for rˆ = ˆez . The derivation ⃗ ⋅ ˆez χκµ (ˆez ) = 2µ χκµ (ˆez ) is left as an exercise to the reader. We of the relation σ arrive at the relationship (2µ) C`jµ0 1/2 µ (

2` + 1 2` + 1 1/2 ) = c2 C jµ ( ) ` 0 1/2 µ 4π 4π

1/2

.

(6.94)

If j = ` + 1/2, then ` = j + 12 = ` + 1, while if j = ` − 1/2, then ` = j − 12 = ` − 1. In both cases, an explicit evaluation of the Clebsch–Gordan coefficient reveals that c2 = −1, completing the proof of Eq. (6.85). Let r⃗1 and r⃗2 be two vectors, r1 and r2 be their magnitudes, and ξ = cos γ =

r⃗2 ⋅ r⃗1 = cos θ2 cos θ1 + sin θ2 sin θ1 cos(ϕ2 − ϕ1 ) r2 r1

(6.95)

be the direction cosine, where (θ1 , ϕ1 ) and (θ2 , ϕ2 ) are the polar and azimuthal angles of r⃗1 and r⃗2 , respectively. The sum over magnetic quantum numbers leads to the following explicit formula, ∣κ∣ i ′ ⃗ ⋅ (ˆ (ξ)} , {12×2 P`(κ) (ξ) + σ r1 × rˆ2 ) P`(κ) 4π κ µ (6.96) where χ+ = χT∗ . The result (6.96) generalizes the well-known addition theorem for spherical harmonics, πκ (ˆ r1 , rˆ2 ) = ∑ χκ µ (ˆ r1 ) ⊗ χ+κ µ (ˆ r2 ) =

`

∗ r1 ) Y`m (ˆ r2 ) = ∑ Y`m (ˆ

m=−`

2` + 1 P` (ˆ r1 ⋅ rˆ2 ) . 4π

(6.97)

The Schr¨ odinger wave function of a bound state has been given as φn`m (⃗ r) = Rn` (r) Y`m (ˆ r) .

(6.98)

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The Schr¨ odinger–Pauli two-component nonrelativistic wave function adds the spin and reads as φn`jµ (⃗ r) = Rn` (r) χκ µ (ˆ r) ,

(6.99)

where κ is given in Eq. (6.82). As we shall discuss in more detail later, one may approximate the Dirac wave function, which is a four-component wave function, by the bispinor (“two-spinor”, i.e., four-component) amplitude 2 ⎛ (1 − p⃗ ) φ r) ⎞ n`jµ (⃗ ⎟ ⎜ 2m ⎟. ψ(⃗ r) ≈ ⎜ (6.100) ⎟ ⎜ σ ⃗ ⋅ p⃗ φ (⃗ r ) ⎠ ⎝ n`jµ 2m This needs to be contrasted with the fully relativistic expression for the wave function, which reads as (see the discussion in Chap. 8)

ψ(⃗ r) = (

fnκ (r) χκ µ (ˆ r) ). i gnκ (r) χ−κ µ (ˆ r)

(6.101)

Here, fnκ (r) and gnκ (r) are radial components of the wave function which will be calculated in Sec. 8.2.2 below. The replacement κ → −κ implies that, in particular, the “lower component” (in the sense of two-component spinors) of a P1/2 state is an S1/2 state, whereas the lower component of a P3/2 state is a D3/2 state. Using the formulas given above, it is not only possible to show the consistency of Eqs. (6.100) and (6.101), but also, to give an approximation for the radial part of the lower component of the wave function, ⃗ κ µ (ˆ ⃗ φn`jµ (⃗ ⃗ n` (r)] χκ µ (ˆ −i⃗ σ⋅∇ r) = − i [(⃗ σ ⋅ ∇)R x) − iRn` (r) (⃗ σ ⋅ ∇)χ r) ′ = − i (⃗ σ ⋅ rˆ)Rn` (r) χκ µ (ˆ r) + i (1 + κ)Rn` (r)χ−κ µ (ˆ r) ′ = i [Rn` (x) + (1 + κ)Rn` (x)] χ−κ µ (ˆ r) .

(6.102)

So, the lower component of the Dirac wave function given in Eq. (6.100) has the right symmetry, and a comparison of Eq. (6.101) to (6.102), with the help of Eq. (6.85) ⃗ reveals that and the identity p⃗ = −i ∇, fnκ (r) ≈ Rn` (r) ,

′ gnκ (r) ≈ Rn` (r) + (1 + κ)Rn` (r) .

(6.103)

In many cases, the approximations given in Eq. (6.103) are suitable approximations for the radial components of the full Dirac–Coulomb wave function given in Eq. (6.101). 6.4.2

Construction of the Vector Spherical Harmonic

The coupling of a spin to an orbital angular momentum is not the only fundamentally important case of a higher-order tensor which is relevant to experiments. Let us consider the coupling of a polarization vector, i.e., of a unit vector e⃗q given in

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the spherical basis, to an orbital angular momentum, described by a spherical harL monic. The resultant object is a vector spherical harmonic Y⃗JM (ˆ x) and reads as (see Sec. 5.4.2 of Ref. [81]) L JM Y⃗JM (ˆ r) = ∑ CLm r) e⃗q , 1q YLm (ˆ

(6.104)

mq

where the basis vectors e⃗q have been given in Eq. (6.22). Vector spherical harmonics are only nonvanishing when the triangularity condition ∆(L, 1, J) = 1 is fulfilled [see Eq. (6.53)], i.e., when J = L − 1, L, L + 1. It is useful to consider the space spanned by vector spherical harmonics with a given value of total angular momentum J. We can transform to a different basis spanned by the vector spherical harmonics (1) (0) (−1) Y⃗JM (θ, ϕ), Y⃗JM (θ, ϕ) and Y⃗JM (θ, ϕ). The transformation reads as follows, √ √ J + 1 J (1) (−1) J−1 Y⃗ (ˆ r) + Y⃗ (ˆ r) , (6.105a) Y⃗J M (ˆ r) = 2J + 1 JM 2J + 1 JM √ √ J J + 1 ⃗ (−1) (1) J+1 ⃗ ⃗ YJ M (ˆ r) = YJM (ˆ r) − Y (ˆ r) , (6.105b) 2J + 1 2J + 1 JM (0) Y⃗ J (ˆ r) = Y⃗ (ˆ r) . (6.105c) JM

JM

In spherical coordinates, the gradient operator can be decomposed into a radial and an angular part as follows, ⃗ = eˆr ∇

∂ 1 ⃗Ω , + ∇ ∂r r

⃗ Ω = eˆθ ∇

∂ 1 ∂ + eˆϕ , ∂θ sin θ ∂ϕ

(6.106)

where the eˆr , eˆθ and eˆϕ are unit vectors in the radial, polar and azimuthal directions. When acting on a function f = f (θ, ϕ) = f (ˆ r) defined on the unit sphere, the ⃗ (ˆ ⃗ Ω f (ˆ radial derivative vanishes, and we have ∇f r) = ∇ r). So, for example, one ⃗ Ω YJM (θ, ϕ) = ∇Y ⃗ JM (θ, ϕ). The following relations illustrate the relation of has ∇ (i) the alternative basis of the vector spherical harmonics Y⃗J M (θ, ϕ) to the spherical harmonics YJM (θ, ϕ), (1) Y⃗J M (θ, ϕ) = √

1

⃗ YJM (θ, ϕ) , ∇ J (J + 1) 1 (0) ⃗ YJM (θ, ϕ) , Y⃗J M (θ, ϕ) = √ L J (J + 1)

(−1) Y⃗J M (θ, ϕ) = rˆ YJM (θ, ϕ) .

(6.107a) (6.107b) (6.107c)

(1) A comparison with formulas given in Chap. 5 of Ref. [81] reveals that Y⃗J M (θ, ϕ) is related to the longitudinal component of emitted radiation, as its curl vanishes. (0) Furthermore, Y⃗J M (θ, ϕ) is related to the electric field encountered in magnetic (−1) multipole components of the radiation. Its divergence vanishes. Finally, Y⃗J M (θ, ϕ) is related to the electric field encountered in electric multipole components of the radiation [see Eq. (5.164) of Ref. [81]].

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Bethe’s gradient formula reads as follows, √ d `+1 ` ⃗ ( + ) u(r) Y⃗``−1 ∇ (u(r) Y`m (θ, ϕ)) = m (θ, ϕ) 2` + 1 dr r √ d ` `+1 − ( − ) u(r) Y⃗``+1 m (θ, ϕ) . 2` + 1 dr r

211

(6.108)

It applies to any function which can be represented as u(r) Y`m (θ, ϕ) and can be used in order to simplify the angular algebra involved in calculations with the Schr¨ odinger–Coulomb propagator, significantly. The two terms on the right-hand side of Eq. (6.108) involve spherical harmonics of rank ` − 1 and ` + 1, respectively. ⃗ itself, seen as a physical enThere is a subtlety, here. Under rotations, the vector ∇ tity, is a scalar according to Eq. (6.27). That is because according to Eq. (6.27), any vector can be written as a tensor of rank 0, with magnetic projection 0, composed of vector components with magnetic projection q, and basis vectors with magnetic 00 projection q ′ , coupled with a Clebsch–Gordan coefficient C1q 1q ′ . While the physical ⃗ is a scalar, its components and the unit vectors in the basis transform operator ∇ as contravariant and covariant components, respectively. Therefore, the total angular momentum J = ` of the vector spherical harmonics on the right-hand side of Eq. (6.108) still is J = `, while the purely angular part involves spherical harmonics of rank ` − 1 and ` + 1, respectively. 6.4.3

Spherical Biharmonic

In Sec. 6.4.1, we have coupled the electron spin to its orbital angular momentum, to obtain the standard spin-angular functions, and in Sec. 6.4.2, the coupling of a unit vector to an orbital momentum has been described. This leads to the vector spherical harmonics. The third possibility of coupling angular momenta, also of fundamental importance, concerns two individual orbital angular momenta (say, of two electrons) coupled together. The total angular momentum operator for the two electrons is given by ⃗ = `⃗1 + `⃗2 , L

(6.109)

and the coupled angular momenta are described by a spherical biharmonic, `2 YL`1M (θ1 , ϕ1 , θ2 , ϕ2 ) = ∑ C`LM Y (θ1 , ϕ1 ) Y`2 m2 (θ2 , ϕ2 ) . 1 m1 `2 m2 `1 m1

(6.110)

m1 m2

Again, using the Clebsch–Gordan coefficients, the two individual angular momenta `1 and `2 are added to produce the total angular momentum L. The spherical biharmonic has the properties `2 ⃗ 2 Y `1 `2 (ˆ L ˆ2 ) = L (L + 1) YL`1M (ˆ r1 , rˆ2 ) , L M r1 , r

`2 `2 Lz YL`1M (ˆ r1 , rˆ2 ) = M YL`1M (ˆ r1 , rˆ2 ) . (6.111)

`2 The function YL`1M (ˆ r1 , rˆ2 ) scalar-valued, rather than a vector (as in the case of the vector spherical harmonic).

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Applications of Racah–Wigner Algebra Tensor Decomposition of the Light Shift

Let us consider the following physical situation: A z-polarized laser illuminates an atom which has been prepared in a specific substate of a hyperfine manifold. In Secs. 3.3.1, 3.3.2, and 3.3.3, we have already encountered three alternative derivations of the dynamic Stark effect. Yet, on the level of nonrelativistic quantum mechanics, we have certainly neglected the electron spin, and certainly neglected the nuclear spin which defines the hyperfine manifold. The task then is to investigate the effect of the dynamic Stark shift operator on a specific substate of the hyperfine manifold. But the question is how to resolve this for a particular hyperfine level. We start by defining the zz, or “33”, component of the ac Stark shift operators (light shift operator) as follows, 1 1 + )z. (6.112) Q33 = e2 z ( E − H − ωL E − H + ωL In our discussion of the dynamic Stark shift in Sec. (3.3.1), we had already encountered the following matrix element of the reference state, given in Eq. (3.53), 1 1 + ) z∣ φn ⟩ , (6.113) ⟨φn ∣Q33 ∣φn ⟩ = e2 ∑ ⟨φn ∣z ( H0 − En + ωL H0 − En − ωL ± where we ignore prefactors which are irrelevant for the following discussion, and specialize the matrix element to a z-polarized laser. The Cartesian components of the light shift operator read as follows, 1 1 Qij = xi ( + ) xj . (6.114) E−H −ω E−H +ω For a spherically symmetric reference state ∣φn ⟩, we can write e2 ⟨φn ∣Q11 + Q22 + Q33 ∣φn ⟩ 3 1 e2 i = xi ∣ φn ⟩ = α(ωL ) , ∑ ⟨φn ∣x 3 ± H0 − En ± ωL

e2 ⟨φn ∣Q33 ∣φn ⟩ =

(6.115)

where ωL is the laser frequency and α(ωL ) is the dynamic dipole polarizability of the reference state ∣φn ⟩, α(ωL ) =

e2 1 i x i ∣ φn ⟩ . ∑ ⟨φn ∣x 3 ± H0 − En ± ωL

(6.116)

We also recall that by the summation convention, the Cartesian coordinates xi are summed over (for a complex atom, one has to sum over the coordinates xia of all electrons labeled by the index a). For a spherically symmetric atom, according to Eq. (3.54), the dynamic Stark shift of the reference state can be expressed as ∆En = −

IL α(ωL ) , 2 0 c

(6.117)

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but for a z-polarized laser field and a reference state which is not spherically symmetric (such as a specific magnetic substate within the hyperfine manifold), one has to manifestly evaluate ∆En = −

IL ⟨φn ∣Q33 ∣φn ⟩ . 2 0 c

(6.118)

The task then is to evaluate the matrix element ⟨φn ∣Q33 ∣φn ⟩ in terms of the quantum numbers, possibly within a hyperfine manifold. According to the discussion in Sec. 6.2.1, one may decompose any tensor with two indices into an ` = 0, three ` = 1, and five ` = 2 components. Our task is to decompose a diagonal matrix element (in an arbitrary reference state) of the light shift operator into irreducible tensor components. We would first like to put the decomposition given in Sec. 6.2.1 on a more solid footing, using the totally antisymmetric tensor. For a general tensor with Cartesian components T ij , we write ij

ij

T ij = ( T ∣`=0 ) + ( T ∣`=1 ) + ( T ∣`=2 )

ij

.

(6.119)

In particular, we have, from Eqs. (6.14)–(6.16), 1 kk ij T δ , 3 1 1 ij ( T ∣`=1 ) = (T ij − T ji ) = ijk k`m T `m , 2 2 1 kk ij 1 ij ij ji ( T ∣`=2 ) = (T + T ) − T δ . 2 3 Let us apply the Wigner–Eckart theorem in the form given in Eq. (6.48), is given in terms of the 3j symbol, ij

( T ∣`=0 ) =

⟨j m∣Tkq ∣j ′ m′ ⟩ = (−1)j−m+2k (

j k j′ ) ⟨j∣∣T⃗(k)∣∣j ′ ⟩ . −m q m′

(6.120a) (6.120b) (6.120c) where it

(6.121)

For the components of the T tensor, this means that ⟨L′ 0 ∣T(`=k)(q=0) ∣ L′′ 0⟩ = (−1)L ( ′

L′ k L′′ ) ⟨L′ ∣∣T⃗(` = k)∣∣ L′′ ⟩ , 0 0 0

(6.122)

where ` = k assumes the values ` = 0, ` = 1, or ` = 2. The above equation needs an explanation. On the left-hand side, we have an expression derived from a Cartesian representation of the tensor, with the ` = k component being isolated. The expressions for the particular components are given in Eq. (6.120). Note that, in this sense, the indices ij on the left-hand side of Eqs. (6.120a), (6.120b), and (6.120c) are not independent Cartesian indices, but merely represent the running Cartesian indices relevant to the ` = k tensor component matrices given in Eq. (6.120). For the ` = k = 2 component of a Cartesian tensor T ij , a priori, we have nine components (i, j = 1, 2, 3). However, ( T ∣`=2 )ij is a symmetric and traceless matrix, which reduces the number of independent components to 9−3−1 = 5, which is exactly the number of available quantum number q = −2, −1, 0, 1, 2 in the spherical basis. In

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order to apply the theorem (6.122), it is still necessary to convert the Cartesian to the spherical components. This is illustrated in some detail in Sec. 2.3.7 of Ref. [81]. In some cases, the conversion from the Cartesian to the spherical basis is particularly easy. For example, it is immediate to show that the Cartesian component ( T ∣`=2 )33 is equal to the q = 0 quadrupole component of the same tensor, 1 1 2 (6.123) ( T ∣`=2 )33 = T 33 − T 11 − T 22 = (T`=2 )q=0 . 3 3 3 The proof can be done on the basis of an inspection of the spherical harmonic Y20 (θ, ϕ), which constitutes a tensor element of a second-rank spherical quadrupole tensor, as follows. One observes that √ √ 5 5 2 2 2 (3 cos θ − 1) = (3z 2 − r2 ) r Y20 (θ, ϕ) = r 16π 16π √ √ 5 5 2 2 1 2 1 2 =3 ( z − x − y )=3 (R`=2 )33 , (6.124) 16π 3 3 3 16π where Rij = xi xj is the second-rank tensor composed of the products of the Cartesian coordinates, and (R`=2 )ij is the matrix describing its quadrupole component. Because the relation of the Cartesian components to the spherical components of a particular tensor is the same for any tensor of the same rank, Eq. (6.123) is obvious. One may thus identify the light-shift operator Q33 as a linear superposition of the k = 0, q = 0 and k = 2, q = 0 tensor components, Q33 =

2 1 1 1 (Q11 + Q22 + Q33 ) + Q33 − Q11 − Q22 = Q0 0 + Q2 0 . 3 3 3 3 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ Q0 0

(6.125)

Q2 0

Using ideas outlined in Sec. 4.3.4, it is possible to evaluate the matrix elements ⟨φn`(m=0) ∣Q0 0 ∣φn`(m=0) ⟩ and ⟨φn`(m=0) ∣Q2 0 ∣φn`(m=0) ⟩ by explicit calculations. Here, φn`(m=0) is a nonrelativistic state with quantum numbers n, ` and m = 0. Using Eq. (6.122), these elements may be related to the reduced matrix elements via the relations ⟨φn`(m=0) ∣Q0 0 ∣φn`(m=0) ⟩ = (−1)` ( =√

`0` ⃗ ) ⟨n`∣∣Q(0)∣∣n`⟩ 000

1 2` + 1

⟨n`(m = 0)∣Q2 0 ∣n`(m = 0)⟩ = (−1)` ( = −√

⃗ ⟨n`∣∣Q(0)∣∣n`⟩ ,

`2` ⃗ ) ⟨n`∣∣Q(2)∣∣n`⟩ 000 √ 2`(` + 1)

(2` − 1)(2` + 1)(2` + 2)(2` + 3)

(6.126a)

⃗ ⟨n`∣∣Q(2)∣∣n`⟩ . (6.126b)

These two relations enable us to find the reduced matrix elements ⃗ ⃗ ⟨n`∣∣Q(0)∣∣n`⟩ and ⟨n`∣∣Q(2)∣∣n`⟩ from the two elements ⟨φn`(m=0) ∣Q0 0 ∣φn`(m=0) ⟩ and

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√ ⟨φn`(m=0) ∣Q2 0 ∣φn`(m=0) ⟩, which may be explicitly calculated. The factor 1/ 2` + 1 for the tensor of rank zero finds a natural explanation in terms of Eq. (6.40). There is an additional problem, however. In ultra-high precision spectroscopy, we are addressing particular hyperfine-structure levels. We must find a way of expressing the light shift for each hyperfine level individually, although the lightshift operator is independent of the nuclear spin as well as the electron spin. In our case, the electron orbital angular momentum ` and its spin S first couple to the total electron angular momentum J. Then, the nuclear angular momentum I couples to J, resulting in the total angular momentum F , which is the sum of the electron and nuclear angular momenta. We start from the outermost level and consider the hyperfine level with quantum numbers n, F , and magnetic projection mF . Using Eq. (6.121), we have ⟨n F mF ∣Q2 0 ∣n F mF ⟩ = (−1)F −mF (

F 2 F ⃗ ) ⟨n F ∣∣Q(2)∣∣n F⟩. −mF 0 mF

(6.127)

We here use a shorthand notation where the quantum numbers alone denote the state. The hyperfine level ∣n F mF ⟩ qualifies itself as ∣n F mF ⟩ = ∣nJIF mF ⟩, because J and I couple to F . We may now apply (6.59) in order to reformulate ⃗ ⟨nF ∣∣Q(2)∣∣nF ⟩, because the light shift operator acts only on J, not on I, and obtain J F I ⃗ ⃗ ⟨nJIF ∣∣Q(2)∣∣nJIF ⟩ = (−1)J+I+F (2F + 1) { } ⟨nJ∣∣Q(2)∣∣nJ⟩ . F J 2

(6.128)

Now, the state ∣nJ⟩ qualifies itself again as ∣nJ⟩ = ∣n`SJ⟩, because ` and S couple to J. So, applying (6.59) one more time, we obtain ⃗ ⃗ ⟨nJ∣∣Q(2)∣∣nJ⟩ = ⟨n`SJ∣∣Q(2)∣∣n`SJ⟩ = (−1)`+S+J (2J + 1) {

` JS ⃗ } ⟨n`∣∣Q(2)∣∣n`⟩ . J ` 2

(6.129)

The two symmetry properties (6.57) and (6.58) allow us to reformulate {

J F I J J 2 J 2J }={ }={ }. F J 2 F F I F IF

(6.130)

{

` ` 2 ` 2 ` ` JS }. } ={ }={ J ` 2 JJS JSJ

(6.131)

Also, we have

The forms on the right-hand sides are more symmetric; they bring the rank (two) of the Q tensor into the middle column. Using Eqs. (6.128), (6.129), (6.130), and (6.131), we now have ⃗ ⟨nF ∣∣Q(2)∣∣nF ⟩ = (−1)I+F +2J+`+S (2J + 1) (2F + 1) ×{

J 2J ` 2 ` ⃗ }{ } ⟨n`∣∣Q(2)∣∣n`⟩ . F IF JSJ

(6.132)

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Together with (6.127), we finally have the relation ⟨nF mF ∣Q2 0 ∣nF mF ⟩ = (−1)I+2F +2J+`+S−mF (2J + 1)(2F + 1) ×(

F 2 F J 2J ` 2 ` ⃗ ){ }{ } ⟨n `∣∣Q(2)∣∣n `⟩ . −mF 0 mF F IF JSJ (6.133)

For the component of rank zero, the analog of Eq. (6.127) is ⟨nF mF ∣Q0 0 ∣nF mF ⟩ = (−1)F −mF ( = (−1)F −mF 1

F 0 F ⃗ ) ⟨n F ∣∣Q(0)∣∣n F⟩ −mF 0 mF

(−1)−F +mF ⃗ √ ⟨n F ∣∣Q(0)∣∣n F⟩ 2F + 1

⃗ ⟨n F ∣∣Q(0)∣∣n F⟩. (6.134) 2F + 1 For the analogs of Eqs. (6.128) and (6.129), we can use the following relations for specific 6j symbols as they enter into the analogs of Eqs. (6.130) and (6.131), =√

{

(−1)−F −I−J J 0J }= √ , F IF (2F + 1)(2J + 1)

(6.135a)

{

(−1)−J−L−S L0L . }= √ J SJ (2J + 1)(2L + 1)

(6.135b)

In the analog of (6.132), the factors of 2J + 1 and 2F + 1 as well as the powers of −1 thus cancel; for the factor 2F + 1, this can be seen in combining the corresponding factors in Eq. (6.134) and (6.135a). Finally, we have 1 ⃗ ⟨n F mF ∣Q0 0 ∣n F mF ⟩ = √ ⟨n `∣∣Q(0)∣∣n `⟩ . (6.136) 2` + 1 We can now use Eqs. (6.133) and (6.136) to write the matrix element of the light shift operator as ⟨nF mF ∣Q33 ∣nF mF ⟩ = ⟨nF mF ∣Q0 0 ∣nF mF ⟩ + ⟨nF mF ∣Q2 0 ∣nF mF ⟩ =√

1 2L+1

×(

I+2F +2J+`+S−mF ⃗ ⟨n`∣∣Q(0)∣∣n`⟩+(−1) (2J +1)(2F +1)

F 2 F J 2J ` 2 ` ⃗ ){ }{ } ⟨n `∣∣Q(2)∣∣n `⟩ . −mF 0 mF F IF JSJ (6.137)

We remember that the absolute normalization of the reduced matrix elements is given by Eqs. (6.126a) and (6.126b). Our result in Eq. (6.137) is consistent with the result given originally in Eq. (12) of Ref. [247] and with the analysis presented in Ref. [117]; it is of considerable importance for practical questions that arise in the analysis of high-precision spectroscopic

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⃗ experiments. Because the quadrupole term ⟨φnS ∣∣Q(2)∣∣φ nS ⟩ vanishes for ` = 0 (i.e., for S states), the light shift is independent of the hyperfine structure components for ⃗ S states (only the term proportional to ⟨n`∣∣Q(0)∣∣n`⟩ survives). For states of a different angular symmetry, say, D states, the situation is different. The result (6.137) implies, in particular, that the light shift of D states does not only depend on the hyperfine component, as manifest in the total angular momentum quantum number F , but also on the projection mF . This dependence is numerically not very large but significant in the analysis of high-precision spectroscopic experiments [117]. In our treatment so far, we have assumed that the light shift operator describes a z polarized laser, and that the hyperfine state also is aligned along the z axis, which is the quantization axis. This raises the question of how to rotate the zpolarized laser into the quantization axis of the atomic total angular momentum via the relation (6.71). We have to use a Wigner D matrix in order to describe the rotation, (2)

D(α β γ) Q2 0 D−1 (α β γ) = ∑ Q2 q′ Dq′ 0 (α β γ) .

(6.138)

q′

This leads to the following, modified formula for the matrix element of the rotated light shift operator, 1 ⃗ ⟨nL∣∣Q(0)∣∣nL⟩ + ∑(−1)I+2F +2J+`+S−mF ⟨nF mF ∣Q′33 ∣nF mF ⟩ = √ 2L + 1 q′ × (2J + 1) (2F + 1) ( ×{

F 2 F (2) ) Dq′ 0 (α β γ) −mF q ′ mF

J 2J L2L ⃗ }{ } ⟨nL∣∣Q(2)∣∣nL⟩ . F IF J SJ

(6.139)

The 3j symbol in this equation vanishes except for the case q ′ = 0, and so (

F 2 F F 2 F )=( ) δq ′ 0 . −mF q ′ mF −mF 0 mF

(2)

(6.140)

(2)

Furthermore, we have D00 (α β γ) = d00 (β) according to Eq. (6.74), and thus 1 ⃗ ⟨nF mF ∣Q′33 ∣nF mF ⟩ = √ ⟨n `∣Q(0)∣n `⟩ + (−1)I+2F +2J+`+S−mF 2` + 1 F 2 F J 2J ` 2 ` (2) ⃗ × (2J + 1)(2F + 1) ( ){ }{ } d00 (β) ⟨n `∣∣Q(2)∣∣n `⟩ . −mF 0 mF F IF JSJ (6.141) Here, β is the angle of the laser polarization axis to the quantization axis of the (2) total angular momentum. The tensor element d00 (β) can be expressed as [see Eq. (4.1.26) of Ref. [14]] 3 1 (2) (0,0) (6.142) d00 (β) = P2 (β) = P2 (β) = (3 cos2 β − 1) = 1 − β 2 + O(β 4 ) . 2 2 This result can be used in order to estimate the effect of a small tilt angle θ of the laser polarization axis against the angular momentum quantization axis.

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6.5.2

Tensorial Decomposition of a Dipole Transition

We have treated the decay rate of an excited atomic state against one-photon dipole decay in Sec. 3.2.3 on the basis of Fermi’s golden rule, and in Sec. 4.7.1 on the basis of the imaginary part of the one-loop self-energy, identifying the decay width as a quantity proportional to the imaginary part of the energy shift. In this treatment, the excited state was treated as a manifestly nonrelativistic state, without fine structure. Our task now is to decompose the transition matrix element for a dipole transition including fine-structure levels. In the length gauge, the dipole transition operator is proportional to xq , where xq is the qth component of the position vector in spherical coordinates [see Eq. (6.21)]. We have q = 0 for the case of a z-polarized laser, and q = ±1 for circular polarization [see Eq. (6.22)]. Taking into account the fine-structure (total angular momentum quantum number J), the matrix element to decompose is ⟨n′ `′ S ′ J ′ µ′ ∣xq ∣ n ` S J µ⟩ ,

(6.143)

′

where in the following we manifestly assume that S = S (for a one-electron state, we have S = 1/2). The Wigner–Eckart theorem in the form (6.121) yields ⎛ J′ 1 J ⎞ ′ ′ ⟨n ` S J ′ ∣∣ r⃗ ∣∣ n ` S J⟩ . (6.144) ⎝ −µ′ q µ ⎠ We recall (6.59) in the following form [with T⃗(k) = r⃗ a tensor of rank k = 1] ⎧ ⎫ ′ ′ ⎪ ⎪ j1 J j2 ⎪ ⎪ ′ ′ ′ ′ ′ ′ j1′ +j2 +J+1 ′ ΠJJ ⎨ ⟨n j1 j2 J ∣∣ r⃗ ∣∣n j1 j2 J⟩ =(−1) ⎬ ⟨n j1 ∣∣ r⃗ ∣∣nj1 ⟩ . (6.145) ⎪ ⎪ ⎪ ⎩ J j1 1 ⎪ ⎭ For a dipole transition, we identify j1′ = `′ , j1 = `, j2′ = j2 = S. Only the electron orbital angular momentum couples to the position operator, and so ⎧ ⎪ `′ J ′ S ⎫ ⎪ ′ ′ ′ ⎪ ⎪ ⟨n′ `′ S J ′ ∣∣⃗ r∣∣n ` S J⟩ = (−1)` +S+J+1 ΠJJ ′ ⎨ ⎬ ⟨n ` ∣∣ r⃗ ∣∣n `⟩ . (6.146) ⎪ ⎪ ⎪ ⎩J ` 1 ⎪ ⎭ We have finally reduced the dipole transition element to a reduced matrix element involving only the angular momentum states with quantum numbers n, `, n′ , `′ , ⟨n′ `′ S J ′ µ′ ∣xq ∣ n ` S J µ⟩ = (−1)J

′

⟨n′ `′ S J ′ µ′ ∣xq ∣ n ` S J µ⟩ = (−1)J

−µ′

′

+J+`′ +S+1−µ′

⎧ `′ J ′ S ⎫ ⎪ ⎛ J′ 1 J ⎞ ⎪ ⎪ ⎪ ′ ′ ×Π ⎨ ⎬ ⟨n ` ∣∣⃗ r∣∣n `⟩ . (6.147) ⎪ ⎝ −µ′ q µ ⎠ ⎪ ⎪ ⎩J ` 1 ⎪ ⎭ For a one-electron state, we can now go a step further and use the fact that S = 1/2. Furthermore, we take into account that the reduced matrix element ⟨n′ `′ ∣∣⃗ r∣∣n `⟩ needs to be evaluated among nonrelativistic states of the form φn`m (⃗ r) = Rn` (r) Y` m (ˆ r). One may use the following result, given previously in Eq. (6.43) for the reduced matrix element of the unit vector operator, sandwiched between spherical harmonics Y`′ m′ (ˆ r) and Y` m (ˆ r), JJ ′

⟨`′ ∣∣ˆ r∣∣`⟩ = (−1)` Π``′ ′

⎛ `′ 1 ` ⎞ . ⎝ 0 0 0⎠

(6.148)

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Then, ⟨n′ `′ ∣∣⃗ r∣∣n `⟩ = (−1)` Π``′ ′

⎛ `′ 1 ` ⎞ (3) R ′′ , ⎝ 0 0 0 ⎠ n ` n`

∞

(j)

Rn′ `′ n` = ∫

0

dr rj Rn′ `′ (r) Rn` (r) . (6.149)

Putting everything together, one finds ⟨n′ `′ 12 J ′ µ′ ∣xq ∣n ` 12 J µ⟩ = (−1)J+J

′

−µ′ −

1 2

ΠJJ ′ ``′

⎧ `′ J ′ 1 ⎫ ⎪ (3) ⎛ J 1 J ⎞ ⎛ `′ 1 ` ⎞ ⎪ ⎪ 2⎪ ⎬ Rn′ `′ n` . (6.150) × ⎨ ⎪ ⎝ −µ′ q µ ⎠ ⎝ 0 0 0 ⎠ ⎪ ⎪ ⎭ ⎩J ` 1⎪ Again, we can describe the effect of a rotation of the laser polarization axis into the quantization axis of the total atomic angular momentum via the relation (6.73). Namely, we have ′

(1)

D(α β γ) xq D−1 (α β γ) = ∑ xq′ Dq′ q (α β γ) ,

(6.151)

q′

leading to the following matrix element for the rotated vector component x′q = D(α β γ) xq D−1 (α β γ), ⟨n′ `′ 12 J ′ µ′ ∣ x′q ∣n ` 12 J µ⟩ = ∑(−1)J+J q′

′

−µ′ −1/2

(1)

ΠJJ ′ ``′ Dq′ q (α β γ)

⎧ `′ J ′ 1 ⎫ ⎪ (3) ⎛ J ′ 1 J ⎞⎛ `′ 1 ` ⎞ ⎪ ⎪ 2⎪ ⎨ ⎬ Rn′ `′ n` . (6.152) ′ ′ ⎪ ⎝ −µ q µ ⎠⎝ 0 0 0 ⎠ ⎪ ⎪ ⎩J ` 1⎪ ⎭ There is no immediate, general simplifying formula available for the 3j symbols in this equation. However, in a concrete case, with all quantum numbers fixed, one may use this formula in order to estimate the effect of a slight tilt of the laser axis against the quantization axis of the angular momentum. ×

6.6 6.6.1

Rydberg Electron and Hydrogenlike Core Physical Foundation

A great deal can be learned about the physics of atoms by considering the interaction of a single Rydberg electron, in a highly excited state, with a hydrogenlike core (where there is an attractive force, even if the hydrogenlike core is electrically neutral). Let us consider a two-electron ion, with the nucleus having charge number Z. One of the electrons is in the 1s ground state, the other electron is in an excited state with principal quantum number n and orbital angular momentum quantum ̵ = c = 0 = 1. We label the inner elecnumber `. Again, we use natural units with h tron by the subscript 1, and the outer electron by the subscript 2. The Hamiltonian reads as H=

p⃗ 2 Zα Zα α p⃗12 + 2 − − + . 2m 2m r1 r2 ∣⃗ r1 − r⃗2 ∣

(6.153)

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The unperturbed Hamiltonian H0 is the sum of an unscreened Hamiltonian (Coulomb potential with charge number Z) for the inner electron, and a fully screened Coulomb potential (charge number Z − 1) for the outer electron, p⃗ 2 Zα (Z − 1) α p⃗12 + 2 − − . 2m 2m r1 r2 The perturbation is just the difference ∆V = H − H0 and reads as α α ∆V = − + . r2 ∣⃗ r1 − r⃗2 ∣ H0 =

(6.154)

(6.155)

From the multipole expansion of the electrostatic interaction (see Chap. 2 of Ref. [81]), we know that ∞ ` 1 r`+1

(6.156)

For our situation, we have r< = r1 and r> = r2 . The first two multipoles thus read as 1 4π r1 1 1 1 ∗ = + ∑ Y1 m (θ1 , ϕ1 ) Y1 m (θ2 , ϕ2 ) + O ( 3 ) . ∣⃗ r2 − r⃗1 ∣ r2 3 r22 m=−1 r2

(6.157)

We now enter with (6.157) into (6.155), we obtain for the sum of the first two terms in the multipole expansion of ∆V , written here very explicitly as ∆V = −

1 α α 4π α r1 1 ∗ + + ∑ Y1 m (θ1 , ϕ1 ) Y1 m (θ2 , ϕ2 ) + O ( 3 ) . r2 r2 3 r22 m=−1 r2

(6.158)

The monopole term cancels, and the first nonvanishing interaction between the outer electron and the core (beyond the screened Coulomb interaction) consists of a dipole interaction. The unperturbed eigenstates of H0 consist of an electron in the ∣1s⟩ ground state, and a second one in an ∣n`⟩ excited Rydberg state. The two-electron state is characterized by the quantum numbers ∣1s, n`⟩. In general, the eigenstates of H0 are ∣n1 `1 , n2 `2 ⟩ with one electron in state ∣n1 `1 ⟩ and the other in the state ∣n2 `2 ⟩. We recall that H0 does not couple the electrons. The second-order energy shift reads as ′ RRR RRR ⎞ ⎛ 1 R RRR ∆V RRRR 1s, n`⟩ ∆E = ⟨1s, n` RR∆V RRR RRR ⎝ E (0) − H0 ⎠ 1s,n` R R ⟨1s, n` ∣∆V ∣ n1 p, n2 (` ± 1)⟩ ⟨n1 p, n2 (` ± 1) ∣∆V ∣ 1s, n`⟩ , (6.159) = ∑ (0) (0) n1 ,n2 ,± E1s,n` − En1 p,n2 (`±1) (0)

where En1 `1 ,n2 `2 is the energy eigenvalue of the state ∣n1 `1 , n2 l2 ⟩ with respect to the unperturbed Hamiltonian H0 as given in Eq. (6.154). It may be expressed as (Zα)2 m . (6.160) 2n2 The dipole selection rule has been used. Both electrons undergo a dipole transition during the interaction. Therefore, the first electron, initially in the ground state, (0)

En1 `1 ,n2 `2 = En1 `1 (Z) + En2 `2 (Z − 1) ,

En` (Z) = En (Z) = −

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gets excited into a p state, and the second electron, initially in a state with orbital angular momentum `, makes a transition into a state with angular momentum ` − 1, or ` + 1. So, the second-order energy shift reads as ∆E =

16π 2 α2 9

1

1

1 n1 ,n2 ,± m=−1 m′ =−1 E1s (Z) − En1 p (Z) + En` (Z − 1) − En2 `±1 (Z − 1) ∑

∑

∑

∗ × ⟨1s ∣r1 Y1m (θ1 , ϕ1 )∣ n1 p⟩ ⟨n1 p ∣r1 Y1m ′ (θ1 , ϕ1 )∣ 1s⟩

× ⟨n` ∣

1 1 ∗ Y (θ2 , ϕ2 )∣ n2 (` ± 1)⟩ ⟨n2 (` ± 1) ∣ 2 Y1m′ (θ2 , ϕ2 )∣ n`⟩ . r22 1m r2

(6.161)

The decisive observation is now as follows: the state ∣n`⟩ is a Rydberg state, and therefore its radial overlap with energetically lower states is small. Therefore, the largest contributions to the sum over intermediate states n2 come from states with n2 ≈ n. Since Rydberg states are very close energetically, we may approximate E1s (Z) − En1 p (Z) + En` (Z − 1) − En2 `±1 (Z − 1) ≈ E1s (Z) − En1 p (Z) ,

(6.162)

essentially ignoring the Rydberg-state energy difference. In that case, the propagator denominator becomes independent of the Rydberg electron, and the energy shift reads as ∆E =

16π 2 α2 9 × ⟨n` ∣

∗ ⟨1s ∣r1 Y1m (θ1 , ϕ1 )∣ n1 p⟩ ⟨n1 p ∣r1 Y1m ′ (θ1 , ϕ1 )∣ 1s⟩ E (Z) − E (Z) ′ 1s n p n1 ,n2 ,± m=−1 m =−1 1

∑

1

∑

1

∑

1 ∗ 1 Y1m (θ2 , ϕ2 )∣ n2 (` ± 1)⟩ ⟨n2 (` ± 1) ∣ 2 Y1m′ (θ2 , ϕ2 )∣ n`⟩ . 2 r2 r2

(6.163)

The sum over n1 and n2 also entails the integral over the continuous part of the spectrum. 6.6.2

Angular Algebra

From now on, without loss of generality, we assume that the Rydberg state has a zero magnetic quantum number. We wish to carry out the angular integration in full detail, assuming the structure ⟨⃗ r∣n`⟩ = Rn` (r) Y`m (ˆ r) for all unperturbed orbitals. Under this assumption, the energy shift may be separated into radial and angular parts as follows, 2 16π 2 α2 T`+1 + T`−1 (3) (R1sn1 p ) 9 E1s (Z) − En1 (Z) n2 m2 =−1 m′ =−1 m3 =−1 1

∆E = ∑ ∑ ∑ n1

× (∫

1

∑

1

∑

2

∗ dΩ1 Y00 (θ1 , ϕ1 ) Y1m2 (θ1 , ϕ1 ) Y1m3 (θ1 , ϕ1 ))

∗ × (∫ dΩ1 Y1m (θ1 , ϕ1 ) (−1)m2 Y1 −m′2 (θ1 , ϕ1 ) Y00 (θ1 , ϕ1 )) . 3 ′

(6.164)

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The sums over m3 and m′3 that are seen explicitly in the above equation have been implicit in Eq. (6.163). The terms T`±1 are given by

T`+1 =

`+1

∑

m′3 =−(`+1)

∗ (∫ dΩ2 Y`0 (θ2 , ϕ2 ) (−1)m2 Y1 −m2 (θ2 , ϕ2 ) Y(`+1) m′3 (θ2 , ϕ2 )) (0)

2

∗ ) , (6.165) × (∫ dΩ2 Y(`+1)m ′ (θ2 , ϕ2 ) Y1 m′ (θ2 , ϕ2 ) Y`0 (θ2 , ϕ2 )) (R n`n2 (`+1) 2 3

T`−1 =

`−1

∑

m′3 =−(`−1)

∗ (∫ dΩ2 Y`0 (θ2 , ϕ2 ) (−1)m2 Y1 −m2 (θ2 , ϕ2 ) Y(`−1)m′3 (θ2 , ϕ2 )) (0)

2

∗ ) . (6.166) × (∫ dΩ2 Y(`−1)m ′ (θ2 , ϕ2 ) Y1 m′ (θ2 , ϕ2 ) Y`0 (θ2 , ϕ2 )) (R n`n2 (`−1) 2 3

The radial integrals have been defined in Eq. (6.149). We now use the result (6.65) in order to express an integral over three spherical harmonics in terms of 3j symbols, with the result 2 16π 2 α2 1 (3) (R1sn1 p ) 9 E1s (Z) − En1 (Z) n2 m2 =−1 m′ =−1 m3 =−1 1

∆E = ∑ ∑ ∑ n1

1

∑

1

∑

2

′ Π110 Π011 0 1 1 0 1 1 110 1 1 0 ×√ ( )( ( )( ) ) (−1)m3 +m2 √ 0 m2 m3 −m3 −m′2 0 4π 0 0 0 4π 0 0 0 ⎡ `+1 ⎢ Π` 1 (`+1) ` 1 ` + 1 ` 1 `+1 × ⎢⎢ ∑ (−1)m2 √ ( )( ) 0 0 0 0 −m2 m′3 ⎢m′3 =−(`+1) 4π ⎣ 2 ′ Π(`+1)1` `+1 1 ` `+1 1 ` (0) ( )( × (−1)m3 √ ) (Rn`n2 (`+1) ) ′ ′ 0 00 −m3 m2 0 4π

+

Π` 1 `−1 ` 1 ` − 1 ` 1 `−1 (−1)m2 √ ( )( ) 0 0 0 0 −m2 m′3 4π m′3 =−(`−1) `−1

∑

×(−1)m3 ′

2 Π(`−1) 1 ` ` − 1 1 ` `−1 1 ` (0) √ ( )( ) (Rn`n2 (`−1) ) ] . ′ ′ 0 00 −m3 m2 0 4π

(6.167)

This formula is not as frightening as it looks. The following special cases must now √ √ √ 011 ` 1 `+1 be considered, Π011 ( ) = − 3, and Π`1(`+1) ( ) = (−1)`+1 3 ` + 1 00 0 000 √ √ ` 1 `−1 ) = (−1)` 3 `. After some algebra, we can rewrite as well as Π`1(`−1) ( 00 0 ∆E as ∆E = U`+1 + U`−1 ,

(6.168a)

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where U`+1 = ∑

n1 n2

2 2 α2 (0) (3) (R1sn1 p ) (Rn`n2 (`+1) ) E1s (Z) − En1 (Z)

×

[(` + 1) (−1)m2 +m2 +m3 +m3 ( ′

∑

′

m2 m′2 m3 m′3

×(

1 0 1 1 ` `+1 )( ) m3 0 m2 −m2 0 m′3

`+1 ` 1 1 0 1 )( )] −m′3 0 m′2 −m′2 0 −m3

(6.168b)

and U`−1 = + ∑

n1 n2

×

2 2 α2 (0) (3) (R1sn1 p ) (Rn`n2 (`−1) ) E1s (Z) − En1 (Z)

∑

[` (−1)m2 +m2 +m3 +m3 ( ′

′

m2 m′2 m3 m′3

1 0 1 1 ` `−1 )( ) m3 0 m2 −m2 0 m′3

`−1 ` 1 1 0 1 )( )] . (6.168c) −m′3 0 m′2 −m′2 0 −m3 In order to carry out the summation over the magnetic projections, we can now apply the following formula, which is found on page 455 of Ref. [16], ×(

p−ψ+q−κ+r−ρ+s−σ ( ∑ (−1)

ψκρσ

p a q q b r r c s sd p )( )( )( ) ψ α −κ κ β −ρ ρ γ −σ σ δ −ψ

= (−1)s−a−d−q ∑(−1)x−ξ (2x + 1) ( xξ

a x d bxc axd bxc )( ){ }{ }. α −ξ δ β ξ γ spq srq

(6.169)

For Eq. (6.168b), this formula implies that (−1)` (−1)m2 +m2 +m3 +m3 ( ′

∑

′

m2 ,m′2 ,m3 ,m′3

×(

1 0 1 1 ` `+1 )( ) m3 0 m2 −m2 0 m′3

`+1 ` 1 1 0 1 )( ) −m′3 0 m′2 −m′2 0 −m3

(−1)` 000 `0` 000 ` 0 ` )( ){ }{ }= . 000 000 111 1 `+1 1 3 (2` + 1) For the term in Eq. (6.168c), this formula means that =(

∑

(−1)` (−1)m2 +m2 +m3 +m3 ( ′

m2 ,m′2 ,m3 ,m′3

′

1 0 1 1 ` `−1 )( ) m3 0 m2 −m2 0 m′3

(−1)` `−1 ` 1 1 0 1 ) ( ) = . −m′3 0 m′2 −m′2 0 −m3 3 (2` + 1) Finally, we obtain a compact expression for ∆E, 2 α2 1 (3) ∆E = (R1sn1 p ) ∑∑ 3 n1 n2 E1s (Z) − En1 (Z) ×(

×[

(6.170)

2 2 `+1 ` (0) (0) (Rn`n2 (`+1) ) + (Rn`n2 (`−1) ) ] . 2` + 1 2` + 1

(6.171)

(6.172)

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One may observe that ∑[ n2

2 2 `+1 ` 1 (0) (0) (Rn`n2 (`+1) ) + (Rn`n2 (`−1) ) ] = ⟨n` ∣ 4 ∣ n`⟩ . 2` + 1 2` + 1 r2

(6.173)

A remark might be in order. One might think that the right-hand side is divergent for S states (` = 0), while the left-hand side of the equation features only convergent matrix elements. However, the divergence of the left-hand side is relegated, in this case, to the divergence introduced when one sums over all virtual states. Finally, one rewrites the energy perturbation as 2

∆E = −

⎛ 2 3 1 ∣⟨1s ∣xi1 ∣ n1 pm⟩∣ ⎞ 1 α2 ⟨n` ∣ 4 ∣ n`⟩ , ∑ ∑ 2 r2 En1 − E1s ⎝ 3 i=1 m=−1 ⎠

(6.174)

where we have re-introduced the sum over the Cartesian coordinates xi1 with i = 1, 2, 3 in order to make the connection to the polarizability (see Chap. 4) explicit. Indeed, we identify the scaled static polarizability of the hydrogenlike core [see Eqs. (4.154) and (5.75a)] as 2

i 9e2 2e2 3 1 ∣⟨1s ∣x1 ∣ n1 pm⟩∣ α1s (ω = 0) = = , ∑ ∑ 3 i=1 m=−1 En1 − E1s 2α4 m3

(6.175)

where we have used the fact that Z = 1 for the hydrogenlike core. The interaction energy can be written as an expectation value, proportional to the core polarizability α(ω = 0) and given in terms of the 1/r4 matrix element, evaluated on the Rydberg state, ∆E = ⟨n` ∣∆V ∣ n`⟩ ,

∆V ≈ −

α(ω = 0) α2 . 2 e2 r4

(6.176) (0)

This energy shift should be added to the unperturbed energy E1s,n` = E1s (Z) + En ` (Z − 1), as given in Eq. (6.160). The total energy thus is the sum of the inner ground-state electron, the outer Rydberg electron (which sees a screened charge Z − 1), and the interaction term, E1s,n` = −

Z 2 α2 m (Z − 1)2 α2 m α(ω = 0) α2 − ⟨n` ∣ 4 ∣ n`⟩ . − 2 2 2 2n 2e r

(6.177)

If desired, an explicit expression for the matrix element of the 1/r4 operator can be found in Eq. (4.346). An obvious generalization of Eq. (6.177) holds for general states with a Rydberg electron orbiting a neutral, polarizable core, and for a general long-range ion–atom interaction.

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Further Thoughts

Here are some suggestions for further thought. (1) Noether Theorem. Let us remember the physical foundation for the angular momentum being a conserved quantity. Let us study the Lagrangian 1 L = m v⃗ 2 − V (∣⃗ r∣) , (6.178) 2 ⃗ We on the level of classical mechanics. It does not depend on the angles ϕ. identify the conserved quantities as 3 ∂L ∂xi ∂L ∂q m → Q = . (6.179) ∑ i m ∂ q˙ ∂ϕm i=1 ∂v ∂ϕ Show that the rotation by a small vector of angles ϕ⃗ can be accounted for by the change of coordinates i

δxi = − (ϕ⃗ × r⃗) = −ijk ϕj xk .

(2)

(3) (4)

(5) (6)

(7)

(6.180)

Furthermore, identify the conserved quantity as the angular momentum, e.g., as follows, ∂L ∂x` m Qm = ` = m v ` (−`mk xk ) = −mk` xk m v ` = − (⃗ x × p⃗) = −Lm . m ∂v ∂ϕ (6.181) Properties of Spin-Angular Functions. Verify that Eq. (6.94) implies that c2 = −1 for both signs in the relation j = ` ± 1/2. Then, prove the second identity listed in Eq. (6.85). Tensorial Decomposition of the Light Shift Operator. Repeat the analysis presented in Sec. 6.5.1 for the xx component of the light-shift operator. Gauge-Invariance of the Light Shift. Carry out the calculation in Sec. 6.5.1 in the velocity as opposed to the length gauge. Show gauge invariance of the angle-resolved matrix elements of the light shift operator. Static Polarizability. Show Eq. (6.175) based on Eqs. (4.154) and (5.75a). Rydberg Electron and Core. Exercise: Convert Eq. (6.177), which is given ̵ = 0 = c = 1), to atomic units and show that all factors of α in natural units (h disappear, i.e., that 1 Z 2 (Z − 1)2 α(ω = 0)∣a.u. − − ⟨n` ∣ 4 ∣ n`⟩∣ , (6.182) E1s,n` ∣a.u. = − 2 2n2 2 r a.u. where “a.u.” denotes a quantity, expressed in atomic units. Find the obvious generalization for general states with a Rydberg electron orbiting a neutral, polarizable core. Multi-Electron Core. Generalize the derivation outlined in Sec. 6.6 to the interaction of a Rydberg electron with a general, multi-electron, core, by adding the interactions with all core electrons. Show, explicitly, as anticipated, that the general result is still equivalent to Eq. (6.177), with an obvious generalization of α(ω = 0) to the core polarizability.

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Chapter 7

Free Dirac Equation

7.1

Overview

The Dirac equation plays a special role in the analysis of free and interacting spin1/2 particles, and in particular, in bound-state problems. The free Dirac equation is naturally obtained as the linearized Klein–Gordon equation, i.e., as a relativistically invariant wave equation describing spin-1/2 particles involving only first-order derivatives. The Dirac equation is separately charge-conjugation (C) invariant, parity (P) invariant, and invariant under time reversal (T ); this generic statement will be supported by concrete calculations in the following. The Dirac equation is Lorentz-covariant, but showing the covariance is a little nontrivial. Namely, one first has to clarify how a four-component bispinor wave function behaves under Lorentz transformations. While the Dirac equation was originally proposed in Refs. [82, 83], its solutions were found later. Good quantum numbers are either the plane-wave momentum and the spin projection in the rest frame, or the plane-wave momentum and the helicity, which measures the spin direction of the particle with respect to its propagation direction. We here start with a brief discussion of the properties of the Dirac equation (Sec. 7.2) and then go on to study the plane-wave solutions in Sec. 7.3.1, in the helicity basis. In the plane-wave and the angular-momentum basis, we find the solutions in Sec. 7.3. Finally, propagators are found in Sec. 7.4. 7.2 7.2.1

Properties of the Free Dirac Equation Dirac Equation as the Linearized Klein–Gordon Equation

̵ = c = 0 = 1. We At this stage, it is useful to recall that we use units with h start from the Schr¨ odinger equation, calculate its conserved probability current, and then proceed to investigate the Klein–Gordon equation which is a relativistic Lorentz-invariant wave equation for spinless particles. The known difficulty of negative-probability states of the Klein–Gordon equation is briefly discussed. The Dirac equation is motivated by the necessity to obtain a first-order time translation operator (Hamiltonian) whose eigenstates nevertheless fulfill a Lorentz-invariant dispersion relation. 227

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The free Schr¨ odinger equation is obtained from the nonrelativistic dispersion 2 ⃗ The corresponrelation E = p⃗ /(2m), via the identifications E → i∂t and p⃗ → −i∇. dence principle implies that we should identify classical physical quantities with quantum mechanical operators. Indeed, the time-dependent Schr¨odinger equation reads ⃗2 ∇ ∂ φ(t, r⃗) , (7.1) i φ(t, r⃗) = − ∂t 2m where φ(t, r⃗) is the time-dependent wave function in the coordinate representation. The charge and current densities ρ and J⃗ read as ↔ ⃗ r⃗) = − i φ∗ (t, r⃗) ∇ φ(t, r⃗) , J(t, 2m where the antisymmetric differential operator acts as follows,

ρ(t, r⃗) = φ∗ (t, r⃗) φ(t, r⃗) = ∣φ(t, r⃗)∣ , 2

↔

⃗ r) − b(⃗ ⃗ r) . a(⃗ r) ∇ b(⃗ r) = a(⃗ r)∇b(⃗ r)∇a(⃗

(7.2)

(7.3)

The Schr¨ odinger equation implies the conservation law ∂ ⃗ r⃗) = 0 , ⃗ ⋅ J(t, ρ(t, r⃗) + ∇ (7.4) ∂t which is a continuity equation, so that we can interpret the positive definite quantity ρ(t, r⃗) = φ∗ (t, r⃗) φ(t, r⃗) as a probability density. Generalizing the treatment to the relativistic domain, in view of the relativistic dispersion relation E 2 = p⃗ 2 + m2 , one is immediately led to the Klein–Gordon equation (

∂2 ⃗ 2 + m2 ) φ(t, r⃗) = 0 , −∇ ∂t2

(∂ µ ∂µ + m2 ) φ = 0 ,

(7.5)

where the latter form is obtained in a relativistic notation (see Chap. 8 of Ref. [81], ⃗ has the µ = 0, 1, 2, 3 is a Lorentz index). The four-vector current J µ = (J 0 , J) components ↔

J 0 (t, r⃗) = i φ∗ (t, r⃗) ∂t φ(t, r⃗) ,

↔

⃗ r⃗) = −i φ∗ (t, r⃗) ∇ φ(t, r⃗) , J(t,

(7.6)

where ↔

a(t) ∂t b(t) = a(t) ∂t b(t) − b(t) ∂t a(t) .

(7.7)

The continuity equation then reads as ∂ ⃗ r⃗) = 0 , ⃗ ⋅ J(t, ρ(t, r⃗) + ∇ ∂µ J µ (x) = 0 . (7.8) ∂t One may easily realize that there is a problem connected with “negative probabilities” if we take √ into account that the Klein–Gordon equation has solutions proportional to (E = p⃗2 + m2 > 0) exp [i(E t − k⃗ ⋅ r⃗)] = exp(ik ⋅ x) ,

exp [−i(E t − k⃗ ⋅ r⃗)] = exp(−ik ⋅ x) .

(7.9)

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√ Here, E = k⃗ 2 + m2 is the modulus of the energy of the particle, and k ⋅ x = k µ xµ = E t − k⃗ ⋅ r⃗. The first factor in Eq. (7.9) gives rise to a negative energy eigenvalue −E when the time derivative operator (Hamilton operator) i ∂t acts on it, i.e., one would have ↔

J 0 (t, r⃗) = i φ∗ (t, r⃗) ∂t φ(t, r⃗) < 0 ,

φ ∝ exp [i(E t − k⃗ ⋅ r⃗)] .

(7.10)

As a consequence, solutions proportional to exp(ik⋅x) correspond to negative probability (negative spatial integral of J 0 ), which is physically unacceptable and suggests that the “probability” should be reinterpreted as a physical quantity proportional to the charge density of antiparticles (which carry the opposite sign of the charge as compared to particles). The Dirac equation was invented because of the necessity to unify the advantages of the Schr¨ odinger equation (namely, positive-definite probability density and existence of a well-defined first-order time translation operator) with the advantages of the Klein–Gordon equation, namely, the relativistic invariance. Let us consider the quantity X = γ 0 E − γ x px − γ y py − γ z pz = γ 0 E − γ 1 p1 − γ 2 p2 − γ 3 p3 = γ 0 E − γ⃗ ⋅ p⃗ = γ µ pµ ,

(7.11)

where p0 = E, and γ µ may be matrix-valued (and hence noncommutative under multiplication). We also distinguish from here on between the covariant (lower) and contravariant (upper) indices. Let us also postulate that the square of X should be equal to X 2 = (E 2 − p⃗ 2 ) 1n×n ,

(7.12)

where we are pedantic in noting that the unit matrix has to be obtained as an overall factor (the dimension n is fixed at n = 4, as we shall see in the following). By inspection, we realize that the following anticommutator relations [see Eq. (2.49)], {γ µ , γ ν } = 2 g µν ,

gµν = diag(1, −1, −1, −1) ,

µ, ν = 0, 1, 2, 3 ,

(7.13)

are sufficient for the expression given in Eq. (7.11) to fulfill Eq. (7.12). By elementary calculations, one can show that the set of matrices (the so-called “Dirac representation”) γ0 = (

12×2 0

0 ), −12×2

γ⃗ = (

⃗ 0 σ ), −⃗ σ0

(7.14)

fulfill the anticommutator relations (7.13). The Pauli spin matrices have been given in Eq. (4.343). For completeness, we also mention the γ 5 matrix, γ5 = i γ0 γ1 γ2 γ3 = (

0 12×2 ), 12×2 0

{γ 5 , γ µ } = 0 .

(7.15)

For reasons to be discussed below, the γ 5 matrix is sometimes called the “fifth current.” It anticommutes with all four other Dirac “current” matrices γ µ (where

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µ = 0, 1, 2, 3) and has the property (γ 5 )2 = 1. This situation raises a pertinent question: The upper index of the Dirac γ matrices is a Lorentz index. So, the vector of γ matrices has to transform like a Lorentz vector under Lorentz transformations. However, one might also counter-argue that the entire formalism is questionable if the explicit form of the γ matrices should change under a Lorentz transformation. One might otherwise ask what the “choice” means if it needs to change upon a change of the Lorentz frame. The solution is that under Lorentz transformation, the Dirac matrices not only transform with regard to their Lorentz index, but also with regard to the internal spinor structure. The combined action of the Lorentz transformation on the vector index and on the spinor structure then leads to the invariance of the Dirac matrices under Lorentz transformations. The Dirac equation is finally written down as (i γ µ ∂µ − m) ψ(x) = 0 ,

x = (t, r⃗) .

(7.16)

The price to pay for a linearized relativistically invariant wave equation is the introduction of a multi-component wave function, i.e., of an “internal spinor space” of the particle. Namely, the wave function ψ necessarily has to be formulated in terms of a four-component entity, which we will later identify as a bispinor (composed of two spinors with two components each). Multiplication by γ 0 from the left brings the Dirac equation (7.17) into noncovariant, Hamiltonian form, ∂ ⃗ + β m) ψ(t, r⃗) = HFD ψ(t, r⃗) , α⋅∇ (7.17) i ψ(t, r⃗) = (−i⃗ ∂t where the free Dirac (FD) Hamiltonian is given as ⃗ +βm=α ⃗ ⋅ p⃗ + β m , HFD = −i⃗ α⋅∇

(7.18)

with β = γ0 = (

12×2 0 ), 0 12×2

αi = γ 0 γ i = (

0 σi ), σi 0

i = 1, 2, 3 .

(7.19)

⃗ is the momentum The free Dirac Hamiltonian HFD is matrix-valued, and p⃗ = −i ∇ operator. From this representation, it is clear that the momentum operator p⃗ com⃗ However, mutes with the Hamiltonian. We denote the momentum eigenvalue as k. a specification of k⃗ is not sufficient in order to characterize the eigenstate. In view of the matrix-valued character of the Dirac Hamiltonian, one needs at least one additional quantum number, which is connected with the spin. One may characterize the spin by its asymptotic value for a particle at rest (see Chap. 2 of Ref. [2]). This is commonly referred to as the rest-mass basis. ̂ for a relativistic particle is the normalized projection of The helicity operator Σ ̂ the particle’s spin onto the propagation direction. We write the helicity operator Σ ⃗ in terms of the (4 × 4)-spin matrices Σ, ⃗ ̂ = Σ ⋅ p⃗ , Σ ∣⃗ p∣

⃗ 0 ⃗ = γ 5 γ 0 γ⃗ = ( σ Σ ). ⃗ 0σ

(7.20)

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̂ operator are the helicities σ = ±1. The vector of spin The eigenvalues of the Σ ⃗ matrices Σ finds the following covariant generalization in terms of the σ µν matrices, i µ ν [γ i , γ j ] = −2 i ijk Σk . [γ , γ ] , σ ij = ijk Σk , (7.21) 2 The helicity operator commutes with the Hamiltonian HFD . This can be seen as follows. Since the momentum operator (all components) commutes with itself, we have to consider the matrix commutators of the Σi with the matrices αj and β, ̂ α ⃗ ⋅ p⃗] reads as which occur in the Dirac Hamiltonian HFD . The commutator [∣⃗ p∣Σ, follows, σ µν =

[Σi pi , αj pj ] = (γ 5 γ 0 γ i γ 0 γ j − γ 0 γ j γ 5 γ 0 γ i ) pi pj = (−γ 5 γ i γ j − γ j γ 5 γ i ) pi pj = 2iγ 5 ijk Σk pi pj = 0 .

(7.22)

Here, the identities {γ 5 , γ µ } = 0 ,

{γ 5 , γ 0 } = {γ 5 , γ i } = 0 ,

(7.23)

have been used repeatedly. Furthermore, one can show the identity [Σi pi , β] = [γ 5 γ 0 γ i , γ 0 ] pi = 0 .

(7.24)

Using the results (7.22) and Eq. (7.24), it follows immediately that ̂ HFD ] = [Σ,

1 ⃗ [Σ ⋅ p⃗, α ⃗ ⋅ p⃗ + β m] = 0 . ∣⃗ p∣

(7.25)

A complete set of good quantum numbers is thus identified as the helicity σ = ±1 and the momentum k⃗ ∈ R3 . One can uniquely characterize states according to these quantum numbers. 7.2.2

Spinor Lorentz Transformation

As outlined in Chap. 8 of Ref. [81], a Lorentz transformation involves the generator matrices (Mαβ )µ ν where α and β designate the matrix, while µ and ν characterize the components, (Mαβ )µ ν = g µ α gνβ − g µ β gνα .

(7.26)

The Lorentz transformation Λµ ν , which has the defining property gµν Λµ ν Λν δ = gρδ , is obtained as the matrix exponential µ 1 1 Λµ ν = exp [ ω αβ (Mαβ )µ ν ] = (exp [ ω αβ Mαβ ]) , 2 2 ν

(7.27)

so that, symbolically, one has Λ = exp [ 12 ω αβ Mαβ ]. Here, the expression exp [ 12 ω αβ (Mαβ )µ ν ] needs to be interpreted as follows. One first multiplies the generalized rotation angles ω αβ with the generator matrices Mαβ , where the components of each individual Mαβ are taken to be the Lorentz components of upper index µ and lower index ν. One then exponentiates the resultant matrix with components 21 ω αβ (Mαβ )µ ν and obtains a Lorentz transformation matrix which can be

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multiplied with a Lorentz vector component xν , to obtain the transformed vector component x′µ . The symbols ω αβ = −ω βα denote the generalized rotation angles in fourdimensional space-time. The generator matrices fulfill the following commutator relations, [Mµν , Mκλ ] = gµλ Mνκ + gνκ Mµλ − gµκ Mνλ − gνλ Mµκ .

(7.28)

A Lorentz vector with components X ν transforms as follows, X ′µ = Λµ ν X ν .

(7.29)

In order to formulate the spinor Lorentz transformation, one simply exchanges the Lorentz matrices Mαβ with spin matrices [−(i/2)σαβ ], i [γα , γβ ] . 2

(7.30)

[ 21 σµν , 12 σκλ ] = i (gµλ 21 σνκ + gνκ 12 σµλ − gµκ 12 σνλ − gνλ 12 σµκ ) ,

(7.31)

i S(Λ) = exp (− ω αβ σαβ ) , 4

σαβ =

The σ matrices fulfill the commutation relations

and it follows that the algebraic relations fulfilled by the matrices [−(i/2)σαβ ] are analogous to those listed in Eq. (7.28). For infinitesimal rotation angles ω µ ν , the Lorentz transformation matrix elements for vectors and spinors therefore read as Λµ ν = g µ ν + ω µ ν ,

S(Λ) = 1 −

i αβ ω σαβ . 4

(7.32)

Under a spinor Lorentz transformation, the four-component wave function ψ(x) does not simply transform as a scalar, which would otherwise imply that ? ψ ′ (x′ ) = ψ(x), but rather transforms as ψ ′ (x′ ) = S(Λ) ψ(x) ,

(7.33)

where S(Λ) is the additional spinor transformation factor. From here on, we will use the shorthand notation x = (xµ ) = (t, r⃗) in order to denote the argument of the Dirac bispinor wave function, and in other contexts as well. In view of the identity + γ 0 (γ µ ) γ 0 = γ µ , it is easy to check that the Dirac adjoint ψ(x) transforms as follows, ψ(x) = ψ + (x) γ 0 ,

′

ψ (x′ ) = ψ(x) S(Λ)−1 .

(7.34)

The net result is that under Lorentz transformations, the Dirac matrix γ µ receives two contributions under a Lorentz transformation, one from the spinor index, and one from the Lorentz index, γ ′µ = Λµ ν S(Λ) γ ν S(Λ)−1 = γ µ .

(7.35)

We here anticipate the result that the matrix actually is shape-invariant under the full Lorentz transformation, γ ′µ = γ µ . This result is perhaps surprising and relies

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on two additional identities, which are shown here as an exercise in spinor algebra. First, the commutator of the γµ matrix with the spin matrix is calculated as follows, [γµ , γα γβ − γβ γα ] = γµ γα γβ + γβ γα γµ − γα γβ γµ − γµ γβ γα = {γµ , γα } γβ − γα γµ γβ + γβ {γα , γµ } − γβ γµ γα − γα {γβ , γµ } + γα γµ γβ − {γµ , γβ } γα + γβ γµ γα = 4 gµα γβ − 4 gµβ γα .

(7.36)

It follows that i (γα γβ − γβ γα )] = 2 i gµα γβ − 2 i gµβ γα . (7.37) 2 Armed with this result, we can easily calculate the transformation of the Dirac matrix under an infinitesimal Lorentz transformation, [γµ , σαβ ] = [γµ ,

γ ′µ (x) = Λµ ν S(Λ) γ ν (x) S(Λ)−1 ≈ Λµ ν (1 −

i αβ i ω σαβ ) γ ν (1 + ω γδ σγδ ) 4 4

i αβ ω [σαβ , γ ν ]) 4 i (7.38) = (g µ ν + ω µ ν ) (γ ν + ω αβ (2 i g ν α γβ − 2 i g ν β γα )) 4 1 1 = (g µ ν + ω µ ν ) (γ ν − ω νβ γβ + ω αν γα ) = (g µ ν + ω µ ν ) (γ ν − ω ν β γ β ) = γ µ . 2 2 In each step, we have expanded to first order in the infinitesimal parameters ω µν . Furthermore, we have used the fact that ω νµ = −ω µν . We can finally supply a well-defined current for the Dirac equation, = (g µ ν + ω µ ν ) (γ ν −

J µ (x) = ψ(x) γ µ ψ(x) ,

J ′µ (x′ ) = Λµ ν J ν (x) ,

∂µ J µ (x) = 0 ,

(7.39)

which transforms as a Lorentz vector. The γ µ matrices extract the components of the current J µ . 7.2.3

Discrete Symmetries

It is interesting to investigate the transformation properties of the Dirac equation under charge conjugation, parity transformation, and time reversal. We start with brief discussions of the transformations associated with parity and time reversal, and then describe the effects of the three transformations on the Dirac equation. Because the Dirac equation entails a coupling to the electromagnetic field (or, electromagnetic four-vector potential), we need to start by investigating the transformation properties of Aµ (x) under parity and time reversal. We recall that the ⃗ and the magnetic induction field B ⃗ are derived as follows: electric field E ⃗ = −∇Φ ⃗ − ∂t A⃗ , E

⃗=∇ ⃗ × A⃗ . B

(7.40)

⃗ → −E ⃗ (as it is a vector), while B ⃗ is invariant (being a pseudoUnder parity, E ⃗ vector). Because ∇ is a vector and changes sign under parity, we must have A⃗ → −A⃗

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under parity, while Φ is invariant. Denoting the quantities transformed under parity with a superscript P, we can formalize this as follows, ΦP (x) = Φ(xP ) ,

⃗ P) , A⃗P (xP ) = −A(x

xP = (t, −⃗ r) .

(7.41)

For time reversal, we consider the Lorentz force law, ⃗ + v⃗ × B). ⃗ F⃗ = q (E

(7.42)

⃗, where a ⃗ = v⃗˙ is the acceleration. The derivative Nonrelativistically, one has F⃗ = m a of v⃗, which is the acceleration, thus is time-reversal invariant. One might otherwise conjecture that since a trajectory flips under time reversal, so should the acceleration. However, one might consider the time reversal of a trajectory which describes an object falling in the gravitational field. Under time reversal, one would see an object shooting upward, but with decreasing upward velocity. This is consistent with the acceleration due to gravity being time-reversal invariant. Hence, under ⃗ → E ⃗ and since v⃗ → −⃗ ⃗ → −B. ⃗ time reversal, E v under time reversal, one has B Using Eq. (7.40), one may formalize these considerations as follows (T denoted the quantities transformed under time reversal), ΦT (x) = Φ(xT ) ,

⃗ T ), A⃗T (xT ) = −A(x

xT = (−t, r⃗) .

(7.43)

We begin our discussion of the transformations of the Dirac equation with charge conjugation. From nonrelativistic quantum mechanics, we know that the electro⃗ where A⃗ magnetic coupling can be described by the covariant coupling p⃗ → p⃗ − e A, is the vector potential of the electromagnetic field. The electromagnetically coupled Dirac equation reads as [γ µ (i∂µ − e Aµ ) − m] ψ = 0 ,

(7.44)

⃗ with the scalar potenwhere the four-vector potential has components A = (Φ, A) ⃗ tial Φ and the vector potential A (see Chap. 8 of Ref. [81]). Under transposition and complex conjugation, one obtains ← Ð + ψ + ((γ µ ) (−i ∂ µ − e Aµ ) − m) = 0 , (7.45) µ

where the differential operator acts on the ψ to the left. Insertion of the γ 0 matrix leads to ← Ð + (ψ + γ 0 ) γ 0 ((γ µ ) (−i ∂ µ − e Aµ ) − m) γ 0 = 0 . (7.46) +

One now uses the familiar identity γ 0 (γ µ ) γ 0 = γ µ to bring the equation into the form ← Ð ψ¯ (γ µ (−i ∂ µ − e Aµ ) − m) = 0 . (7.47) Under an additional transposition, one obtains T ((γ µ ) (−i∂µ − e Aµ ) − m) ψ¯T = 0 .

(7.48)

One introduces the charge conjugation matrix C with the defining property C (γ µ ) C −1 = −γ µ , T

(7.49)

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so that the charge conjugated spinor is ψ C (x) = C ψ¯T (x). The charge conjugated Dirac equation fulfilled by ψ C (x) thus reads as [γ µ (i∂µ + e Aµ ) − m] ψ C (x) = 0 .

(7.50)

It differs from the ordinary Dirac equation (7.44) only in the sign of the physical charge e of the particle. In the Dirac representation, a possible choice is C = i γ 2 γ 0 . Charge conjugation dictates that the sign of this charge, too, flips under charge conjugation, so we can write the charge-conjugated Dirac equation in terms of eC = −e as follows, [γ µ (i∂µ − eC Aµ ) − m] ψ C (x) = 0 .

(7.51)

This form of the equation is formally identical to the original Dirac equation, but formulated in terms of the charge-conjugated quantities, and hence shows the chargeconjugation invariance of the Dirac equation. In order to discuss parity, we start from the Dirac equation, [γ µ (i∂µ − e Aµ ) − m] ψ(x) = [γ 0 (i∂0 − e Φ) + γ i (i∂i − e Ai ) − m] ψ(x) = 0 , (7.52) where i = 1, 2, 3 is a spatial index. Under parity, x → xP = (t, −⃗ r), while ∂i → −∂i , and so the parity-transformed wave function ψ P (x) needs to fulfill the equation P [γ 0 (i∂0 − e ΦP (x)) + γ i (−i∂i + e AP i (x)) − m] ψ (x) = 0 .

(7.53)

One introduces a parity transformation matrix P , with the property P γ 0 P −1 = γ 0 ,

P γ i P −1 = −γ i ,

ψ P (x) = P ψ(xP ) ,

(7.54)

where again i = 1, 2, 3 is spatial. In the Dirac representation, we may choose P = γ 0 with P −1 = P . The Dirac equation may thus be parity-transformed as follows, −1 P [γ 0 (i∂0 − e ΦP (x)) + γ i (−i∂i + e AP P ψ(xP ) = 0 . i (x)) − m] P

(7.55)

P

The parity-transformed bispinor ψ (x) = P ψ(xP ) fulfills the parity transformed Dirac equation, which in view of Eq. (7.41) reads as [γ 0 (i∂0 − e Φ(xP )) + γ i (i∂i − e Ai (xP )) − m] P ψ(xP ) = 0 ,

(7.56)

which can be written as [γ µ (i∂µ − e Aµ (xP )) − m] ψ P (x) = 0 .

(7.57)

The Dirac equation is thus seen to be P invariant. The time reversal of the Dirac equation is treated next. Again, we start from the ordinary Dirac equation, [iγ µ (∂µ − e Aµ ) − m] ψ(x) = 0 . Transposition and complex conjugation leads to ← Ð + ψ + (x) ((γ µ ) (−i ∂ µ − e Aµ ) − m) = 0 .

(7.58)

(7.59)

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The insertion of the γ 0 matrix leads to the Dirac adjoint [see Eq. (7.34)], ← Ð + (ψ + (x) γ 0 ) [γ 0 ((γ µ ) (−i ∂ µ − e Aµ ) − m) γ 0 ] = 0 ,

(7.60)

which is equivalent to ← Ð ψ(x) [γ µ (−i ∂ µ − e Aµ ) − m] = 0 .

(7.61)

We here use the fact that a proper formulation of time reversal must respect the fact that the in and out states interchange under T , so that it becomes indispensable to consider the adjoint of Dirac equation (see Sec. 4.4.2 of Ref. [248]). Again, we transpose and write T [(γ µ ) (−i∂µ − e Aµ (x)) − m] ψ¯T (x) = 0 .

(7.62)

We introduce the time reversal matrix T , with the properties (T means time reversal, T is the time reversal matrix, and T denotes the transpose), T

T

T (γ 0 ) T −1 = γ 0 ,

T (γ i ) T −1 = −γ i ,

ψ T (x) = T ψ¯T (xT ) ,

(7.63)

where the time-reversed bispinor is denoted as ψ T (x). We now carry out an explicit time-reversal operation with xT = (−t, r⃗) on Eq. (7.62), use the transformation ∂t → −∂t (under time reversal) and write T T T [(γ 0 ) (i∂t − e ΦT (x)) + (γ i ) (−i∂i − e ATi (x)) − m] T −1 T ψ¯T (xT ) = 0 . (7.64)

Using Eq. (7.43), one obtains [γ 0 (i∂t − e Φ(xT )) + (−γ i ) (−i∂i + e Ai (xT )) − m] T ψ¯T (xT ) = 0 ,

(7.65)

and thus [γ µ (i∂µ − e Aµ (xT )) − m] ψ T (x) = 0 ,

(7.66)

which is formally equal to the original Dirac equation. Thus, we derive the T invariance. In the Dirac representation, we may choose T = i γ 5 γ 2 . It is extremely instructive to calculate the time reversal of the fifth current, based exclusively on the property (7.63), T

T

T

T

T

T (γ 5 ) T −1 = T i (γ 3 ) (γ 2 ) (γ 1 ) (γ 0 ) T −1 T

T

T

T

= iT (γ 3 ) T −1 T (γ 2 ) T −1 T (γ 1 ) T −1 T (γ 0 ) T −1 = i (−γ 3 ) (−γ 2 ) (−γ 1 ) (+γ 0 ) = −iγ 3 γ 2 γ 1 γ 0 = −iγ 0 γ 1 γ 2 γ 3 = −γ 5 . (7.67) The literature is not always consistent in regard to the minus sign, and it is instructive to clarify its origin.

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7.2.4

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Overview of the Symmetry Properties

It is instructive to present an overview. We recall the charge conjugation matrix C, the parity transformation matrix P , and the time reversal matrix T , C (γ µ ) C −1 = −γ µ , T

C = i γ2 γ0 , P =γ ,

Pγ P

T = i γ5 γ2 ,

0 T

0

0

−1

(7.68a)

=γ ,

−1

= −γ ,

(7.68b)

T (γ ) T −1 = −γ i .

(7.68c)

0

i

Pγ P

i

i T

T (γ ) T −1 = γ 0 ,

The matrices C, P and T fulfill the relations C T = C −1 , P T = P −1 , and T T = T −1 , where we denote the transpose by T. In view of the property {γ µ , γ 5 } = 0, their product can be written as CP T = −γ 5 .

(7.69)

(CP T )γ µ (CP T )−1 = −γ µ

(7.70)

This implies that

[see page 239 of Ref. [5] and Eq. (40.48) of Ref. [249]]. The transformation of the wave function reads as follows, ψ C (x) = C ψ¯T (x) ,

ψ P (x) = P ψ(xP ) ,

ψ T (x) = T ψ¯T (xT ) ,

(7.71)

where xT = −xP = (−t, r⃗). This implies that xPT = xCPT = −x, because charge conjugation does not affect the coordinate. Based on the result ψ PT (x) = iγ 0 γ 5 γ 2 γ 0 ψ ∗ (xPT ) = iγ 5 γ 2 ψ ∗ (xPT ) ,

(7.72)

where the first γ 0 is due to P, the second due to T , one can show that ∗

ψ CPT (x) = (iγ 2 γ 0 ) γ 0 (−i γ 5 (γ 2 ) ψ(xCPT )) 2

= γ 2 γ 5 (−γ 2 )ψ(xCPT ) = (γ 2 ) γ 5 ψ(xCPT ) = − γ 5 ψ(xCPT ) = −γ 5 ψ(−x) .

(7.73)

The combined action of CPT involves two complex conjugations. The CPT invariance of the Dirac equation can thus be shown as follows. We first carry out the transformation x → −x = xPT in the Dirac equation, [γ µ (−i∂µ − e APT µ (x)) − m] ψ(−x) = 0 .

(7.74)

APT µ

We then identify, according to Eqs. (7.41) and (7.43), (x) = Aµ (−x), and e = −eC = −eCPT , in view of the charge conjugation, observe that the coordinate does not change under charge conjugation, and insert a γ 5 matrix, γ 5 [γ µ (−i∂µ + eCPT Aµ (xCPT )) − m] γ 5 (−γ 5 )ψ(−x) = 0 .

(7.75)

Because γ 5 anti-commutes with all γ µ matrices, we have [γ µ (i∂µ − eCPT Aµ (xCPT )) − m] ψ CPT (x) = 0 .

(7.76)

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This shows the CPT invariance of the Dirac equation and explains the minus sign in Eq. (7.70). Furthermore, it is instructive to recall the transformation properties of the fifth current, T

C (γ 5 ) C −1 = γ 5 ,

P γ 5 P −1 = −γ 5 ,

T

T (γ 5 ) T −1 = −γ 5 .

(7.77)

We note that the transformation property for time reversal depends on our precise formulation of the time reversal operation as detailed in Sec. 7.2.3. It is also useful to remark that all above considerations need to be adjusted when formulated in terms of the (second-quantized) Dirac field operator as opposed to the (first-quantized) Dirac wave function ψ, with potential confusion because both quantities are usually denoted by the same letter ψ. Regarding this point, we also refer to Chaps. 2 and 3 of Ref. [2] and Chap. 7 of Ref. [250], and to Ref. [251], as well as Sec. 7.5. Roughly speaking, it is interesting to observe that it is only due to the anticommutativity of the second-quantized field operators that the overall sign of the Dirac Lagrangian is restored under CPT . Otherwise, the minus sign in Eq. (7.70) would lead to a counter-intuitive reversal of the overall sign of the Dirac Lagrangian under CPT . 7.3 7.3.1

Solutions of the Free Dirac Equation Plane-Wave Solutions of the Dirac Equation

The Lorentz-invariant action for the Dirac particle reads as S = ∫ d4 x ψ(x) (i γ µ ∂µ − m) ψ(x) ,

(7.78)

with the Lagrangian density L(x) = ψ(x) (i γ µ ∂µ − m) ψ(x) .

(7.79)

Variation with respect to ψ(x) leads to the Dirac equation (i γ ∂µ − m) ψ(x). Upon ⃗ exp(−ik ⋅ x), the free Dirac using a plane-wave ansatz of the form ψ(x) = U (k) ⃗ = 0, equation is transformed into the form of an algebraic equation, (k
− m) U (k) µ ⃗ where the Feynman slash notation is introduced as k
= γ kµ . Here, U (k) is an (as yet) unspecified four-component spinor. One observes the algebraic identity µ

where

(k
− m) (k
+ m) = k 2 − m2 = 0 ,

(7.80)

√ ⃗ = ( k⃗2 + m2 , k) ⃗ k = (k 0 , k)

(7.81)

⃗ = (k
+ m) χ, is the momentum four-vector. The Dirac equation is solved by U (k) where χ is any constant four-component vector. Canonically, one constructs a representative set of solutions by selecting for χ the vectors which have one of the four entries equal to one, and the other entries equal to zero. This is described in Eqs. (2.36) and (2.37) of Ref. [2]. However, the use of this approach leads to a somewhat unclear situation with regard to the interpretation of the spin degree

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of freedom. Namely, while the solutions are “boosted” from the rest frame by the factor (k
+ m), the interpretation of the spin index of the solution in the laboratory frame becomes obscure. In the helicity basis, solutions of the free Dirac equation are labeled by the planewave momentum vector k⃗ and the helicity quantum number σ = ±1. We explore ̂=Σ ⃗ ⋅ p⃗/∣⃗ the action of the helicity operator Σ p∣, defined in Eq. (7.20). The helicity spinors are given by θ θ −i ϕ ⎛ cos ( 2 ) ⎞ ⎛ − sin ( 2 ) e ⎞ ⃗ ⃗ a+ (k) = , a− (k) = , (7.82) θ ⎝ sin ( θ ) ei ϕ ⎠ ⎝ ⎠ cos ( 2 ) 2 ⃗ The (2 × 2)-version where θ and ϕ are the polar and azimuth angles of the vector k. of the helicity operator acts on the helicity spinors as follows, ⃗ ⋅ k⃗ σ ⃗ = σ aσ (k) ⃗ , a (k) σ = ±1 . (7.83) ⃗ σ ∣k∣ One observes the property ⃗ ⋅ p⃗ ⃗ ⋅ k⃗ σ σ ⃗r ⃗r ⃗r ⃗ e±ik⋅⃗ ⃗ e±ik⋅⃗ ⃗ e±ik⋅⃗ aσ (k) = ±σaσ (k) , (7.84) aσ (k) =± ⃗ ∣⃗ p∣ ∣k∣ ⃗ Particle states are proportional to exp(−ik⋅x) = exp(−ik 0 t) exp(ik⋅ ⃗ r⃗), where p⃗ = ±k. 0 ⃗ while antiparticle waves are proportional to exp(ik ⋅ x) = exp(ik t) exp(−ik ⋅ r⃗). ⃗ is equal to the normalized projecHence, the helicity quantum number σ of aσ (k) tion of the momentum operator onto the spin of the particle, for positive-energy (particle) states, while a minus sign is incurred in the corresponding relation for negative-energy (anti-particle) states. After invoking the reinterpretation principle, which will be discussed in more detail in Sec. 7.4, one sees that σ is equal to the normalized projection of the physical momentum onto the spin, for both particles and antiparticles. The helicity spinors fulfill the following relations, ⃗ ⋅ k⃗ ⃗ ⊗ a+ (k) ⃗ =σ ⃗ ⊗ a+ (k) ⃗ = 12×2 , ∑ σ aσ (k) , (7.85) ∑ aσ (k) σ σ ⃗ ∣k∣ σ σ where the sum in each case runs over the possible values σ = ±1. The fundamental spinor solutions to the Dirac equation read as ⃗ exp(−ik ⋅ x) , ⃗ exp(ik ⋅ x) , ψ(x) = U± (k) φ(x) = V± (k) (7.86) where the four-vector k is of the structure given in Eq. (7.81). Here, U± describes a positive-energy solution, whereas V± describes a negative-energy solution. In view of the Dirac equation, the following algebraic relations have to be fulfilled by the ⃗ and V± (k), ⃗ bispinor amplitudes U± (k) ⃗ = 0, ⃗ = 0. (k
− m) U± (k) (k
+ m) V± (k) (7.87) The positive-energy solutions can be projected out using the identity (k
−m)(k
+m) = k 2 − m2 = 0. Let us define the massless bispinors ⃗ ⃗ ⎞ ⎛ a−√(k) ⎞ ⎛ a+√(k) 2 2 ⃗ ⃗ = −u+ (k), ⃗ ⃗ = −u− (k) ⃗ . ⃗ ⎟ , u− (k) = ⎜ a (k) ⎟ , v+ (k) u+ (k) = ⎜ a (k) v− (k) ⃗ ⃗ + − ⎝ √2 ⎠ ⎝ − √2 ⎠ (7.88)

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Denoting the energy as E, we can then write ⃗ = U+ (k)

⃗ = U− (k)

√

√

E m

E m

√ ⎛ E+m ⃗ ⎞ a+ (k) ⎜ ⎟ ⃗ 2m (k
+ m) u+ (k) ⎜ ⎟ ⎟, = ⎜√ √ ⎜ ⎟ 2 2 ⃗ +m ⎜ E−m (E − ∣k∣) ⃗ ⎟ a+ (k) ⎝ ⎠ 2m √ E+m ⎛ ⃗ ⎞ a− (k) ⎜ ⎟ ⃗ 2m (k
+ m) u− (k) ⎜ ⎟ ⎟. =⎜ √ √ ⎜ ⎟ ⃗ 2 + m2 ⎜ E−m (E − ∣k∣) ⃗ ⎟ − a ( k) − ⎝ ⎠ 2m

(7.89a)

(7.89b)

The negative-energy eigenstates are given as ⃗ = V+ (k)

⃗ = V− (k)

√

√

E m

E m

√ E−m ⎛ ⃗ ⎞ − a+ (k) ⎜ ⎟ ⃗ 2m (m − k
) v+ (k) ⎜ ⎟ ⎟, =⎜ √ √ ⎜ ⎟ 2 2 ⃗ +m ⎜ E+m (E − ∣k∣) ⃗ ⎟ a+ (k) ⎝− ⎠ 2m √ E−m ⎛ ⃗ ⎞ − a− (k) ⎜ ⎟ ⃗ 2m (m − k
) v− (k) ⎜ ⎟ ⎟. =⎜ √ √ ⎜ ⎟ ⃗ 2 + m2 ⎜ E+m (E − ∣k∣) ⃗ ⎟ a ( k) − ⎝ ⎠ 2m

(7.90a)

(7.90b)

The normalization conditions are ⃗ Uσ (k) ⃗ = 1, U σ (k)

⃗ Vσ (k) ⃗ = −1 V σ (k)

(7.91)

and ⃗ γ 0 Uσ′ (k) ⃗ = E, ⃗ γ 0 Vσ′ (k) ⃗ = −E . (7.92) U σ (k) V σ (k) m m The Dirac adjoint has been defined in Eq. (7.34). The following projector sum formulas are instrumental in the derivation of the propagator (see Sec. 7.4), ⃗ ⊗ U σ (k) ⃗ = ∑ Uσ (k) σ

k
+ m , 2m

⃗ ⊗ V σ (k) ⃗ = ∑ Vσ (k) σ

k
− m . 2m

(7.93)

A massless particle is described by the Dirac equation iγ µ ∂µ ψ(x) = 0. The free massless Dirac Hamiltonian reads as ⃗ ⋅ p⃗) = ∣⃗ ̂, ⃗ ⋅ p⃗ = γ 5 (γ 5 α ⃗ ⋅ p⃗) = γ 5 (Σ H =α p∣ γ 5 Σ

(7.94)

̂ is the helicity projection operator defined in Eq. (7.20). The fundamental where Σ bispinor solutions to the massless Dirac equation read as ⃗ exp(−ik ⋅ x) , ψ(x) = u± (k)

⃗ exp(ik ⋅ x) , φ(x) = v± (k)

(7.95)

where we refer to Eq. (7.88). The dispersion relation for the massless equation ⃗ In order for the energy to be negative, the eigenvalues of is E = k 0 = ∣⃗ p∣ = ∣k∣.

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̂ operator) have to have opposite sign, while chirality (γ 5 matrix) and helicity (Σ the same sign is incurred for positive energy. As an example, one may observe ⃗ exp(ik ⋅ x) has negative energy and negative chirality, but that the solution v− (k) ̂ while it is still composed of the a positive eigenvalue of the helicity operator Σ, ⃗ fundamental helicity spinors a− (k) which have negative eigenvalue in regard to the ⃗ k∣. ⃗ One may thus conclude that the quantum number ⃗ ⋅ k/∣ projection operator σ σ in the fundamental bispinors uσ and vσ is a quantum number which is equal to the (physical) helicity for particles, and equal to the negative of the (physical) helicity for antiparticles. This observation is crucial in recent attempts to formulate a consistent quantum theory for tachyonic neutrinos, in which case a certain set of curious identities helps one to quantize the theory so that neutrinos of negative helicity, and antineutrinos of positive helicity, are suppressed in view of Gupta– Bleuler conditions [252, 253]. The massless solutions (7.95) fulfill two projector sum formulas, which constitute analogues of those given in Eq. (7.93), ⃗ = k
, ⃗ uσ (k) ⃗ ⊗ uσ (k) ∑ 2 ∣k∣

⃗ vσ (k) ⃗ ⊗ v σ (k) ⃗ = k
, ∑ 2 ∣k∣

(7.96)

σ

σ

as well as ⃗ (−σ) uσ (k) ⃗ ⊗ uσ (k) ⃗ γ 5 = k
, ∑ 2 ∣k∣ σ

⃗ (−σ) vσ (k) ⃗ ⊗ v σ (k) ⃗ γ 5 = k
. (7.97) ∑ 2 ∣k∣ σ

⃗ uσ (k) ⃗ = In contrast to the normalization conditions (7.91), one has uσ (k) ⃗ ⃗ v σ (k) vσ (k) = 0 for the massless case. 7.3.2

Dirac Angular Quantum Number

In order to discuss the angular-momentum resolved eigenstates of the free Dirac equation, we recall from Eq. (4.348) the Dirac angular quantum number κ, which unifies the orbital angular momentum ` and the total angular momentum j. The eigenfunctions of the free Dirac Hamiltonian are neither eigenfunctions of the orbital ⃗ nor of the spin angular momentum S⃗ = Σ/2, ⃗ angular momentum L because neither of these operators commutes with the Hamiltonian (7.18). Indeed, one has ⃗ = −i (⃗ ⃗ . [HFD , L] α × p⃗) = −[HFD , 21 Σ]

(7.98)

⃗ 1Σ ⃗ thus constitutes a good quantum The total angular momentum operator J⃗ = L+ 2 number, ⃗ = [HFD , L ⃗ + 1 Σ] ⃗ = 0. [HFD , J] 2

(7.99)

Alternatively, we (re-)define the Dirac angular operator K as the (4 × 4)-matrix generalization of the operator originally given in Eq. (6.81) of Sec. 6.4.1, ⃗ ⋅L ⃗ + 1) . K = β (Σ

(7.100)

It is crucial to observe that K commutes with the Hamiltonian (7.18), and with any Hamiltonian in which a spherically symmetric potential is added to the Dirac

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Hamiltonian. However, the proof is a little tricky; it is helpful to decompose the commutator [HFD , K] into 2 × 2 matrix structures, ⃗ ⋅L ⃗ + 1)] −[HFD , K] = − [⃗ α ⋅ p⃗, β (Σ ⃗ ⋅L ⃗ + 1) + (Σ ⃗ ⋅L ⃗ + 1) β α ⃗ ⋅ p⃗ (Σ ⃗ ⋅ p⃗ =βα =(

⃗ + 2σ ⃗ ⋅ L} ⃗ ⋅ p⃗ 0 {⃗ σ ⋅ p⃗, σ ) ⃗ ⃗ ⋅ L} − 2 σ ⃗ ⋅ p⃗ −{⃗ σ ⋅ p⃗, σ 0

=(

⃗ ⋅ p⃗ + 2 σ ⃗ ⋅ p⃗ 0 −2 σ ) = 0. ⃗ ⋅ p⃗ − 2 σ ⃗ ⋅ p⃗ 2σ 0

(7.101)

⃗ = −2 σ ⃗ ⋅ L} ⃗ ⋅ p⃗ which deserves a further explanaWe have used the relation {⃗ σ ⋅ p⃗, σ tion, given in Eqs. (7.103)–(7.105). The Pauli spin matrices [Eq. (4.343)] fulfill the relation σ i σ j = δ ij

12×2 + i ijk σk .

(7.102)

⃗ is easily calculated, ⃗ ⋅ L} The anticommutator {⃗ σ ⋅ p⃗, σ ⃗ = σ i pi σ j Lj + σ j Lj σ i pi ⃗ ⋅ L} {⃗ σ ⋅ p⃗, σ = pi Li + Li pi + i ijk σ k pi Lj + i jik σ k Lj pi = ijk xj pi pk + ijk xj pk pi + i ijk σ k Li pj − i ijk σ k pj Li = i ijk σ k (Li pj − pj Li ) .

(7.103)

For the commutator of the angular momentum and the spatial momentum operator, one obtains [Li , p` ] = [ijk xj pk , p` ] = ijk [xj , p` ] pk = i i`k pk .

(7.104)

Finally, we obtain the relation ⃗ σ ⃗ ⋅ p⃗} = i ijk σ k i ij` p` = −2 δ k` σ k p` = −2 σ ⃗ ⋅ p⃗ , {⃗ σ ⋅ L,

(7.105)

⃗ σ ⃗ ⋅ p⃗} = −2 σ ⃗ ⋅ p⃗. In order to fix ideas, we recall that the which proves that {⃗ σ ⋅ L, eigenvalues of the Dirac operator K are denoted as −κ. One may thus construct the bispinor (four-component) solutions of the Dirac equation in the angular-momentum basis from two two-component spinors, which carry the same total angular momentum j but are constructed from orbital angular momenta ` = j − 1/2 and ` = j + 1/2, adding or subtracting the electron spin to or from the orbital angular momentum. The spin-angular functions χκ µ (⃗ r), which had originally been defined in Eq. (6.80), read as follows, ⃗ + 1) χκ µ (ˆ (⃗ σ ⋅L r) = −κ χκ µ (ˆ r) ,

χκµ (ˆ r) = ∑ C jµ m

` µ−m

Y` µ−m (ˆ r) ∣ 12 1 m 2

m⟩ . (7.106)

Here, ∣ 12 m⟩ is a fundamental spinor pointing up or down [see Eq. (6.42)], and we use rˆ instead of θ and ϕ as the arguments of χκµ . The Dirac quantum number κ

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summarizes both the orbital quantum number ` as well as the spin and total angular momentum quantum number j into one single integer. We recall Eq. (6.82), −(j + 1/2) 1 κ = (−1)j+`+1/2 (j + ) = { 2 (j + 1/2)

for j = ` + 1/2 for j = ` − 1/2

.

(7.107)

The identification of κ for the atomic states with the lowest angular momenta is as follows, S1/2 ∶

` = 0,

j = 12 ,

κ = −1 ,

(7.108a)

P1/2 ∶

` = 1,

j=

κ = 1,

(7.108b)

P3/2 ∶

` = 1,

j=

κ = −2 ,

(7.108c)

D3/2 ∶

` = 2,

j=

κ = 2,

(7.108d)

D5/2 ∶

` = 2,

j=

κ = −3 ,

(7.108e)

F5/2 ∶

` = 3,

j=

κ = 3,

(7.108f)

1 , 2 3 , 2 3 , 2 5 , 2 5 , 2

and so on. The Dirac angular quantum number κ assumes all positive and negative integer values except zero. 7.3.3

Angular Momenta and Massless Dirac Equation

In Eqs. (7.95) and (7.88), we had already obtained the solutions of the massless Dirac equation, in the helicity basis. However, for a number of problems, e.g., the radiative orbital electron capture by a nucleus [254], it is advantageous to formulate the bispinor solution of the massless Dirac equation in an angular-momentum-resolved, spherical basis. The bispinor ansatz for the wave function reads as follows, ψ(⃗ r) = (

f (r) χκ µ (ˆ r) ). ig(r) χ−κ µ (ˆ r)

(7.109)

It is convenient to introduce the imaginary unit as a prefactor in the lower component, because of another imaginary unit present in the momentum operator which couples upper and lower components. The action of the free massless Dirac Hamil⃗ ⋅ p⃗ on the wave function can be decomposed into 2 × 2 matrices as tonian HFMD = α follows, ⃗ ⋅ p⃗ ψ(⃗ HFMD ψ(⃗ r) = α r) = (

f (r) χκ µ (ˆ r) ⃗ ⋅ p⃗ 0 σ )( ). ⃗ ⋅ p⃗ 0 σ ig(r) χ−κ µ (ˆ r)

(7.110)

⃗ ⋅ p⃗ into angular and radial parts. We now need to decompose the scalar product σ This is achieved by writing 1 1 ⃗ ⋅ p⃗ = 2 (⃗ (⃗ σ ⋅ r⃗) (⃗ σ ⋅ r⃗) σ σ ⋅ r⃗) (⃗ r ⋅ p⃗ + i ijk σ k xi pj ) r2 r ⃗ ⃗⋅L ∂ σ = (⃗ σ ⋅ rˆ) (−i +i ). ∂r r

⃗ ⋅ p⃗ = σ

(7.111)

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We now use several properties of the spinors discussed in Sec. 6.4.1 and take ad⃗ ⋅ p⃗ as vantage of Eq. (7.106), to rewrite the action of the free Hamiltonian H = α follows, ⃗ ⎞ ⃗⋅L σ ∂ ⎛ + i )⎟ 0 (⃗ σ ⋅ r ˆ ) (−i ⎜ r) ∂r r ⎟ ( f (r) χκ µ (ˆ HFMD ψ(⃗ r) = ⎜ ) ⎜ ⎟ ig(r) χ ⃗ ⃗ ∂ σ ⋅ L (ˆ ⎜ ⎟ −κ µ r ) (⃗ σ ⋅ r ˆ ) (−i + i ) 0 ⎝ ⎠ ∂r r 1 ∂ ⎛ (⃗ r) ⎞ σ ⋅ rˆ) ( g(r) − g(r) (κ − 1)) χ−κ µ (ˆ ⎟ ⎜ ∂r r ⎟ =⎜ ⎟ ⎜ ⎟ ⎜ ∂ 1 σ ⋅ rˆ) ( f (r) − f (r) (−κ − 1)) χκ µ (ˆ r) ⎠ ⎝ −i (⃗ ∂r r ⎛ (− ∂ + 1 (κ − 1)) g(r) χκ µ (ˆ r) ⎞ r) ⎞ ⎛ f (r) χκ µ (ˆ ⎟ ⎜ ∂r r ⎟=E ⎜ ⎟ , (7.112) =⎜ ⎟ ⎜ ⎟ ⎜ ∂ 1 ⎝ i g(r) χ−κ µ (ˆ ⎠ r ) r) ⎠ ⎝ i ( ∂r + r (κ + 1)) f (r) χ−κ µ (ˆ where the last equality is postulated if ψ(⃗ r) is an eigenstate of the Hamiltonian. ⃗ κ µ (ˆ ⃗ + 1) − For the angular part, we have used the identities (⃗ σ ⋅ L)χ r) = [(⃗ σ⋅L 1]χκ µ (ˆ r) = (−κ − 1) χκ µ (ˆ r) and (⃗ σ ⋅ rˆ) χκ µ (ˆ r) = −χ−κ µ (ˆ r) from Sec. 6.4.1. The radial eigenvalue equations are thus given by (

1+κ ∂ + ) f (r) = E g(r) , ∂r r

(

∂ 1−κ + ) g(r) = −E f (r) . ∂r r

(7.113)

Given a total angular momentum j, we can choose either one of two possibilities for the Dirac angular quantum number κ, namely, κ = ±(j + 1/2). For negative κ, the solution to the free Dirac equation for massless particles is given as follows, E fEκ (r) = √ jj−1/2 (E r) , π

E gEκ (r) = − √ jj+1/2 (E r) , π

κ = −(j + 1/2) < 0 , (7.114a)

whereas for positive κ = (j + 1/2), E fEκ (r) = √ jj+1/2 (E r) , π

E gEκ (r) = √ jj−1/2 (E r) , π

κ = (j + 1/2) > 0 . (7.114b)

These formulas are easily combined as follows, E fEκ (r) = √ j∣κ∣−1/2+sgn(κ)/2) (E r) , π

E gEκ (r) = sgn(κ) √ j∣κ∣−1/2−sgn(κ)/2 (E r) , π (7.115) where sgn(κ) = κ/∣κ∣ is the sign function. The defining differential equation for spherical Bessel functions reads as (

∂2 2 ∂ ` (` + 1) + + 1) j` (x) = 0 . − ∂x2 x ∂x x2

(7.116)

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The recursion relations are j`−1 (x) + j`+1 (x) =

2` + 1 j` (x) , x

` j`−1 (x) − (` + 1) j`+1 (x) = (2` + 1) j`′ (x) .

(7.117a) (7.117b)

Alternatively, one can express the derivative as follows, `+1 ` j` (x) = −j`+1 (x) + j` (x) x x 1 1 = (j`+1 (x) + j`−1 (x)) − j` (x) . (7.118) 2 2x Let us verify, in the spirit of Eq. (7.113), the solutions of the massless Dirac equation for κ = −(j + 1/2) < 0. Using Eq. (7.118), one verifies two relations, j`′ (x) = j`−1 (x) −

E2 E 2 j − 1/2 ∂ E2 ′ (E r) = − √ jj+1/2 (E r) + √ fEκ (r) = √ jj−1/2 jj−1/2 (E r) , ∂r π π π Er (7.119a) E 2 j − 1/2 κ+1 fEκ (r) = − √ jj−1/2 (E r) , r π Er

κ = −(j + 21 ) .

(7.119b)

Adding these, one obtains ∂ κ+1 E fEκ (r) + fEκ (r) = − E √ jj+1/2 (k r) = E gEκ (r) , ∂r r π

(7.120)

verifying the first entry in Eq. (7.113). Again, using Eq. (7.118), the differentiation of the radial function gEκ (r) proceeds as follows, −

E2 E 2 j + 3/2 E2 ′ ∂ (E r) = √ jj−1/2 (E r) − √ gEκ (r) = √ jj+1/2 jj+1/2 (E r) , ∂r π π π Er (7.121a)

κ−1 E 2 j + 3/2 gEκ (r) = √ jj+1/2 (E r) , r π Er

κ = −(j + 1/2) .

(7.121b)

So, one verifies the second entry in Eq. (7.113), −

E ∂ κ−1 E2 gEκ (r) + gEκ (r) = √ jj−1/2 (E r) = E ( √ jj−1/2 (E r)) = E fEκ (r) . ∂r r π π (7.122)

This concludes the verification of the solution of the massless Dirac equation for negative κ, as given in Eq. (7.114a). The normalization integrals are given as follows, ∫ ∫

∞

E′ 1 E dr r2 ( √ jj−1/2 (E r)) ( √ jj−1/2 (E ′ r)) = δ(E − E ′ ) , 2 π π

(7.123a)

∞

E E′ 1 dr r2 ( √ jj+1/2 (E r)) ( √ jj+1/2 (E ′ r)) = δ(E − E ′ ) . 2 π π

(7.123b)

0

0

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These relations follow from the orthogonality properties of the spherical Bessel functions, which are discussed in detail in Sec. 2.4.3 of Ref. [81]. The bispinor wave function given in Eq. (7.109) with the radial components given in Eq. (7.114), is normalized to the energy scale, ψEκµ (⃗ r) = (

fEκ (r) χκ µ (ˆ r) ), igEκ (r) χ−κ µ (ˆ r)

+ ⟨E ′ κ′ µ′ ∣Eκµ⟩ = ∫ d3 r ψE r) ψEκµ (⃗ r) = δ(E − E ′ ) δκκ′ δµµ′ . ′ κ′ µ′ (⃗

(7.124a) (7.124b)

A final remark is in order. In principle, the wave function given in Eq. (7.109) can be used for the description of a Dirac particle in a process in which its mass is negligible. However, a slight generalization is necessary for neutrinos. Because of the properties of the weak interaction, neutrinos are born in helicity and chirality eigenstates within the zero-mass approximation. The massless Dirac Hamiltonian ⃗ ⋅ p⃗ commutes with the chirality operator γ 5 . Furthermore, any simultaHFMD = α ̂ neous eigenstate of HFMD and γ 5 also is an eigenstate of the helicity operator Σ,

⃗ ⋅ p⃗. In the helicity basis, the appropriate because the latter is proportional to γ 5 α eigenstates have been given in Eqs. (7.88). The situation is a little more complicated in the angular-momentum-resolved basis. It helps to observe that for given j, one may choose two possible values of κ, namely, κ1 = −(j + 1/2) and κ2 = (j + 1/2), and form the following linear combination of solutions, r) + ifEκ2 (r) χκ2 µ (ˆ r) ⎞ 1 ⎛ fEκ1 (r) χκ1 µ (ˆ (ν) ψEjµ (⃗ r) = √ 2 ⎝ i [gEκ1 (r) χ−κ1 µ (ˆ r) + i gEκ2 (r) χ−κ2 µ (ˆ r)] ⎠ r) + i jj+1/2 (E r) χ(j+1/2) µ (ˆ r) ⎞ E ⎛ jj−1/2 (E r) χ−(j+1/2) µ (ˆ =√ . (7.125) 2π ⎝ −i jj+1/2 (E r) χ(j+1/2) µ (ˆ r) − jj−1/2 (E r) χ−(j+1/2) µ (ˆ r) ⎠

This wave function is an eigenstate of chirality, and has negative helicity, as required for neutrinos, (ν)

ψEjµ (⃗ r) = (

νEjµ (⃗ r) ), −νEjµ (⃗ r)

(7.126a)

E (jj−1/2 (E r) χ−(j+1/2) µ (ˆ νEjµ (⃗ r) = √ r) + i jj+1/2 (E r) χ(j+1/2) µ (ˆ r)) . (7.126b) 2π The normalization integral for the neutrino wave function is given as follows, (ν)+

(ν)

3 r) ψE ′ j ′ µ′ (⃗ r) = δ(E − E ′ ) δjj ′ δµµ′ . ∫ d r ψEjµ (⃗

(7.127)

This wave function is normalized to the energy scale, just like the one given in Eq. (7.124). 7.3.4

Angular Momenta and Free Dirac Equation

⃗ ⋅ p⃗ + βm, where βm is the mass The free massive Dirac Hamiltonian reads as H = α term. In the angular-momentum representation, the action of the free Hamiltonian

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can be written as ⃗ ⎞ ⃗⋅L σ ∂ ⎛ +i )⎟ m (⃗ σ ⋅ rˆ) (−i ⎜ r) ∂r r ⎟ ( f (r) χκ µ (ˆ Hψ(⃗ r) = ⎜ ) ⎜ ⎟ ig(r) χ ⃗ ⃗⋅L σ ∂ r) ⎜ ⎟ −κ µ (ˆ +i ) −m σ ⋅ rˆ) (−i ⎝ (⃗ ⎠ ∂r r ⎛ [(− ∂ + κ − 1 ) g(r) + m f (r)] χκ µ (ˆ r) ⎞ r) ⎞ ⎛ f (r) χκ µ (ˆ ⎟ ⎜ ∂r r ⎟=E ⎜ ⎟. =⎜ ⎟ ⎜ ⎟ ⎜ κ+1 ∂ ⎝ i g(r) χ−κ µ (ˆ r) ⎠ r) ⎠ ⎝ i [( ∂r + r ) f (r) − m g(r)] χ−κ µ (ˆ (7.128) The radial equations differ from Eq. (7.113) by the inclusion of the mass term, 1−κ ∂ + ) g(r) = (m − E) f (r) , ∂r r ∂ 1+κ ( + ) f (r) = (m + E) g(r) . ∂r r (

(7.129a) (7.129b)

The solutions of the free Dirac equation follow as a generalization of the massless case [Eq. (7.114a)], with prefactors adjusted to account for the mass. For negative κ, one has √ (E 2 − m2 )1/4 √ √ E + m jj−1/2 ( E 2 − m2 r) , π 2 √ (E − m2 )1/4 √ √ gEκ (r) = − E − m jj+1/2 ( E 2 − m2 r) , π

fEκ (r) =

(7.130a) κ = −(j + 1/2) . (7.130b)

For positive κ, the solution is given as √ (E 2 − m2 )1/4 √ √ E + m jj+1/2 ( E 2 − m2 r) , π 2 √ (E − m2 )1/4 √ √ gEκ (r) = E − m jj−1/2 ( E 2 − m2 r) , π

fEκ (r) =

(7.131a) κ = j + 1/2 .

(7.131b)

The normalization integral is finally found as follows, ψEκµ (⃗ r) = (

fEκ (r) χκ µ (ˆ r) ), igEκ (r) χ−κ µ (ˆ r)

3 + r) ψEκµ (⃗ r) = δ(E − E ′ ) δκκ′ δµµ′ . ∫ d r ψE ′ κ′ µ′ (⃗

(7.132) The bispinor solutions, for both the massive as well as the massless free Dirac equation, combine an upper spinor component of orbital angular momentum ` = j ∓ 1/2 with a lower spinor component of orbital angular momentum ` = j ± 1/2. The solutions ψk˜κµ (⃗ r), normalized to the momentum scale, ψk˜κµ (⃗ r) = (

fk κ (r) χκ µ (ˆ r) ), igk κ (r) χ−κ µ (ˆ r)

(7.133)

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are easily obtained on the basis of the formula √ E 2 − m2 dE ′ ′ δ(E − E ) = δ(E − E ′ ) , (7.134) δ(k − k ) = dk E √ where E = k 2 + m2 . Hence, the wave functions normalized to the momentum scale are obtained from those normalized to the energy scale [see Eqs. (7.130) and (7.131)], via multiplication by a factor (E 2 − m2 )1/4 /E 1/2 , i.e., (E 2 − m2 )1/4 fEκ (r) , E 1/2 (E 2 − m2 )1/4 gEκ (r) . gkκ (r) = E 1/2 The normalization integral for the latter solutions is fkκ (r) =

ψkκµ (⃗ r) = (

fkκ (r) χκ µ (ˆ r) ), igkκ (r) χ−κ µ (ˆ r)

(7.135) (7.136)

3 + r) ψkκµ (⃗ r) = δ(k − k ′ ) δκκ′ δµµ′ . ∫ d r ψk′ κ′ µ′ (⃗

(7.137) For the massless case, one has E = k, and so the distinction between the two normalizations is not necessary. 7.4 7.4.1

Quantized Dirac Field and Propagators Free Dirac Progator in Feynman’s Formulation

In the calculation of classical electromagnetic fields, the use of the retarded, Feynman and advanced propagator depends on the physical interpretation of positiveand negative-frequency components of the incoming signal, and is discussed in detail in Chap. 4 of Ref. [81]. If we interpret the charge and current distributions as sources of radiation, and the fields are to be calculated, then the most adequate representation involves the retarded Green function. The physical interpretation is that both positive-frequency as well as negative-frequency components of the oscillating charges and currents act as sources of radiation. However, if we wish to calculate the emission and absorption of virtual photons within quantum field theory (quantum electrodynamics), then the most adequate representation involves the use of the Feynman Green function, which is naturally obtained from the time-ordered products of field operators. Here, negative-frequency components are interpreted as sinks of radiation; this corresponds to the time-ordered product of the electromagnetic field operators. In the case of fermion fields, only one interpretation is possible. Namely, the negative-energy solutions of the Dirac theory have to be (re-)interpreted as positiveenergy solutions, using the following procedure: One writes the propagator in such a way that negative-energy solutions are automatically propagated into the past, and reinterpreted as positive-energy solutions propagating into the future, which describe the trajectories of antiparticles. Let us suppose that t′′ is the time at the endpoint of a trajectory, and t′ is the time at the “beginning.” The antiparticle phase

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factor is exp[iE (t′′ − t′ )]; if the particle propagates into the past, with t′′ = t< , and t′ = t> , then the accumulated phase factor is exp[iE (t< −t> )] = exp[−iE (t> −t< )], and is equal to the phase factor accumulated during a positive-energy trajectory, propagating into the future, from t< to t> . So, in a Feynman diagram, we can describe an incoming positron by an outgoing line, propagating into the past, decorated with a four-momentum which is the opposite of the physical, incoming positron momentum. In particular, the energy is negative. This is equivalent to an incoming positron with a positive energy, but with the arrow of propagation in space-time reversed. To put these statements into perspective, a positron falling to the ground in a gravitational field [255] is interpreted as a positron falling upward, but backward in time, so that, effectively, it falls down just like any other particle. In our formulas, we have already taken the reinterpretation principle into account, labeling the solu⃗ where k⃗ is the physical momentum along the antiparticle trajectory, tion by Vσ (k) even if the phase factor is exp(ik ⋅ x) = exp(iEt − ik⃗ ⋅ r⃗) with a manifestly positive energy variable E. The reinterpretation principle, which reinterprets trajectories propagating into the past, is key in abandoning the hole theory, which would otherwise necessitate us to think about antiparticles as “holes” in a Dirac “sea” of filled negative-energy states. Let us illustrate these statements by calculating the time-ordered vacuum expectation value of the field operators of a spin-1/2 theory, which is supposed to yield the time-ordered (Feynman) propagator. In the momentum representation, it is equal to the inverse of the Hamiltonian (upon multiplication with γ 0 ). The Lorentz-invariant integration measure used in its derivation is d4 k d3 k 1 =∫ δ(k 2 − m2 ) ∣ 0 , 3 k >0 (2π) 2 E (2π)4 where E=

√

k⃗2 + m2 − i  ,

(7.138)

(7.139)

⃗ and the energy-momentum four-vector is k µ = (E, k). We start from the field operator ψ(x) = ∫

d3 k m ⃗ e−i k⋅x + d+ (k) Vσ (k) ⃗ ei k⋅x } , ∑ {bσ (k) Uσ (k) σ (2π)3 E σ=±

(7.140)

where bσ annihilates particles and d+σ creates antiparticles. This operator is denoted by the symbol ψ although it is manifestly distinct from the first-quantized Dirac wave function ψ, which is usually denoted by the same symbol. The fundamental ⃗ and Vσ (k) ⃗ have been given in Eqs. (7.89) and (7.90). The Dirac bispinors Uσ (k) adjoint of the field operator is ψ(x) = ∫

d3 k m + ⃗ ei k⋅x + dρ (k) V ρ (k) ⃗ e−i k⋅x ] . ∑ [b (k) U ρ (k) (2π)3 E ρ=± ρ

(7.141)

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Formally, these formulas are consistent with the following reinterpretation, bσ (−k) → d+σ (k) ,

b+σ (−k) → dσ (k) ,

(7.142)

for negative-energy states, in agreement with the above reinterpretation principle. The anticommutators of annihilation and creation operators vanish, {bσ (k), bρ (k ′ )} = {b+σ (k), b+ρ (k ′ )} = 0 ,

(7.143a)

{dσ (k), dρ (k ′ )} = {d+σ (k), d+ρ (k ′ )} = 0 .

(7.143b)

However, anticommutators of creation and annihilation operators do not vanish, E (3) ⃗ ⃗′ δ (k − k ) δσρ , m E {dσ (k), d+ρ (k ′ )} = (2π)3 δ (3) (k⃗ − k⃗′ ) δσρ . m {bσ (k), b+ρ (k ′ )} = (2π)3

(7.144a) (7.144b)

We have the following expression for the time-ordered field propagator, with x1 = (t1 , r⃗1 ), x2 = (t2 , r⃗2 ), and k ′µ = (E ′ , k⃗′ ), ⟨0 ∣T ψξ (x1 ) ψ ξ′ (x2 )∣ 0⟩ = Θ(t1 − t2 ) ⟨0 ∣ψξ (x1 ) ψ ξ′ (x2 )∣ 0⟩ − Θ(t2 − t1 ) ⟨0 ∣ψ ξ′ (x2 ) ψξ (x1 )∣ 0⟩ .

(7.145)

Here, ∣0⟩ denotes the vacuum ground state of the underlying field theory. The negative sign in front of the second term is introduced for fermion field operators, by convention, for the time-ordered product of fermionic field operators. The indices in the expressions ψξ (x1 ) and ψ ξ′ (x2 ) denote the spinor components. The timeordered product of fermion field operators is a tensor product in spinor space, ⟨0 ∣T ψ(x1 ) ψ(x2 )∣ 0⟩ ≡ ⟨0 ∣T ψ(x1 ) ⊗ ψ(x2 )∣ 0⟩, of which we calculate the component with spinor indices ξ, ξ ′ . We can go on to calculate ⟨0 ∣T ψξ (x1 ) ψ ξ′ (x2 )∣ 0⟩ d3 k m ⃗ e−i k⋅x1 + d+ (k) Vσ,ξ (k) ⃗ ei k⋅x1 ] ∑ [bσ (k) Uσ,ξ (k) σ (2π)3 E σ=± RR d3 k ′ m + ′ ⃗′ ) ei k′ ⋅x2 + dρ (k ′ ) V ρ,ξ′ (k⃗′ ) e−i k′ ⋅x2 ]RRRR 0⟩ ′ (k × ∫ [b (k ) U ∑ ρ,ξ ρ RRR (2π)3 E ′ ρ=± RR 3 ′ ′ d k m + ′ ′ i k′ ⋅x2 + dσ (k ′ ) V σ,ξ′ (k⃗′ ) e−i k ⋅x2 ] − Θ(t2 − t1 ) ⟨0 ∣∫ ∑ [b (k ) U σ,ξ′ (k⃗ ) e (2π)3 E ′ σ=± σ RR d3 k m −i k⋅x1 + i k⋅x1 RRR ⃗ ⃗ [b ] × ∫ (k) U ( k) e + d (k) V ( k) e RRR 0⟩ . (7.146) ∑ ρ ρ,ξ ρ,ξ ρ (2π)3 E ρ=± RRR

= Θ(t1 − t2 ) ⟨0 ∣∫

It is very instructive to leave the spinor indices in place. Noting that annihilation operators lead to a vanishing contribution when acting on the vacuum, we can

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ascertain that the expression (7.146) becomes d3 k m d3 k ′ m ∫ (2π)3 E (2π)3 E ′ + ′ ⃗ ⊗ U ρ (k⃗′ )] ′ e−i k⋅x2 +i k′ ⋅x2 × ∑ ∑ ⟨0 ∣bσ (k) bρ (k )∣ 0⟩[Uσ (k) ξξ

⟨0 ∣T ψξ (x1 ) ψ ξ′ (x2 )∣ 0⟩ = Θ(t1 − t2 ) ∫ σ=± ρ=±

d3 k ′ m d3 k m ∫ 3 (2π) E (2π)3 E ′ ⃗ ⊗ V σ (k⃗′ )] ′ ei k⋅x2 −i k′ ⋅x2 . × ∑ ∑ ⟨0 ∣dσ (k ′ ) d+ρ (k)∣ 0⟩[Vρ (k) ξξ − Θ(t2 − t1 ) ∫ σ=± ρ=±

(7.147) Here, ⃗ U ρ,ξ′ (k⃗′ ) , ⃗ ⊗ U ρ (k⃗′ )] ′ = Uσ,ξ (k) [Uσ (k) ξξ

(7.148)

⃗ ⊗ V σ (k )] ′ = Vρ,ξ (k) ⃗ V σ,ξ′ (k ) . [Vρ (k) ξξ

(7.149)

⃗′

⃗′

Using the anticommutators given in Eq. (7.144) and the sum rule (7.93) in order to convert the double integral into a single one, one may write a much more compact expression, d3 k m −ik⋅(x1 −x2 ) γ 0 E − γ⃗ ⋅ k⃗ + m e (2π)3 E 2m 3 0 d k m ik⋅(x1 −x2 ) γ E − γ⃗ ⋅ k⃗ − m − Θ(t2 − t1 ) ∫ e , (7.150) (2π)3 E 2m

⟨0 ∣T ψ(x1 ) ψ(x2 )∣ 0⟩ = Θ(t1 − t2 ) ∫

where the tensor product in spinor space is understood, and k ⋅ x = E t − k⃗ ⋅ r⃗. The following contour integral representations are introduced for the Heaviside step functions, 1 dk0 −ik0 (t1 −t2 ) e f (k0 ) , 2π k0 − E + i dk0 −ik0 (t1 −t2 ) 1 Θ(t2 − t1 ) f (E) eiE(t1 −t2 ) = − i ∫ e f (−k0 ) . 2π k0 + E − i

Θ(t1 − t2 ) f (E) e−iE(t1 −t2 ) = i ∫

(7.151a) (7.151b)

The disturbing relative minus sign in Eq. (7.150) is eliminated, and we obtain ⟨0 ∣T ψ(x1 )ψ(x2 )∣ 0⟩ 0 ⃗ ⋅ k⃗ + m d4 k m −ik0 (t1 −t2 )+ik⋅(⃗ 1 ⃗ r1 −⃗ r2 ) γ k 0 − γ e 4 (2π) E 2m k0 − E + i 0 ⃗ ⋅ k⃗ − m d4 k m −ik0 (t1 −t2 )+ik⋅(⃗ 1 ⃗ r1 −⃗ r2 ) −γ k0 + γ + i∫ e . (2π)4 E 2m k0 + E − i

= i∫

(7.152)

Here, we have replaced k⃗ → −k⃗ in the second term, without affecting the result of the integration (the integral extends over all k⃗ ∈ R3 ). The integration symbol is d4 k = dk 0 d3 k. There is, actually, a (tacitly assumed) convention [2, 256] that in any integrals ∫ d3 k, the component k0 is set equal to E, as given in Eq. (7.139),

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wherever it occurs, but if the integral is over the full d4 k, then the integration interval is the full k0 ∈ (−∞, ∞). With this notion in mind, we have ⟨0 ∣T ψ(x1 ) ψ(x2 )∣ 0⟩ = i ∫

1 d4 k −ik⋅(x1 −x2 ) k
+ m m e 4 (2π) 2m E k0 − E + i d4 k −ik⋅(x1 −x2 ) −k
− m m 1 e (2π)4 2m E k0 + E − i

+ i∫ = i∫

d4 k −ik⋅(x1 −x2 ) k
+ m 1 1 1 e ( − ) (2π)4 2 E k0 − E + i k0 + E − i

= i∫

d4 k −ik⋅(x1 −x2 ) k
+ m e = i SF (x1 , x2 ) . 4 2 (2π) k − m2 + i

(7.153)

In the last line, we implicitly define the Feynman propagator SF . In energymomentum space, we can read off the representation of the Feynman propagator as follows, SF (k) =

k
+ m 1 = 2 . k
− m + i k − m2 + i 

(7.154)

This is, of course, a well-known, classic result. In addition to the propagator, it is extremely instructive to also calculate the field anticommutator {ψ(x1 ), ψ(x2 )}, which is of course equal to its own vacuum expectation value. Simplifying the notation by leaving out the spinor index as in Eq. (7.147), and using the anticommutator (7.144), we have d3 k m −ik⋅(x1 −x2 ) ⃗ ⊗ U σ (k) ⃗ Uσ (k) ∑ [e (2π)3 E σ ⃗ . ⃗ ⊗ V σ (k)] + eik⋅(x1 −x2 ) Vσ (k)

{ψ(x1 ), ψ(x2 )} = ∫

(7.155)

With the help of the spin sum (7.93), it follows that {ψ(x1 ), ψ(x2 )} = ∫ =∫

d3 k m −ik⋅(x1 −x2 ) k
+ m k
− m (e + eik⋅(x1 −x2 ) ) 3 (2π) E 2m 2m d3 k m i ∂
1 + m −ik⋅(x1 −x2 ) −i∂
1 − m ik⋅(x1 −x2 ) ( e + e ) (2π)3 E 2m 2m

= (i∂
1 + m) ∫

d3 k 1 (e−ik⋅(x1 −x2 ) − eik⋅(x1 −x2 ) ) (2π)3 2E

= (i ∂
1 + m) i ∆(x1 − x2 ) ,

(7.156)

where ∂
1 = γ µ ∂/∂xµ1 . Finally, ∆(x1 − x2 ) is the expression [see Eqs. (3.56) and (3.77) of Ref. [2]], i ∆(x1 − x2 ) = ∫

d3 k 1 (e−ik⋅(x1 −x2 ) − eik⋅(x1 −x2 ) ) . (2π)3 2E

(7.157)

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Free Dirac Equation

The time derivative of the ∆ distribution at equal times reads as [we denote xµ1 = (t1 , r⃗1 ) and xµ2 = (t2 , r⃗2 )] ∂ d3 k 1 ∂ (e−ik⋅(x1 −x2 ) − eik⋅(x1 −x2 ) )∣ = −i ∆(x1 − x2 )∣ ∫ 3 2E ∂t1 ∂t (2π) 1 t1 =t2 x0 =y0 d3 k 1 ik⋅(⃗ ⃗ ⃗ (e r1 −⃗r2 ) + eik⋅(⃗r1 −⃗r2 ) ) = −δ (3) (⃗ r1 − r⃗2 ) . (2π)3 2 (7.158)

= −∫

For x0 = y0 , the only contribution to Eq. (7.156) is the one from the time derivative, and we have {ψ(x1 ), ψ(x2 )}∣t

1 =t2

= −γ 0 ∂0 ∆(x1 − x2 ) = γ 0 δ (3) (⃗ r1 − r⃗2 ) .

(7.159)

This result guarantees the locality of the theory. Furthermore, considering the Lagrangian (7.79), one can actually show that the conjugate field momentum is π(x) ≡

∂L = i ψ(x) γ 0 = i ψ + (x). ∂(∂0 ψ)

(7.160)

Hence, Eq. (7.159) can also be interpreted as follows, {ψ(x1 ), π(x2 )}∣t1 =t2 = i δ (3) (⃗ r1 − r⃗2 ) ,

(7.161)

which is the canonical equal-time anticommutation relation for fermionic fields. This relation should be compared to Eq. (10.282). 7.4.2

Feynman Propagator and Green Function

Green functions for the electron-positron Dirac field have, in general, many uses, including the calculation of time propagation and the calculation of propagator denominators in perturbative diagram calculations. In consequence, it seems that the Dirac Green function appears, in the literature, in at least three slightly different forms. All three versions are adapted, in each case, to the application one has in mind. In the following, we will attempt to summarize the formulas for a generic Dirac Hamiltonian ⃗ + e A0 + βm , ⃗ ⋅ (⃗ H =α p − e A) ⃗, p⃗ = − i∇

⃗ r) , A⃗ = A(⃗

(7.162a) A = Φ = Φ(⃗ r) . 0

(7.162b)

For the Coulomb field, one often denotes V (⃗ r) = e Φ(⃗ r) as the potential. It is useful to distinguish ● the time evolution Green function GF = GF (t1 − t2 , r⃗1 , r⃗2 ), ● the Feynman propagator SF = SF (t1 − t2 , r⃗1 , r⃗2 ), ● the perturbative Green function GF = GF (t1 − t2 , r⃗1 , r⃗2 ).

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Here, we have used the fact that the static nature of the Hamiltonian in Eq. (7.162) allows us to express the Green functions in terms of the time difference t1 − t2 only. ⃗ spatial translation However, for nontrivial scalar and vector potentials Φ and A, invariance could be lost, and we need to indicate the two arguments r⃗1 and r⃗2 separately. Let us assume that a complete set of eigenfunctions exists, H ψn (⃗ r) = En ψn (⃗ r) ,

(7.163)

which may include a continuum, and let us also define the summation over eigenfunctions to be generalized to include the continuum [see also the remarks following Eq. (3.45)]. The completeness of the spectrum is summarized in the formula r1 ) ⊗ ψn (⃗ r2 ) = 14×4 δ ∑ ψn (⃗ +

n

(3)

(⃗ r1 − r⃗2 ) .

(7.164)

Here, the outer product is denoted as ⊗ and is taken in the four-component bispinor space. One might ask how one can understand the completeness relation in the nonrelativistic limit. This is because the Dirac equation has “two spectra” in this limit, one for the particle, and one for the antiparticle solutions. In the nonrelativistic limit, the particle solutions represent two-component Schr¨odinger–Pauli spinors which will provide the “upper” components of the completeness relation, i.e., a term 12×2 δ (3) (⃗r1 − r⃗2 ), while the antiparticle solution will provide the “lower” completeness, also proportional to 12×2 δ (3) (⃗ r1 − r⃗2 ). Their sum gives the unit matrix 14×4 in bispinor space. The time evolution Green function GF = GF (t1 −t2 , r⃗1 , r⃗2 ) fulfills the differential equation (i

∂ − H) GF (t1 − t2 , r⃗1 , r⃗2 ) = i δ(t1 − t2 ) δ (3) (⃗ r1 − r⃗2 ) , ∂t

(7.165)

with boundary conditions chosen so that positive-energy states propagate into the future, which negative-energy state propagator into the past. The function GF can be written as GF (t1 − t2 , r⃗1 , r⃗2 ) = Θ(t1 − t2 ) ∑ ψn (⃗ r1 ) ⊗ ψn+ (⃗ r2 ) e−i [En −i] (t1 −t2 ) En >0

+ − Θ(t2 − t1 ) ∑ ψm (⃗ r1 ) ⊗ ψm (⃗ r2 ) e−i [Em +i] (t1 −t2 ) , (7.166a) Em 0

and t1 > t2 , then without any further integration over t2 . It is still very instructive to consider what would happen if we were to integrate over t2 . Let us assume that ψ(t2 , r⃗2 ) = ψk (⃗ r2 ) exp(−iEk t2 ) is the wave function at time t2 , where ψk (⃗ r2 ) is one of the eigenfunctions of H (where Ek > 0). The function obtained by convolution is φk (t1 , r⃗1 ) = ∫ d3 r2 dt2 GF (t1 − t2 , r⃗1 , r⃗2 ) ψk (t2 , r⃗2 ) = 2πδ(0) ψk (⃗ r1 ) exp(−iEk t1 ) .

(7.169)

We have set ∫ dt2 = 2πδ(0), for obvious reasons, where the Dirac-δ, as usual, denotes the corresponding distribution. Since Eq. (7.169) denotes a convolution with respect to the time t2 , which is equivalent to multiplication in Fourier space, we can alternatively multiply in Fourier (frequency) space and then, transform back into the time domain. Hence, we investigate the function φk (E, r⃗1 ) = ∫ d3 r2 GF (E, r⃗1 , r⃗2 ) ψk (E, r⃗2 ) , ψk (E, r⃗2 ) = 2π δ(E − Ek ) ψk (⃗ r2 ) .

(7.170)

Hence, φk (E, r⃗1 ) = ∫ d3 r2 =

ψn (⃗ r1 ) ⊗ ψn+ (⃗ r2 ) 1 2π δ(E − Ek ) ψk (⃗ r2 ) ∑ i n En (1 − i) − E

1 1 2π δ(E − Ek ) ψk (⃗ r1 ) , i Ek − E − i

(7.171)

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where we assume that Ek > 0. The Fourier backtransform of φk (E, r⃗1 ) should confirm the result (7.169). Closing the integration contour in the lower half of the complex plane, we find that dE exp(−iEt1 ) φk (E, r⃗1 ) 2π 1 1 dE exp(−iEt1 ) (− ) 2π δ(E − Ek ) ψk (⃗ r1 ) =∫ 2π i E − (Ek − i) dE 1 =∫ exp(−iEt1 ) [− (−2πiδ(E − Ek ))] 2π δ(E − Ek ) ψk (⃗ r1 ) 2π i = 2πδ(0) ψk (⃗ r1 ) exp(−iEk t1 ) . (7.172)

φk (t1 , r⃗1 ) = ∫

This confirms the result (7.169). The Green function GF , adapted to perturbative calculations, is defined so that its Fourier transform yields the expression 1/(H − E), where, in writing 1/(H − E), we ignore infinitesimal imaginary parts. The expression is slightly different from Eq. (7.166a) and reads as follows, r1 ) ⊗ ψn+ (⃗ r2 ) e−i [En −i] (t1 −t2 ) GF (t1 − t2 , r⃗1 , r⃗2 ) = iΘ(t1 − t2 ) ∑ ψn (⃗ En >0

+ − iΘ(t2 − t1 ) ∑ ψm (⃗ r1 ) ⊗ ψm (⃗ r2 ) e−i [Em +i] (t1 −t2 ) , (7.173a) Em 0

r2 ) e−i [Em +i] (t1 −t2 ) , (7.174a) + iΘ(t2 − t1 ) ∑ ψm (⃗ r1 ) ⊗ ψ m (⃗ Em = max(r1 , r2 ) ,

r< = min(r1 , r2 ) .

(7.182)

The details of the derivation are analogous to the other Green functions discussed in Chap. 4, but with one important difference which is mentioned below. In agreement with Eq. (A.20) in Ref. [257], one finds for r1 = r< = min(r1 , r2 ) and r2 = r> = max(r1 , r2 ) (1)

Fκ11 (r< , r> , E) = − (E + m) C j`(κ) (i C r< ) h`(κ) (i C r> ) , κ (1) Fκ12 (r< , r> , E) = − i C 2 j`(κ) (i C r< ) h`(−κ) (i C r> ) , ∣κ∣ κ (1) Fκ21 (r< , r> , E) = − i C 2 j`(−κ) (i C r< ) h`(κ) (i C r> ) , ∣κ∣ (1)

Fκ22 (r< , r> , E) = − (E − m) C j`(−κ) (i C r< ) h`(−κ) (i C r> ) ,

(7.183a) (7.183b) (7.183c) (7.183d)

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Free Dirac Equation

where C=

√

m2 − E 2 ,

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259

(7.183e)

(1)

and the h` (x) = j` (x) + i y` (x) are spherical Hankel function of the first kind [159]. The definitions of the F functions are completed by the symmetry relations Fκ11 (r> , r< , E) = Fκ11 (r< , r> , E) ,

(7.183f)

Fκ12 (r> , r< , E) Fκ21 (r> , r< , E) Fκ22 (r> , r< , E)

(7.183g)

= = =

Fκ21 (r< , r> , E) , Fκ12 (r< , r> , E) , Fκ22 (r< , r> , E) .

(7.183h) (7.183i)

Note that the functions Fκ11 (r1 , r2 , E) and Fκ22 (r1 , r2 , E) are continuous at the cusp r1 = r2 , while the functions Fκ12 (r1 , r2 , E) and Fκ21 (r1 , r2 , E) have a kink (discontinuity). We recall from the derivation of the Schr¨odinger–Coulomb Green function [the result has been given in Eq. (4.108)] that the radial component is continuous at the cusp r1 = r2 . The same applies to the photon Green functions discussed in Chap. 4 of Ref. [81]. Here, we encounter radial Green functions which are discontinuous at the cusp. This is due to the fact that the Dirac Hamiltonian, as opposed to the Schr¨ odinger Hamiltonian, actually is a first-order differential operator, when expressed in coordinate space. The presence of the discontinuity is required by the necessity to generate a Dirac-δ, by a single differentiation. We recall the variable C defined in Eq. (7.183e). Due to the continuum spectrum, the free Dirac Green √ function has cuts for E > m and E < −m; these are encoded in the variable C = m2 − E 2 . The Feynman prescription can either be implemented by lending to E an infinitesimal imaginary part along a suitable integration contour, or, by the substitution m → m − i. The latter ensures that the branch cuts occur for E = ReE − i, for ReE > m, and E = ReE + i, for ReE < −m. The branch cut of the square root is chosen along the negative real axis, the square root is calculated on the principal branch, which results in √ (7.184) Re(C) = Re( m2 − E 2 ) > 0 . The sum over κ can be performed for the free Dirac Green function, with the result F(⃗ r1 , r⃗2 , E) = [(

C 1 e−C r12 ⃗ ⋅ r⃗12 + βm + E] + 2 )iα , r12 r12 4πr12

(7.185)

where r12 = ∣⃗ r1 − r⃗2 ∣. However, the radial decomposition (7.179) is more useful in many practical calculations. Higher powers of the free Dirac Green Function F 2 , which are sometimes needed in bound-state calculations, can be generated upon differentiation. E.g., one finds that F 2 (⃗ r1 , r⃗2 , E) = ⟨⃗ r1 ∣(

2 1 ∂ 1 ∂ ) ∣ r⃗2 ⟩ = ⟨⃗ r1 ∣ ∣ r⃗2 ⟩ = F(⃗ r1 , r⃗2 , E) . H −E ∂E H −E ∂E (7.186)

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The angular-momentum decomposition reads as F 2 (⃗ r1 , r⃗2 , z) = ∑( κµ

(F 2 )11 r1 , rˆ2 ) (F 2 )12 σ ⋅ rˆ1 π−κ (ˆ r1 , rˆ2 ) κ (r1 , r2 , E) πκ (ˆ κ (r1 , r2 , E) i⃗ ). 2 21 2 22 −(F )κ (r1 , r2 , E) i⃗ σ ⋅ rˆ1 πκ (ˆ r1 , rˆ2 ) (F )κ (r2 , r1 , E) π−κ (ˆ r1 , rˆ2 ) (7.187)

Explicit expressions for the radial components can be derived by explicit differentiation of the functions given in Eq. (7.183) with respect to the energy. The following result is then easily obtained, E m + 2 E κ 11 (r2 Fκ12 + r1 Fκ21 ) , Fκ − C2 m−E E E (F 2 )12 r1 Fκ22 − r2 Fκ11 , κ (r1 , r2 , E) = − m−E m+E E E (F 2 )21 r2 Fκ22 − r1 Fκ11 , κ (r1 , r2 , E) = − m−E m+E m + 2 E κ 22 E (r2 Fκ21 + r1 Fκ12 ) . (F 2 )22 Fκ − κ (r1 , r2 , E) = − C2 m+E (F 2 )11 κ (r1 , r2 , E) =

(7.188a) (7.188b) (7.188c) (7.188d)

We here suppress the arguments of the components on the right-hand sides. This result is easily obtained by direct differentiation of the coordinate space representation (7.183), and by using Eq. (7.118). Often, in bound-state calculations, we need to extract divergences connected with the insertion of a Coulomb interaction into the propagator. This amounts to ⃗ ⋅ p⃗ + β m → α ⃗ ⋅ p⃗ + β m + V in the propagator denominator, or, to a replacement α the expansion F → G = F − F V F + F V F V F + ⋯ for the Dirac–Coulomb Green function G. In many cases, the divergent terms can be described as we commute the potential outside of the free Green function, which, in turn, leads to a much simpler expression for the Green function that needs to be used in typical subtraction terms. Indeed, it is often useful to define GV = F −

1 {V, F 2 } , 2

V =−

Zα . r

(7.189)

Here, recall that the coordinate-space representation of the term V F 2 is V (⃗ r1 )F 2 (⃗ r1 , r⃗2 ), while that of F 2 V is F 2 (⃗ r1 , r⃗2 ) V (⃗ r2 ); the two terms are not equal, 2 and so V does not commute with F . In the angular-momentum decomposition, one has GV (⃗ r1 , r⃗2 , z) = ∑( κµ

11 GV,κ (r1 , r2 , E) πκ (ˆ r1 , rˆ2 ) 12 −GV,κ (r1 , r2 , E) i⃗ σ ⋅ rˆ1 πκ (ˆ r1 , rˆ2 )

12 GV,κ (r1 , r2 , E) i⃗ σ ⋅ rˆ1 π−κ (ˆ r1 , rˆ2 ) ). 22 GV,κ (r1 , r2 , E) π−κ (ˆ r1 , rˆ2 )

(7.190)

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Free Dirac Equation

The components of GV read as follows [we suppress the arguments (r1 , r2 , E)], r1 + r2 11 {(m + 2 E κ) Fκ11 − E (E + m) [r2 Fκ12 + r1 Fκ21 ]} , GV,κ = Fκ11 + (Zα) 2 r1 r2 C 2 (7.191a) r1 + r2 12 12 11 22 {E (E − m) r2 Fκ − E (E + m) r1 Fκ } , GV,κ = Fκ + (Zα) (7.191b) 2 r1 r2 C 2 r1 + r2 21 {E (E − m) r1 Fκ11 − E (E + m) r2 Fκ22 } , GV,κ = Fκ21 + (Zα) (7.191c) 2 r1 r2 C 2 r1 + r2 22 {− (m + 2 E κ) Fκ22 + E (E − m) [r2 Fκ21 + r1 Fκ12 ]} . GV,κ = Fκ22 + (Zα) 2 r1 r2 C 2 (7.191d) The structure for the angular-momentum expansion is the same as in Eq. (7.179). A Green function related to the one given in Eq. (7.191) has been calculated in Eq. (5.4) of Ref. [258]. 7.5

Further Thoughts

Here are some suggestions for further thought. (1) Lorentz Group. In view of Eqs. (7.27) and (7.30), explain the correspondence 1 i (7.192) − σαβ ↔ Mαβ 4 2 on the basis of the commutator relations fulfilled by the matrices. (2) Lorentz Transformation and Dirac Adjoint. Verify Eq. (7.34) by an explicit calculation. (3) Dirac Current. Show Eq. (7.39), i.e., convince yourself that J µ (x) = ψ(x) γ µ ψ(x) ,

J ′µ (x′ ) = Λµ ν J ν (x) ,

∂µ J µ (x) = 0 .

(7.193)

(4) CPT Transform. Verify all intermediate steps in Eq. (7.73) in detail. (5) C Transform. Show that C

ψ C (x) = ψ (x) .

(7.194)

Convince yourself that the Dirac current is invariant under charge conjugation, J µ (x) = eψ(x) γ µ ψ(x) = eψ C (x) γ µ ψ C (x) .

(7.195)

This result is counter-intuitive because the charge-conjugated current characterizes antiparticles, which have the opposite charge as compared to particles. In the second-quantized formalism, ψ(x) denotes the field operator (7.140). Let us temporarily denote the second-quantized field operator by the symbol ˆ ψ(x). Show that, in the second-quantized formulation, one has ˆ C (x) γ µ ψˆC (x) , ˆ ˆ γ µ ψ(x) = −e ψ Jˆµ (x) = e ψ(x)

(7.196)

i.e., that the second-quantized current operator changes sign, in accordance with intuition. In the calculation, pay close attention to the sequence of the anticommuting field operators.

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(6) PT Transform. On the basis of Eq. (7.71), show that ψ PT (x) = i γ 5 γ 2 ψ ∗ (xPT ) ,

ψ C (x) = i γ 2 ψ ∗ (x) ,

(7.197)

so that ψ PT (x) = γ 5 ψ C (−x) .

(7.198)

C

(7) (8)

(9)

(10)

Knowing that ψ (x) fulfills the charge-conjugated Dirac equation, attempt to determine whether ψ PT (x) fulfills the charge-conjugated, or the original, Dirac equation. Dirac Algebra. Show Eq. (7.80). Dirac Field and Anticommutators. Consider the canonical anticommutation relations given in Eq. (7.144) and the quantized Lagrangian given in Eq. (7.78). From the quantized Lagrangian (7.78), derive the field momentum, and show that the equal-time anticommutator given in Eq. (7.159) is compatibility with the canonical anticommutation relation of the field and its conjugate momentum. For the anticommutator, consult Eqs. (7.161) and (10.282). Radial Dirac Equation for a Massless Particle. Consider the radial equations governing a Dirac eigenstate, given in Eq. (7.113). Solve the first of these equations for g(r), and insert the result into the second equation. You will find a second-order differential equation for f (r). Also, solve the second of the given differential equations for f (r), and insert the result into the first equation. You should find a second-order differential equation for g(r). Verify that both obtained second-order differential equations are compatible with the explicit solutions given in Eq. (7.114), and with Eq. (7.116). Neutrino Solutions of the Radial Dirac Equation. Consider Eqs. (7.110), (7.117), (7.118) and (7.126). Show, by explicit calculation, that (ν)

(ν)

HFMD ψEjµ (⃗ r) = E ψEjµ (⃗ r) = (

⃗ ⋅ p⃗ 0 σ (ν) ) ψEjµ (⃗ r) . ⃗ ⋅ p⃗ 0 σ

(7.199)

Show that, in view of the structure (ν)

ψEjµ (⃗ r) = (

νEjµ (⃗ r) ), −νEjµ (⃗ r)

(7.200)

⃗ p⃗ ψ the property (7.199) implies that Σ⋅ r) = −∣⃗ p∣ ψEjµ (⃗ r), i.e., that ψEjµ (⃗ r) Ejµ (⃗ is a negative-helicity eigenstate. (11) Antineutrino Wavefunction. Write the analogy of Eq. (7.126) for an antineutrino solution. Observe that antineutrinos have negative energy, negative chirality, but positive helicity. (12) Discrete Symmetries and Second Quantization. Consider the Lagrangian for the Dirac field, given, in Eq. (7.79), (ν)

(ν)

L(x) = ψ(x) (i γ µ ∂µ − m) ψ(x) ,

(ν)

(7.201)

but this time interpreting the variables ψ(x) and ψ(x) as second-quantized field operators, in the sense of Eqs. (7.140) and (7.141). Show that, upon

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(13)

(14) (15)

(16)

(17)

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subsequent parity (P), time-reversal(T ), and charge-conjugation (C) transformations, the Lagrangian restores its original sign, because of an exchange in the sequence of field operators. In view of the anticommutators given in Eqs. (7.143) and (7.144), show that various minus signs encountered in the derivation of the CPT transformation in the first-quantized theory are compensated, and the overall sign of Lagrangian is fully restored. Show that this conclusion is consistent with arguments presented in Chap. 7 of Ref. [250], and, under the inclusion of gravitational coupling, with the conclusions of Ref. [251]. Dirac Propagator. Perform an explicit Fourier transformation of Eq. (7.150) with respect to time, and show that you obtain Eq. (7.153) after formally introducing the Fourier backtransformation. Tachyon Propagator. Study pertinent literature references [252, 253, 259] and repeat the derivation in Sec. 7.4.1 for the tachyonic spin-1/2 field operators. Field Anticommutator. In regard to Eq. (7.159), one might ask if the spatial derivatives in the expression ∂
∆(x − y) contribute to the field anticommutator at x0 = y 0 . Convince yourself that for x0 = y0 , the expression e−ik⋅(x−y) − eik⋅(x−y) = 2i sin(k⃗ ⋅ r⃗) generates a factor proportional to k i cos(k⃗ ⋅ r⃗) upon differentiation with respect to xi , and the integral vanishes upon sym⃗ metrization with respect to the directions of k. Green Function and Physical Dimension. Perform a quick check of the physical dimension of Eq. (7.183). Convince yourself that in natural units, the physical dimension of F in Eq. (7.176) has to be equal to the mass squared, because the Hamiltonian on the left-hand side of Eq. (7.176) has the dimension of mass, whereas the Dirac-δ on the right-hand side has the dimension of mass to the power of three. Then, convince yourself that the expressions in Eq. (7.183) have exactly this mass dimension. Green Function and Physical Dimension. Show Eqs. (7.188) and (7.191).

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8.1

Overview

In the current chapter, we explore the Dirac equation in situations where the particle is coupled to electromagnetic fields, or, is moving in a curved space-time. The unifying element here is the concept of the covariant derivative. Namely, the application of a local gauge transformation multiplies the wave function by an unobservable phase factor, which can be different at any point in space-time. The application of the differential operator yields two terms, one of which amounts to the derivative, multiplied by said phase factor, while the other is a residual term which is absorbed in a gauge-transformed four-vector potential (electromagnetic coupling) or a gauge-transformed connection matrix (gravitational coupling). The former gauge transformation has been discussed, e.g., in Chap. 1 of Ref. [81], and is recalled in Sec. 8.2.1. Dirac particles in Coulomb fields are discussed in Sec. 8.2. The laserdressed propagator is discussed in Sec. 8.3. Gravitational coupling is discussed in Sec. 8.4. In the latter context, a gauge transformation amounts to a change of the local Lorentz frame. For Dirac particles in strong, plane-wave laser fields (see Sec. 8.3), we discuss a case of electromagnetic coupling, yet different from Sec. 8.2, in that the coupling concerns a time-varying electromagnetic (plane-wave) field. There is a certain analogy between the Dirac–Coulomb propagator (see Sec. 8.2.5) and the Dirac–Volkov propagator (see Sec. 8.3.2). Note that the corresponding problem of evaluating the dressed propagator, for gravitational coupling, has not yet been solved yet. Throughout this chapter, the Dirac wave function ψ should be taken as the wave function coming from the first-quantized theory, not as the second-quantized field operator. 8.2 8.2.1

Dirac Equation and Coulomb Field Electromagnetic Covariant Derivative

⃗ The four-vector potential is Aµ = (Φ, A). Under a gauge transformation of the electromagnetic scalar and vector potential (see Chap. 1 of Ref. [81]), they transform 265

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as follows, ⃗ , A⃗′ = A⃗ + ∇Λ

Φ′ = Φ − ∂t Λ ,

A′µ = Aµ − ∂ µ Λ ,

(8.1)

̵ = c = 0 = 1. where again, throughout this entire chapter, we use natural units with h The gauge transformation of the wave function proceeds as follows, ψ ′ = exp(i e Λ) ψ .

(8.2)

One defines the covariant derivative Dµ with the properties, Dµ = ∂µ + ie Aµ . Dµ′ ψ ′

=

(8.3)

(∂µ + ie A′µ ) ψ ′

= (∂µ + ie Aµ − ie ∂µ Λ) exp(i e Λ) ψ

= exp(i e Λ) (∂µ + ie Aµ ) ψ = (Dµ ψ)′ .

(8.4)

The covariant derivative of the gauge-transformed wave function is equal to the gauge transform of the covariant derivative. Introducing the gauge transformation G(Λ), we can write this property as G(Λ) = exp (i e Λ) ,

G(Λ)−1 Dµ′ G(Λ) ψ = Dµ ψ .

(8.5)

Z e2 Zα =− . 4πr r

(8.6)

For the Coulomb potential, one has ⃗ = (− Aµ = (Φ, A)

Ze ⃗ , 0) , 4πr

eA0 = −

⃗ then the electromagnetically coupled Dirac equation (7.44), which we recall If A⃗ = 0, for convenience, [γ µ (i∂µ − e Aµ ) − m] ψ = 0 ,

(8.7)

can thus be brought into the following form [see Eq. (7.17)] ⃗ − m] ψ = 0 , γ⋅∇ γ 0 [γ 0 (i∂0 − e A0 ) − i⃗

(8.8)

from which the Dirac–Coulomb Hamiltonian, which has the property i∂t ψ = HDC ψ, can easily be derived, Zα , HDC ψ = Eψ . (8.9) r The Dirac–Coulomb Hamiltonian is extremely important for bound-state problems. ⃗ ⋅ p⃗ + β m + V , HDC = α

8.2.2

V =−

Dirac–Coulomb Bound-State Wave Functions

The Dirac–Coulomb Hamiltonian, which we have just encountered in Eq. (8.9), should still be recalled at the start of the current discussion, ⃗ ⋅ p⃗ + β m + V , HDC = α

V =−

Zα . r

(8.10)

⃗ and β matrices have been defined in Eq. (7.19). We now need to find the The α result of the Hamiltonian operator acting onto the standard solution with upper

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component f (r) χκ µ (ˆ r) and lower component ig(r) χ−κ µ (ˆ r), where we recall the discussion in Sec. 6.4.1 and Sec. 7.3.3 [see Eq. (7.111)]. Our standard ansatz is ψ=(

f (r) χκµ (θ, ϕ) ). i g(r) χ−κµ (θ, ϕ)

(8.11)

We give this derivation in great detail, following our earlier treatment of the massive but free Dirac equation, ⃗ ⎞ ⃗⋅L ∂ σ Zα ⎛ (⃗ σ ⋅ rˆ) (−i +i )⎟ m− ⎜ r) r ∂r r ⎟ ( f (r) χκ µ (ˆ HDC ψ(⃗ r) = ⎜ ) ⎟ ⎜ ⃗ ⃗⋅L ∂ σ Zα r) ⎟ ig(r) χ−κ µ (ˆ ⎜ (⃗ σ ⋅ r ˆ ) (−i + i ) −m − ⎠ ⎝ ∂r r r ⎛ [(− ∂ + 1 (κ − 1)) g(r) + (m − Zα ) f (r)] χκ µ (ˆ r) ⎞ ⎟ ⎜ ∂r r r ⎟ = E ψ(⃗ r) . =⎜ ⎟ ⎜ ⎜ ⎟ ∂ 1 Zα r) ⎠ ⎝ i [( ∂r + r (κ + 1)) f (r) − (m + r ) g(r)] χ−κ µ (ˆ (8.12) We have used Eq. (6.81). The radial Dirac–Coulomb equations are thus given as follows, 1 Zα ∂ + (κ − 1)) g(r) − (E − m + ) f (r) = 0 , ∂r r r ∂ 1 Zα ( + (κ + 1)) f (r) − (E + m + ) g(r) = 0 . ∂r r r

(−

(8.13a) (8.13b)

In this form, the equations are amenable to further manipulation and simplification. Inserting one equation into the other, we could obtain a second-order differential equation for either f (r) or g(r). However, these equations do not have immediate analytic solutions. The radial wave functions f (r) and g(r) in the bound-state solutions are actually composed of two, not one, Laguerre polynomials in the terminating hypergeometric functions which describe the bound states. Therefore, we proceed in two steps. In the first, we perform a linear transformation on the f (r) and g(r) to write them in terms of two other functions h1 (r) and h2 (r). The linear transformation depends on a free parameter X. The parameter X is fixed so that the two resulting equations for h1 and h2 take a particularly simple form: One of the functions appears only with a simple prefactor, independent of the radial variable, while the other function has both a differential operator acting on it, as well as an r-dependent prefactor. From these two equations, one can immediately form two second-order equations which take the form of Whittaker’s differential equations, and the solution can be written in compact form. We then find the criterion for the two Laguerre polynomials to terminate, which is equivalent to the relativistic quantization condition and leads to the relativistic Dirac formula for the energy levels. The second step involves the calculation of normalization integrals and will be discussed in the following.

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We thus introduce a free parameter X, and write the functions h1 (r) and h2 (r) as follows, h (r) 1 X f (r) ( 1 )=( )( ), h2 (r) X 1 g(r)

1 f (r) 1 −X h1 (r) ( )= ( )( ), g(r) h2 (r) 1 − X 2 −X 1

(8.14)

where the latter equation is obtained by matrix inversion. Inserting these new functions into Eq. (8.13), and leaving X as a free parameter, we obtain (

(

κ − 1 + XZα Zα + X(κ − 1) + X(E −m) − ∂r ) h2 (r) + (m−E − + X∂r ) h1 (r) = 0 , r r (8.15a)

κ + 1 + XZα Zα + X(κ + 1) + X(E +m) + ∂r ) h1 (r) − (m+E + + X∂r ) h2 (r) = 0 . r r (8.15b)

We now multiply both Eqs. (8.15a) and (8.15b) by X, solve them for the expressions X∂r h1 (r) and X∂r h2 (r), respectively, and re-insert these results into Eq. (8.15). The resulting are found as [(1 − X 2 )m − (1 + X 2 )E − +[ [

(1 + X 2 )Zα + 2Xκ ] h1 (r) r

(1 + X 2 )κ + 2XZα − (1 − X 2 ) ∂ + 2XE − (1 − X 2 ) ] h2 (r) = 0 , r ∂r

(8.16a)

∂ (1 + X 2 )κ + 2XZα + (1 − X 2 ) + 2XE + (1 − X 2 ) ] h1 (r) r ∂r − [(1 + X 2 )E + (1 − X 2 )m +

(1 + X 2 )Zα + 2Xκ ] h2 (r) = 0 . r

(8.16b)

Dividing both equations by (1 − X 2 ), one obtains two equations with somewhat simplified prefactors, 1 + X2 (1 + X 2 )Zα + 2Xκ E − ] h1 (r) 1 − X2 (1 − X 2 )r ⎡ 1+X 2 ⎤ 2X ⎢ 2X ∂ ⎥⎥ 2 κ + 1−X 2 Zα − 1 + ⎢⎢ 1−X + E − h2 (r) = 0 , r 1 − X2 ∂r ⎥⎥ ⎢ ⎣ ⎦ ⎡ 1+X 2 ⎤ 2X ⎢ 1−X 2 κ + 1−X 2X ∂ ⎥⎥ 2 Zα + 1 ⎢ + E + h1 (r) ⎢ r 1 − X2 ∂r ⎥⎥ ⎢ ⎣ ⎦ (1 + X 2 )Zα + 2Xκ 1 + X2 E +m+ ] h2 (r) = 0 . −[ 1 − X2 (1 − X 2 )r

[m −

(8.17a)

(8.17b)

We now use our freedom to choose X so that (1 + X 2 )Zα + 2Xκ = 0. One of the two solutions of the quadratic equation for X reads as √ √ 1 κ κ−γ X= (8.18) (γ − κ) = − , γ = κ2 − (Zα)2 . Zα ∣κ∣ κ+γ

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The variable X is positive when κ is negative, and negative when κ is positive. The following relations fulfilled by the variable X are not completely trivial to derive, X2 = 1 + X2 = 1 − X2 2X = 1 − X2

2κγ 2κ2 − − 1, (Zα)2 (Zα)2

1 − X2 =

2κ (κ − γ) κ(κ2 − γ 2 ) κ (Zα)2 = = 2γ (Zα)2 γ γ κ+γ 2 (γ − κ) γ 2 − κ2 Zα Zα =− . = 2γ (Zα)γ γ κ+γ

2γ , κ+γ

,

(8.19a) (8.19b) (8.19c)

With these useful relations, we can directly replace the corresponding terms in Eq. (8.17) and obtain κ κ2 − (Zα)2 − γ Zα ∂ E] h1 (r) + [ − E− ] h2 (r) = 0 , γ γr γ ∂r

(8.20a)

κ2 − (Zα)2 + γ Zα ∂ κ − E+ ] h1 (r) − [ E + m] h2 (r) = 0 . γr γ ∂r γ

(8.20b)

[m − [

These relations can trivially be simplified upon observing that κ2 − (Zα)2 = γ 2 . As envisaged, the function h1 (r) in Eq. (8.20a) carries only a constant prefactor, while the same is true for the function h2 (r) in Eq. (8.20b). We can now solve Eq. (8.20a) for h1 (r), with the result h1 (r) = (m −

(γ − 1) ∂ κ −1 Zα E) [ E− + ] h2 (r) , γ γ r ∂r

(8.21)

and use the result in Eq. (8.20b). Likewise, we can solve Eq. (8.20b) for h2 (r), with the result h2 (r) = (m +

∂ κ −1 (γ + 1) Zα E) [ − E+ ] h1 (r) , γ r γ ∂r

(8.22)

and use the result in Eq. (8.20a). We thus find the uncoupled second-order differential equations fulfilled by h1 (r) and h2 (r) separately, [m − +

γ(γ + 1) Zα Zα 2 − 2E + ( E) ]h1 (r) = 0 , r2 r γ

(m − +

κ κ −1 ∂2 γ − 1 (γ + 1) ∂ E] h1 (r) + (m + E) [− 2 + ( − ) γ γ ∂r r r ∂r (8.23a)

κ −1 ∂ 2 γ + 1 (γ − 1) ∂ γ(γ − 1) E) [ 2 + ( − ) − γ ∂r r r ∂r r2

2 Zα Zα κ 2E − ( E) ]h2 (r) − [ E + m] h2 (r) = 0 . r γ γ

(8.23b)

We have simplified the equations by recalling that γ 2 = κ2 − (Zα)2 . Furthermore, it is useful to multiply Eq. (8.23a) by m + κγ E and Eq. (8.23b) by m − κγ E. This

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yields [

∂2 2 ∂ γ(γ + 1) + − ∂r2 r ∂r r2 +

[

Zα 2 κ2 Zα 2E − ( E) + ( 2 E 2 − m2 )] h1 (r) = 0 , r γ γ

(8.24a)

∂2 2 ∂ γ(γ − 1) + − 2 ∂r r ∂r r2 +

2 Zα Zα κ2 2E − ( E) + ( 2 E 2 − m2 )] h2 (r) = 0 . r γ γ

(8.24b)

Finally, one arrives at two very compact equations, (

2 ∂ γ (γ + 1) Zα ∂2 + − + 2E − (m2 − E 2 )) h1 (r) = 0 , ∂r2 r ∂r r2 r

(8.25a)

(

∂2 2 ∂ γ (γ − 1) Zα + − + 2E − (m2 − E 2 )) h2 (r) = 0 . 2 ∂r r ∂r r2 r

(8.25b)

After all the manipulations, we have arrived at differential equations that can be seen as a generalization of the Schr¨odinger equation. We now transform hi (r) → υi (r)/r for i = 1, 2 in order to get rid of the first-order derivative. We additionally define, with Eq. (7.183e), √ (8.26) C = m2 − E 2 , and find (

∂2 γ (γ + 1) Zα − + 2E − C 2 ) υ1 (r) = 0 , ∂r2 r2 r

(8.27a)

(

∂2 γ (γ − 1) Zα − + 2E − C 2 ) υ2 (r) = 0 . ∂r2 r2 r

(8.27b)

We now rescale the variables according to the relations Zα E E =√ . (8.28) C m2 − E 2 The functions are transformed according to υi (r) = υi (ρ/(2C)) = υ i (ρ) ≡ υi (ρ). In the last step, we simply redefine υ i (ρ) to be equal to υi (ρ). The rescaling turns Eq. (8.27) into a dimensionless equation of the form ρ = 2C r ,

k = Zα

[

γ (γ + 1) k 1 ∂2 1 k ∂2 − + − ] υ (ρ) = [ − + + 1 2 2 2 ∂ρ ρ ρ 4 ∂ρ 4 ρ

1 4

− (γ + 12 )2 ] υ1 (ρ) = 0 , (8.29a) ρ2

[

∂2 γ (γ − 1) k 1 ∂2 1 k − + − ] υ2 (ρ) = [ 2 − + + 2 2 ∂ρ ρ ρ 4 ∂ρ 4 ρ

1 4

− (γ − 12 )2 ] υ2 (ρ) = 0 . (8.29b) ρ2

These equations are related to each other by the replacement γ → γ − 1. This replacement indirectly accounts for the fact that a state of defined j (total angular momentum of the electron, defined as the sum of the orbital angular momentum

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and the spin) is made up of a combination of ` and ` − 1. The component with j = ` − 1/2 is obtained by subtracting the spin, and the component with j = ` + 1/2 is obtained by adding the spin. The solution to Eq. (8.29a) above is given by a Whittaker function, which was originally discussed in Sec. 4.2.2, υ1 (ρ) = M

1 (ρ) k,γ+ 2

1

= e− 2 ρ ργ+1 1 F1 (γ + 1 − k, 2γ + 2, ρ) ,

(8.30)

whereas the solution of Eq. (8.29b) is obtained after the replacement γ → γ − 1, namely, υ2 (ρ) = M

k,γ−

1 (ρ) 2

1

= e− 2 ρ ργ 1 F1 (γ − k, 2γ, ρ) .

(8.31)

If at least one of the solutions is supposed to terminate (we can set the other one equal to zero), then we must demand that the first argument of the hypergeometric functions be equal to zero, or to a negative integer nr , where nr could be interpreted as a relativistic principal quantum number. This already determines the relativistic spectrum of hydrogenlike ions, m Zα E , E=¿ − nr = γ − k = γ − √ , (8.32) 2 2 2 m −E Á Á À1 + (Zα) (nr + γ)2 √ with nr = 0, 1, 2, 3, . . . and γ = κ2 − (Zα)2 . In the special case nr , we set υ1 equal to zero and convince ourselves that the energy of the ground state of hydrogenlike ions reads as √ nr = 0 , κ = −1, E = m 1 − (Zα)2 . (8.33) A Taylor expansion for Zα → 0 gives rise to the expansion E ≈ m − (Zα)2 m/2, where the first term is the rest-mass energy and the second term is equal to the nonrelativistic (Schr¨ odinger) binding energy. We can thus replace k → nr + γ in the Whittaker functions (8.30) and (8.31), Γ(nr ) Γ(2γ + 2) 2γ+1 L (ρ) , (8.34a) Γ(nr + 2γ + 1) nr −1 1 − ρ γ Γ(nr + 1) Γ(2γ) 2γ−1 υ2 (ρ) = M Lnr (ρ) , (8.34b) 1 (ρ) = e 2 ρ nr +γ,γ− Γ(nr + 2γ) 2 where we convert to a notation with associated Laguerre polynomials. The functions h1 (r) and h2 (r) are obtained from υ1 (ρ) and υ2 (ρ) by a simple division through ρ. Recalling that ρ = 2 C r, we have υ1 (ρ) = M

1 (ρ) nr +γ,γ+ 2

1

= e− 2 ρ ργ+1

h1 (r) = N1 e−C r (2Cr)γ L2γ+1 nr −1 (2Cr) ,

(8.35a)

h2 (r) = N2 e−C r (2Cr)γ−1 L2γ−1 (8.35b) nr (2Cr) , √ where we recall that C = m2 − E 2 . The Γ functions in Eqs. (8.34a) and (8.34b) are absorbed in the normalization constants N1 and N2 implicitly defined in Eqs. (8.35a) and (8.35b).

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In the second step of the calculation, we now have to calculate the normalization constants. This is nontrivial, because we have already exhausted the freedom of choice with regard to the parameter X in the first step. Now, we have two free “relative” normalization constants N1 and N2 in front of the two independent solutions h1 (r) and h2 (r), and we also have an overall normalization constant in front of the f (r) and g(r), which, in turn, constitute linear combinations of the h1 (r) and h2 (r). In order to fix the “relative normalization constants”, we have to go back to the defining first-order, coupled differential equations which link the h1 (r) to the h2 (r). This leads to a first relation which needs to be fulfilled by N1 and N2 . The overall normalization condition for the bound-state wave function then leads to a condition which finally fixes the constants N1 and N2 to defined values. We start by observing that an expansion of Eq. (8.22) about the point r = 0 leads to the condition nr (2γ + nr ) C N1 = (E κ + m γ) N2 .

(8.36)

In view of the relation f (r)2 + g(r)2 =

4X 1 + X2 (h1 (r)2 + h2 (r)2 − h1 (r) h2 (r)) , 2 2 (1 − X ) 1 + X2

(8.37)

we can write with the normalization integral with the help of Eq. (8.19), ∞

2

3 r)∣ = ∫ ∫ d r ∣ψ(⃗

=

0

dr r2 [f (r)2 + g(r)2 ]

(8.38)

∞ κ (κ + γ) 2Zα h1 (r) h2 (r) + h2 (r)2 ] . (8.39) dr r2 [h1 (r)2 + ∫ 2 2γ κ 0

This integral has three terms, proportional to N12 , N22 , and N1 N2 , and evaluates to κ (κ + γ) Γ(2γ + nr ) [(γ + nr ) (2γ + nr ) nr N12 8γ nr ! C 3 2Zα − (2γ + nr ) nr N1 N2 + (γ + nr ) N22 ] = 1 . κ

(8.40)

Let us do a dimensional analysis. All factors are dimensionless with the exception of the factor C 3 in the denominator, which carries a mass dimension of power three in natural units. This implies that N1 and N2 should have physical dimension of mass to the power 3/2. The two equations (8.36) and (8.40) can be solved for the two unknowns N1 and N2 . However, the algebraic simplification is tricky. One needs two identities, γ2C 2 C2 = , Zα (m γ)2 + Eκ2 ((nr + γ) C − E Zα) Zα m2 (κ E − γ m) (κ E + γ m) = nr (nr + 2 γ) . C2

(8.41) (8.42)

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Using these identities, one can finally bring the result for N1 and N2 into a compact form, ¿ κ 2C 2 Á Á À (nr − 1)! (E κ + γ m) , √ N1 = ∣κ∣ Zα m Γ(2 γ + nr + 1) (κ + γ) ¿ 2C 2 Á Á À nr ! (E κ − γ m) . N2 = √ Γ(2 γ + nr ) (κ + γ) Zα m

(8.43a)

(8.43b)

Here, we might think that we have reached our goal, but, in fact, we have not. The above formulas for N1 and N2 are not universally applicable and have to be modified when nr = 0. The expression e−C r (2Cr)γ L2γ+1 nr −1 (2Cr) vanishes when nr = 0. However, N1 also becomes singular in this case, in view of the factor (nr − 1)! which tends to infinity for nr → 0. In this case, we are free to choose the normalization factor N1 , because the function multiplied by it vanishes anyway. Let us now investigate the case nr = 0 in greater detail. In this case, we have nr = 0 ,

E=

mγ , ∣κ∣

E κ + mγ = γ m

κ + ∣κ∣ . ∣κ∣

(8.44)

The latter factor refers to the condition in Eq. (8.36). It implies that when nr = 0, only negative values of κ are allowed. The normalization factors are then given as follows, N1 = 0 ,

κ = −∣κ∣ ,

N2 = 4 γ (

Zα m ) ∣κ∣

E=

3/2

√

mγ , ∣κ∣

nr = 0 ,

1 Γ(2γ + 1) κ (κ + γ)

.

(8.45a)

(8.45b)

The first few states for which nr = 0 and κ = −∣κ∣ are as follows, E(1S1/2 ) =

√

1 − (Zα)2 = 1 −

√ E(2P3/2 ) =

E(3D5/2 ) =

nr = 0 ,

κ = −1 , (8.46)

4 − (Zα)2 2

√

(Zα)2 (Zα)4 − + O(Zα)6 , 2 8

=1−

(Zα) (Zα) − + O(Zα)6 , 8 128 2

4

nr = 0 ,

κ = −2 , (8.47)

9 − (Zα)2 (Zα)2 (Zα)4 =1− − + O(Zα)6 , 3 18 648

nr = 0 ,

κ = −3 . (8.48)

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Let us summarize once more the decisive formulas. The wave function has the form r) ⎞ ⎛ f (r) χκ µ (ˆ ⎟, ψ(⃗ r) = ⎜ ⎝ i g(r) χ−κ µ (ˆ r) ⎠ f (r) =

κ+γ 1 = , 2 1−X 2γ

1 X h1 (r) − h2 (r) , 2 1−X 1 − X2

h1 (r) = N1 e−C r (2Cr)γ L2γ+1 nr −1 (2Cr) , The parameters are given as √ √ γ = κ2 − (Zα)2 , C = m2 − E 2 ,

E = m f (n, j) ,

X Zα =− , 2 1−X 2γ

g(r) = −

(8.49a)

X 1 h1 (r) + h2 (r) , 2 1−X 1 − X2 (8.49b)

h2 (r) = N2 e−C r (2Cr)γ−1 L2γ−1 nr (2Cr) . (8.49c)

1 κ = (−1)j+`+1/2 (j + ) , 2 −

(Zα)2 f (n, j) = (1 + ) (nr + γ)2

1 2

,

(8.49d)

nr = n − ∣κ∣ = n − j − 12 . (8.49e)

We have used Eq. (6.82), and defined nr to be a “reduced” principal quantum number. The normalization constants N1 and N2 are given in Eqs. (8.43a) and (8.43b) for a general case and in Eqs. (8.45a) and (8.45b) for the case nr = 0. The latter case is relevant only to the 1S state. 8.2.3

Dirac–Coulomb Continuum-State Wave Functions

The calculation of the continuum states of the Dirac equation, with E > 0, is an interesting task in view of the more involved nature of the eigenvalue problem, as compared to the nonrelativistic Schr¨odinger equation, for which we had discussed the continuum states in Sec. 4.2.5. We use the standard ansatz (8.11) for the Dirac bispinor wave function, eigenstate of the Hamiltonian given in Eq. (8.10), with the spin-angular function χκµ (θ, ϕ) being defined in Eq. (6.84). Following the same steps as were used for the bound-state problem in Sec. 8.2.2, we obtain the radial equations previously given in Eq. (8.13), and write them in a slightly different form, κ+1 ∂ f (r) = − f (r) + (E + m − V ) g(r) , ∂r r ∂ κ−1 g(r) = g(r) − (E − m − V ) f (r) . ∂r r

(8.50a) (8.50b)

One can rescale the problem by introducing the functions u1 and u2 , u1 (r) = r f (r) ,

u2 (r) = r g(r) .

(8.51)

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The radial equations become ∂ u1 (r) −κ/r E+m−V u (r) ( )=( )( 1 ). −(E − m − V ) κ/r u2 (r) ∂r u2 (r)

(8.52)

For a continuum state, it is advantageous to set, following Chap. 5 of Ref. [17], u1 = (E + m)

1/2

u2 = i (E − m)

(Φ1 + Φ2 ) ,

1/2

(Φ1 − Φ2 ) .

(8.53a) (8.53b)

Here, we remember that E > m, as we are talking about a continuum state. These equations entail that the radial parts u1 and u2 are real rather than imaginary if we set Φ∗1 = Φ2 .

(8.54)

We will be using this observation later. One defines the variable p so that in the asymptotic limit of a free particle, the quantity p would be the modulus of the momentum (in view of the free dispersion relation E 2 = p2 + m2 ). One also defines the imaginary radial variable x = 2ipr. We summarize, √ x = 2 i pr . (8.55) p = E 2 − m2 , Defining the functions Φj (x) ≡ Φj (r) (j = 1, 2), we “overload” the symbols Φj (x) with an additional definition, which is consistent as we observe that x is a dimensionless variable (in natural units), while r is dimensionful. This implies that ∂ ∂ Φj (r) = 2ip Φj (x) , ∂r ∂x The radial equations (8.50) take the form

j = 1, 2 .

1 iZαE κ iZα ∂ Φ1 (x) = ( + ) Φ1 (x) − ( − ) Φ2 (x) , ∂x 2 px x px ∂ κ iZα 1 iZαE Φ2 (x) = − ( + ) Φ1 (x) − ( + ) Φ2 (x) . ∂x x px 2 px

(8.56)

(8.57a) (8.57b)

The variable γ has been defined in Eq. (8.18). One solves Eq. (8.57a) for Φ2 , which expresses Φ2 in terms of Φ1 (x) and ∂Φ1 (x)/∂x. Plugging in the resulting expression into Eq. (8.57b), one obtains a first-order, and a second-order derivative term, and writes the equation 1 ∂ 1 1 iZαE 1 γ 2 ∂2 Φ + Φ (x) − [ + ( + ) + 2 ] Φ1 (x) = 0 . 1 1 ∂x2 x ∂x 4 2 p x x

(8.58)

With the substitution M(x) = x1/2 Φ1 (x) ,

(8.59)

one finds the differential equation ∂2 1 1 iZαE 1 M(x) + [− − ( + ) + ∂x2 4 2 p x

1 4

− γ2 ] M(x) = 0 . x2

(8.60)

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This matches the Whittaker differential equation, given in Eq. (4.34), which we recall for convenience, 1 κ d2 w + (− + + dρ2 4 ρ

1 4

− µ2 ) w = 0, ρ2

(8.61)

with the identification ZαE , µ=γ. (8.62) p Here, we note the difference of the generic κ parameter in Whittaker’s differential equation, and the Dirac angular quantum number κ in the spin-angular function. The solution of Eq. (8.61) regular at the origin is given as κ = − 12 − iy ,

y=

w = Mκ,µ (x) = e−x/2 xµ+1/2 1 F1 (µ +

1 2

− κ, 2µ + 1, x) .

(8.63)

For the parameters given in Eq. (8.62), it takes the form M(x) = M−1/2−iy,γ (x) = e−x/2 xγ+1/2 1 F1 (γ + 1 + iy, 2γ + 1, x) .

(8.64)

Our solution Φ1 (x) for the radial part of the Dirac–Coulomb continuum wave function therefore has to be equal to x−1/2 M(x). Restoring the radial variable r according to x = 2ipr, and decorating the function with a normalization factor N and some other convenient prefactors, as well as a phase factor exp(iη), one has Φ1 (x) = Φ1 (r) = N (γ + iy) eiη (2pr)γ e−ipr 1 F1 (γ + 1 + iy, 2γ + 1, 2ipr) .

(8.65)

It is useful to define the function Φ(r), Φ(r) = rγ e−ipr 1 F1 (γ + 1 + iy, 2γ + 1, 2ipr) ,

(8.66)

Φ1 (x) = N (γ + iy) eiη (2p)γ Φ(r) .

(8.67)

so that The complex conjugates of the derivatives of the radial wave functions Φj (x) take the form ∗ ∂Φ∗j (x) ∂Φj (x∗ ) ∂ ( Φj (x)) = = − , j = 1, 2 , (8.68) ∂x ∂x∗ ∂x because x∗ = −x. So, the complex-conjugated radial equations are ∂ ∗ 1 iZαE κ iZα Φ (x) = ( + ) Φ∗2 (x) − ( − ) Φ∗1 (x) , (8.69a) ∂x 2 2 px x px ∂ ∗ κ iZα 1 iZαE Φ1 (x) = − ( + ) Φ∗2 (x) − ( + ) Φ∗1 (x) . (8.69b) ∂x x px 2 px The equation systems (8.57) and (8.69) are identical upon the replacement Φ1 → Φ∗2 ,

(8.70)

justifying the ansatz (8.54). Using the property (8.70), Eq. (8.57a) therefore becomes 1 iZαE κ iZα ∂ Φ1 (x) = ( + ) Φ1 (x) − ( − ) Φ∗1 (x) . (8.71) ∂x 2 px x px

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With the help of the definitions (8.65) and (8.66), this equation can be rewritten in terms of a relation for the phase factor exp(−2iη), exp(−2iη) = −

r 1 ∂Φ y Φ γ + iy [ − ip (1 + ) ∗ ] . γ − iy κ − iy/E Φ∗ ∂r pr Φ

(8.72)

Quite surprisingly, one can show that the right-hand side is independent of r. To this end, one writes the complex conjugate of the function Φ(r) as Φ∗ (r) = rγ eipr 1 F1 (γ + 1 − iy, 2γ + 1, −2ipr) = rγ e−ipr 1 F1 (γ + iy, 2γ + 1, 2ipr) , (8.73) where the Kummer transformation (see page 253 of Ref. [154]) has been used, 1 F1 (a, b, z)

= exp(z) 1 F1 (b − a, b, −z).

(8.74)

In the numerator of Eq. (8.72), one generates confluent hypergeometric functions according to the formula ∂ a (8.75) 1 F1 (a, b, z) = 1 F1 (a + 1, b + 1, z) . ∂z b Both the a as well as the b parameters have to be lowered in order to lead to a cancelation of the r-dependent terms in Eq. (8.72), i.e., to a cancelation of the hypergeometric functions in the numerator and denominator. One achieves this by the use of the contiguous relations (see page 254 of Ref. [154]) b 1 F1 (a + 1, b, z) − b 1 F1 (a, b, z) − z 1 F1 (a + 1, b + 1, z) = 0 , (8.76) (a − b) 1 F1 (a − 1, b, z) + a 1 F1 (a + 1, b, z) + (b − 2a − z) 1 F1 (a + 1, b + 1, z) = 0 . (8.77) The first of these equations, given in (8.76), can be used in order to lower both the a as well as b parameters, when applied to the derivative of the Kummer function given in Eq. (8.75). The second equation, given in (8.77), can be used in order to lower the a parameter of the function Φ(r) given in Eq. (8.66), in the sense of the consecutive replacements γ +2+i y → γ +1+i y → γ +i y. These operations eventually lead to a complete elimination of the r-dependence in Eq. (8.72). One finally arrives at the following formula e2iη = −

κ − i y/E . γ + iy

(8.78)

Assembling the information collected in Eqs. (8.51), (8.53) and (8.65), one arrives at the following formulas for the radial components, r f (r) = (E + m)1/2 N (2pr)γ [(γ + iy) e−ipr+iη 1 F1 (γ + 1 + i y, 2γ + 1, 2ipr) + c.c.] , (8.79a) r g(r) = i (E − m)1/2 N (2pr)γ [(γ + iy) e−ipr+iη 1 F1 (γ + 1 + i y, 2γ + 1, 2ipr) − c.c.] . (8.79b) We now need to calculate the normalization factor N , with the notion of normalizing the radial wave functions to a Dirac-δ on the energy scale. To this end, we need

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to study the asymptotic behavior for r → ∞, of the wave function. The relevant formula for the 1 F1 function is as follows [see page 278 of Ref. [154]], 1 F1 (a, b, z)

→

Γ(b) a−c x x e , Γ(a)

x → ∞.

(8.80)

Using the recurrence relation of the Γ function, (γ + iy)/Γ(γ + 1 + i y) = 1/Γ(γ + i y), as well as the formula Γ(γ + i y) = ∣Γ(γ + i y)∣ exp(i arg(Γ(γ + i y))], one finds (γ + iy) eiη 1 F1 (γ + 1 + i y, 2γ + 1, 2ipr) → (2pr)−γ

Γ(2γ + 1) −πy/2 iδ e e exp(2ipr) , ∣Γ(γ + i y)∣ (8.81)

for large r, where the phase factor is δ = − 21 πγ − arg[Γ(γ + i y)] + y ln(2pr) + η .

(8.82)

Its dependence on r is only logarithmic. The asymptotics of the radial upper and lower component wave functions take the following forms, √ Γ(2γ + 1) cos(pr + δ) = E + m C cos(pr + δ) , ∣Γ(γ + i y)∣ (8.83a) √ Γ(2γ + 1) r g(r) → − 2 (E − m)1/2 e−πy/2 N sin(pr + δ) = − E − m C sin(pr + δ) , ∣Γ(γ + i y)∣ (8.83b)

r f (r) → 2 (E + m)1/2 e−πy/2 N

with an obvious definition of C, C = 2 e−πy/2 N

Γ(2γ + 1) . ∣Γ(γ + i y)∣

(8.84)

From these asymptotics, we can derive the normalization integral of two Dirac eigenstates ∣1⟩ and ∣2⟩, composed of radial wave functions f1 and f2 , and g1 and g2 , of different energies E1 and E2 , and in consequence, different asymptotic free momenta p1 and p2 , in the limit E1 → E2 , as follows, ⟨1∣2⟩ = ∫ dr [(r f1 (r))(r f2 (r)) + (r g1 (r))(r g2 (r))] → C2 ∫

∞ 0

dr [(E1 + m) cos(p1 r + δ) cos(p2 r + δ)]

+ [(E1 − m) sin(p1 r + δ) sin(p2 r + δ)] .

(8.85)

The radial integral with the cosines is ∫

∞ 0

dr cos(p1 r + δ) cos(p2 r + δ) = ∫

∞ 0

dr

ei(p1 r+δ) + e−i(p1 r+δ) ei(p2 r+δ) + e−i(p2 r+δ) 2 2

∞ 1 → ∫ dr [ei(p1 r−p2 r) + e−i(p1 r−p2 r) ] 4 0 ∞ 1 = ∫ dr ei(p1 r−p2 r) = πδ(p1 − p2 ) . 2 −∞

(8.86)

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Analogously, for the sine functions, we have ∫

∞ 0

dr sin(p1 r + δ) sin(p2 r + δ) = πδ(p1 − p2 ) .

(8.87)

Hence, ⟨1∣2⟩ = 2π C 2 E1 δ(p1 − p2 ) ≡ δ(E1 − E2 ) ,

(8.88)

where the last equality holds if the wave functions are normalized to the energy scale. Using known properties of the Dirac-δ function, one can easily show that δ(p1 − p2 ) =

p1 1 ∣ δ(E1 − E2 ) = δ(E1 − E2 ) , ∣dp/dE∣ E=E1 E1

(8.89)

where we recall Eq. (8.55). Hence, we must demand that 2π C 2 =

1 , p

for a general state with asymptotic free momentum p and energy E. Eq. (8.84), we have C = 2 e−πy/2 N

Γ(2γ + 1) 1 . ≡√ ∣Γ(γ + i y)∣ 2π p

(8.90) Using

(8.91)

So, the normalization factor has to be chosen as ∣Γ(γ + i y)∣ 1 eπy/2 N= √ . Γ(2γ + 1) 2 2π p

(8.92)

With Eq. (8.79), it means that the final result for the radial components of the wave functions is as follows, r f (r) = (E + m)1/2

(2pr)γ eπy/2 ∣Γ(γ + i y)∣ √ Γ(2γ + 1) 2 2π p

× [(γ + iy) e−ipr+iη 1 F1 (γ + 1 + i y, 2γ + 1, 2ipr) + c.c.] , r g(r) = i (E − m)1/2

(2pr) e √ 2 2π p γ

πy/2

(8.93a)

∣Γ(γ + i y)∣ Γ(2γ + 1)

× [(γ + iy) e−ipr+iη 1 F1 (γ + 1 + i y, 2γ + 1, 2ipr) − c.c.] .

(8.93b)

These wave functions are normalized to the energy scale [see Eq. (8.88)]. 8.2.4

Dirac–Coulomb Virial Theorem

We had discussed the virial theorem for the Schr¨odinger–Coulomb Hamiltonian in Sec. 4.2.4. For the Dirac–Coulomb case, the Hamiltonian is given by Eq. (8.9), Zα ⃗ = i [HDC , r⃗] . , α (8.94) r For an eigenstate of the Dirac–Coulomb Hamiltonian, (HDC − E)∣ψ⟩ = 0, we have ⃗ ⋅ p⃗ + β m + V , HDC = α

V =−

[HDC − E, A] ∣ψ⟩ = (HDC − E)A∣ψ⟩ .

(8.95)

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One of the virial theorems for the Dirac–Coulomb Hamiltonian, which reads as ⟨⃗ α ⋅ p⃗⟩ = − ⟨V ⟩ ,

(8.96)

⟨⃗ α ⋅ p⃗⟩ = ⟨i [HDC − E, r⃗] ⋅ p⃗⟩ = − ⟨i r⃗ ⋅ (HDC − E) p⃗⟩ ⃗ )⟩ . = − ⟨i r⃗ ⋅ [HDC − E, p⃗]⟩ = ⟨i r⃗ ⋅ [⃗ p, V ]⟩ = ⟨⃗ r ⋅ ∇(V

(8.97)

can be derived as follows,

For an eigenstate of the Dirac–Coulomb Hamiltonian, with V = −Zα/r, one has Zα (Zα)⃗ r Zα )⟩ = ⟨⃗ r⋅ ⟩=⟨ ⟩ = − ⟨V ⟩ . 3 r r r This result is consistent with Eq. (7) of Ref. [260]. A consequence is ⃗ )⟩ = ⟨⃗ ⃗ (− ⟨⃗ r ⋅ ∇(V r⋅∇

E = ⟨⃗ α ⋅ p⃗⟩ + ⟨V ⟩ + ⟨βm⟩ = ⟨βm⟩ .

(8.98)

(8.99)

This virial theorem reflects certain symmetries of the Schr¨odinger–Coulomb, and Dirac–Coulomb Hamiltonian. Let us briefly explore the calculation of matrix elements by the Hellmann– Feynman theorem [261, 262], applying scaling arguments to the calculation of two important matrix elements. Only two parameters enter the Dirac Hamiltonian HDC , and these are the nuclear charge number Z and the electron mass m. We thus consider the eigenfunction ψ = ψ(Z, m) formally as a function of Z and m and differentiate ∂ ⟨ ψ(Z, m) ∣HDC (Z, m)∣ ψ(Z, m) ⟩ Z ∂Z = 2 ⟨Z ∂Z ψ(Z, m)∣HDC (Z, m)∣ψ(Z, m)⟩+⟨ψ(Z, m)∣Z∂Z HDC (Z, m)∣ψ(Z, m)⟩ = E(Z, m) Z

∂ Zα ⟨ψ(Z, m)∣ψ(Z, m)⟩ + ⟨ψ(Z, m)∣ (− ) ∣ψ(Z, m)⟩ ∂Z r

Zα ) ∣ψ(Z, m)⟩ = ⟨ψ(Z, m) ∣ V ∣ ψ(Z, m)⟩ . (8.100) r This derivation is akin to the canonical derivation of the Hellmann–Feynman theorem [261, 262]. So, we have ∂ Z E(Z, m) = ⟨ψ(Z, m) ∣ V ∣ ψ(Z, m)⟩ , (8.101) ∂Z to all orders in the Zα expansion. The normalization ⟨ψ(Z, m)∣ψ(Z, m)⟩ = 1 holds independent of Z and m. Let us now differentiate with respect to m, ∂ ⟨ ψ(Z, m) ∣HDC (Z, m)∣ ψ(Z, m) ⟩ ∂m ∂ ∂ = 2⟨ ψ(Z, m)∣HDC (Z, m)∣ψ(Z, m)⟩ + ⟨ψ(Z, m)∣ HDC (Z, m)∣ψ(Z, m)⟩ ∂m ∂m = ⟨ψ(Z, m)∣ (−

= E(Z, m)

∂ ⟨ψ(Z, m)∣ψ(Z, m)⟩ + ⟨ψ(Z, m) ∣β∣ ψ(Z, m)⟩ ∂m

= ⟨ψ(Z, m) ∣β∣ ψ(Z, m)⟩ .

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So, we have E(Z, m) ∂ E(Z, m) = = ⟨ψ(Z, m) ∣ β ∣ ψ(Z, m)⟩ (8.102) ∂m m to all orders in the Zα expansion. Using the explicit form of the Dirac–Hamiltonian, this identity is seen to be equivalent to the virial theorem Eq. (8.97) for the Dirac– Coulomb equation. The last equality can independently be derived from the fact that the Dirac energy E(Z, m) actually is directly proportional to the electron mass m, i.e. it has the functional form E(Z, m) = m f (Z) .

(8.103)

In order to show this, we consider the Dirac–Coulomb Hamiltonian once more and write it as ∂ Zα Zα ⃗⋅ ⃗ ⋅ p⃗ + β m − = −i α +βm− HDC = α r ∂ r⃗ r Zα ∂ Zα ∂ ⃗⋅ ⃗⋅ +β− ) = m (−i α +β− ). (8.104) = m (−i α ⃗ ∂(m⃗ r) mr ∂ρ ρ Here, we have defined ρ⃗ = m⃗ r. The electron mass scales out as a prefactor, and ⃗ ρ⃗ + β − Zα/ρ is free of the mass. Further relativistic ⃗⋅∇ the scaled Hamiltonian −i α sum rules and integral properties of the Dirac equation have been discussed by Goldman1 and Drake2 in Ref. [263]. Useful generalizations of virial relations have been derived by Shabaev3 in Refs. [264, 265]. Additional relevant considerations, also in regard to the Hellmann–Feynman theorem, can be found in Ref. [266]. 8.2.5

Dirac–Coulomb Propagator

The calculation of the Dirac–Coulomb propagator follows essentially the same lines as the calculation of the Schr¨ odinger–Coulomb Green function in Sec. 4.3. However, because the Dirac–Coulomb Hamiltonian is a (4 × 4)-matrix in spinor space, the Dirac–Coulomb propagator also must be of the same structure. We thus investigate ⃗ ⋅ p⃗ + β m − Zα/r. The Green the inverse of the Dirac–Coulomb Hamiltonian HDC = α function has the following structure [see Eqs. (7.173) and (7.176)] GDC (⃗ r1 , r⃗2 , E) = ⟨⃗ r1 ∣

1 ∣ r⃗2 ⟩ , HDC − E

(HDC − E) GDC (⃗ r1 , r⃗2 , E) = δ (3) (⃗ r1 − r⃗2 )

(8.105a)

14×4 .

(8.105b)

A remark is in order. When using the Feynman prescription for the displacement of the poles, the expression for the Green function gets modified into ⟨⃗ r1 ∣1/[HDC (1 − i) − E]∣ r⃗2 ⟩, i.e., the energies of virtual positive-energy states obtain an infinitesimal negative imaginary part, while the energies of virtual 1 Samuel

Pedro Goldman (b. 1949). William Frederic Drake (b. 1944). 3 Vladimir Moiseevich Shabaev (b. 1959). 2 Gordon

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negative-energy states obtain an infinitesimal positive imaginary part. In boundstate calculations, the energy variable E typically is equal to a reference-state energy, minus the energy of one or more virtual photons [257, 258, 267–270]. As is well known, one can implement the Feynman prescription either by infinitesimally displacing the virtual-state poles in the Dirac–Coulomb propagator, or, by a judicious choice of the photon integration contours [257, 258, 267–270]. In the latter case, which is relevant to practical calculations [257, 258, 267–270], it would be redundant to also displace the poles infinitesimally, and this approach is implemented in the following. If one were to insist on the former approach, then one would need to separate the contributions of positive-energy, and negative-energy states, to the Dirac–Coulomb Green function. This separation is immediately accomplished in a finite-basis-set representation (see Sec. 2.2 of Ref. [270]). If one were to attempt a corresponding separation of the contributions for the analytic approach below, inspiration could be drawn from Ref. [271]. Let us also point out that Ref. [271] treats the calculation of the reduced Dirac–Coulomb Green function. Returning to the solution of Eq. (8.105), we observe that these equations constitute the generalization of Eq. (4.82) to the Dirac–Coulomb case. In (2 × 2)-matrix notation, we have Zα ⃗ r1 ⎞ ⎛(− ⃗⋅∇ + m − E) 12×2 −i σ ⎟ GDC (⃗ ⎜ r1 r1 , r⃗2 , E) = δ (3) (⃗ r1 − r⃗2 ) 14×4 . ⎟ ⎜ Zα ⃗ ⃗ − m − E) 1 −i σ ⋅ ∇ (− 2×2 r ⎠ ⎝ 1 r1 (8.106) In full analogy to the Schr¨ odinger–Coulomb Green function, the Dirac–Coulomb Green function also has an expansion in terms of angular-momentum components, which are characterized by the Dirac angular momentum quantum number κ defined in Eq. (6.82), as κ = (−1)j+`+1/2 (j + 21 ). In analogy to the free Dirac Green function (7.179), the expansion of the Dirac–Coulomb Green function has the expansion GDC (⃗ r1 , r⃗2 , E) = ∑( κ

11 12 GDC,κ (r1 , r2 , E)πκ (ˆ r1 , rˆ2 ) GDC,κ (r1 , r2 , E)i⃗ σ ⋅ rˆ1 π−κ (ˆ r1 , rˆ2 ) ). 21 22 −GDC,κ (r1 , r2 , E)i⃗ σ ⋅ rˆ1 πκ (ˆ r1 , rˆ2 ) GDC,κ (r2 , r1 , E)π−κ (ˆ r1 , rˆ2 ) (8.107)

Here, the sum over κ runs over the positive and negative integers, with the exception of zero, and the sum over the magnetic projections runs over µ = ⃗ ⋅ rˆ χκ µ (ˆ −(∣κ∣ − 1/2), . . . , ∣κ∣ − 1/2. We have used the identity σ r) = −χ−κ µ (ˆ r) in the reformulation of the angular projection. Otherwise, the angular part involves the projection operator, originally defined in Eq. (6.96). The angular decomposition of the Dirac–Coulomb Green function follows the same pattern as the free Green function, given in Eq. (7.179). The result is given in Eq. (A.16) of Ref. [257]. A compact representation of the radial Dirac–Coulomb Green functions can be given in the following functional form (for r1 = r< = min(r1 , r2 ) and r2 = r> = max(r1 , r)2),

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Gκ11 (r< , r> , E) = Q (1 + z) M− (λ, ν, 2cr< ) W+ (λ, ν, 2cr> ) ,

(8.108a)

Gκ12 (r< , r> , E) Gκ21 (r< , r> , E)

= Q c M− (λ, ν, 2cr< ) W− (λ, ν, 2cr> ) ,

(8.108b)

= Q c M+ (λ, ν, 2cr< ) W+ (λ, ν, 2cr> ) ,

(8.108c)

Gκ22 (r< , r> , E)

= Q (1 − z) M+ (λ, ν, 2cr< ) W− (λ, ν, 2cr> ) .

(8.108d)

The symmetry relations are completely analogous to Eq. (7.183f), Gκ11 (r> , r< , E) = Gκ11 (r< , r> , E) ,

(8.109a)

Gκ12 (r> , r< , E) Gκ21 (r> , r< , E) Gκ22 (r> , r< , E)

(8.109b)

= = =

Gκ21 (r< , r> , E) , Gκ12 (r< , r> , E) , Gκ22 (r< , r> , E) .

(8.109c) (8.109d)

The functions denoted as M± and W± are related to linear combinations of two Whittaker functions, M± (λ, ν, y) = (λ − ν) Mν−1/2,λ (y) ± p− Mν+1/2,λ (y) ,

(8.110a)

W± (λ, ν, y) = p+ Wν−1/2,λ (y) ± Wν+1/2,λ (y) ,

(8.110b)

where we recall the definition of the Whittaker functions given in Eqs. (4.35a) and (4.35b). Various symbols enter these expressions; these are given as follows, Q=

1

Γ(λ − ν) , Γ(1 + 2 λ)

4 c2 (r1 r2 )3/2 √ λ = κ2 − (Zα)2 = ∣κ∣ − ζ ,

c= ζ=

√

m2 − E 2 ,

(Zα)2 , ∣κ∣ + λ

p± = κ ±

Zα , c

(Zα) E . ν=√ m2 − E 2

(8.111a) (8.111b)

In practical calculations [257,258,267], the energy variable E may assume manifestly complex values. For practical calculations, it is sometimes useful to rewrite the Whittaker functions M and W in terms of more fundamental hypergeometric U and 1 F1 functions, profiting from the isolation of the integer part of λ as given in Eq. (8.111). We have the following formula for the M function, 1 1 1 1 Mν±1/2,λ (y) = e− 2 y y λ+ 2 1 F1 ( ∓ + λ − ν, 2 λ + 1, y) 2 2

= e− 2 y y ∣κ∣+ 2 −ζ 1 F1 (∣κ∣ + 1 − δ± − ζ − ν, 2 ∣κ∣ + 1 − 2 ζ, y) , 1

1

(8.112)

whereas the W functions can be rewritten as follows, 1 1 1 1 Wν±1/2,λ (y) = e− 2 y y λ+ 2 U ( ∓ + λ − ν, 2 λ + 1, y) 2 2

= e− 2 y y ∣κ∣+ 2 −ζ U (∣κ∣ + 1 − δ± − ζ − ν, 2 ∣κ∣ + 1 − 2 ζ, y) . (8.113) 1

1

We here define the variable δ± as δ± = δ±1,1 = {

1 for the case + . 0 for the case −

(8.114)

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For the W function, it is possible to rewrite the U function as a 2 F0 hypergeometric, 1 1 1 1 1 1 Wν±1/2,λ (y) = e− 2 y y ν±1/2 2 F0 ( ∓ + λ − ν, ∓ − λ − ν, − ) 2 2 2 2 y 1 1 = e− 2 y y ν±1/2 2 F0 (∣κ∣ + 1 − δ± − ζ − ν, −∣κ∣ + 1 − δ± + ζ − ν, − ) . (8.115) y A last representation useful for small y is the decomposition of the W function in terms of two confluent hypergeometric functions, ⎡ 1 1 ⎢ 1 1 Γ(−2 λ) Wν±1/2,λ (y) = e− 2 y y λ+ 2 ⎢ 1 1 F ( ∓ + λ − ν, 2 λ + 1, y) ⎢ Γ ( ∓ − λ − ν) 1 1 2 2 ⎣ 2 2 ⎤ Γ(2 λ) 1 1 ⎥ ⎥ + F ( ∓ − λ − ν, 1 − 2 λ, y) 1 1 ⎥ 2 2 Γ ( 12 ∓ 12 + λ − ν) ⎦ 1 1 Γ(−2 ∣κ∣ + 2 ζ) = e− 2 y y ∣κ∣+ 2 −ζ [ 1 F1 (∣κ∣ + 1 − δ± − ν − ζ, 2 ∣κ∣ + 1 − 2 ζ, y) Γ (−∣κ∣ + 1 − δ± − ν + ζ) Γ(2 ∣κ∣ − 2 ζ) 1 F1 (−∣κ∣ + 1 − δ± − ν + ζ, −2 ∣κ∣ + 1 + 2 ζ, x)] . Γ (∣κ∣ + 1 − δ± − ν − ζ) This concludes our discussion of the Dirac–Coulomb Green function. +

8.3 8.3.1

(8.116)

Dirac–Volkov Equation for Laser Fields Dirac–Volkov Solutions for Laser Fields

In the presence of electromagnetic background fields, characterized by a vector potential Aµ (φ), the Dirac equation takes the form (iγ µ ∂µ − eγ µ Aµ − m) ψ(x) = 0 ,

(8.117)

where we assume minimal coupling in the form of the replacement pµ = i ∂µ → pµ − e Aµ . A particularly interesting case is obtained when Aµ assumes the form of a vector potential for a plane-wave field, encompassing both linear as well as circular polarization. The Dirac–Volkov wave functions ψ(x) which fulfill the Dirac equation with an external potential can be written in the form (we use the Feynman slash notation k
= γ µ kµ ) (i ∂
− eA(φ) − m) ψ(x) = 0 , φ = k ⋅ x = ω t − k⃗ ⋅ r⃗ . (8.118) In general, Dirac–Volkov solutions can be characterized by a constant four-vector p, which is equal to the asymptotic momentum of the Dirac–Volkov state in the absence of the laser field, and by a spin projection σ. The Dirac–Volkov solutions differ from the solutions of the free Dirac equation in a matrix-valued phase-dependent prefactor, and in a laser-phase factor which is obtained by integrating over the laser-field vector potential [3]. We have √ √ m e k
A (φ) m ψp,σ (x) = (1 + ) Uσ (p) exp (i Sp [φ]) ≡ E(p, x) Uσ (p) . QV 2 (k ⋅ p) QV (8.119)

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Here, E(p, x) is a laser-dressed prefactor which still needs to be evaluated more precisely and expanded into plane waves, and Q = q 0 is the timelike component of the effective momentum within the laser field [see also Eq. (8.122) below]. The bispinor Uσ (p) is defined for an electron in the helicity basis, according to Eq. (7.89). The normalization volume is V . The phase of the laser field is given as φ = k ⋅ x = ω t − k⃗ ⋅ r⃗ . (8.120) The phase factor in the exponential is given as follows, Sp [φ] = −p ⋅ x − ∫

φ 0

dφ′ (e

p ⋅ A (φ′ ) A2 (φ′ ) − e2 ). k⋅p 2 (k ⋅ p)

(8.121)

After averaging over a laser period, one obtains the effective four-momentum q for the electron in the laser field, q =p+

e2 ∣a2 ∣ k, 2k ⋅ p

a2 =

2π 1 A2 (φ′ ) dφ′ < 0 . ∫ 2π 0

(8.122)

The quantity a2 is the average of the square of the vector potential over a laser period, and it is negative for a space-like vector potential (radiation gauge). Here, we write the modulus ∣a2 ∣ = −a2 > 0. Because k 2 = 0 for the laser photon, we can define the effective mass of the electron in the laser field as the four-vector product of q with itself, q 2 = m2 + e2 ∣a2 ∣ = m2∗ .

(8.123)

q 2 = (q 0 )2 − q⃗2 = Q2 − q⃗2 = m2∗ ,

(8.124)

Observing that

we end up with a modified relativistic energy-momentum relation. The solution (8.119) is not written in terms of plane waves, which are otherwise useful in order to carry out the integrations over time in S-matrix calculations of laser-dressed processes. In order to be able to carry out these integrations, we thus need to expand the Dirac–Volkov state into plane waves. We first turn our attention to linear polarization, in which case the vector potential reads as Aµ (φ) = aµ cos φ .

(8.125)

One could otherwise define the vector potential as Aµ (φ) = a µ cos φ, where a is the amplitude and µ is a polarization vector, but the above notation is more compact. Usually, in laser-physics problems, one works in the radiation gauge, where 0 = 0, and the four-vector is  = (0, ⃗i ) (for i = 1, 2). Of course, the four-vector square is 2 = µ µ = −1. The four-vector square of the four-vector potential of the laser is given as Aµ (φ)Aµ (φ) = aµ aµ cos2 φ = a2 cos2 φ = −∣a2 ∣ cos2 φ .

(8.126)

After integrating out the laser phase, the phase factor evaluates to iSp [φ] = −i (p +

e2 ∣a2 ∣ ea ⋅ p e2 ∣a2 ∣ k) ⋅ x − i sin φ − i sin(2φ) . 4(k ⋅ p) k⋅p 8 (k ⋅ p)

(8.127)

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The first term in round brackets is easily identified as the laser-dressed momentum in the linearly polarized field, q =p+

e2 ∣a2 ∣ k, 4(k ⋅ p)

q 2 = p2 +

e2 ∣a2 ∣ e2 ∣a2 ∣ = m2 + = m2∗ . 2 2

(8.128)

We also define the parameters α=

ea ⋅ p , k⋅p

β=−

e2 ∣a2 ∣ . 8 (k ⋅ p)

(8.129)

We need to expand the phase factor exp(i Sp [φ]). When comparing with the literature, one has to be very careful about the phase conventions used. Also, we should point out one little subtlety. Namely, in Eq. (50) of Ref. [272], we have the expres2 2 a sion y = 8e(k⋅p) , but the quantity denoted as a2 in Ref. [272] is our −∣a2 ∣. There, we have Aµ = aµ cos φ and not Aµ = a µ cos φ [consult the text before Eq. (47) of Ref. [272]]. The definitions of α here and of the corresponding variable in Ref. [272] are in mutual agreement. The phase factor in Eq. (8.121) therefore evaluates to iSp [φ] = −q ⋅ x − i α sin φ + i β sin(2φ) .

(8.130)

In order to expand this phase factor into plane waves, we have to define generalized Bessel functions which are given as π 1 ∫ exp[i s θ − i α sin(θ) + i β sin(2θ)] dθ 2π −π π 1 = ∫ cos[s θ − α sin(θ) + β sin(2θ)] dθ . 2π −π

A0 (s, α, β) =

(8.131)

The general theorem of Fourier decomposition then implies that ∞

exp (−iα sin φ + iβ sin(2φ)) = ∑ A0 (s, α, β) e−isφ .

(8.132)

s=−∞

It is useful to also define the following generalized A functions, ∞

exp (−iα sin φ + iβ sin(2φ)) = ∑ A0 (s, α, β) e−isφ ,

(8.133a)

cosL φ exp (−iα sin φ + iβ sin(2φ)) = ∑ AcL (s, α, β) e−isφ ,

(8.133b)

sinL φ exp (−iα sin φ + iβ sin(2φ)) = ∑ AsL (s, α, β) e−isφ .

(8.133c)

s=−∞ ∞ s=−∞ ∞ s=−∞

The generalized Bessel functions with cosine and sine prefactors, given by AcL (1, α, β) and AsL (1, α, β), fulfill the following relations, A0 (s, α, β) = Ac0 (s, α, β) = As0 (s, α, β) ,

(8.134a)

AcL (s, α, β) AsL (s, α, β)

(8.134b)

= =

1 (AcL−1 (s + 1, α, β) + AcL−1 (s − 1, α, β)) , 2 1 (AsL−1 (s + 1, α, β) − AsL−1 (s − 1, α, β)) 2i

,

(8.134c)

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as is easily shown by inserting the parameters s+1 and s−1 into the right-hand side of Eqs. (8.133b) and (8.133c). The exponentials in the Dirac–Volkov state (8.119) can be expanded into Bessel functions in view of the Jacobi–Anger expansions, ∞

exp (iz cos φ) = ∑ is Js (z) eisφ , s=−∞

∞

exp (iz sin φ) = ∑ Js (z) eisφ .

(8.135)

s=−∞

We have the following relations and sum rules, ∞

A0 (s, α, β) = ∑ J2n+s (α) Jn (β) , n=−∞

∞

∞

s=−∞

s=−∞

2

∑ A0 (s, α, β) = ∑ [A0 (s, α, β)] = 1 .

(8.136) The Volkov solutions for linear polarization can thus be expanded into plane waves, with expansion coefficients being given by generalized Bessel functions, √ e k
a m ∞
Ac (s, α, β)) ψp,σ (x) = ∑ (A0 (s, α, β) + QV s=−∞ 2 (k ⋅ p) 1 √ m × Uσ (p) exp [−i(q + s k) ⋅ x] ≡ F (p, x) Uσ (p). (8.137) QV Here, F (p, x) is a prefactor which constitutes a 4 × 4 matrix, and q is given by Eq. (8.128). The factor F (p, x) constitutes a specialization of the prefactor E(p, x) given in Eq. (8.119). The four-vector potential for a circularly polarized plane-wave field can be given in a unified notation as Aµ (φ) = aµ1 cos φ + aµ2 sin φ ,

aµ1 = ̃ a µ1 ,

aµ2 = ̃ a µ2 .

(8.138)

The two polarization four-vectors 1 and 2 have zero timelike components 0λ = 0 (for λ = 1, 2), i.e., λ = (0, ⃗λ ), and k⃗ ⋅ ⃗λ = 0, so that λ ⋅ ρ = −δλρ (λ, ρ = 1, 2). For circular polarization, we have A2 (φ) = Aµ (φ)Aµ (φ) − ̃ a2 independent of φ (̃ a2 > 0). The phase factor evaluates to Sp [φ] = − (p +

e a1 ⋅ p e a2 ⋅ p e2 ̃ a2 k) ⋅ x − sin φ + cos φ , 2(k ⋅ p) k⋅p k⋅p

(8.139)

where the first term in round brackets in easily identified as the laser-dressed momentum in the circularly polarized field, q =p+

e2 ̃ a2 k, 2(k ⋅ p)

q 2 = m 2 + e2 ̃ a2 = m2∗ .

(8.140)

We can thus write the argument in the exponential in terms of the two parameters q ⋅ 1 q ⋅ 2 ξ = ẽ a , η = ẽ a , (8.141) k⋅q k⋅q so that Sp [φ] = −q ⋅ x − ξ sin φ + η cos φ .

(8.142)

Hence, we can easily expand the exponentials into plane waves, as follows, +∞

exp (−iξ sin φ + iη cos φ) = ∑ B0 (s, ξ, η) e−isφ . s=−∞

(8.143)

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Here, we may define the generalized Bessel functions as Fourier components of the function which needs to be expanded, 2π 1 exp (i s θ − iξ sin θ + iη cos θ) dθ . (8.144) B0 (s, ξ, η) = ∫ 2π 0 With the help of the Jacobi–Anger expansions (8.135), we can write ∞

B0 (s, ξ, η) = ∑ in Js+n (ξ) Jn (η) .

(8.145)

n=−∞

It is useful to also define the following generalized B functions, ∞

exp (−iξ sin φ + iη cos φ) = ∑ B0 (s, α, β) e−isφ ,

(8.146a)

cosL φ exp (−iξ sin φ + iη cos φ) = ∑ BLc (s, α, β) e−isφ ,

(8.146b)

sinL φ exp (−iξ sin φ + iη cos φ) = ∑ BLs (s, α, β) e−isφ .

(8.146c)

s=−∞ ∞

s=−∞ ∞

s=−∞

The generalized Bessel functions with cosine and sine prefactors, given by BLc (s, α, β) and BLs (s, α, β), fulfill the following relations, B0 (s, ξ, η) = B0c (s, ξ, η) = B0s (s, ξ, η) ,

(8.147a)

BLc (s, ξ, η) BLs (s, ξ, η)

(8.147b)

= =

c c 1 (BL−1 (s + 1, ξ, η) + BL−1 (s − 1, ξ, η)) , 2 s s 1 (BL−1 (s + 1, ξ, η) − BL−1 (s − 1, ξ, η)) 2i

,

(8.147c)

in full analogy with Eq. (8.134). The Volkov solutions for circular polarization are thus expanded into so-called generalized Bessel functions, as follows, √ m ∞ e k
s ψp,σ (x) = (a1 B c (s, ξ, η) + a ∑ (B0 (s, ξ, η) +
2 B1 (s, ξ, η))) QV s=−∞ 2 (k ⋅ p)
1 × Uσ (p) exp [−i(q + s k) ⋅ x] √ m ≡ G(p, x) Uσ (p). QV

(8.148)

The last line implicitly defines the (4 × 4)-matrix G(p, x), which tends to the unit matrix as the laser field is being switched off. In the circular case, we should point out that a simplification of the generalized Bessel function is possible, which leads to a full reduction to ordinary Bessel functions. Graf’s addition theorem (see Sec. 7.6.1 of Ref. [156]) states that s/2

∞

inϕ = Js (z) ( ∑ Js+n (x) Jn (y) e

n=−∞

1/2

z = (x2 + y 2 − 2xy cos ϕ)

x − y e−iϕ ) x − y eiϕ

,

= [(x − y exp(−iϕ)) (x − y exp(iϕ))]1/2 .

(8.149a) (8.149b)

±iϕ

This is valid for ∣y e

∣ < ∣x∣. If we apply this formula to Bs , we get √ −ξ − i η s/2 Bs (0, ξ, η) = J−s ( ξ 2 + η 2 ) ( ) . −ξ + i η

(8.150)

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Using well-known properties of Bessel functions, it is easy to prove that Bs ∗ (0, ξ, η) = Bs (0, ξ, −η) and B−s (0, ξ, η) = Bs (0, −ξ, η). Similar relations also hold for Bsc (0, ξ, η) and Bss (0, ξ, η). 8.3.2

Dirac–Volkov Propagator

We start the discussion by recalling the Dirac–Volkov solution, which has been given in Eq. (8.119) as follows, √ √ e k
A (φ) m m (1 + ) Uσ (p) exp (i Sp [φ]) ≡ E(p, x) Uσ (p) . ψp,σ (x) = QV 2 (k ⋅ p) QV (8.151) Here, p is the asymptotic energy-momentum four-vector of the electron, outside the laser field (it fulfills the relation p2 = m2 ), and the bispinor Uσ (p) is defined for an electron in the helicity basis, according to Eq. (7.89). The matrix E(p, x) can be given as φ e k
A(φ) e p ⋅ A(φ′ ) e2 A2 (φ′ ) ] exp (−i p ⋅ x − i ∫ dφ′ [ − ]) . 2k ⋅ p k⋅p 2(k ⋅ p) 0 (8.152) Assuming that Aµ (φ) is a real rather than complex field, we define the adjoint by

E(p, x) = [1 +

E(p, x) = γ 0 E + (p, x) γ 0 = [1 +

φ e A(φ) k
e p ⋅ A(φ′ ) e2 A2 (φ′ ) ] exp (i p ⋅ x + i ∫ dφ′ [ − ]) . (8.153) 2k ⋅ p k⋅p 2k ⋅ p 0

Here, E + (p, x) denotes the Hermitian conjugate of the matrix E(p, x). By the standard rules for Dirac γ matrices, this means that in E, one forms the complex conjugate of all scalar quantities and reverses the order of all Dirac matrices. We generalize the E matrix, which initially has been defined for on-mass-shell momenta (p2 = m2 ), to momenta off the mass shell. The E matrices can be shown to have the following properties, (i∂
− e A(φ)) E(p, x) = E(p, x) p,
← Ð E(p, x) (−i ∂
− e A(φ)) = p
E(p, x) .

(8.154) (8.155)

They also fulfill the following orthogonality properties, ∫

d4 x E(p, x)E(p′ , x) = δ (4) (p − p′ ) 14×4 , (2π)4

(8.156)

∫

d4 p E(p, x)E(p, x′ ) = δ (4) (x − x′ ) 14×4 . (2π)4

(8.157)

We recall that both E as well as E are 4 × 4 matrices, and that the products E(p, x)E(p′ , x) as well as E(p, x)E(p, x′ ) include matrix multiplication operations.

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From the properties (8.154) and (8.157), we can easily find the form of the laserdressed Green function [273–275], d4 p p+m E(p, x1 ) 2
2 E(p, x2 ) (2π)4 p − m + i

G(x1 , x2 ) = ∫

d4 p e k
A(φ1 ) e A(φ2 ) k
p+m [1 + ] 2
2 [1 + ] (2π)4 2 (k ⋅ p) p − m + i 2 (k ⋅ p)

=∫

× exp (−ip ⋅ (x1 − x2 ) − i ∫

φ1

φ2

dφ′ [

e p ⋅ A(φ′ ) e2 A2 (φ′ ) − ]) , (8.158) k⋅p 2 (k ⋅ p)

where φ1 = k ⋅ x1 and φ2 = k ⋅ x2 . Note that this Green function is the generalization of the Feynman propagator SF (x1 , x2 ), given in Eq. (7.153), to the laser-dressed setting. Per our discussion in Chap. 7, the Feynman propagator differs from the time-ordered expectation value of the Dirac field operators by an overall prefactor which is equal to the imaginary unit. The Dirac–Volkov propagator fulfills the time-dependent equation (i∂
1 − e A(φ1 ) − m) G(x1 , x2 ) = δ (4) (x − x′ ) 14×4 , (8.159) µ where ∂
1 = γ µ ∂/∂x1 , and φ1 = k ⋅ x1 = ωt1 − k⃗ ⋅ r⃗1 . In contrast to Eqs. (7.176) and (8.105), one cannot eliminate the time dependence which is inherent to the oscillating laser field. One can show Eq. (8.159) by acting with the differential operator on the left-hand side of Eq. (8.158), using Eq. (8.154) and the property 2 2 (p
− m) (p
+ m) = p − m , canceling the propagator denominator in Eq. (8.158), and finally carrying out the integral over the momenta, using Eq. (8.157). From now on, we assume linear polarization, where we recall the Dirac–Volkov solution from Eq. (8.137) as follows, √ m ∞ e k
a
Ac (s, α, β)) ψp,σ (x) = ∑ (A0 (s, α, β) + QV s=−∞ 2 (k ⋅ p) 1 √ m × Uσ (p) exp [−i (q + s k) ⋅ x] ≡ F (p, x) Uσ (p) . (8.160) QV In this case, the matrix E(p, x) specializes to the matrix F (p, x) defined in Eq. (8.137). Equation (8.158) can be rewritten using the expansion into generalized Bessel functions, assuming linear polarization Aµ = aµ cos φ, as G(x1 , x2 ) = ∫

d4 p e2 ∣a2 ∣ exp [−i (p + k) ⋅ (x1 − x2 ) − ik ⋅ (sx1 − s′ x2 )] ∑ (2π)4 s,s′ 4k ⋅ p

× [A0 (s, α, β) +

ek
a eak
p
Ac (s, α, β)]
+m [A0 (s′ , α, β) +
Ac1 (s′ , α, β)] , 2k ⋅ p 1 p2 − m2 + i 2k ⋅ p (8.161)

where the sums over s and s′ extend from −∞ to ∞ and we recall from Eq. (8.129) 2 ∣a2 ∣ a⋅p , and β = − e8k⋅p . The Green function can thus be expressed in terms that α = ek⋅p of the free propagator, inserted between the Volkov-like matrix-valued functions E and E.

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In Eq. (8.161), the dependence on the integration variable p is such that the Fourier components of the propagator are not uniquely identified. It would be better if one could write the propagator as an expansion over the Fourier components exp(−i q ⋅ x), i.e., in terms of q =p+

e2 ∣a2 ∣ k. 4k ⋅ p

(8.162)

The Jacobian of this transformation is J = det (

0 1 2 3 ∂q µ e2 ∣a2 ∣ k ν kα δ αµ µν µναβ ∂q ∂q ∂q ∂q ) = det (δ − ) =  ∂pν 4(k ⋅ p)2 ∂pµ ∂pν ∂pα ∂pβ

= µναβ (δ 0µ − × (δ 2α −

e2 a2 k ν kγ δ γ1 e2 a2 k µ kγ δ γ0 1ν ) (δ − ) 4(k ⋅ p)2 4(k ⋅ p)2

e2 a2 k α kγ δ γ2 e2 a2 k β kγ δ γ3 3β ) (δ − ) = 1. 4 (k ⋅ p)2 4 (k ⋅ p)2

(8.163)

Here, µναβ is the usual anti-symmetric symbol with 0123 = 1. The last step can be seen most easily by choosing an explicit coordinate system, e.g., by letting k = (ω, ω, 0, 0). We then have dp0 dp1 dp2 dp3 = J −1 dq 0 dq 1 dq 2 dq 3 = dq 0 dq 1 dq 2 dq 3 .

(8.164)

Written in the new integration variable, and noting that k ⋅ p = k ⋅ q and a ⋅ p = a ⋅ q, the expression for the propagator reads d4 q G(x1 , x2 ) = ∫ (2π)4 × (A0 (s′ , α, β) +

q − e ∣a ∣ k + m e k
a
Ac (s, α, β))
4 (k⋅q)
∑ (A0 (s, α, β) + 2k ⋅ q 1 q 2 − m2∗ + i s,s′ =−∞ 2

∞

2

ea
k
Ac (s′ , α, β)) exp (−iq ⋅ (x − x ) − ik ⋅ (s x − s′ x )) . 1 2 1 2 2 (k ⋅ q) 1 (8.165)

We see that the poles are shifted to the effective mass shell, which for linear polarization reads as follows [see Eq. (8.128)] p2 = m2∗ = m2 +

e2 ∣a2 ∣ . 2

(8.166)

Finally, for circular polarization, we have from Eq. (8.148), √ m ∞ ψp,σ (x) = ∑ exp [−i(q + s k) ⋅ x] QV s=−∞ × (B0 (s, ξ, η) + ≡

√

e k
s (a1 B c (s, ξ, η) + a
2 B1 (s, ξ, η))) Uσ (p) 2 (k ⋅ p)
1

m H(p, x) Uσ (p) , QV

(8.167)

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which defines the matrix-valued function H(p, x). We recall the definitions ξ = 2 1 and η = e ̃ a q⋅ from Eq. (8.141), and the laser-dressed mass in the circularly ẽ a q⋅ k⋅q k⋅q polarized field, ∣̃ a2 ∣ = ̃ a2 > 0 .

m2∗ = m2 + e2 ∣̃ a2 ∣ ,

(8.168)

The propagator thus reads as G(x1 , x2 ) = ∫

d4 p p+m H(p, x2 ) H(p, x1 ) 2
2 (2π)4 p − m + i

(8.169)

and can be expressed as follows, G(x1 , x2 ) = ∫ ×

d4 p (2π)4 p2

∞

∑

s,s′ =−∞

(B0 (s, ξ, η) +

e k
s (a1 B c (s, ξ, η) + a
2 B1 (s, ξ, η))) 2 (k ⋅ p)
1

p e k
c s
+m (B0 (s, ξ, η) + (a ) 1 B1 (s, ξ, η) + a 2 B1 (s, ξ, η))

2 − m + i 2 (k ⋅ p)

× exp [−i (p +

e2 ∣̃ a2 ∣ k) ⋅ (x1 − x2 ) − ik ⋅ (s x1 − s′ x2 )] . 4k ⋅ p

(8.170)

After the change of variable p → q in the Green function, governed by Eqs. (8.162) and (8.163), we can express G(x1 , x2 ) as follows, G(x1 , x2 ) = ∫

d4 q (2π)4

s,s′ =−∞

(B0 (s, ξ, η) +

e k
s (a1 B c (s, ξ, η) + a
2 B1 (s, ξ, η))) 2 (k ⋅ q)
1

e ∣̃ a ∣ ek

q − 4 (k⋅q) k
+ m c s (B0 (s, ξ, η) + (a ) 1 B1 (s, ξ, η) + a 2 B1 (s, ξ, η))

2 2 q − m∗ + i 2(k ⋅ q) 2

×

∞

∑

2

× exp [−iq ⋅ (x1 − x2 ) − ik ⋅ (s x1 − s′ x2 )] .

(8.171)

In the limit√of a vanishing laser field, the Volkov solution asymptotically tends to ψp,σ (x) → m/(EV )uσ (p) exp(−i p ⋅ x), which is a the solution of the free Dirac equation, in view of the property B0 (s, 0, 0) = δs0 . Also, in this limit, we have H(p, x) → exp(−ip ⋅ x), which is the laser-field-free expression. For circular polarization, one can simplify the amplitudes as described in Ref. [276], using identities related to Eq. (8.149). Furthermore, in the limit a → 0, the effective four-momentum (8.122) takes the same value as the free momentum, q µ = pµ . Obviously, the arguments of the generalized Bessel functions (8.141) are equal to zero in this case. One can then show that the laser-dressed electron propagator approaches the expression a→0

G(x1 , x2 ) ÐÐ→ ∫ i.e., the free electron propagator.

d4 p e−i p⋅(x1 −x2 ) , (2π)4 p
− m + i

(8.172)

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Dirac Equation with Coupling to Gravitational Fields Metric and Covariant Derivative

In Sec. 4.2.1, we had already encountered the flat-space-time metric in “West-Coast” conventions, {γ µ , γ ν } = 2 g µν , where g µν = diag(1, −1, −1, −1), as adopted in the modern particle physics literature [2,256] [see also Eq. (2.49)]. In general relativity, the metric tensor can locally be chosen to be identical to the flat-space metric, but globally, space-time curvature induces a deviation from the flat-space metric, according to the replacement g µν → g µν (x). In order to describe the coupling of Dirac particles to curved space-times, an interesting and quite sophisticated formalism will need to be discussed in detail. The presence of the coordinate-dependent metric implies that the partial derivative needs to be replaced by a covariant derivative. For tensors, the definition of the covariant derivative ∇µ is well established [see Exercise 8.4 on page 211 of Ref. [277]]. Let us still include a pertinent discussion. From differential geometry, we know that Christoffel symbols enter the definition of the covariant derivative. For a positiondependent metric g µν (x), the covariant components Aµ (x) are connected to the contravariant components Aµ (x) of a vector field by the relations xµ = g µν (x) xν , xµ = g µν (x) xν , Aµ (x) = g µν (x) Aν (x) . (8.173) Aµ (x) = g µν (x) Aν (x) , From now on, we suppress the dependence of the metric on the coordinates and just write g µν for g µν (x) for the curved-space metric, in order to distinguish it from the flat-space metric given in Eq. (2.49). Let us briefly motivate the definition of the covariant derivative. First of all, we recall that a vector A⃗ actually is independent of a coordinate system. A local basis consists of the set (“tetrad” basis) of unit vectors e⃗µ , A⃗ = Aµ e⃗µ , e⃗µ ⋅ e⃗ν = g µν , e⃗µ ⋅ e⃗ν = g µ ν = δ µ ν . (8.174) The basis vectors e⃗µ are non-constant, and an additional term emerges as one calculates the derivative of the vector A⃗ with respect to a coordinate, ∂ ⃗ A = ∂ν A⃗ = (∂ν Aµ ) e⃗µ + Aµ (∂ν e⃗µ ) = (∂ν Aµ ) e⃗µ + Aµ e⃗λ g λρ (⃗ eρ ⋅ ∂ν e⃗µ ) ∂xν = (∂ν Aµ ) e⃗µ + Aµ e⃗λ g λρ Γρ µ ν = (∂ν Aµ ) e⃗µ + Aµ e⃗λ Γλµ ν , (8.175) where Γρ µ ν = e⃗ρ ⋅ ∂ν e⃗µ is a Christoffel symbol, and the Einstein summation convention has been used (all the Greek indices are summed from zero to three). We now swap the dummy indices (summation) in the second term according to (λ ↔ µ) and identify the components of the covariant derivative, ∂ν A⃗ = e⃗µ ∇ν Aµ , ∇ν Aµ = ∂ν Aµ + Γµλ ν Aλ . (8.176) µ While the Christoffel symbols Γλ ν are not tensors, one can still raise the first index with the metric, ∂⃗ eµ (8.177) Γρµν = e⃗ρ ⋅ ν , Γβµν = g βρ Γρµν . ∂x

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A rather straightforward calculation shows that 1 ∂g βµ ∂g βν ∂g µν Γβµν = ( + − ). 2 ∂xν ∂xµ ∂xβ By inspection, one can easily derive the symmetry properties Γβµν = Γβνµ ,

Γβµν = Γβνµ .

The covariant derivative of a vector field A (x) has the properties, ∂ ∇ν Aµ = ∂ν Aµ + Γµνβ Aβ , ∂ν ≡ , ∂xν

(8.178)

(8.179)

µ

∇ν Aµ = ∂ν Aµ − Γβνµ Aβ ,

(8.180a) (8.180b)

g βν ∇µ Aν = ∇µ (g βν Aν ) = ∇µ Aβ ,

(8.180c)

i.e., the covariant derivative transforms constitutes a tensor (unlike the partial derivative), and its indices can therefore be raised and lowered with the metric, both before and after the covariant differentiation. The property (8.180b) can be shown as follows, ∂ ⃗ A = ∂ν A⃗ = (∂ν Aµ ) e⃗µ + Aµ (∂ν e⃗µ ) = (∂ν Aµ ) e⃗µ + Aµ ∂ν (g µρ e⃗ρ ) ∂xν = (∂ν Aµ ) e⃗µ + Aµ (∂ν g µρ ) g ρσ e⃗σ + Aρ ∂ν e⃗ρ = (∂ν Aµ ) e⃗µ − Aµ g µρ (∂ν g ρσ ) e⃗σ + Aρ e⃗σ (⃗ eσ ⋅ ∂ν e⃗ρ ) eρ ⋅ ∂ν e⃗σ ) eσ ⋅ e⃗ρ ) −Aρ e⃗σ (⃗ = (∂ν Aµ ) e⃗µ − Aρ (∂ν g ρσ ) e⃗σ + Aρ e⃗σ ∂ν (⃗ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¶ =g σρ

= (∂ν Aµ ) e⃗µ − Aρ e⃗σ Γρσν = (∂ν Aµ ) e⃗µ − Aµ Γµσν e⃗σ .

(8.181)

We can now use the property (8.179) and write ∇ν Aµ = ∂ν Aµ − Γβνµ Aβ .

(8.182)

The equations (8.176) and (8.182) are complementary. The Riemann curvature tensor Rρ σµν is of rank four and can be expressed in terms of the Christoffel symbols, Rρ σµν = ∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ .

(8.183)

For the Ricci tensor Rµν , one contracts the first and the third index of the Riemann curvature tensor, Rµν = Rα µαν ,

R = g µν Rµν = Rµ µ .

(8.184)

The scalar curvature R of space-time is obtained as the contraction of the Ricci tensor. The Einstein–Hilbert action SEH is given by √ δSEH √ 1 SEH = ∫ d4 x − det g R , = − det g (Rαβ − g αβ R) , (8.185) 2 δg αβ

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and the variational derivative of the action with respect to the metric tensor leads to the condition Rαβ − 12 g αβ R = 0, which is the Einstein–Hilbert equation for a region of the Universe, in which there is no matter and no other fields which give rise to a nonvanishing energy-momentum tensor. The variational condition δSEH = 0 is equivalent to saying that the global integral over the curvature of space-time R acts like a soap film whose surface tension integral is being minimized by the shape assumed by space-time geometry, which is four-dimensional but might be embedded in a higher-dimensional manifold which would defy our intuitive understanding. The √ integration measure ∫ d4 x − det g is Lorentz-invariant. The coupling of the spacetime curvature to the matter density in the Universe can be written in terms of a sum of the Einstein–Hilbert action SEH and the matter action SM , where the latter can be expressed in terms of the matter Lagrangian LM , √ 1 SGR = SEH + SM , SM = ∫ d4 x − det g LM , (8.186a) 16π G √ 2 δSM ∂( − det g LM ) 2 ∂LM √ Tµν = − √ = − = −2 µν + g µν LM . µν µν ∂g ∂g − det g δg − det g (8.186b) The latter functional derivative acts as a definition of the energy-momentum tensor Tµν . The form given in Eq. (8.186b) is valid if the matter Lagrangian does not depend on derivatives of the metric. Examples of the energy-momentum tensor include a collection of particles, T µν = ∑ ∫ dτa a

δ (4) (x − x(τa )) √ ma vaµ vaν . − det g

(8.187)

Here, the vaµ are the four-vector components of the velocity va of particle a, and τa is the proper time of the trajectory of particle a. For an electromagnetic field, the stress-energy tensor is given as follows (see Ref. [81]), 1 µν αβ g F Fαβ ) , (8.188) 4 where we recall that 0 = µ0 = 1 for the current discussion. Under the inclusion of the energy-momentum tensor, the variational equations read as follows, T µν = − (F µα F ν α −

1 R g αβ = 8πG T αβ . (8.189) 2 These equations have all been derived under a variation of the action with respect to the metric. They still do not clarify how the Maxwell equations need to be generalized to curved space-time, and this will be explored next. It is not hard to guess that for a free electromagnetic field, one simply has √ to use the Lorentz-invariant measure ∫ d4 x − det g in the action integral for the electromagnetic field, √ 1 SEM = ∫ d4 x − det g (− F µν Fµν ) . (8.190) 4 Rαβ −

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The field-strength tensor, in the curved space-time, is given as follows, ∇µ Aν − ∇ν Aµ = ∂µ Aν − ∂ν Aµ .

(8.191)

Here, we have used the definition (8.180) of the covariant derivative and the symmetry properties (8.179) of the Christoffel symbols. Variation with respect to the metric leads to the energy-momentum tensor already given in Eq. (8.188), modulo some total divergence. However, the variational principle also dictates that the action SEM should be invariant under a variation of the four-vector field Aµ . The corresponding Euler–Lagrange equation gives rise to the condition √ ∂ µν (F − det g) = 0 , ∂xµ

∇µ F µν = 0 ,

(8.192)

where ∇µ is the covariant derivative. It might be useful to remind the reader of the identity √ 1 ∂ (F µν − det g) = ∇µ F µν , (8.193) µ − det g ∂x √ which follows from ∂µ ln( −∣g∣) = 21 g αβ ∂µ gαβ = Γκµκ (we recall the Einstein summation convention one more time) and the fact that F µν is antisymmetric. The covariant derivative of the field-strength tensor can be expressed in terms of Christoffel symbols, √

∇α F βγ = ∂α F βγ + Γβαµ F µγ + Γγαρ F βρ , ∇µ Fαβ =

∂µ Fαβ − Γραµ Fρβ

− Γρβµ Fαρ .

(8.194a) (8.194b)

If we consider a source term, then Eq. (8.190) gets modified to read SEM = δ ∫ d4 x

√

1 − det g (− F µν Fµν − Jµ Aµ ) 4

√ √ ∂ δSEM νµ µ = (F − det g) − J − det g = 0 . δAµ ∂xν

(8.195)

In component form, we have √

√ 1 ∂ νµ (F − det g) = J µ , − det g ∂xν

∇µ F µν = J ν ,

(8.196)

where ∇µ is the covariant derivative. The equations (8.196) are the generalizations of the inhomogeneous Maxwell equations (see Ref. [81]), namely, ∂µ F µν = J ν . Finally, we should remark that the covariant derivative of the metric is obtained as a generalization of Eq. (8.194b), ∇µ g αβ = g αβ,µ − Γλαµ g λβ − Γλβµ g αλ = g αβ,µ − Γβαµ − Γαβµ = 0 . It vanishes according to Eq. (8.178).

(8.197)

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8.4.2

Tetrad Basis and Affine Connection Matrix

One might think that the Dirac Hamiltonian could be coupled to a central gravitational field by simply adding the Newtonian gravitational potential to the free Dirac Hamiltonian, leading to the expression mM ⃗ ⋅ p⃗ + β m − G . (8.198) HG = α r One might argue that, in the same right, we had simply added the static Coulomb potential to the free Dirac Hamiltonian, which leads to the Dirac–Coulomb Hamiltonian given in Eq. (8.9). However, the Hamiltonian given in Eq. (8.198) does not constitute a valid description of a Dirac particle in a central gravitational field. The reason is that the Coulomb potential is not added to the free Dirac Hamiltonian just by chance. Rather, this additional term results from the minimal coupling i ∂µ → i ∂µ − e Aµ

(8.199)

which is applicable to electromagnetic gauge theory (see Sec. 8.2.1). Setting µ = 0 and e A0 = V = −Zα/r, one obtains the Dirac–Coulomb Hamiltonian. No such U (1)-gauge theory inspired prescription is available for gravitational coupling. The gravitational potential of a planet cannot be inserted at face value into the Dirac equation, like the Coulomb potential. Rather, the full formalism of general relativity needs to be employed, and the gravitational coupling involves both a redefinition of the algebra of Dirac matrices, adjusted to curved space-time. The coupling is formulated covariantly under local spinor Lorentz transformations; the covariance principle is derived from general relativity, not from electromagnetic U (1) gauge theory. We recall the Dirac equation in flat space according to Eq. (7.17), (i γ µ ∂µ − m) ψ(x) = 0 ,

x = (t, r⃗) .

(8.200)

The γ matrices fulfill the (anti-)commutation relations {γ , γ } = 2 g , where g µν = diag(1, −1, −1, −1) is the flat-space-time metric in “West-Coast” conventions adopted in the modern particle physics literature [2, 256]. In general relativity, the metric tensor can locally be chosen to be identical to the flat-space metric, but globally, space-time curvature induces a deviation from the flat-space metric, according to the replacement g µν → g µν (x). The curved-space-time Dirac matrices fulfill the commutation relations µ

µ

{γ µ (x), γ ν (x)} = 2 g µν (x) ,

ν

µν

(8.201)

where the local curved-space metric is g µν (x) with µ, ν = 0, 1, 2, 3. Just to fix ideas, we should stress that the Dirac equation in curved space-time cannot simply be obtained by replacing the γ µ matrices in Eq. (8.200) by the equivalent γ µ (x) matrices. This perhaps surprising fact has already been observed in Refs. [278–280]. We here follow the conventions used in Refs. [255, 281] for the flat-space and curved-space Dirac gamma matrices. The flat-space-time Dirac gamma matrices

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are denoted with a tilde (̃ γ ) while the curved-space-time Dirac gamma matrices are γ matrices used in the following written with an overline (γ). This means that the ̃ are the same as the γ matrices (without tilde) used for the flat-space Dirac equation. The notation for the curved-space-time matrices is inspired by the covariant structure of their anticommutator, expressed in Eq. (8.201), and the tensor (“vector”) structure is denoted by the overline. By contrast, from the point of view of general relativity, the flat-space matrices can be regarded as “modified” γ matrices, hence the tilde. Via integration by parts, we can modify the Lagrangian of the free Dirac field, given by the Lagrangian (7.79), L(x) = ψ(x) (i γ µ ∂µ − m) ψ(x) ,

(8.202)

→ i ← S0 = ∫ d4 x ψ(x) ( γ µ ∂ µ − m) ψ(x) , 2

(8.203)

← → A(x) ∂ µ B(x) ≡ A(x) ∂µ B(x) − B(x) ∂µ A(x) .

(8.204)

as follows,

where

The generalization of the Dirac action (8.203) to curved space-time involves two steps: (i) an obvious generalization of the anticommutator relations (8.201) to curved space, {γ µ (x), γ µ (x)} = 2 g µν (x), and (ii) the introduction of a covariant derivative, ∂µ → ∇µ , which is defined as ∇µ = ∂µ − Γµ (x) ,

(8.205)

where ∇µ is the covariant derivative and Γµ (x) is a (4 × 4)-matrix that specifies the affine connection. The action for the Dirac particle in curved space-time is obtained as a generalization of Eq. (7.78) √ S = ∫ d4 x − det g ψ(x) (i γ µ (x)∇µ − m) ψ(x) , (8.206) √ where det g < 0 is the determinant of the local space-time metric, and − det g d4 x is a Lorentz-invariant measure. Because the local Dirac γ matrices fulfill the relation {γ µ (x), γ ν (x)} = 2 g µν (x), it is natural to postulate that the covariant derivative of a single curved-space Dirac matrix γ µ (x) needs to vanish, individually, ∇ν γ µ (x) = 0 . Under the assumption (8.207), we can symmetrize Eq. (8.206) as follows, √ i ← → S = ∫ d4 x − det g ψ(x) ( γ µ (x) ∇ µ − m) ψ(x) . 2 By variation, the gravitationally coupled Dirac equation is obtained as (i γ µ ∇µ − m) ψ(x) = [i γ µ (∂µ − Γµ ) − m] ψ(x) = 0 .

(8.207)

(8.208)

(8.209)

µ

For a Lorentz vector T , we recall that according to Eq. (8.182), ∇ν Tµ = ∂ν Tµ − Γλνµ Tλ ,

(8.210)

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where the Γµρσ (x) are the Christoffel symbols defined according to Eq. (8.178). Based on Eq. (8.210), we write the covariant derivative [278–280] of the Dirac matrix γ µ (x) in terms of the Christoffel symbols Γρµν (x) and a commutator with the affine connection matrices Γν (x), ∇ν γ µ (x) = ∂ν γ µ (x) − Γρνµ (x) γ ρ (x) − [Γν (x), γ µ (x)] = 0 .

(8.211)

The reason for this ansatz will be discussed in the following. We need to find a representation of the curved-space Dirac algebra {γ µ (x), γ ν (x)} = 2 g µν (x). This is most easily done in a “tetrad” formalism, where γ A matrices, multione writes the curved-space γ ρ (x) in terms of the free-space ̃ plied by a “square root” of the metric, namely, the tetrad (or vielbein) [see also Eqs. (8.174) and (8.218)], γ ρ (x) = eA γA , ρ (x) ̃

̃ γA = eα A (x) γ α (x) ,

γ α (x) = eα γA , A (x) ̃

ρ ̃ γ A = eA ρ (x) γ (x) ,

(8.212a) µ eµA (x) eA ρ (x) = δ ρ ,

(8.212b)

where {̃ γA, ̃ γ B } = 2̃ g AB .

(8.213)

We recall that we use the “West-Coast” convention for the flat-space metric, which we denote as ̃ g AB = ̃ gAB = diag(1, −1, −1, −1). In our notation, we draw inspiration from the book [282] and denote space-time indices related to a local, freely falling Lorentz frame (“anholonomic basis”) with capital Latin indices A, B, C, . . . = 0, 1, 2, 3. By contrast, lowercase Latin indices starting with i, j, k, . . . = 1, 2, 3 are reserved for “spatial” global coordinates. Our notation of the Dirac matrices addresses some ambiguities which could otherwise result from other approaches [278–280, 283–285]. For example, unless the Dirac matrices are distinguished by overlining or tildes, the explicit expression γ 1 could be associated with a flat-space matrix ̃ γ I=1 or with a curved-space matrix γ i=1 . The metric is recovered as B B {γ ρ (x), γ σ (x)} = eA γA , ̃ γB } = 2 ̃ gAB eA ρ (x) eσ (x) {̃ ρ (x) eσ (x) = 2 g ρσ (x) .

(8.214)

In case we raise the indices, the result is γA, ̃ γB } = 2 ̃ g AB eρA (x) eσB (x) = 2 g ρσ (x) . (8.215) {γ ρ (x), γ σ (x)} = eρA (x) eσB (x) {̃ The matrix with components g ρσ (x) is the inverse of the matrix with entries g ρσ (x), and the matrix with components eµA (x) is the inverse of the matrix with components eA µ (x). From now on, we will suppress the dependence of the curved-space-time symbols on the space-time coordinate x. A remark on our notation is in order. We have denoted the flat-space Dirac matrices ̃ γ A and the curved-space matrices as γ µ . An alternative notation would be to denote all the indices from the non-coordinate basis in brackets, so that vector components in a non-coordinate basis would be denoted as V (A) (such a basis is used for a local Lorentz frame). This is in contrast to a vector component in a

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global coordinate (holonomic) basis, which would read V µ . This would imply an alternative notation for the metric too; one might leave out the overline and denote the curved-space metric as gµν = g µν , while g(A)(B) = g(AB) = ̃ gAB . However, all things considered, we found that the additional brackets to be supplemented for the flat-space metric would probably rather clutter the notation, while the differentiation between the curved- and flat-space matrices using the overline and tilde requires no extra space. 8.4.3

Ricci Rotation Coefficients

Just as much as the Christoffel symbols are naturally obtained upon a differentiation of a vector field, in the holonomic basis spanned by the e⃗µ , with e⃗µ ⋅ e⃗ν = g µν ,

(8.216)

we can define a local, orthonormal basis spanning the “local Lorentz frame” as follows, eˆA ⋅ eˆB = ̃ gAB .

(8.217)

Here, the eˆA may be chosen locally; their dependence on x is not explicitly indicated. Of course, e⃗µ = eµA eˆA ,

⃗µ . eˆA = eA µ e

(8.218)

One can expand a Lorentz vector, in the orthonormal basis, as follows, V = V A eˆA .

(8.219)

In analogy with Eq. (8.175), one may differentiate a vector, as follows, ∂ν V⃗ = ∂ν [V A eˆA ] = (∂ν V A ) eˆA + V A ∂ν eˆA = (∂ν V B ) eˆB + V A ∂ν (eµA eˆµ ) . = (∂ν V B ) eˆB + V A eµA (∂ν eˆµ ) + V A (∂ν eµA ) eˆµ .

(8.220)

The second term on the right-hand side can be rewritten in terms of a Christoffel symbol, while the third term is re-expanded in the orthonormal basis, µ ∂ν V⃗ = (∂ν V B ) eˆB + V A eµA [ˆ eρ ⋅ ∂ν eˆµ ] eˆρ + V A eB ˆB µ (∂ν eA ) e ´¹¹ ¹ ¹¸µ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ρ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ =V

=Γνµ

µ = (∂ν V B ) eˆB + V µ Γρνµ eB ˆB + V A (eB ˆB ρ e µ ∂ν eA ) e µ A µ ρ A = (∂ν V B + (eB + eB ˆB µ ∂ν eA ) V µ Γνρ eA V ) e µ A = (∂ν V B + (eB ˆB = (∇ν V B + ωνBA V A ) eˆB = (Dν V B ) eˆB . (8.221) µ ∇ν e A ) V ) e

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Here Dν V B is a covariant derivative for vectors, Dν V B = ∇ν V B + ωνBA V A ,

(8.222)

with respect to local Lorentz transformations, and constitutes a definition. The connection 1-form is given as follows, νB ωµAB = eA . ν ∇µ e

(8.223)

The Ricci rotation coefficient is obtained via multiplication by an additional tetrad coefficient, ω ABC = eµC ωµAB .

(8.224)

However, we shall use the terminology of a spin-connection 1-form and a Ricci rotation coefficient interchangeably, here. In particular, indices in the orthonormal basis can be raised and lowered with the flat-space metric ̃ g AB . 8.4.4

Covariant Derivative of a Spinor

Let us consider a change of the tetrad basis, which alternatively can be interpreted as a Lorentz transformation in the “internal” flat space, e′µA = ΛA B eµB .

(8.225)

We define eµA = eνA g νµ = eB gBA , µ ̃

νµ eµA = eA = eµB ̃ g BA . ν g

(8.226)

We can then use ̃ gCA ΛA B ̃ g BD = (Λ−1 )D C (which is a consequence of the fact that Lorentz transformations preserve the four-dimensional inner product of vectors) and multiply this expression by eµD to find D

µ −1 e′µ C = eD (Λ )

C

.

(8.227)

We now turn our attention to the curved space Dirac γ matrices, defined in (8.212). These carry a Lorentz index and spinor indices. The transformation of ̃ γ A can be inferred by recognizing that A is a Lorentz index, and that the matrix additionally has two spinor indices which also transform. Hence, just like in free space, one has ̃ γ ′A = ΛA B S(Λ) ̃ γ B S(Λ−1 ) .

(8.228)

Just like in free space, the spinor transformation is defined such as to reverse the effect of the Lorentz index transformation, S(Λ) ̃ γ B S(Λ−1 ) = (Λ−1 )B C ̃ γC .

(8.229)

So, in agreement of Eq. (7.35), we have ̃ γ ′A = ΛA B S(Λ) ̃ γ B S(Λ−1 ) = ΛA B (Λ−1 )B C ̃ γC = ̃ γA .

(8.230)

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It means that just like in free space, we can choose one particular representation of the Clifford algebra, and then calculate all effects in that representation. The flat-space limit is reached smoothly. A Lorentz transform will thus alter the tetrad, and in consequence, the γ ′µ are different from the γ µ , A

γ ′µ = e′µ γ A = eµA (Λ−1 ) A ̃

B

̃ γ B = eµA S(Λ) ̃ γ A S(Λ)−1 = S(Λ) γ µ S(Λ)−1 .

(8.231)

The spinor Lorentz transformation S(Λ) has been defined in Eq. (8.233b). Note that in comparison to Eq. (7.38), the Lorentz index of the Dirac γ matrix is not being transformed, hence the change γ µ → γ ′µ ≠ γ µ . We can then show that although the curved–space Dirac γ matrices change under a Lorentz transformation, the metric g remains invariant, i.e., 1 1 A C g ′µν = {γ ′µ , γ ′ν } = {eµA (Λ−1 ) B ̃ γ B , eνC (Λ−1 ) D ̃ γD} 2 2 C A 1 γB , ̃ γ D } (Λ−1 ) D = eµA eνC (Λ−1 ) B {̃ 2 D

A

= eµA eνC (Λ−1 )

B

̃ g BD (Λ−1 T )

C

= eµA eνC ̃ g AC = g µν .

(8.232)

We can formulate the local vector and spinor Lorentz transformations as follows [see Eqs. (7.27) and (7.30)], C 1 ΛC D = (exp [ AB MAB ]) ≈ δ C D + C D , (8.233a) 2 D i AB i AB S(Λ) = exp (− ̃ σ AB ) ≈ 1 − ̃ σ AB , (8.233b) 4 4 where the matrices MCD have been defined in Eq. (7.26) [see also Eq. (8.301)], and the spin matrices ̃ σ AB have been defined in Eq. (7.30). Furthermore, using the relation (7.31), we have already shown in Eq. (7.38) that the ̃ γ A matrices are shapeinvariant under Lorentz transformations, provided one also transforms the Lorentz index. The (conceivably infinitesimal) generators AB = −BA are formulated in terms of upper-case Latin instead of lower-case Greek letters, in order to distinguish them from the Ricci rotation coefficients. Under a local Lorentz transformation (change of the tetrad), the Lagrangian density for the gravitationally coupled Dirac particle transforms as follows [see Eqs. (7.79), (8.206) and (8.208)], ′

L = ψ (i γ µ ∇µ − m) ψ → L′ = ψ (i γ ′µ ∇′µ − m) ψ ′ .

(8.234)

We suppress the arguments here, and recall that ψ ′ = S(Λ) ψ ,

′

ψ = ψ S(Λ)−1 ,

(8.235)

in which case (8.234) becomes L′ = ψ S(Λ)−1 [i (S(Λ) γ µ S(Λ)−1 ) ∇′µ − m] S(Λ) ψ = ψ [i γ µ (S(Λ)−1 ∇′µ S(Λ)) − m] ψ .

(8.236)

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Then, for the curved-space Lagrangian density to be Lorentz invariant, we must require that S(Λ)−1 ∇′µ S(Λ)ψ = ∇µ ψ ,

(8.237)

i.e., that the covariant derivative of a spinor after Lorentz transformation is equal to the Lorentz-transformed covariant derivative, ∇′µ ψ ′ = ∇′µ S(Λ) ψ = S(Λ) ∇µ ψ = (∇µ ψ)′ .

(8.238)

Furthermore, we do not want the local change of the tetrad basis to alter the fundamental structure of the covariant derivative acting on a spinor, i.e., ∇′µ = ∂µ − Γ′µ .

(8.239)

So, (∂µ − Γ′µ ) S(Λ)ψ = (∂µ S(Λ)) ψ + S(Λ) ∂µ ψ − Γ′µ S(Λ) ψ = S(Λ) [∂µ + S(Λ)−1 ∂µ S(Λ) − S(Λ)−1 Γ′µ S(Λ)] ψ .

(8.240)

Comparing this result with the right-hand side of Eq. (8.238), we conclude that Γµ = S(Λ)−1 Γ′µ S(Λ) − S(Λ)−1 ∂µ S(Λ) ,

(8.241)

Γ′µ = S(Λ) Γµ S(Λ)−1 + [∂µ S(Λ)] S(Λ)−1 .

(8.242)

or alternatively

The spinor Lorentz transformation has been given in Eq. (8.233b). Also, we have ̃ σAB = 2i [̃ γA , ̃ γB ]. In this case, one can reformulate Eqs. (8.241) and (8.242) in terms of a simplification of the second term on the right-hand side, i (∂µ AB ) ̃ σAB , 4 i Γ′µ = S(Λ) Γµ S(Λ)−1 − (∂µ AB ) ̃ σAB . 4 Γµ = S(Λ)−1 Γ′µ S(Λ) +

(8.243a) (8.243b)

The generators AB are antisymmetric, and despite the fact that the connection matrix Γµ is changed by a Lorentz transformation, one might conjecture that their overall mathematical structure, including the symmetry properties with respect to the anholonomic indices (A and B) and Lorentz indices (µ), should be conserved. This suggests the ansatz i Γµ = CµAB ̃ σAB , 4

(8.244)

where CµAB is antisymmetric in A and B and carries another Lorentz index µ. Furthermore, because the covariant derivative of a spinor is formulated only in regard to local Lorentz transformations (a spinor does not constitute a vector), we may conjecture that the CµAB coefficients might be connected with the Ricci rotation coefficients defined in Eq. (8.223), rather than the Christoffel symbols.

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Covariant Derivative of the Dirac Matrices

A last preparatory step in the calculation of the Γµ matrix is the motivation of the covariant derivative of the curved-space γ µ matrix that we already encountered in Eq. (8.211), ∇µ γ ν = ∂µ γ ν + Γνµρ γ ρ − [Γµ , γ ν ] .

(8.245)

We begin with the Dirac Lagrangian density in curved space, which we recall from Eq. (8.234), L = ψ [iγ µ (∂µ − Γµ ) − m] ψ .

(8.246)

← Ð + + L+ = ψ + [−i ( ∂ µ − Γ+µ ) (γ µ ) − m] ψ .

(8.247)

ψ ≡ ψ+̃ γ0 ,

(8.248)

Its adjoint reads as

We define ψ as

with the flat-space ̃ γ 0 , ensuring that ψ transforms with the inverse of the local Lorentz transform. With this in mind, and the fact that (̃ γ 0 )2 = 1, Eq. (8.247) becomes ← Ð + γ 0 [−i ( ∂ µ − Γ+µ ) (γ µ ) − m] ̃ γ 0̃ L+ = ψ + ̃ γ0ψ ← Ð + 0 = ψ [−i ( ∂ µ − ̃ γ 0 Γ+µ ̃ γ − m] ψ . (8.249) γ0) ̃ γ 0 (γ µ ) ̃ γ0 = γµ. Since ̃ γ 0 (̃ γ µ )+ ̃ γ0 = ̃ γ µ , it is trivial to show that ̃ γ 0 (γ µ )+ ̃ We assume for Γµ , the ansatz given in Eq. (8.244), and write + i i + 0 + 0 ̃ γ 0 (Γµ ) ̃ γ =̃ γ 0 ( CµAB ̃ σAB ) ̃ γ 0 = − CµAB ̃ γ 0 (̃ σAB ) ̃ γ . 4 4 One can show that +

̃ γ 0 (̃ σAB ) ̃ γ0 = ̃ σAB .

(8.250)

(8.251)

Plugging this into (8.250), we find i ̃ γ 0 Γ+µ ̃ σAB = −Γµ . γ 0 = − CµAB ̃ 4 Thus, Eq. (8.249) becomes ← Ð L+ = ψ [−i ( ∂ µ + Γµ ) γ µ − m] ψ . The Lagrange density has to be Hermitian; hence L = L+ and ← Ð ψ [i γ µ (∂µ − Γµ ) − m] ψ = ψ [−i ( ∂ µ + Γµ ) γ µ − m] ψ .

(8.252)

(8.253)

(8.254)

This can be reformulated as follows, ψ γ µ (∂µ ψ) + (∂µ ψ) γ µ ψ = ψ (γ µ Γµ − Γµ γ µ ) ψ .

(8.255)

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Adding the term ψ (∂µ γ µ ) ψ to both sides of this equation, we have ∂µ (ψ γ µ ψ) = ψ (∂µ γ µ ) ψ − ψ [Γµ , γ µ ] ψ . The curved-space probability current is given as J µ = ψ γµ ψ , ∇µ J µ = 0 . Thus, ∂µ J µ + Γµµρ J ρ = ∂µ (ψ γ µ ψ) + Γµµρ (ψ γ ρ ψ) = 0 .

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(8.256) (8.257) (8.258)

µ

In this expression, we replace the term ψ γ ψ with the help of Eq. (8.256), yielding ψ (∂µ γ µ ) ψ − ψ [Γµ , γ µ ] ψ + ψ Γµµρ γ ρ ψ = 0 . (8.259) Introducing, artificially, a Kronecker symbol δν µ = g ν µ , one has ψ g ν µ ((∂µ γ ν ) + Γνµρ γ ρ − [Γµ , γ ν ]) ψ = ψ (g ν µ ∇µ γ ν ) ψ = 0 , (8.260) where we identify the covariant derivative ∇µ γ ν of a Dirac γ matrix in accordance with Eq. (8.211). 8.4.6

Spin Connection

It remains to calculate the explicit form of the CµAB in Eq. (8.244). We impose the γ A ), and find that restriction that ∇µ γ ν = 0, use the tetrad basis (γ µ = eµA ̃ ρ γ B ] = 0 . (8.261) ∇µ γ ν = (∂µ eνB + Γνµρ eB ) ̃ γ B ] = (∇µ eνB ) ̃ γ B − eνB [Γµ , ̃ γ B − eνB [Γµ , ̃ We recall that the Ricci rotation coefficient [the “spin connection” of Eq. (8.223)] is νB ωµAB = eA , (8.262) ν ∇µ e A and multiply Eq. (8.261) by eν to obtain ν ν (8.263) γB − [Γµ , ̃ γA] = 0 . γ B ] = ωµAB ̃ eA γ B − eA ν (∇µ eB ) ̃ ν eB [Γµ , ̃ In order to proceed, we use our ansatz for Γµ given in Eq. (8.244), and Eq. (7.37), to find i i [Γµ , ̃ γ A ] = CµBC [̃ σBC , ̃ γ A ] = CµBC [2 i (g A C ̃ γB − g A B ̃ γC )] 4 4 1 = (CµAB ̃ γB + CµAC ̃ γC ) = CµAB ̃ γB , (8.264) 2 AB AB BA where we used the assumption that Cµ is antisymmetric (Cµ = −Cµ ). Combining this result with (8.263), we obtain that quite simply, CµAB is a rotation coefficient, CµAB = ωµAB , (8.265) and i νB Γµ = ωµAB ̃ σAB , ωµAB = eA . (8.266) ν ∇µ e 4 Finally, we can write the covariant derivative operating on a spinor as i ∇µ ψ = (∂µ − Γµ ) ψ = (∂µ − ωµAB ̃ σab ) ψ . (8.267) 4 In principle, this result is well known [255, 278, 280–282, 284, 286–289], including a missing prefactor in Ref. [278], as pointed out in Ref. [281]. However the derivations in the literature are not very detailed, and it is thus useful to recall the basis of the formalism here.

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Transformation Properties of Rotation Coefficients

The transformation properties of Ricci rotation coefficients, or connection 1-forms, deserve a special discussion. We have, in the unprimed frame, according to Eq. (8.223), ρB ωµAB = eA . ρ ∇µ e

(8.268)

This translates into the primed frame as follows, ′ρB ωµ′AB = e′A . ρ ∇µ e

(8.269)

According to Eqs. (8.233a) and (8.233b), infinitesimal transformations read as follows, i σAB . (8.270) S(Λ) = 14×4 − AB ̃ 4 The transformation properties of the Ricci rotation coefficients under infinitesimal local Lorentz transformations can be inferred directly, Λ A B ≈ g A B + A B ,

B B ρD ] ωµ′AB ≈ (g A C + A C ) eC ρ ∇µ [(g D +  D ) e ρD ρB B C A ) + eA = (eA ρ ∇µ ( D e ρ +  C e ρ ) ∇µ e ρD = ωµAB + A C ωµCB + (∇µ B D ) ̃ g AD + B D eA ρ ∇µ e

= ωµAB + A C ωµCB + ∇µ BA + B D ωµAD .

(8.271)

Summarizing, one has ωµ′AB ≈ ωµAB − ∂µ AB + A C ωµCB − ωµAD D B .

(8.272)

AB

One particular observation is very important. Because  is invariant under global AB AB Lorentz transformations, we have ∇µ  = ∂µ  , commensurate with the transformation properties of a Lorentz scalar. From AC = −CA , one gets the relation A B = −B A , via multiplication by ̃ gCB = ̃ gBC . On the other hand, the transformation property of the spin connection matrix has been given in Eq. (8.243b), i AB ω ̃ σAB . (8.273) 4 µ This formula has to be compatible with the result (8.272) [see also Eq. (8.266)]. This is because we can otherwise calculate the transformation properties of the Γµ matrices separately. Indeed, Γ′µ has to read as follows, Γ′µ = S(Λ) Γµ S(Λ)−1 + [∂µ S(Λ)] S(Λ)−1 ,

Γ′µ =

Γµ =

i ′AB ′ ̃ ω σAB , 4 µ

(8.274)

′ with Lorentz-transformed coefficients ωµ′AB and Lorentz-transformed matrices ̃ σAB . However, in view of Eq. (8.230), one has

i ′A ′B i A B ′ ̃ σAB = [̃ γ ,̃ γ ] = [̃ γ ,̃ γ ]=̃ σAB , 2 2

(8.275)

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and so Γ′µ =

i ′AB ̃ ω σAB . 4 µ

(8.276)

We recall from Eq. (8.270) the relation i S(Λ)−1 ≈ 14×4 + AB ̃ σAB . 4

(8.277)

So, the derivative term in Eq. (8.273) can be expressed as follows, i [∂µ S(Λ)] S(Λ)−1 = − ∂µ AB ̃ σAB . 4

(8.278)

It remains to calculate the following expression, to first order in the  parameters, i i i σAB ) ωµCD ̃ σEF ) σCD (14×4 + EF ̃ S(Λ) Γµ S(Λ)−1 ≈ (14×4 − AB ̃ 4 4 4 i i i ≈ ωµCD ̃ σCD + (− AB ̃ σAB ) ( ωµCD ̃ σCD ) 4 4 4 i i + ( ωµCD ̃ σEF ) σCD ) ( EF ̃ 4 4 i i i i i σAB ωµCD ̃ σAB = ωµCD ̃ σCD − AB ̃ σCD + ωµCD ̃ σCD AB ̃ 4 4 4 4 4 i 1 AB CD = ωµCD ̃  ωµ [̃ σAB , ̃ σCD ] . (8.279) σCD + 4 16 Using the commutator relation [see Eq. (7.31) and Chap. 8 of Ref. [81]] [̃ σAB , ̃ σCD ] = 2i (̃ gAD ̃ σBC + ̃ gBC ̃ σAD − ̃ gAC ̃ σBD − ̃ gBD ̃ σAC ) ,

(8.280)

one can derive the result 1 AB CD i σAB .  ωµ [̃ σAB , ̃ σCD ] = (A C ωµCB − ωµAD D B ) ̃ 16 4

(8.281)

In the derivation of Eq. (8.281), the identity f AB ̃ σAB =

1 AB (f − f BA ) ̃ σAB 2

(8.282)

may be useful. Adding, now, the results given in Eqs. (8.278) and (8.279), one easily obtains Γ′µ = S(Λ) Γµ S(Λ)−1 + [∂µ S(Λ)] S(Λ)−1 i i i σAB + (A C ωµCB − ωµAD D B ) ̃ = ωµAB ̃ σAB − ∂µ AB ̃ σAB 4 4 4 i i = (ωµAB − ∂µ AB + A C ωµCB − ωµAD D B ) ̃ σAB ≈ ωµ′AB ̃ σAB , 4 4 compatible with Eq. (8.272).

(8.283)

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Applications of Gravitational Coupling

8.5.1

Dirac–Schwarzschild Hamiltonian

For a space-time geometry independent of time, it is possible to invoke a Hamiltonian formalism. The Schwarzschild metric for a non-rotating black hole in isotropic coordinates reads (see Sec. 43 of Chap. 3 of Ref. [290]), g µν = diag (w2 (r), −v 2 (r), −v 2 (r), −v 2 (r)) ,

(8.284)

and the coefficients are given as rs −1 rs 2 rs ) (1 + ) , v(r) = (1 + ) . (8.285) 4r 4r 4r The Schwarzschild radius is given as rs = 2 G M , where G is Newton’s gravitational constant, and M is the mass of the black hole (or planet). The gravitationally coupled Dirac equation is written out as follows, w(r) = (1 −

[iγ µ (∂µ − Γµ ) − m] ψ = [i γ 0 ∂0 + iγ i ∂i − iγ µ Γµ − m] ψ = 0 .

(8.286)

The Hamiltonian form thus is 2

i (γ 0 ) ∂0 ψ = (−i γ 0 γ i ∂i + iγ 0 γ µ Γµ + γ 0 m) ψ ,

(8.287)

with γ = ̃ γ /w(r) and γ = ̃ γ /v(r) for i = 1, 2, 3. An explicit calculation establishes that ̃ 2 v ′ (r) w(r) + v(r) w′ (r) γ0 ̃ γ⃗ ⋅ rˆ G(r) , G(r) = , (8.288) γ 0 γ µ Γµ = − v(r) w(r) 2 v(r) w(r) where rˆ is the position unit vector. In the Schwarzschild-type metric, we have 0

γ0 =

i

0

̃ γ0 , w(r)

γi =

i

̃ γi , v(r)

γ0 γi =

̃ γ0 ̃ γi αi = , w(r) v(r) w(r) v(r)

(8.289)

0

where αi = ̃ γ ̃ γ i . By convention, the Dirac matrix β is denoted as β = ̃ γ 0 in the Hamiltonian formalism, and thus i∂t ψ = (

w(r) w(r) ⃗ ⋅ p⃗ − i ⃗ ⋅ rˆ G(r) + w(r) β m) ψ . α α v(r) v(r)

(8.290)

Using the explicit result for the G function from Eq. (8.288), one can identify the gravitationally coupled Dirac Hamiltonian H for a static space-time metric as follows, w i iw ⃗ − ⃗ + βmw . ⃗ ⋅ p⃗ − ⃗ ⋅ ∇w ⃗ ⋅ ∇v α α (8.291) i∂t ψ = H ψ , H= α v 2v v2 We now scale space according to the scaling ψ ′ = v 3/2 ψ ,

H ′ = v 3/2 H v −3/2 .

(8.292)

This leads to a Hermitian gravitational Hamiltonian, which acts on the Hilbert space of square-integrable functions with the scalar product ⟨φ, ψ⟩ = ∫ d3 r φ∗ (⃗ r) ψ(⃗ r). It reads as [281, 287] 1 w HGR = {⃗ α ⋅ p⃗, F} + βmw , F= . (8.293) 2 v

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In order to write this equation in terms of the radial components of an energy eigenstate ψ, one uses an ansatz analogous to Eq. (7.109), ψ(⃗ r) = (

f (r) χκ µ (ˆ r) ), ig(r) χ−κ µ (ˆ r)

HGR ψ(⃗ r) = EGR ψ(⃗ r) .

(8.294)

Using various identities fulfilled by the fundamental spin-angular functions χκ µ (ˆ r), as discussed in Sec. 6.4.1, together with the identity (7.111), one finally arrives at the following set of radial equations, (

∂ 1 − κ 2 v ′ (r) w(r) + v(r) w′ (r) EGR + + ) g(r) = v(r) (m − ) f (r) , (8.295a) ∂r r 2 v(r) w(r) w(r)

(

1 + κ 2 v ′ (r) w(r) + v(r) w′ (r) EGR ∂ + + ) f (r) = v(r) (m + ) g(r) . (8.295b) ∂r r 2 v(r) w(r) w(r)

Because of the rather involved structure of the w and v functions for the Schwarzschild metric, as given in Eq. (8.285), the exact calculation of the bound and continuum solutions of Eq. (8.295) still constitutes an open problem, even for the relatively simple Schwarzschild geometry of space-time. The particle-antiparticle symmetry E ↔ −E,

f (r) ↔ g(r) ,

κ ↔ −κ ,

(8.296)

which leaves the radial equations (8.295) invariant, ensures that both particles as well as antiparticles are attracted by the gravitational field, and establishes the equivalence principle for antiparticles. For the Schwarzschild metric in isotropic coordinates, given in Eq. (8.285), we have to first order in the Schwarzschild radius rs , w = (1 −

rs rs −1 4r − rs rs ) (1 + ) = ≈1− , 4r 4r 4r + rs 2r

(8.297a)

v = (1 +

rs 2 rs ) ≈1+ , 4r 2r

(8.297b)

w 16 r2 (4r − rs ) rs = ≈1− . v (4r + rs )3 r

(8.297c)

The Schwarzschild radius is given as rs = 2 G M , where G is Newton’s gravitational constant, and M is the mass of the planet. To first order in rs , we have to analyze the Dirac–Schwarzschild Hamiltonian HDS , which is given by HDS =

1 rs rs {⃗ α ⋅ p⃗, (1 − )} + βm (1 − ) . 2 r 2r

It is amenable to a Foldy–Wouthuysen transformation (see Chap. 11).

(8.298)

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8.5.2

Dirac Adjoint for Curved Space-Times

Let us try to derive the particle-antiparticle symmetry of the gravitationally and electromagnetically coupled Dirac equation on the basis of the Lagrangian of the field. To this end, we first need to generalize the concept of the Dirac adjoint to curved space-times. We recall that the Dirac adjoint transforms with the inverse of the Lorentz transform as compared to the original Dirac spinor. A general spinor Lorentz transformation S(Λ) is given as follows, i S(Λ) = exp (− AB ̃ σAB ) , 4

̃ σAB =

i [̃ γA, ̃ γB ] . 2

(8.299)

Note that the generator parameters AB , for local Lorentz transformations, can be coordinate-dependent. The spin matrices ̃ σAB are, however, the flat-space spin matrices. The spin matrices fulfill the commutation relations [see also Eq. (7.31)] g CF 12 ̃ g DE 21 ̃ g CE 21 ̃ g DF 21 ̃ σ CD , 12 ̃ σ EF ] = i (̃ σ DE + ̃ σ CF − ̃ σ DF − ̃ σ CE ) . [ 12 ̃

(8.300)

It is highly instructive to compare these relations to those fulfilled by the matrices MAB that generate (four-)vector Lorentz transformations, C C ̃ ̃ (MAB )C D = ̃ gA gDB − ̃ gB gDA .

(8.301)

The Lorentz transformation ΛC D is obtained as the matrix exponential C 1 ΛC D = (exp [ AB MAB ]) . 2 D

The algebra fulfilled by the [M

CD

,M

EF

]=̃ g

CF

(8.302)

M matrices is well known to be

MDE + ̃gDE MCF − ̃gCE MDF − ̃gDF MCE .

(8.303)

The two algebraic relations (7.31) and (8.303) are equivalent if one replaces i (8.304) 2 which exactly leads from Eq. (8.299) to Eq. (8.302). Under a local Lorentz transformation, a Dirac spinor transforms as

MCD → − σCD ,

ψ ′ (x′ ) = S(Λ) ψ(x) .

(8.305)

In order to write the Lagrangian, one needs to define the Dirac adjoint in curved space-time. In order to address this question, one has to remember that in flatspace-time, the Dirac adjoint ψ(x) is defined in such a way that it transforms with the inverse of the spinor Lorentz transform as compared to ψ(x), ′

ψ (x′ ) = ψ(x) S(Λ−1 ) = ψ(x) [S(Λ)]−1 .

(8.306)

The problem of the definition of ψ(x) in curved space-time is sometimes treated in the literature in a rather cursory fashion [278]. Let us see if in curved space-time, we can use the ansatz ψ(x) = ψ + (x) ̃ γ0 ,

(8.307)

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with the same flat-space ̃ γ 0 as in the flat-space Dirac adjoint. In this case, ′

ψ (x′ ) = ψ + (x′ ) S + (Λ) ̃ γ 0 = (ψ + (x′ ) ̃ γ 0 ) [̃ γ 0 S + (Λ) ̃ γ0] .

(8.308)

To first order in the Lorentz generators AB , we have indeed, i AB 0 + i  ̃ γ ̃ σAB ̃ γ 0 = 1 + AB ̃ σAB = [S(Λ)]−1 , 4 4 where we have used the identity ̃ γ 0 S + (Λ) ̃ γ0 = 1 +

i + + i 0 0 + 0 0 + 0 0 [̃ γB , ̃ γA ] = − ̃ γ [̃ γ ̃ γB ̃ γ ,̃ γ ̃ γA ̃ γ ]̃ γ 2 2 i 0 γ [̃ γB , ̃ γA ] ̃ γ 0 = −̃ γ0 ̃ σBA ̃ γ0 = ̃ γ0 ̃ σAB ̃ γ0 . = − ̃ 2

(8.309)

+ ̃ σAB = −

(8.310)

It is easy to show that Eq. (8.309) generalizes to all orders in the AB parameters, which justifies our ansatz given in Eq. (8.307). 8.5.3

Lagrangian and Charge Conjugation

Equipped with an appropriate form of the Dirac adjoint in curved space-time, we start from the Lagrangian density L = ψ(x) [γ µ {i (∂µ − Γµ ) − e Aµ } − m] ψ(x) ,

(8.311)

and attempt to derive the particle-antiparticle symmetry on the level of a transformation of the Lagrangian. The Lagrangian is Hermitian, and so ← Ð + + L = L+ = ψ + (x) [(γ µ )+ {−i ∂ µ − e Aµ } − (−i) (Γµ ) (γ µ )+ − m] [ ψ(x) ] . (8.312) An insertion of ̃ γ 0 matrices and use of the identity (̃ γ 0 )2 = 1 leads to the relation ← Ð L+ = ψ + (x) ̃ γ 0 [̃ γ 0 (γ µ )+ ̃ γ 0 {−i ∂ µ − e Aµ } +

+

+ i {̃ γ 0 (Γµ ) ̃ γ0} ̃ γ 0 (γ µ )+ ̃ γ 0 − m] ̃ γ 0 [ ψ(x) ] .

(8.313)

Also, we recall that ̃ γ 0 (Γµ )+ ̃ γ 0 = −Γµ , because i i + Γ+µ = − ωµAB ̃ σAB = − ωµAB ̃ γ0 ̃ σAB ̃ γ 0 = −̃ γ 0 Γµ ̃ γ0 . 4 4

(8.314)

So, ← Ð + L+ = ψ + (x) ̃ γ 0 [γ µ {−i ∂ µ − e Aµ } − i Γµ γ µ − m] ̃ γ 0 [ ψ(x) ] .

(8.315)

+

Now, we use the relations ψ + (x) ̃ γ 0 = ψ(x) and ̃ γ 0 [ ψ(x) ] = ψ(x), and arrive at the form ← Ð L+ = ψ(x) [γ µ {−i ∂ µ − e Aµ } − i Γµ γ µ − m] ψ(x) . (8.316) Because L is a scalar, a transposition again does not change the Lagrangian, and we have Ð → T T T T T (L+ ) = ψ T (x) [(γ µ ) {−i ∂ µ − e Aµ } − i (γ µ ) (Γµ ) − m] [ψ(x)] . (8.317)

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An insertion of the charge conjugation matrix C = ĩ γ2 ̃ γ 0 leads to Ð → T T (L+ ) = ψ T (x) C −1 [C (γ µ ) C −1 {−i ∂ µ − e Aµ } T

−1 − m] C [ψ(x)] . − i C (γ µ ) C −1 C ΓT µ C T

(8.318)

We have used the identities C (γ µ ) C −1 = −γ µ , and C (Γµ ) C −1 = −Γµ . The latter of these can be shown as follows, T

−1 C ΓT = µ C

T

i i AB i i T T ] C −1 } = { ωµAB [−̃ { ωµ C [̃ γB ,̃ γA γB , −̃ γA ]} = −Γµ . 4 2 4 2

(8.319)

The result is the expression Ð → T T (L+ ) = ψ T (x) C −1 [(−γ µ ) {−i ∂ µ − e Aµ } − i (−γ µ ) (−Γµ ) − m] C [ψ(x)] . (8.320) Now we express the result in terms of the charge-conjugated spinor and its adjoint (further remarks on this point are presented in the following), T

ψ C (x) = C [ψ(x)] ,

ψ C (x) = −ψ T (x) C −1 ,

(8.321)

to write Ð → T L = (L+ ) = −ψ C (x) [γ µ {i ∂ µ + e Aµ } − i γ µ Γµ − m] ψ C (x) = − ψ C (x) [γ µ {i(∂µ − Γµ ) + e Aµ } − m] ψ C (x) .

(8.322)

The Lagrangian (8.322) differs from (8.311) only with respect to the sign of electric charge, as is to be expected, and with respect to the replacement of the Dirac spinor ψ(x) by its charge conjugation ψ C (x). The overall minus sign is a consequence of the fact that we here investigate the Lagrangian on the level of the (first-quantized) relativistic quantum theory, but not on the level of the (second-)quantized quantum field theory, for which we refer the reader to Ref. [251]. We had already seen that the charge conjugation of the Dirac current only yields the intuitive result in the second-quantized formalism [see Eqs. (7.195) and (7.196)]. The gravitational covariant derivative ∂µ −Γµ has retained its form in going from (8.312) to (8.322), in agreement with the particle-antiparticle symmetry of the gravitational interaction. We should now supplement the derivation of Eq. (8.321). Very explicitly, the charge-conjugated spinor is given by T

T

γ0] = C ̃ γ 0 ψ ∗ (x) . ψ C (x) = C [ψ(x)] = C [ψ + (x) ̃

(8.323)

Hence, we have the relation +

+

ψ C (x) = (C ̃ γ 0 ψ ∗ (x)) ̃ γ 0 = [ψ ∗ (x)] ̃ γ0 C+ ̃ γ0 = ψ T (x) ̃ γ0 C+ ̃ γ 0 = ψ T (x) C = −ψ T (x) C −1 ,

(8.324)

where the two identities (i) ̃ γ0 C+ ̃ γ 0 = C and (ii) C −1 = −C have been used. These will be derived in the following. The explicit form of the ̃ γ 2 matrix in the Dirac

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+

representation implies that (̃ γ 2 ) = −̃ γ 2 . Based on this relation, we can easily show that +

+

C + = (i ̃ γ2 ̃ γ 0 ) = −i ̃ γ 0 (̃ γ2) = i ̃ γ0 ̃ γ 2 = −i ̃ γ2 ̃ γ 0 = −C .

(8.325)

The first identity ̃ γ0 C+ ̃ γ 0 = C can now be shown as follows, ̃ γ0 C+ ̃ γ0 = ̃ γ 0 [−i ̃ γ2 ̃ γ0] ̃ γ 0 = −i ̃ γ0 ̃ γ2 = i ̃ γ2 ̃ γ0 = C .

(8.326)

Furthermore, one has 2

C C + = C (−C) = i ̃ γ2 ̃ γ0 i ̃ γ0 ̃ γ 2 = − (̃ γ 2 ) = − (−14×4 ) = 14×4 ,

(8.327)

so that C −1 = C + = −C ,

(8.328)

which proves, in particular, that C −1 = −C. One consequence of the derivation outlined here is that the equivalence principle holds for antimatter [251, 291]. This conclusion is supported by preliminary experimental results [292]. 8.6

Further Thoughts

Here are some suggestions for further thought. (1) Dirac Ground State. We investigate whether or not it is possible to “guess” the exact functional form of the 1S1/2 ground state wave function of the hydrogen atom within the fully relativistic formalism. To this end, we start from Eq. (8.12), ⃗ ⎞ ⃗⋅L σ ∂ Zα ⎛ (⃗ σ ⋅ r ˆ ) (−i + i )⎟ m − ⎜ r) r ∂r r ⎟ ( f (r)χκ µ (ˆ ⎜ HDC ψ(⃗ r) = ⎜ ) ⎟ ig(r)χ ⃗ ⃗ ∂ σ ⋅ L Zα (ˆ ⎟ ⎜ −κ µ r ) (⃗ σ ⋅ r ˆ ) (−i + i ) −m − ⎠ ⎝ ∂r r r ⎛ [(− ∂ + 1 (κ − 1)) g(r) + (m − Zα ) f (r)] χκ µ (ˆ r) ⎞ ⎜ ⎟ ∂r r r ⎟ = E ψ(⃗ =⎜ r) ⎜ ⎟ ⎜ ⎟ ∂ 1 Zα r) ⎠ ⎝ i [( ∂r + r (κ + 1)) f (r) − (m + r ) g(r)] χ−κ µ (ˆ (8.329) where one has used ⃗ = K − 1, σ⋅L

K χκµ = −κ χκµ ,

⃗ ⋅ rˆ χκ µ (ˆ σ r) = −χκ µ (ˆ r) .

(8.330)

The upper and lower component equations are Zα ∂ 1 − E) f (r) + (− + (κ − 1)) g(r) = 0 , r ∂r r ∂ 1 Zα ( + (κ + 1)) f (r) − (m + + E) g(r) = 0 . ∂r r r

(m −

(8.331) (8.332)

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Our ansatz for the wave function is f (r) = A rγ exp(−β r) ,

g(r) = B rγ exp(−β r) .

(8.333)

Comparing coefficients of different powers of r, one should obtain the system of equations (m − E) A + β B = 0 ,

(8.334a)

(−2 − γ) B − Z α A = 0 ,

(8.334b)

(E + m) B + A β = 0 ,

(8.334c)

−Z α B + A γ = 0 .

(8.334d)

Furthermore, one may ascertain that all of these equations are satisfied if we set √ √ γ = 1 − (Zα)2 − 1 , E = m 1 − (Zα)2 , √ 1 − (Zα)2 − 1 γA = A. (8.335) β = Zα m , B= Zα Zα Finally, try to determine if the overall normalization factor A could read as √

√

√

1/2

⎛ 22 1−(Zα)2 −1 m2 1−(Zα)2 +1 (Zα)2 1−(Zα)2 +3 ⎞ ⎟ A = ⎜√ √ √ √ ⎝ 1 − (Zα)2 1 − 1 − (Zα)2 Γ(2 1 − (Zα)2 ) ⎠

.

(8.336)

(2) Dirac Continuum and Bound States. Generalize the treatment, inspired by the second-order differential equation given in (8.58) for the continuum wave functions (see Sec. 8.2.3), to the case of bound-state wave functions (see Sec. 8.2.2). Instead of Eq. (8.53), try the ansatz u1 = (m + E)

1/2

(Φ1 + Φ2 ) ,

(8.337a)

u2 = (m − E)

1/2

(Φ1 − Φ2 ) .

(8.337b)

Compare with Eqs. (18) and (19) of Ref. [293]. (3) Momentum Normalization for Continuum States. Work out the Dirac continuum wave functions in the momentum normalization [as opposed to the energy normalization, see Eq. (8.88)] ⟨1∣2⟩ = δ(p1 − p2 ) .

(8.338)

Go through all intermediate steps of the derivation and convince yourself of the validity of the calculation of the normalization integral. (4) Positron Continuum States. The derivation in Sec. 8.2.3 concerned electron continuum solutions. Work out the positron continuum solutions in explicit form. (5) Nonrelativistic Limit. Calculate the nonrelativistic limit of the relativistic bound-state wave functions (see Sec. 8.2.2) and compare with the derivation of the nonrelativistic wave function calculated in Sec. 4.2.3. Which parameters

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are relevant to the expansion, and which ones have to be kept constant? On the basis of the considerations, or otherwise, show the validity of the approximation given in Eq. (6.100), whose leading term can be found in Eq. (6.99). Then, attempt to do the same for the continuum wave functions discussed in Sec. 8.2.3 and compare to Sec. 4.2.5. (6) Virial Theorems. Study the formalism introduced in Ref. [264]. Investigate whether or not the formalism allows one to derive closed-form analytic expressions for the correction to the wave function induced by an external magnetic field. These would be relevant for numerical calculations of radiative corrections to the bound-electron g factor, e.g., in Ref. [39]. (7) Maxwell–Dirac–Mohr Equations. A very interesting work by Mohr4 concerns candidate wave functions for the photon [294]. We shall dwell on related questions in terms of potential further thoughts that might be invested into this intriguing problem, and start with the spin operators of the photon which are given by the matrices Sk (k = 1, 2, 3) [see Eq. (5.139) on page 194 of Ref. [81]], (Sk )ij = −i kij ,

(8.339)

where ijk =  is the Levi-Civit`a tensor. The explicit representation for k = 1, 2, 3 reads as ijk

⎛0 0 0 ⎞

S1 = ⎜ 0 0 −i ⎟ ⎝0 i 0 ⎠

⎛0 0 i⎞

S2 = ⎜ 0 0 0 ⎟ , ⎝ −i 0 0 ⎠

⎛ 0 −i 0 ⎞

S3 = ⎜ i 0 0 ⎟ . ⎝0 0 0⎠

(8.340)

These spin operators follow the conventions proposed in Ref. [295] for the generators of the global SU (2) symmetry in the basis of the π 0,± mesons. The commutation relation is [ Si , Sj ] = i ijk Sk .

(8.341)

For the discussion of hyperfine effects for spin-1 particles, e.g., the deuteron nucleus in the context of hyperfine splitting in deuterium, it is often advantageous to define a basis where the τ 3 matrix is diagonal, i.e., ⎛1 0 0 ⎞ τ3 = ⎜ 0 0 0 ⎟ , ⎝ 0 0 −1 ⎠

(8.342)

supplemented by 01 1 ⎛ τ1 = √ ⎜ 1 0 2 ⎝0 1

0⎞ 1⎟ , 0⎠

0 −1 0 ⎞ i ⎛ τ 2 = √ ⎜ 1 0 −1 ⎟ . 2 ⎝0 1 0 ⎠

(8.343)

These conventions are adopted in Ref. [294]. (In Ref. [294], the electric and magnetic fields are used in the spherical basis, but we here advocate a modified representation which allows for Cartesian coordinates to be used for the 4 Peter

Joseph Mohr (b. 1943).

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description of the fields.) We recall the basis vectors in the spherical basis from Chap. 6, 1 1 1 ⎛ ⎞ e⃗+1 = − √ (ˆ ex + i eˆy ) = − √ ⎜ i ⎟ , 2 2 ⎝0⎠ In the spherical basis,

e⃗0 = eˆz ,

1 ex − i eˆy ) . e⃗−1 = √ (ˆ 2

S3 is diagonal, i.e., one has

S3 e⃗+1 = e⃗+1 ,

S3 e⃗0 = 0⃗ ,

S3 e⃗−1 = −⃗e−1 .

(8.344) (8.345)

⃗ one can show that For an arbitrary vector field B, ⃗ ⋅ ∇) ⃗ = −∇ ⃗, ⃗ B ⃗ ×B i (S

(8.346)

⃗ = Bx eˆx + By eˆy + Bz eˆz , given in the Cartesian basis. One may observe where B that, taking the divergence of the Ampere–Maxwell law, one obtains the time derivative of Gauss’s law. The law of the absence of magnetic monopoles, ⃗ = 0, can be interpreted as an intrinsic property of the magnetic induction ⃗ ⋅B ∇ field. Hence, the only Maxwell equations which need to be addressed, are the Ampere–Maxwell law, and the Faraday law. In the source-free case, with 0 = µ0 = c = 1, they read as ∂ ⃗ E = 0, ∂t ⃗ = 0, ⃗ ⋅ ∇) ⃗− ∂ B ⃗ E i (S ∂t ⃗ ⋅ ∇) ⃗+ ⃗ B i (S

(8.347) (8.348)

which can be grouped as ⎛ 12×2 ∂ ⎜ ∂t ⎜ ⎜ ⃗ ⋅∇ ⃗ ⎝ −S

⃗ ⎞⎛E ⃗ ⎞ ⎛ ⃗0 ⎞ S⃗ ⋅ ∇ ⎟

⎟⎜ ⎟=⎜ ⎟. (8.349) ∂ ⎟ ⎝iB ⃗ ⎠ ⎝ ⃗0 ⎠ ⎠ ∂t One may define modified Dirac–Mohr matrices ̂ γ according to the following prescription, −12×2

⎛ 12×2 0 ⎞ ⎟, ̂ γ0 = ⎜ ⎝ 0 −12×2 ⎠

̂ γi = (

0 Si ), −Si 0

(8.350)

and write the Maxwell–Dirac–Mohr equation (8.349) as ̂ γ µ ∂µ Ψ = 0 ,

(8.351)

⃗ iB) ⃗ T is a candidate wave function of the photon. One of the where Ψ = (E, advantages of Eq. (8.351) is that it is possible to define a Hamilton operator for the photon, ⃗ ⋅ p⃗ , ̂ H =α

⃗, p⃗ = −i∇

̂i = ̂ α γ0 ̂ γi .

(8.352)

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Longitudinal photons can be identified as zero-energy states of the photon field [294]. A good exercise is to study the formalism introduced in Ref. [294]. Also, in Ref. [294], the Green functions of the electromagnetic field are calculated based on a first-order differential equation, ̂ γ µ ∂µ Ψ = Ξ(x), where Ξ(x) is a ⃗ ⃗ T [see Eq. (52) of Ref. [294]]. six-component source term Ξ(x) = (−J(x), 0) The formalism introduced in Ref. [294] has been used in Ref. [296]. One advantage of the formalism introduced in Ref. [294] is that the Maxwell Green function, given in Chap. 9 of Ref. [294], directly connects the sources to the fields, without any intermediate recourse to potentials. The advantages of the formalism introduced in Ref. [294] for practical applications still remain to be analyzed, and one is encouraged to expand on the formalism introduced in Ref. [296]. (8) Numerical Evaluation of Generalized Bessel Functions. For the numerical evaluation of an array of (ordinary) Bessel functions of increasing index, one may use recursive methods, described in Refs. [258, 297, 298] and in Sec. III A of Ref. [272]. Key to the solution of the numerical problem is the transformation of a five-term recurrence relation fulfilled by the generalized Bessel functions, given in Eq. (11) of Ref. [272], into a four-term, and a threeterm recurrence relation, given in Eqs. (24)–(28) of Ref. [272]. The numerical behavior of the expansion coefficients of a Dirac–Volkov state, as a function of the index, is analyzed in Sec. V B of Ref. [272]. In this context, one considers the asymptotic behavior of the expansion coefficients, which constitute generalized Bessel functions, as illustrated in Fig. 5 of Ref. [272]. Investigate to which extent the observed behavior allows us to establish a classical-quantum correspondence relation for the Dirac–Volkov states. (9) Covariant Derivative. Consider the analogy of Eqs. (8.4) and (8.238), Dµ′ ψ ′ = (Dµ ψ)′

⇔

∇′µ ψ ′ = (∇µ ψ)′ .

(8.353)

Furthermore, consider the analogy, on the basis of Eqs. (8.3) and (8.239), i e Aµ

⇔

−Γµ .

(8.354)

Then, on the basis of the analogy, interpret the identity (8.242), Γ′µ = S(Λ) Γµ S(Λ)−1 + [∂µ S(Λ)] S(Λ)−1 ,

(8.355)

in terms of the gauge transformation of the electromagnetic four-vector potential (8.1), i.e., in terms of the relation of A′µ to Aµ . (10) Christoffel Symbols. Show Eq. (8.178) and verify all intermediate steps in Eq. (8.181). (11) Lorentz Transformation of a Dirac Matrix. Show, in analogy to Eq. (8.230), that the Dirac matrix with covariant index, ̃ γA , is invariant under ′ =̃ γA . local Lorentz transformations, i.e., that ̃ γA

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(12) Basic Properties of the Tetrad. Show that (eρB )

−1

= eB ρ ,

(eB ρ)

−1

= eρB ,

µ eµA eA ν = δν ,

µ A eA µ eB = δB .

(8.356)

Using the properties eµA = eνA g νµ = eB gBA , µ ̃

νµ eµA = eA = eµB ̃ g BA , ν g

(8.357)

convince yourself that eA µ eνA = g µν , eµA

eµB = ̃ gAB ,

eµA eνA = g µν , eA µ

e

µB

(8.358)

=̃ g

AB

.

(8.359)

(13) Lorentz Group. Attempt to show the commutator relation (8.280) using the Clifford algebra of the flat-space Dirac matrices. (14) Comparison with Other Literature. Study the formalism introduced in Ref. [289], as well as in Ref. [299], and Ref. [300], and, last, but not least, in Ref. [284]. Compare with Refs. [278, 280, 301–304], and try to decide if the formalism introduced in the cited references is compatible with our formulas for the gravitational coupling of a Dirac particle. (15) Spin Connection: Alternative Form. Consider the following alternative form of the affine connection matrix [278, 281], ∂eB i µν , Γρ (x) = − g µα (x) [ νρ eα B − Γα νρ ] σ 4 ∂x

(8.360)

where the local spin matrix is σ µν = σ µν (x) = 2i [γ µ (x), γ ν (x)]. Show the equivalence of Eq. (8.360) and Eq. (8.266). Then, use the equation [γ µ (x), σ αβ (x)] = 2 i g µα (x) γ β (x) − 2 i g µβ (x) γ α (x) ,

(8.361)

in order to verify that the functional form (8.360) fulfills Eq. (8.211). (16) Spin Connection: Lorentz Transformation. The Ricci rotation coefficient, or “spin connection”, was defined in Eq. (8.223) as νB ωµAB = eA . ν ∇µ e

(8.362)

So, it is intuitively clear that under a Lorentz transformation of the local Lorentz frame [see Eq. (8.225)], e′µA = ΛA B eµB ,

(8.363)

′νB ωµ′AB = e′A . ν ∇µ e

(8.364)

one has Verify that this transformation property is consistent with Eq. (8.242), Γ′µ = S(Λ) Γµ S(Λ)−1 + [∂µ S(Λ)] S(Λ)−1

(8.365)

where, in contrast to the discussion of Sec. 8.4.7, one uses full, exponentiated Lorentz transformations, A

ΛA B = (exp [CD MCD ])

B

i AB S(Λ) = exp (− ̃ σ AB ) . 4

,

(8.366a) (8.366b)

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(17) Dirac–Schwarzschild Propagator. Try to generalize the treatment of the Dirac–Coulomb propagator, outlined in Sec. (8.2.5), to the Dirac– Schwarzschild Hamiltonian given in Eq. (8.298). One needs to solve the equation (HDS − E) G(⃗ r1 , r⃗2 , E) = δ (3) (⃗ r1 − r⃗2 )

14×4 ,

(8.367)

where HDS is given in Eq. (8.298). Show how the particle-antiparticle symmetry is reflected in the spin-angular decomposition of the propagator.

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Chapter 9

Electromagnetic Field and Photon Propagators

9.1

Overview

In the current chapter, we investigate the photon propagator from different angles. Initially, we had discussed the quantization of the electromagnetic field in Sec. 2.4. However, we restricted the discussion to the transverse components of the electric field. We proceed as follows. In the previous chapter, our focus was laid on the Feynman propagator of the fermion field, which was obtained as the time-ordered product of the fermion field operators, as we recall Eq. (7.153), ⟨0 ∣T ψ(x1 ) ψ(x2 )∣ 0⟩ = i SF (x1 , x2 ) .

(9.1)

Here, we recall that, because of the bispinor structure of the Dirac field operator, the notation adopted in Eq. (9.1) is a short-hand notation for a matrix-valued function, where on the left-hand side, we suppress the spinor indices of the fermion field operators. In the case of the photon field, we shall encounter the time-ordered product of field operators [see Eq. (9.120)] ⟨0 ∣T Aµ (x) Aν (x′ )∣ 0⟩ = iDFµν (x − x′ ) = i g µν GF (x − x′ ) , DFµν (x − x′ )

(9.2)

where is the Feynman propagator of the photon field, and GF (x − x′ ) is the scalar Green function of the wave equation, which fulfills the equation ∂ ⃗ 2 ) GF (t − t′ , r⃗ − r⃗′ ) = δ(t − t) δ (3) (⃗ ( 2 −∇ r − r⃗′ ) . (9.3) ∂t ̵ = c = 0 = 1 This is discussed in greater detail in Sec. 9.2.1. Natural units with h are used throughout this chapter. The Green function of the wave equation has been discussed at a number of places in the literature (see, e.g., Chap. 4 of Ref. [81]). We note that the retarded, advanced and Feynman Green functions all fulfill the wave equation (9.3), while being subject to different boundary conditions. Here, we derive a few properties of the Green functions of the wave equation, by a careful consideration of the Diracδ distributions entering the expressions in both coordinate as well as momentum space. This is done in Sec. 9.2. 321

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As we had seen in Chap. 2, the timelike component of the four-vector potential (the so-called “scalar potential”) is not quantized in Coulomb gauge, due to the lack of a conjugate field momentum. However, the spatial components of the vector-field operator are indeed quantized in Coulomb gauge. In general, the four-vector field ⃗ composed of the scalar potential Φ and the vector potential A, ⃗ fulfills Aµ = (Φ, A) different gauge conditions in different gauges. In turn, the time-ordered product of the photon field operators enters the expression for the photon propagator in Eq. (9.2). Gauge conditions such as ⃗ ⋅ A⃗ = 0 ∇

(Coulomb Gauge)

(9.4)

(Lorenz Gauge)

(9.5)

[see Eq. (2.43)] and ∂µ Aµ = 0

[see Eq. (9.75)] are primarily imposed on the classical four-vector potential, while the expression (9.2) is constructed with the quantized vector-field operators. We shall learn that the gauge conditions imposed on the classical potentials can often be imposed on the quantized field operators only in the sense of expectation values of the physical states of the fields. Still, it becomes clear that the explicit form of the photon propagator will depend on the gauge being used. In Coulomb gauge, we shall discuss the photon propagator in Sec. 9.3. The calculation is made more interesting by the fact that the “scalar potential” is not quantized in Coulomb gauge, per our discussion in Chap. 2. However, in the Hamiltonian of the electromagnetically coupled currents, one finds a term [see Eq. (9.50)] which contains the timelike component of the current J 0 and needs to be matched against the D00 component of the photon propagator [see Eq. (9.60)]. This matching is done against the second-order (as opposed to first-order) S matrix element, which defines the spatial components Dij of the photon propagator [see Eq. (9.58)]. The result of the matching is the photon propagator in Coulomb gauge [see Eq. (9.73)], obtained as a result of adding the instantaneous, non-quantized Coulomb interaction (which couples to the scalar component of the current) to the retarded interaction which couples to the spatial components of the current. In Lorenz gauge (see Sec. 9.4), one can “solve” the problem connected with the absence of a canonical momentum conjugate to A0 , by introducing a gauge-fixing term [see Eq. (9.79)]. This procedure enables us to write down a nonvanishing, canonical field momentum Π0 , conjugate to the scalar potential [see Eq. (9.102)], resulting in the photon propagator in Lorenz gauge [see Eq. (9.121)]. However, the artificial “rabbit drawn from the hat” (the nonvanishing Π0 ), comes at a price: In writing the modified Lagrangian (9.79), we have introduced, at will, a term proportional to (∂ ⋅ A)2 which vanishes in Lorenz gauge (in the classical theory), but not necessarily in other gauges. Furthermore, a closer inspection reveals that ∂ ⋅ A = ∂µ Aµ actually does not vanish in the quantized theory, i.e., if Aµ is a field operator as opposed to a classical field. Hence, we have introduced nonphysical degrees of freedom, which will turn out to be scalar and longitudinal photons.

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323

Physical states of the electromagnetic field cannot contain unphysical degrees of freedom. The intuitive feeling is that if we have introduced artificial degrees of freedom, then we somehow need to eliminate them. This is done by imposing the Lorenz-gauge condition ∂ ⋅ A = 0 not as an operator identity, but in terms of a condition ∂ ⋅ A(+) ∣Ψ⟩ = 0 which is fulfilled for any physical state ∣Ψ⟩ of the photon field. Here, A(+) contains the positive-frequency (annihilation) contribution to the photon-field operator. This is called the Gupta–Bleuler formalism (see Sec. 9.5.3), even if it could be characterized as a “Lorenz gauge condition in the mean.” The transition to other gauges is discussed in Sec. 9.5. Finally, in Sec. 9.6, we explore the interactions of quantized and classical fields. The notion is that the quantized description allows for the description of field quanta, while a classical description is adequate for strong fields for which we can neglect the influence of quantum fluctuations. For example, if (Dirac) currents are quantized and electromagnetic fields are not, then we may describe (virtual) fermion pair creation processes. This formalism is used in Sec. 18.2 in order to derive the effective Heisenberg–Euler Lagrangian, which is valid for very strong electromagnetic fields. Conversely, if we employ a quantized photon field and classical currents, then we may describe photon creation processes. This latter formalism is applied in Sec. 9.6.1. The way of calculation essentially relies on the Wick theorem, which is given (in various, equivalent forms) in Eqs. (9.163), (9.164), (9.165) and (9.166). Quantizing only one of the two involved fields leads us to a much more manageable theoretical description. 9.2 9.2.1

Time Orderings, Field Commutators and Green Functions Miscellaneous Fundamental Relations for Green Functions

We write the defining equation (9.3) for the Green function of the wave equation as ∂ ⃗2 . (9.6) ◻ G(x) = δ (4) (x) , ◻= 2 −∇ ∂t Here, ◻ is the “quabla” operator. We use a relativistically covariant formulation. The coordinates are [see also Eq. (2.58)] x0 = t, x1 = x, x2 = y, x3 = z. (9.7) As is common, we use Latin indices for spatial vectors (i, j ∈ {1, 2, 3}), whereas Greek indices are used for space-time four-vectors, µ, ν ∈ {0, 1, 2, 3}. The space-time metric is g µν = diag(1, −1, −1, −1). We work in flat Minkowski space. The derivatives with respect to the coordinates are ∂ µ = ∂/(∂xµ ) and ∂µ = ∂/(∂xµ ). The scalar product ⃗ = v µ wµ . For the spatial components of a of four-vectors is v ⋅ w = v 0 w0 − v⃗ ⋅ w general four-vector, we will sometimes use the notation v 1 = v x , v 2 = v y , v 3 = v z . A somewhat nonstandard notation used here concerns the spatial components of the gradient operator, which we recall from Eq. (2.56), ∂ (9.8) ∇i ≡ ∂xi with upper indices.

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In momentum space, with k 2 = k µ kµ , we have for the retarded, advanced and Feynman Green functions, GR (k) = −

1 (k 0

+ i )2

− k⃗2

,

GA (k) = −

1 (k 0

− i )2

− k⃗2

,

GF (k) = −

k2

1 . (9.9) + i

All three Green functions GR , GA and GF fulfill the relation ◻ GR (x) = ◻GA (x) = ◻GF (x) = δ (4) (x) .

(9.10)

A summary of important properties of the retarded, advanced, and so-called Feynman solutions can be found in Sec. 4.1.5 of Ref. [81], but we here wish to advance a ⃗ − i , little further. The poles of the retarded Green function are located at k 0 = ±∣k∣ i.e., the lower half of the complex plane. In the Fourier backtransform integral, the contour for the k 0 integration needs to be closed in the upper half of the complex plane when t < 0. Hence, the retarded Green function vanishes for negative t. Analogously (see Chap. 4 of Ref. [81]), one can show that GA (x) vanishes for t > 0. A special role is taken by the Feynman Green function GF , whose poles are √ 0 2 ⃗ at k = ± k − i , positive-frequency (positive-k 0 ) poles are picked up for t > 0, whereas negative-frequency poles are relevant for t < 0. As explained in detail in Chap. 4 of Ref. [81], a Fourier transformation to coordinate space leads to the following formulas. One has for the retarded Green function, GR (x) = − ∫ =

e−ik⋅x d4 k (2π)4 (k 0 + i )2 − k⃗2

Θ(t) 1 [δ(t − r) − δ(t + r)] ≈ Θ(x0 ) δ(x2 ) . 4πr 2π

(9.11)

Here, x = (t, r⃗), with x0 = t and r = ∣⃗ r∣. The representations indicated after the “≈” sign indicate a form obtained after we ignore the overlap term of the step function Θ and the representation of the Dirac-δ function. For the advanced Green function, one finds d4 k e−ik⋅x (2π)4 (k 0 − i )2 − k⃗2 Θ(−t) 1 = − [δ(t − r) − δ(t + r)] ≈ Θ(−x0 ) δ(x2 ) . 4πr 2π

GA (x) = − ∫

(9.12)

Finally, the Feynman Green function reads as follows, in space-time coordinates, GF (x) = − ∫

d4 k e−ik⋅x i 1 =− 2 2 . 4 2 (2π) k + i  4π x − i

(9.13)

The derivation of this latter result will be discussed in more detail in the following. For completeness, we now consider the Fourier transformations with respect to time, to obtain a mixed photon energy-coordinate representation which is convenient

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especially for bound-state calculations, ∞

ei ωr , 4π r −∞ ∞ e−i ωr dt eiωt GA (x) = GA (ω, r⃗) = ∫ , 4π r −∞

GR (ω, r⃗) = ∫

GF (ω, r⃗) = ∫

∞

dt eiωt GR (x) =

dt eiωt GF (x) =

ei

√

ω 2 +i r

(9.14) (9.15) √ Im ω 2 + i > 0 .

(9.16) 4π r The condition in Eq. (9.16) ensures that the modulus of the Feynman Green function remains bounded in the entire complex ω plane. 9.2.2

−∞

,

Distributions and Fourier Transforms

In dealing with Green functions, one often has to treat expressions which contain infinitesimal quantities, Dirac-δ functions and other singular distributions. Let us try to compile a few important formulas and in particular, investigate in detail the derivation of the Fourier backtransform (9.13). We start with the following formula, which is relevant to the splitting of an integral into a principal-value term and a pole term, in the distributional sense, 1 1 = (P.V.) + i π δ(z) . (9.17) z − i z Here, (P.V.) denotes the principal value, and  denotes an infinitesimal imaginary part. We now investigate a slight generalization of Eq. (9.17), where η = (, 0, 0, 0)

(9.18)

is a time-like infinitesimal four-vector. Then, we have (x − iη)2 = x2 − 2ix0  = x2 − i sgn(x0 ) ,

(9.19)

where we have absorbed 2∣x0 ∣ into a redefined , which we denote as . The sign function is denoted sgn(t) = t/∣t∣; it equals unity for positive argument and −1 for negative argument. So, we can write 1 1 = (P.V.) 2 + i π sgn(x0 ) δ(x2 ) . (x − i η)2 x

(9.20)

We shall use this result in Eq. (9.26). On another issue, collecting the zeros of the argument of a Dirac-δ, using the well-known formula 1 δ(f (z)) = ∑ ′ δ(z − zi ) , f (zi ) = 0 , (9.21) ∣f (z i )∣ i we can write δ(x2 ) = δ((x0 )2 − r⃗ 2 ) =

1 [δ(x0 − ∣⃗ r∣) + δ(x0 + ∣⃗ r∣)] , 2∣⃗ r∣

(9.22)

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and

Θ(x0 ) δ(x0 − ∣⃗ r∣) , (9.23) 2∣⃗ r∣ where we ignore a “dangerous” term proportional to Θ(x0 ) δ(x0 + ∣⃗ r∣) [see the discussion following Eq. (4.35) of Ref. [81] and Eqs. (9.11) and (9.12)]. The latter term can be important if we take into account the infinitesimal overlap of the representation of the Dirac-δ function with the step function Θ. This overlap can be neglected in most calculations but one should be aware of exceptions [305]. A Lorentz-invariant integration measure is given as d3 k 1 d4 k Θ(k 0 )δ(k 2 ) = , (9.24) 4 ⃗ (2π) (2π)3 2∣k∣ Θ(x0 ) δ(x2 ) ≈

where the integration proceeds on the full R4 for the left-hand side, while it is restricted to R3 for the right-hand side. The following integral is of rather fundamental importance, 1 1 1 1 d3 k 1 ik⋅x d3 k 1 −ik⋅x e =− 2 e =− 2 , . ∫ ∫ 3 2 ⃗ ⃗ (2π) 2∣k∣ 4π (x + iη) (2π)3 2∣k∣ 4π (x − iη)2 (9.25) The derivation is a little tricky, which is why we present it in some detail [we use ⃗ r = ∣⃗ the convention that k = ∣k∣, r∣, the relation k ⋅ x = ω t − k⃗ ⋅ r⃗, and a convergent factor exp(−k)], 1 ∞ 1 d3 k 1 i k⋅x e = dk k ei (k t−k r u) ∫ du ∫ ∫ 3 2 ⃗ (2π) ∣k∣ (2π) −1 0 ∞ 1 1 dk (ei k (t+r+i) − ei k (t−r+i) ) = lim ∫ 2 →0 (2π) ir 0 1 1 1 1 ( − ) (2π)2 ir −i (t + r + i) −i (t − r + i) 2r 1 = ((P.V.) 2 2 − iπ δ(t + r) + iπ δ(t − r)) (2π)2 r r −t 1 1 1 1 = − 2 [(P.V.) 2 − i π sgn(t) δ(x2 )] = − 2 . (9.26) 2π x 2π (x + iη)2 We have used the identity sgn(t) [δ(t + r) − δ(t − r)] , (9.27) sgn(t) δ(x2 ) = sgn(t) δ(t2 − r2 ) = 2r and infinitesimal convergence-generating factors. Analogously, one can derive the results d3 k 1 −ik⋅x 1 1 e = − 2 , (9.28a) ∫ 3 ⃗ (2π) 2∣k∣ 4π (x − iη)2 = lim →0

∫

d3 k 1 1 1 (eik⋅x + e−ik⋅x ) = − 2 (P.V.) 2 , ⃗ (2π)3 2∣k∣ 2π x

(9.28b)

∫

d3 k 1 i (eik⋅x − e−ik⋅x ) = sgn(x0 ) δ(x2 ) . 3 ⃗ (2π) 2∣k∣ 2π

(9.28c)

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As anticipated, we now consider the Fourier backtransform (9.13) of the Feynman propagator, using the identity 1 1 1 1 1 = = 2 ( − ). (9.29) 2 2 0 ⃗ ⃗ ⃗ ⃗ − i k + i k0 − k + i 2∣k∣ k0 − ∣k∣ + i k + ∣k∣ Starting from the momentum representation GF (k) = −1/(k 2 + i), and using Eq. (9.29), one obtains GF (x) = − ∫

⃗

d3 k eik⋅⃗x 1 ⃗ 0 ⃗ 0 {Θ(x0 )e−i∣k∣x (−2πi) − Θ(−x0 )ei∣k∣x (+2πi)} , (9.30) 3 ⃗ 2π (2π) 2∣k∣

where one closes the contour either in the lower complex half-plane (for x0 > 0), leading to the term proportional to Θ(x0 ), or in the upper complex half-plane (for x0 < 0), leading to the term proportional to Θ(−x0 ). Because one integrates over d3 k, the two terms can be summarized as follows, d3 k 1 −i∣k∣ ⃗ ∣x0 ∣+ik⋅⃗ ⃗x e GF (x) = i ∫ 3 ⃗ (2π) 2∣k∣ =

i d3 k 1 −ik⋅x i d3 k 1 ik⋅x 0 e + e Θ(x0 ) ∫ Θ(−x ) ∫ ⃗ ⃗ 2 (2π)3 ∣k∣ 2 (2π)3 ∣k∣

i 1 1 1 i 1 Θ(x0 ) [− 2 ] + Θ(−x0 ) [− 2 ], (9.31a) 2 2π (x − iη)2 2 2π (x + iη)2 where use has been made of Eq. (9.25). Finally, using Eq. (9.20), one obtains 1 i GF (x) = − 2 [Θ(x0 ) (P.V.) 2 + i π sgn(x0 ) Θ(x0 ) δ(x2 ) 4π x 1 + Θ(−x0 ) (P.V.) 2 − i π sgn(x0 ) Θ(−x0 ) δ(x2 )] x 1 1 i i . (9.31b) = − 2 [(P.V.) 2 + iπδ(x2 )] = − 2 2 4π x 4π x − i The final result is surprisingly compact. For the transformations to the mixed frequency-coordinate representation, (t, r⃗) → (ω, r⃗), we use the identity 1 1 1 1 = ( − ). (9.32) t2 − r⃗ 2 − i 2 r t − r − i t + r + i The Fourier transformation (9.16) of GF (x) can thus be calculated easily, ∞ ∞ 1 1 GF (ω, r⃗) = ∫ dt ei ω t GF (x) = ∫ dt ei ω t 2 2 4π i t − r⃗ 2 − i −∞ −∞ ∞ 1 1 1 1 = 2 ∫ ( − ) dt ei ω t 4π i −∞ 2 r t − r − i t + r + i 1 (2iπ) eiω r (−2iπ) e−iω r ei∣ω∣r = 2 { Θ(ω) − Θ(−ω)} = . (9.33) 4π i 2r 2r 4πr This derivation holds for real ω, i.e., Im ω = 0. For complex ω, as discussed in Chap. 4 of Ref. [81], one has √ exp (i ω 2 + i r) exp (−b r) √ GF (ω, r⃗ − r⃗′ ) = (9.34) = , b = −i ω 2 + i . 4π r 4π r =

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The fulfillment of the condition Re(b) ≥ 0 is achieved by laying the branch cut of the square root function along the positive real axis, thus ensuring that one has Re(b) > 0, irrespective of the complex phase of ω. Some final remarks are in order. (i) For the backtransform of the Feynman propagator, we can state two equivalent prescriptions, GF (x) = − ∫

d4 k e−ik⋅x d4 k e−ik⋅x = − , ∫ (2π)4 k 2 + i  k2 CF (2π)4

(9.35)

where CF is the Feynman contour, which provides for the poles with Re ω < 0 to be encircled in the mathematically positive direction, while poles with Re ω > 0 are encircled in the mathematically negative direction, i.e., the pole with Re ω > 0 is infinitesimally below the real axis (see Chap. 4 of Ref. [81]). The indication of both an infinitesimal part i and the Feynman contour is a tautology; one should decide for either one. Here, unless otherwise indicated, we always choose the explicit indication of the infinitesimal imaginary part in the denominator. (ii) The distributions GR (x) and GA (x) are centered on the light cone and vanish outside of it, but GF (x) is a distribution which is manifestly nonvanishing outside of the light cone. (iii) For t = 0, our result for the Feynman Green function simplifies to GF (t = 0, r⃗) =

i , (2π)2 r2

(9.36)

in agreement with Eq. (1.180) of Ref. [2]. (iv) We have stressed that all Green functions we discussed are solutions of Eq. (9.10), while their difference is a solution of the homogeneous equation. The distribution G− (x), which is the difference of the retarded and advanced Green functions, 1 1 [Θ(x0 ) − Θ(−x0 )] δ(x2 ) = sgn(x0 ) δ(x2 ) , (9.37) 2π 2π fulfills the homogeneous equation ◻G− (x) = 0. The retarded and advanced Green functions can be reconstructed according to G− (x) = GR (x) − GA (x) ≈

GR (x) = Θ(x0 ) G− (x) , 9.3 9.3.1

GA (x) = −Θ(−x0 ) G− (x) .

(9.38)

Photon Propagator in Coulomb Gauge Legendre Transformation and Hamiltonian

As already discussed in Sec. 2.4.1, one of the main difficulties in the quantization of the electromagnetic field lies in the absence of a canonical momentum for the zeroth component A0 of the four-vector potential, namely, the scalar potential. One can overcome the difficulties along two lines of argument. In Coulomb gauge (see Sec. 2.4.1), one simply refrains from quantizing the scalar potential. This implies that the scalar potential A0 enters the QED Lagrangian via two terms, namely, ⃗ 2 , which is related to the the so-called longitudinal part of the field energy, 21 E ∥ ⃗∥ of the electric field, E ⃗∥ = −∇A ⃗ 0 , and also, by the explicit longitudinal component E

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term −J0 A0 , which is part of the covariant expression −Jµ Aµ . After the Legendre transformation, the QED Hamiltonian contains a non-quantized, nonretarded, term due to the scalar potential, which can be matched against the scattering amplitude generated by the quantized, spatial components of the photon propagator. A complementary approach is based on gauge-fixing terms which can be added to the QED Lagrangian; these generate a nonvanishing canonical momentum Π0 which allows us to quantize the theory. However, one pays a price: The alternative quantization procedure adds unphysical degrees of freedom which later have to be eliminated from the physical states of the photon field. The latter elimination proceeds by imposing additional conditions, e.g., the Gupta–Bleuler condition discussed in Sec. 9.5.3. Here and in the following, we use the notations Φ ≡ A0 and J0 = ρ interchangeably, and set x = (t, r⃗) as well as x′ = (t′ , r⃗′ ). We recall the Coulomb gauge condition (2.43) from Sec. 2.4.1, ⃗ r⃗) = 0 , ⃗ ⋅ A(t, ∇ A⃗ (t, r⃗) = A⃗⊥ (t, r⃗) , A⃗∥ (t, r⃗) = 0 , (9.39) and the potentials are coupled to the sources as follows, [see Eq. (2.44)], ⃗ 2 Φ (t, r⃗) = − ρ (t, r⃗) , ∇ 2

(9.40a)

∂ ⃗ 2 ) A⃗⊥ (t, r⃗) = J⃗⊥ (t, r⃗) , −∇ (9.40b) ∂t2 ∂ ⃗ Φ (t, r⃗) = J⃗∥ (t, r⃗) . ∇ (9.40c) ∂t The action-at-a-distance solution to Eq. (9.40a), 1 Φ (t, r⃗) = ∫ d3 r′ ρ (t, r⃗′ ) + Φhom (t, r⃗) , (9.41) 4π∣⃗ r − r⃗′ ∣ could be interpreted as being at variance with the causality principle; however, according to Chap. 4 of Ref. [81], the constraint (9.40c) actually forces the homogeneous solution Φhom (t, r⃗) (solution of the Laplace equation) to be chosen so that the full Φ (t, r⃗) is connected to the sources only via retarded Green functions, within the classical theory. Furthermore, we shall show that the contribution of the homogeneous solution to the Hamiltonian density is not physically significant. This can be seen as follows. We start from the QED Lagrangian, from which we exclude, for simplicity, the free Dirac density LD = ψ(iγ µ ∂µ − m)ψ (reverting to flat space, γ µ ≡ ̃ γ µ , in comparison to Chap. 8) and interpret Jµ = eψγµ ψ as a classical current distribution. The resulting Lagrangian is 1 L = − F µν Fµν − Jµ Aµ 4 1 ⃗2 ⃗2 ⃗2 = (E∥ + E⊥ − B ) − J0 A0 + J⃗ ⋅ A⃗ , (9.42) 2 where we have split the electric field into longitudinal and transverse parts, according to Chap. 1 of Ref. [81]. In Coulomb gauge, one has ⃗∥ = −∇Φ ⃗⊥ = − ∂ A⃗⊥ . ⃗ , E E (9.43) ∂t (

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⃗⊥ (t, r⃗) The conjugate variable of the vector potential is the transverse electric field E (up to a minus sign), ⃗⊥ (t, r⃗) . ⃗ r⃗) = ∂ A⃗⊥ (t, r⃗) = −E Π(t, ∂t

(9.44)

⃗ vector to be composed of the conjugate variables of the Here, we define the Π A⃗ vector, in the sense of the remarks included after Eq. (2.49). We had seen in Eq. (2.74) that [E⊥i (t, r⃗), Aj⊥ (t, r⃗′ )] = i δ ⊥,ij (⃗ r − r⃗′ ) ,

(9.45)

where we recall the transverse Dirac-δ from Eq. (2.54), δ ⊥,ij (⃗ r − r⃗′ ) = ∫

′ ki kj d3 k ⃗ ij (δ − ) ei k⋅(⃗r−⃗r ) . 3 2 ⃗ (2π) k

(9.46)

For the Legendre transformation of the Lagrangian density, though, we have to use the formula 1 ⃗2 ⃗2 1 ⃗2 ∂Ak − L = (E E + J0 A0 − J⃗ ⋅ A⃗ , ⊥ +B )− ∂t 2 2 ∥ in order to obtain the Hamiltonian. We now investigate the space integral H = Πk

1 ⃗2 H∥ (t) = ∫ d3 r (− E (t, r⃗) + J0 (t, r⃗) A0 (t, r⃗)) 2 ∥ 1 2 ⃗ r⃗)] + ρ(t, r⃗) Φ(t, r⃗)) = ∫ d3 r (− [∇Φ(t, 2 1 ⃗ 2 Φ(t, r⃗) + ρ(t, r⃗) Φ(t, r⃗)) = ∫ d3 r ( Φ(t, r⃗) ∇ 2 1 1 = ∫ d3 r ρ(t, r⃗) Φ(t, r⃗) = ∫ d3 r J0 (t, r⃗) A0 (t, r⃗) . 2 2

(9.47)

(9.48)

⃗ 2 Φ0 = −ρ. Now, let Φ0 be a particular solution of the inhomogeneous equation ∇ Then, the general solution to Eq. (9.40a) can be written as Φ = Φ0 + Φhom , where Φhom is a solution to the homogeneous equation, and so, after integrations by parts, one has 3 0 3 ∫ d r J0 (t, r⃗) A (t, r⃗) = ∫ d r ρ(t, r⃗) (Φ0 (t, r⃗) + Φhom )

⃗ 2 Φ0 (t, r⃗)] (Φ0 (t, r⃗) + Φhom ) = ∫ d3 r [−∇ ⃗ 2 (Φ0 (t, r⃗) + Φhom ) = − ∫ d3 r Φ0 (t, r⃗) ∇ = ∫ d3 r Φ0 (t, r⃗) ρ(t, r⃗) .

(9.49)

In the end, H∥ (t) finds a much simpler form, H∥ (t) =

1 1 3 3 ′ J0 (t, r⃗′ ) . ∫ d r ∫ d r J0 (t, r⃗) 2 4π∣⃗ r − r⃗′ ∣

(9.50)

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This implies that we can replace 1 1 1 ⃗2 ⃗) + J0 (t, r⃗) A0 (t, r⃗) → ∫ d3 r′ J0 (t, r⃗) J0 (t, r⃗′ ) ≡ H∥ (t, r⃗) − E ∥ (t, r 2 2 4π∣⃗ r − r⃗′ ∣ (9.51) in Eq. (9.47), where H∥ (t) = ∫ d3 r H∥ (t, r⃗). Here, H∥ (t, r⃗) is a suitable expression for the electrostatic self-energy density of the charge distribution. Furthermore, we write 1 ⃗2 ⃗2 H = H0 + Hint , H0 = (E +B ) , Hint = H∥ − J⃗ ⋅ A⃗ , (9.52) 2 ⊥ where all entities are functions of t and r⃗. 9.3.2

Matching and Photon Propagator

In Chap. 2, we had discussed the quantization of H0 , which resulted in the quantized-field Hamiltonian given in Eq. (2.71), 3 2 ⃗ (aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ) , ⃗ r⃗) = ∑ ∫ √ d k ˆλ (k) A(t, λ 2ω (2π)3 λ=1

(9.53)

with the fundamental field commutators given in Eq. (2.70), ⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ . [aλ (k), λ

(9.54)

We now calculate the photon propagator in Coulomb gauge by matching the firstorder and second-order contributions to the S matrix, generated by the Hamiltonian (9.52). From H∥ (t), as given in Eq. (9.50), one has H∥ (t) =

1 1 3 3 ′ J0 (t, r⃗′ ) . ∫ d r ∫ d r J0 (t, r⃗) 2 4π∣⃗ r − r⃗′ ∣

(9.55)

The first-order contribution to the S matrix, still of second order in the currents, thus is given by 1 i 3 3 ′ J0 (x′ ) ∫ dt ∫ d r ∫ d r J0 (x) 2 4π∣⃗ r − r⃗′ ∣ i δ(t − t′ ) = − ∫ d4 x ∫ d4 x′ J0 (x) J0 (x′ ) . (9.56) 2 4π∣⃗ r − r⃗′ ∣

S [1] = −i ∫ dtHC (t) = −

The second-order contribution to the S matrix contains the term (−i)2 4 4 ′ i j ′ ′ ∫ d x ∫ d x Ji (x) T⟨0∣A (x)A (x )∣0⟩ Jj (x ) 2! 1 ij (x − x′ )) Jj (x′ ) = − ∫ d4 x ∫ d4 x′ Ji (x) (iDC 2

S [2] =

(9.57)

where we identify the Feynman (time-ordered) propagator (spatial components) in Coulomb gauge as ij iDC (x − x′ ) = T⟨0∣Ai (x)Aj (x′ )∣0⟩ .

(9.58)

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The sum S [1] + S [2] can be matched against the covariant form [(ij) → (µν)], 1 µν S [2] = − ∫ d4 x ∫ d4 x′ Jµ (x) (iDC (x − x′ )) Jν (x′ ) , (9.59) 2 by writing the expression for S [1] as i δ(t − t′ ) S [1] = − ∫ d4 x ∫ d4 x′ J0 (x) J0 (x′ ) 2 4π∣⃗ r − r⃗′ ∣ 1 00 (x − x′ )) J0 (x′ ) , (9.60) = − ∫ d4 x ∫ d4 x′ J0 (x) (iDC 2 where δ(t − t′ ) 00 DC (x − x′ ) = . (9.61) 4π∣⃗ r − r⃗′ ∣ Transformation to momentum space yields 00 00 DC (k) = ∫ d4 (x − x′ ) eik⋅(x−x ) DC (x − x′ ) ′

⃗

1 e−ik⋅(⃗r−⃗r ) = . (9.62) 4π∣⃗ r − r⃗′ ∣ k⃗2 It remains to perform the calculation of the time-ordered product of field operators in Eq. (9.58). To this end, we recall Eq. (2.71), = ∫ d3 (r − r′ )

′

3 2 ⃗ (aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ) , ⃗ r⃗) = ∑ ∫ √ d k A(t, ˆλ (k) λ 2ω (2π)3 λ=1

⃗ with the fundamental field commutator (2.70), where ω = ∣k∣, ⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ , [aλ (k), λ

(9.63)

(9.64)

as well as Eq. (9.46) for the transverse Dirac-δ function, d3 k ki kj ∇ i ∇j ⃗ r −⃗ ij i k⋅(⃗ r′ ) ij (δ − ) δ (3) (⃗ r − r⃗′ ) . (9.65) ) e = (δ − ⃗2 (2π)3 ∇ k⃗2 For the sum over polarizations, we have according to Eq. (2.63), δ ⊥,ij (⃗ r − r⃗′ ) = ∫

i j 2 i ⃗ ⃗ = δ ij − k k . (9.66) jλ′ (k) ∑ λ (k) k⃗2 λ=1 The vacuum expectation value of Coulomb-gauge field operators for the vector potential reads as follows,

⃗ ⋅ ˆλ′ (k) ⃗ = δλλ′ , ˆλ (k)

⃗ = 0, k⃗ ⋅ ˆλ (k)

iDij (x − x′ ) = ⟨0∣Ai (t, r⃗)Aj (t′ , r⃗′ )∣0⟩ 2

= ∑ ∫ d3 k ∫ λ,λ′ =1

(9.67)

−i k⋅(x−x′ ) ⃗ j ′ (k⃗′ )⟨0∣aλ (k)a ⃗ +′ (k)∣0⟩e ⃗ i (k) λ √ √λ d3 k ′ λ 2ω(2π)3 2ω ′ (2π)3

= ∑∫

d3 k ⃗ j (k) ⃗ e−i k⋅(x−x′ ) i (k) λ 2ω (2π)3 λ

= (δ ij −

∇ i ∇j ei k⋅(x −x) 1 ∇i ∇j 1 ) ∫ d3 k = − 2 (δ ij − ) , 3 2 ⃗ ⃗2 2ω (2π) 4π (x − x′ − iη)2 ∇ ∇

2

λ=1

′

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where we have used Eqs. (9.25) and (9.54). For the time-ordered product of field operators, one obtains ij iDC (x − x′ ) = ⟨0∣T Ai (t, r⃗)Aj (t′ , r⃗′ )∣0⟩

= Θ(t − t′ ) ⟨0∣T Ai (t, r⃗)Aj (t′ , r⃗′ )∣0⟩ + Θ(t′ − t) ⟨0∣T Aj (t′ , r⃗′ )Ai (t, r⃗)∣0⟩ = Θ(t − t′ ) Dij (x − x′ ) + Θ(t′ − t) Dij (x′ − x) ,

(9.68)

and thus ij iDC (x − x′ ) = −

= −

Θ(t − t′ ) ∇i ∇j Θ(t′ − t) 1 )[ (δ ij − + ′ ] 2 ′ 2 2 ⃗ 4π (x − x − iη) (x − x − iη)2 ∇ 1 ∇i ∇j 1 ij ) [Θ(t − t′ ) {(P.V.) (δ − 2 2 ⃗ 4π (x − x′ )2 ∇

+ i π sgn(t − t′ ) δ((x − x′ )2 )} + (x ↔ x′ )] = −

1 1 ∇i ∇j ij ) (δ − ′ )2 − i  2 ⃗ 4π 2 (x − x ∇

= − i (δ ij −

∇i ∇j ) GF (x − x′ ) , ⃗2 ∇

(9.69)

where we have used the result (9.13), 1 i 1 i [(P.V.) 2 + iπδ(x2 )] = − 2 2 . (9.70) 4π 2 x 4π x − i The time-ordered product of vector potential field operators in coordinate space (Coulomb gauge, spatial components) thus reads as GF (x) = −

ij DC (x − x′ ) = − (δ ij −

∇i ∇j ) GF (x − x′ ) . ⃗2 ∇

(9.71)

Using the momentum representation GF (k) = −1/(k 2 + i), one finds 1 ki kj (δ ij − ). (9.72) + i k⃗2 So, in the Coulomb gauge, we can express the time-order expectation value of the components of the four-vector potential as follows, ij DC (k) =

k2

µν iDC (x − x′ ) = ⟨0∣T Aµ (t, r⃗)Aµ (t′ , r⃗′ )∣0⟩ , ′ d4 k Dµν (k) e−ik⋅(x−x ) , (2π)4 C 1 ⎧ ⎪ µ=ν=0 ⎪ ⎪ ⃗2 ⎪ ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ µν ki kj DC (k) = ⎨ 1 . (δ ij − ) µ = i, ν = j ⎪ ⎪ 2 2 ⃗ ⎪ k + i k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ≠ν ⎩0

µν (x − x′ ) = ∫ DC

(9.73)

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We reemphasize that the result (9.73) has been obtained without formally quantizing the Coulomb interaction, thereby remedying the lack of a suitable canonical momentum Π0 , to supplement the scalar potential A0 , which persists in Coulomb gauge. The matching procedure described here leads to a successful quantization of the theory in Coulomb gauge and constitutes an alternative to the covariant procedure which will be outlined in the following. 9.4

Photon Propagator in Lorenz Gauge

9.4.1

Gauge Invariance and Mass of Photon

As a prelude to the calculation of the photon propagator in Lorenz gauge, let us briefly recall a few aspects of classical electrodynamics, with an emphasis on the properties of the Lorenz (covariant) gauge. We had briefly discussed gauge transformations in Sec. 4.6.2 and Sec. 8.2.1, in the context of the covariant derivative. A gauge transformation Λ changes the scalar and vector potentials as follows [see Eq. (8.1)], ⃗ , A⃗ → A⃗′ = A⃗ + ∇Λ

A0 → (A0 )′ = A0 −

∂ Λ, ∂t

A′µ = Aµ − ∂ µ Λ .

(9.74)

Here, Λ is an arbitrary gauge function of time and space, and µ = 0, 1, 2, 3 is a space-time index. The Lorenz gauge condition is ⃗ ⋅ A⃗ + ∇

∂ 0 A = ∂µ Aµ = 0 . ∂t

(9.75)

Let us briefly mention why the gauge invariance of QED protects the masslessness of the photon. Without sources, the Maxwell Lagrangian reads as 1 L = − F µν Fµν , 4

Fµν = ∂µ Aν − ∂ν Aµ .

(9.76)

The variational field equations ◻Aµ − ∂µ (∂ ⋅ A) = ∂ν F νµ = 0 are invariant under a gauge transformation, Aµ → Aµ − ∂ µ Λ. The gauge invariance of the field equations can be traced to the trivial identity ◻∂µ Λ − ∂µ (∂ ν ∂ν Λ) = 0. However, if we assume the field to be massive, 1 1 L = − F µν Fµν + µ2 A2 , 4 2

(9.77)

then the field equations become ◻ Aµ − ∂ µ (∂ ⋅ A) = ∂ν F νµ = −µ2 Aµ .

(9.78)

While the field-strength tensor Fνµ is gauge invariant, the expression −µ2 Aµ is not; if the photon were massive then gauge invariance would be lost.

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Gauge-Fixing Term and Quantization

Let us briefly outline the quantization procedure for the electromagnetic field in Lorenz gauge. The main problem is the lack of a canonical momentum conjugate to the scalar potential, as outlined in Eq. (2.48). In the first step, we add a term −(λ/2) (∂ ⋅ A)2 to the Lagrangian, with a Lagrange multiplier λ. This leads to a nonvanishing canonical momentum Π0 , complementing the time-like component A0 of the four-vector potential, and makes it possible to formulate the canonical commutation relations. These relations, for the fundamental creation and annihilation operators (in Lorenz gauge), will be different from those for the Coulomb gauge. The reason lies in our somewhat artificial introduction of a nonvanishing canonical momentum, conjugate to A0 . It implies the introduction of a quantized representation (which does not vanish) for the quantity ∂ ⋅ A which, according to the classical formulation, should vanish (but does not do so in the fully quantized formalism). In the course of the calculation, we have introduced two “unphysical” degrees of freedom of the photon field: the scalar and longitudinal polarizations, augmenting the two polarizations orthogonal to the propagation vector, to four. One might ask: Why do we have to deal with two additional degrees of freedom, not one? Finally, we have only introduced the canonical variables A0 and Π0 into the formalism, which should give us one, not two, additional polarizations. The reason is a subtle difference of the four-vector potential operators in the Lorenz gauge. In Coulomb gauge, our operator A⃗ given in Eq. (2.71) fulfills the gauge condition ⃗ ⋅ A⃗ = 0, even in the fully quantized setting (because of the properties of the ∇ polarization vectors contained therein). This gauge condition reduces the number of available photon polarizations from three to two, within Coulomb gauge. The quantized Lorenz gauge operators which we shall use do not fulfill the Lorenz gauge condition ∂⋅A = 0. As already mentioned, we have introduced two additional degrees of freedom, the first for the introduction of the A0 (and Π0 ) into the quantization procedure, the second for the lack of a gauge condition which would be fulfilled by the quantized operators. In fact, in the quantized formalism, the Lorenz gauge condition is implemented in a subtle way, as discussed in the following. The addition of the two “unphysical” degrees of freedom allows us to carry out the quantization of the electromagnetic field in a fully covariant form. However, we now have to deal with the two unphysical photon polarizations. In fact, the two problems are intertwined. Namely, it will turn out that the canonical momentum Π0 is proportional to ∂ ⋅ A, so that the equation ∂ ⋅ A = 0 cannot be fulfilled as an operator equation in the quantized theory anyway. We simply create “too many” dynamical variables, some of which belong to “unphysical” degrees of freedom (these are “unphysical” scalar and longitudinal photons, specifically). Some of the states turn out to have a negative norm, leading to a Fock space with an indefinite metric. A natural idea is to demand that if the condition ∂ ⋅ A = 0 cannot be fulfilled as an operator identity, then at least, it should be fulfilled in terms of an expectation

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value. Or, we could postulate that ∂ ⋅ A should vanish when acting on the physical states of the electromagnetic field, i.e., ∂ ⋅ A(+) ∣Ψ⟩ = 0 for any physical state ∣Ψ⟩ which contains real photons. Here, A(+) is the “annihilation part” of the fourvector potential operator, to be discussed in Eq. (9.113). This latter condition is called the Gupta–Bleuler condition and is discussed in Sec. 9.5.3. Let us now cast this program into mathematical form. We start from the following Lagrangian for the source-free photon field with a gauge-fixing term, λ 1 (9.79) L = − F µν Fµν − (∂ ⋅ A)2 . 4 2 This Lagrangian is not gauge-invariant any more; upon a transformation Aµ → Aµ − ∂ µ Λ, it changes as λ 2 L → L − (−2 (◻Λ) (∂ ⋅ A) + (◻Λ) ) . (9.80) 2 The Lagrangian (9.79) thus is gauge-invariant only within the family of Lorenz gauges; any gauge transformation Λ which fulfills ◻Λ = 0 is still permissible in the sense that a “gauge always shoots twice” (see Sec. 1.2.4 of Ref. [81]). A variation with respect to λ of (9.79) yields back the gauge condition ∂ ⋅ A = 0. However, we shall rather interpret λ as a coupling constant whose value can be fixed in order to cast the field equations into a desired form. In fact, we shall arrange for λ to be chosen so that the field equations have the same form as in Lorenz gauge, or, more precisely, the same as we would obtain if we were to impose the Lorenz gauge condition on the four-vector potential, after the variation of the action. Still, the formalism, in the first place, knows nothing about the Lorenz gauge condition, and treats Aµ as a normal four-vector quantum field with all available degrees of freedom and polarizations at one’s disposal. Fixing λ to the appropriate value required to reproduce the Lorenz gauge field equations is referred to as the “Feynman gauge” while strictly speaking, it is not a gauge condition imposed on the fourvector potential, but a condition on an artificially introduced coupling parameter in the Lagrangian (9.79), fixed to cast the field equations into a desired form. The variational equations read as follows, ∂L ∂L )− = 0. ∂(∂ν Aµ ) ∂Aµ The only nonvanishing partial derivatives are ∂L = − (∂ ν Aµ − ∂ µ Aν ) − λ g νµ (∂ ⋅ A) , ∂(∂ν Aµ ) where we use the representation ∂ν (

∂ ⋅ A = ∂ρ Aρ = g νµ ∂ν Aµ .

(9.81)

(9.82)

(9.83)

The variational equations, in the source-free case, therefore read as follows, ∂ν (

∂L ) = − (∂ ν ∂ν Aµ − ∂ µ ∂ν Aν ) − λ ∂ µ (∂ ⋅ A) ∂(∂ν Aµ ) = − [◻Aµ − (1 − λ) ∂ µ (∂ ⋅ A)] = 0 .

(9.84)

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The resulting equation, ◻ Aµ − (1 − λ) ∂ µ (∂ ⋅ A) = 0 = ∂ν F νµ + λ ∂ µ (∂ ⋅ A) ,

(9.85)

turns into the Lorenz gauge source-free Maxwell equations for λ = 1. Having calculated (9.82), we now evaluate the canonically conjugate momenta Πρ =

∂L = − (∂ 0 Aρ − ∂ ρ A0 ) − λ g 0ρ (∂ ⋅ A) = F ρ0 − λ g ρ0 (∂ ⋅ A) . ∂(∂0 Aρ )

(9.86)

The canonical momentum Π0 conjugate to A0 reads as Π0 = −λ (∂ ⋅ A) .

(9.87)

From now on, we shall assume that λ = 1,

Π0 = −∂ ⋅ A ,

Πi = F i0 = E i ,

(9.88)

⃗ is the electric-field operator. We now postulate the following canonical where E quantization relations at equal times, [Πµ (t, r⃗), Aν (t, r⃗′ )] = −i g µ ν δ (3) (⃗ r − r⃗′ ) ,

(9.89)

which is consistent with the indefinite metric of the Hilbert space implied by the commutation relation Eq. (9.92) discussed in the following. We note that these commutation relations are valid at equal times. When specialized to the spatial components, Eq. (9.89) reads as follows, [Πi (t, r⃗), Aj (t, r⃗′ )] = −i g i j δ (3) (⃗ r − r⃗′ ) ,

(9.90)

where g i j = δ ij is equal to the Kronecker delta. We notice that Πi = −Ei , not Πi , is the canonically conjugate variable corresponding to Ai . (The correspondence to the quantum mechanical relation [pi , xj ] = −i δ ij is maintained.) It is instructive to compare Eq. (9.89) with Eq. (2.59) in Sec. 2.4.1. In the latter case, the transverse delta function is obtained, in momentum space, after a summation over only two polarizations, not four. The covariant photon vector potential operator reads as 3

Aµ (x) = ∑ ∫ λ=0

d3 k 1 µ ⃗ −i k⋅x i k⋅x },  (k) {akλ + a+kλ ⃗ e ⃗ e ⃗ λ (2π)3 2 ∣k∣

(9.91)

where we work in a fully relativistic setting and aim to use the Lorentz-invariant integration measure (9.24). This necessitates a change in the normalizations of the fundamental field operators, as already indicated in Sec. 2.5.2. We therefore distinguish the “relativistically” transformed operators by their subscripts akλ ⃗ from ⃗ In accordance with Eq. (2.117), we postulate the “nonrelativistic” operators aλ (k). the following commutation relations, + 3 (3) ⃗ ⃗ ′ ⃗ [akλ (k − k ) . ⃗ , ak ⃗ ′ λ′ ] = −gλλ′ (2 ∣k∣) (2π) δ

(9.92)

Here, gλ λ′ is determined by the photon polarizations, which we augment by scalar and longitudinal photons. The symbol gλ λ′ would otherwise indicate a space-time metric, which we use in the West-Coast conventions gµν = diag(1, −1, −1, −1), but

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its meaning in Eq. (9.92) really comes from the indefinite metric associated with the Fock space of photons, where the two “additional” photon polarizations are suppressed because their state acquire a negative norm, as per the commutation relation (9.92). Details are discussed in the following. The sum over polarizations is carried out according to 3

3

µ ⃗ ν ⃗ λ′ (k) gλλ′ = g µν . ∑ ∑ λ (k)

(9.93)

λ=0 λ′ =0

This formula is in need of an explanation and illustration. We first augment the two transverse photons by scalar and longitudinal photons, to arrive at four virtual photon polarizations. For the polarization with λ = 0 (scalar photons), the polarization vector is parallel to the time-like unit vector ⃗ =n ⃗ ⋅ 0 (k) ⃗ = 1. n ˆ = (1, 0, 0, 0) , 0 (k) ˆ, 0 (k) (9.94) The physically allowed polarizations (for real photons) are λ = 1, 2. We have a ⃗ = 0 and k⃗ ⋅ ⃗1,2 (k) ⃗ = 0. Of course, just like for vanishing time-like component 01,2 (k) any two orthogonal space-like vectors normalized to unity, we have ⃗ ⋅ j (k) ⃗ = −δ ij , i (k) i, j = 1, 2 . (9.95) The third polarization vector (longitudinal photons) is in the propagation direction of the photon, but has a vanishing time-like component. In terms of the wave vector k µ , it can be written as follows, µ ⃗ ⋅ 3 (k) ⃗ = −1 . (9.96) ⃗ , ⃗ = √ν , 3 (k) ν µ = k µ − (k ⋅ n) nµ = (0, k) µ3 (k) −ν ⋅ ν We convince ourselves that ν ⋅ ν = k 2 − 2(k ⋅ n)2 + (k ⋅ n)2 = ω 2 − k⃗2 − ω 2 = −k⃗2 . (9.97) Furthermore, ⃗ = ∣k∣ ⃗ = −k ⋅ 3 (k) ⃗ , k ⋅ 0 (k)

⃗ = 0. k ⋅ 1,2 (k) ⃗ ˆez , we have For a photon propagating in the z direction, with k⃗ = ∣k∣ ⎛1⎞ ⎜ ⎟ ⃗ = ⎜0⎟ , 0 (k) ⎜0⎟ ⎜ ⎟ ⎝0⎠

⎛0⎞ ⎜ ⎟ ⃗ = ⎜1⎟ , 1 (k) ⎜0⎟ ⎜ ⎟ ⎝0⎠

⎛0⎞ ⎜ ⎟ ⃗ = ⎜0⎟ , 2 (k) ⎜1⎟ ⎜ ⎟ ⎝0⎠

⎛0⎞ ⎜ ⎟ ⃗ = ⎜0⎟ . 3 (k) ⎜0⎟ ⎜ ⎟ ⎝1⎠

(9.98)

(9.99)

⃗ ˆez , we can now verify Eq. (9.93) as follows, even by inspection, For the case k⃗ = ∣k∣ ⃗ ˆez ) = gλλ′ , ⃗ ˆez ) ⋅ λ′ (∣k∣ λ (∣k∣

3

3

µ ⃗ ⃗ ˆez )gλλ′ = g µν . (9.100) ˆez ) νλ′ (∣k∣ ∑ ∑ λ (∣k∣

λ=0 λ′ =0

Let us verify the compatibility of the field operator, in the convention used in Eq. (9.91), with the explicit form of the commutators given in Eq. (9.92), and the canonical commutation relation (9.89). One has according to Eq. (9.91), 3

Aµ (x) = ∑ ∫ λ=0

d3 k 1 µ ⃗ −i k⋅x i k⋅x },  (k) {akλ + a+kλ ⃗ e ⃗ e ⃗ λ (2π)3 2 ∣k∣

(9.101)

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so that 3

Π0 (x) = − ∂ ⋅ A = ∑ ∫ λ=0

d3 k 1 ⃗ {a⃗ e−i k⋅x − a+⃗ ei k⋅x } (ik ⋅ λ (k)) kλ kλ ⃗ (2π)3 2 ∣k∣

d3 k 1 ⃗ −i k⋅x + i k⋅x }, ∣k∣ {(ak0 − (a+k0 = i∫ ⃗ − ak3 ⃗ )e ⃗ − ak3 ⃗ )e ⃗ (2π)3 2 ∣k∣

(9.102)

where Eq. (9.98) has been used. Also, the expression for A0 may be simplified, d3 k 1 −i k⋅x i k⋅x {ak0 } + a+k0 ⃗ e ⃗ e ⃗ (2π)3 2 ∣k∣

A0 (x) = ∫

(9.103)

⃗ = 1, and 0 (k) ⃗ = 0, for λ = 1, 2, 3. From now on, within the in view of 00 (k) λ discussion of the field commutators, we shall assume that t = t′ , ′

(9.104)

′

so that x = (t, r⃗) and x = (t, r⃗ ). The commutator is d3 k ′ 1 ⃗ d3 k 1 −i k⋅x ∣k∣ [{(ak0 ⃗ − ak3 ⃗ )e ∫ ⃗ (2π)3 2 ∣k∣ (2π)3 2 ∣k⃗′ ∣

[Π0 (x), A0 (x′ )] = i ∫

+ i k⋅x } , {ak⃗′ 0 e−i k ⋅x + a+k⃗′ 0 ei k ⋅x }] . − (a+k0 ⃗ − ak3 ⃗ )e ′

′

′

′

(9.105)

According to Eq. (9.92), + 3 (3) ⃗ ⃗ ′ ⃗ [ak0 (k − k ) , ⃗ , ak ⃗ ′ 0 ] = −(2 ∣k∣) (2π) δ

(9.106)

+ while [ak0 ⃗ , a ⃗ ′ ] and other commutators vanish. Then, k3

[Π0 (x), A0 (x′ )] = i ∫

d3 k 1 d3 k ′ 1 ⃗ ∣k∣ ∫ ⃗ (2π)3 2 ∣k∣ (2π)3 2 ∣k⃗′ ∣

−i k⋅x+ik ⋅x i k⋅x−ik ⋅x + − [a+k0 ) × ([ak0 ⃗ , ak ⃗ , ak⃗′ 0 ] e ⃗′ 0 ] e ′

= −

′

′

′

′ ′ i d3 k [e−i k⋅x+ik⋅x + ei k⋅x−ik⋅x ] = −ig 0 0 δ (3) (⃗ r − r⃗′ ) . ∫ 2 (2π)3 (9.107)

It remains to calculate [Πi (x), Aj (x′ )]. We have Πi (x) = ∂ i A0 − ∂ 0 Ai = −∂i A0 − ∂0 Ai d3 k 1 −i k⋅x i k⋅x } (+ik i ) {ak0 − a+k0 ⃗ e ⃗ e ⃗ (2π)3 2 ∣k∣

= −∫ 3

−∑∫ λ=1

d3 k 1 ⃗ i (k) ⃗ {a⃗ e−i k⋅x − a+⃗ ei k⋅x } , (−i∣k∣) λ kλ kλ ⃗ (2π)3 2 ∣k∣

(9.108)

while 3

Aj (x) = ∑ ∫ λ=1

d3 k 1 j ⃗ −i k⋅x i k⋅x },  (k) {akλ + a+kλ ⃗ e ⃗ e ⃗ λ (2π)3 2 ∣k∣

(9.109)

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where we carefully take note of the summation limits, which extend from λ = 1 to λ = 3. So, the commutator among the spatial components evaluates to [Πi (x), Aj (x′ )] = − [Πi (x), Aj (x′ )] = − (T ij + U ij ) , T ij = [−∂i A0 , Aj ] = 0 , 3

U ij = [ ∑ ∫ λ=1 3

∑∫

λ′ =1 3

= ∑ ∫ λ,λ′ =1

d3 k 1 ⃗ i ⃗ −i k⋅x i k⋅x }, i∣k∣ λ (k) {akλ − a+kλ ⃗ e ⃗ e ⃗ (2π)3 2 ∣k∣ ′ ′ ′ ′ d3 k ′ 1 j ⃗′ λ′ (k ) {ak⃗′ λ′ e−i k ⋅x + a+k⃗′ λ′ ei k ⋅x }] 3 ′ ⃗ (2π) 2 ∣k ∣

d3 k ′ 1 d3 k 1 3 ⃗ ⃗ i (k) ⃗ j ′ (k⃗′ ) 2∣k∣(2π) i∣k∣ ∫ λ λ ⃗ (2π)3 2 ∣k∣ (2π)3 2 ∣k⃗′ ∣

′ ′ ′ ′ × (δλλ′ δ (3) (k⃗ − k⃗′ ) e−i k⋅x+ik ⋅x + δλλ′ δ (3) (k⃗ − k⃗′ ) ei k⋅x−ik ⋅x )

= iδ ij δ (3) (⃗ r − r⃗′ ) .

(9.110)

We have used the polarization sum formula [appealing to Eq. (9.99)] 3

i ⃗ j ⃗ λ (k) = δ ij , ∑ λ (k)

(9.111)

λ=1

which is, notably, different from Eq. (2.63) because of the addition of the third polarization. The calculations described in Eqs. (9.101)–(9.111) are different from the Coulomb gauge [see Eq. (2.4.2)] because of the different normalizations of the fundamental commutators [Eq. (9.89) versus Eq. (2.70)] and because of the addition of the scalar and longitudinal photons. 9.4.3

Representations of the Photon Propagator

It is instrumental to calculate the vacuum expectation value of the product of two field operators, iDµν (x − x′ ) = ⟨0 ∣Aµ (x) Aν (x′ )∣ 0⟩ 3

= ∑ ∫ λ,λ′ =0

d3 k 1 d3 k ′ 1 µ ⃗ ν ⃗′  (k) λ′ (k ) ⃗ ∫ (2π)3 2∣k⃗′ ∣ λ (2π)3 2∣k∣

−i k⋅x + × ⟨0 ∣akλ ak⃗′ λ′ ei k ⋅x ∣ 0⟩ ⃗ e ′

= − g µν ∫

′

d3 k 1 −i k⋅(x−x′ ) g µν 1 e = 2 , 3 ′ ⃗ (2π) 2∣k∣ 4π (x − x − i η)2

(9.112)

where we have used Eqs. (9.25), (9.92), and (9.93), and recall that η is an infinitesimal time-like vector, with η = (, 0, 0, 0). The introduction of the imaginary unit in the definition of the photon propagator follows the conventions used in Eqs. (7.153) and (7.154) for the fermionic case. We now decompose the field operator Aµ (x)

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into a negative-frequency, creation part A(−)µ (x) and positive-frequency, annihilation part A(+)µ (x), 3 d3 k 1 µ ⃗ −i k⋅x λ (k) akλ . Aµ (x) = A(+) µ (x) + A(−) µ (x) , A(+) µ (x) = ∑ ∫ ⃗ e 3 ⃗ (2π) 2 ∣ k∣ λ=0 (9.113) Of course, one has A(−) µ (x) = Aµ (x) − A(+) µ (x) = [A(+) µ (x)]+ , illustrating the fact that Aµ (x) is a Hermitian operator. Because the sign of a frequency (of an energy) does not change under a Lorentz transformation, the separation into A(+) and A(−) is Lorentz-invariant. This implies that we can write Dµν (x − x′ ) as follows, iDµν (x − x′ ) = ⟨0 ∣A(+)µ (x) A(−)ν (x′ )∣ 0⟩ = ⟨0 ∣[A(+)µ (x), A(−)ν (x′ )]∣ 0⟩ (9.114) d4 k −i k⋅(x−x′ ) e Θ(k 0 ) δ(k 2 ) , (2π)4 where we have used the results (9.112) and (9.24). The commutator itself is a complex number, free of operators, and therefore equal to its own vacuum expectation value. Thus, the vacuum expectation value of the field commutator, iD−µν (x − x′ ) = ⟨0 ∣ [Aµ (x), Aν (x′ )] ∣ 0⟩, actually is centered on the light cone, iD−µν (x − x′ ) = ⟨0 ∣ [Aµ (x), Aν (x′ )] ∣ 0⟩ (9.115) = − [A(−)µ (x′ ), A(+)ν (x)] = −g µν ∫

= iDµν (x − x′ ) − iDνµ (x′ − x) = i g µν

sgn(t − t′ ) δ((x − x′ )2 ) . 2π

Also, in the Lorenz gauge, we have D−µν (x − x′ ) = g µν G− (x − x′ ) , (9.116) where G− (x − x′ ) is the Green function defined in Eq. (9.37). The retarded and advanced propagators are centered on the light cone; they can be obtained from the field commutator as follows, µν DR (x − x′ ) = Θ(t − t′ ) D−µν (x − x′ )

Θ(t − t′ ) δ((x − x′ )2 ) = g µν GR (x − x′ ) . 2π For the advanced propagator, we have µν DA (x − x′ ) = − Θ(t′ − t) D−µν (x − x′ ) = −g µν Θ(t′ − t) G− (x − x′ ) = g µν

(9.117)

Θ(t′ − t) δ((x − x′ )2 ) = g µν GA (x − x′ ) . (9.118) 2π Here, we have used the result (9.38) in the form Θ(−(t − t′ )) GA (x − x′ ) = −Θ(−(t − t′ )) G− (x − x′ ) = δ((x − x′ )2 ) . (9.119) 2π Finally, the time-ordered vacuum expectation value of field operators leads to the Feynman propagator of the photon field, iDFµν (x − x′ ) = ⟨0 ∣T Aµ (x) Aν (x′ )∣ 0⟩ = − g µν

= Θ(t − t′ ) ⟨0 ∣Aµ (x) Aν (x′ )∣ 0⟩ + Θ(t′ − t) ⟨0 ∣Aν (x′ ) Aµ (x)∣ 0⟩ = Θ(t − t′ ) Dµν (x − x′ ) + Θ(t′ − t) Dνµ (x′ − x) =

g µν 1 = i g µν GF (x − x′ ) , 4π 2 (x − x′ )2 − i 

(9.120)

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where we encounter the same combination of infinitesimal parts as in Eq. (9.31). In the last step, the relation (9.13) has been used. Using Eq. (9.9), we can find the form g µν (9.121) DFµν (k) = − 2 k + i for the Feynman propagator of the photon field in momentum space. 9.5 9.5.1

Photon Propagator in General Gauges Construction Principle

Starting from Eq. (9.120), we can define the photon propagator Dµν (x − x′ ) as the vacuum expectation value of the time-ordered product of four-vector field operators, ⟨0 ∣T Aµ (x) Aν (x′ )∣ 0⟩ = iDµν (x − x′ ) ,

(9.122)

′

where D (x − x ), in the Lorenz gauge, is given in Eq. (9.120). In Coulomb gauge, the space-time expressions are given in Eqs. (9.61) and (9.71), while in momentum space, the formulas can be found in Eqs. (9.62), (9.72), and (9.73). In order to derive the photon propagator in an arbitrary gauge, one proceeds in the following steps. First, one takes a step back and matches the second-order S matrix element, calculated using classical current distributions but with quantized field operators, against the first-order S matrix element, calculated using the same classical current distribution and the classical (retarded) four-vector potential generated by the same current distribution. This leads to a relation of the time-ordered product of field operators and the retarded Green function. One concludes that, in order to derive the photon propagator in an arbitrary gauge, it is sufficient to establish the equations which couple the classical four-vector potential Aµ to the four-current density J µ , then, invert these relations, and place the poles of the photon propagator according to the prescriptions pertinent to the Feynman, retarded, or advanced propagators. Let us start with the first step. In second-order perturbation theory, with a classical current distribution J µ (x), the S-matrix element for forward scattering of the vacuum state of the electromagnetic field can be expressed in terms of the retarded Green function, (−i)2 4 4 ′ µ ν ′ ′ S [2] = ∫ d x ∫ d x Jµ (x) T⟨0∣A (x)A (x )∣0⟩ Jν (x ) 2! 1 = − ∫ d4 x ∫ d4 x′ Jµ (x) (Θ(t − t′ ) ⟨0∣Aµ (x)Aν (x′ )∣0⟩ 2 + Θ(t′ − t) ⟨0∣Aν (x′ )Aµ (x)∣0⟩) Jν (x′ ) µν

= − ∫ d4 x ∫ d4 x′ Jµ (x)Θ(t − t′ )⟨0∣Aµ (x)Aν (x′ )∣0⟩Jν (x′ ) µν = − i ∫ d4 x ∫ d4 x′ Jµ (x) DR (x − x′ ) Jν (x′ ) ≡ −i ∫ d4 xJµ (x) AµR (x) .

(9.123)

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So, the matching of the retarded Green function with the second-order S-matrix is successful if the retarded photon propagator is given by µν iDR (x − x′ ) = Θ(t − t′ )⟨0∣Aµ (x)Aν (x′ )∣0⟩ ,

(9.124)

and in the classical sense µν AµR (x) = ∫ d4 x′ DR (x − x′ ) Jν (x′ ) .

(9.125)

Because the latter equation can be given in momentum space as µν AµR (k) = DR (k) Jν (k) ,

(9.126)

we may derive the photon propagator in various gauges by considering the coupling of the vector potential to its source, ◻ Aµ (x) − ∂ µ (∂ ⋅ A(x)) = J µ (x) .

(9.127)

After imposing gauge conditions, these equations generally take a simpler form, e.g., in the Lorenz gauge where one would neglect the term ∂ ⋅ A(x). This would entail the second step outlined above. One then transforms this equation to energymomentum space, and inverting the matrix to solve for Aµ . In Lorenz gauge, for example, one would impose ∂ ⋅ A(x) = 0, resulting in µν Aµ (k) = DR (k) Jν (k) ,

−k 2 Aµ (k) = J µ (k) , µν DR (k) = −

g

(k 0

µν

+ i) − k⃗2

→−

g k2

(9.128a)

µν

+ i

,

(9.128b)

where the second equation illustrates the transition from the retarded to the Feynman Green function via a simple reassignment of the positions of the poles. One thus verifies Eq. (9.121). Indeed, one should remember that the solution of the equation (9.126) does not yet specify the prescription to place the poles; these must be placed differently for the Feynman and retarded propagators, as discussed. 9.5.2

Most General Form and Weyl Gauge

An inspection of Eqs. (9.127) and (9.128) shows that one can add arbitrary terms of the form 1 µν µν (9.129) DR → DR + 2 (f µ k ν + f ν k µ ) 2k to the retarded or Feynman photon propagators, without changing the physical fields. The continuity equation for the current paves the way in the derivation. E.g., the transition from the Feynman to the Coulomb gauge is accomplished using f0 =

k0 , k⃗2

fi = −

ki . k⃗2

(9.130)

With a gauge transformation Λ = ∫ dt A0 ,

(9.131)

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the transformed scalar potential can be made to vanish, (A0 )′ = A0 −

∂ Λ = A0 − A0 = 0 , ∂t

⃗ = A⃗ + ∫ dt ∇A ⃗ 0, A⃗′ = A⃗ + ∇Λ

(9.132)

at the expense of adding an additional term to the vector potential. This is called the temporal or Weyl1 gauge. In this gauge, one has DF00 (k) = 0 ,

DFij (k) =

ki kj 1 (δij − 2 ) . k 2 + i ω

(9.133)

Other choices for the functions f µ in Eq. (9.129) lead to the so-called Landau and Fried–Yennie gauges (see also Sec. 9.7). 9.5.3

Gupta–Bleuler Condition

We now have to carry out the program sketched in Sec. 9.4.2 and reestablish the consistency of the quantization for the electromagnetic field in the covariant formalism, by implementing the condition ∂ ⋅ A → 0 in a “weak” sense, as a condition on any state which contains real photons. This reestablishes the Lorenz gauge condition equation fulfilled by the physical state vectors. The “physical” Hilbert space then is assured to have a positive definite metric. Let us consider a state ∣s⟩ which constitutes a superposition of scalar photon states, ⃗ a+ (k) ⃗ ∣0⟩ . ∣s⟩ = ∫ d3 k f (k) 0

(9.134)

Its norm is ⃗ ⟨0∣[a0 (k⃗′ ), a+ (k)]∣0⟩ ⃗ ⟨s∣s⟩ = ∫ d3 k ′ ∫ d3 k f ∗ (k⃗′ ) f (k) 0 ⃗ f (k) ⃗ = − ∫ d3 k ∣f (k)∣ ⃗ 2 < 0, = − g00 ∫ d3 k f ∗ (k)

(9.135)

exemplifying the Fock space with negative norm (or, “indefinite norm” because the other polarizations actually have positive norm). Let us first investigate the consequences of postulating that the physical states of the electromagnetic field should fulfill a variation of the condition ∂µ Aµ = 0 in the mean, e.g., in the sense of quantum mechanical expectation values. This would otherwise imply that ⟨Ψ′ ∣∂µ Aµ ∣Ψ⟩ = 0

(9.136)

for any physical states ∣Ψ⟩ and ∣Ψ′ ⟩ of the photon field. Decomposing this condition into creation and annihilation parts, we see that the above equation is equivalent to ⟨Ψ′ ∣∂µ A(+) µ ∣Ψ⟩ + ⟨Ψ′ ∣∂µ A(−) µ ∣Ψ⟩ = 0 , 1 Hermann

Klaus Hugo Weyl (1885–1955).

(9.137)

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where the “positive-frequency” (annihilation) part [see Eq. (9.113)] has the divergence 3

⃗) = −i ∑ ∫ ∂µ A(+) µ (t, x λ=0

d3 k 1 ⃗ ⃗ − a3 (k)] ⃗ e−i k⋅x . ∣k∣ [a0 (k) ⃗ (2π)3 2 ∣k∣

(9.138)

In view of Eqs. (9.94), (9.97) and (9.98), only scalar and longitudinal photons (λ = 0, 3) contribute to the result. We now formulate the Gupta–Bleuler condition in a way which is a little stricter than Eq. (9.136), namely, as the operator equation ∂µ A(+) µ ∣Ψ⟩ = 0

(9.139)

for all physical states. This means that ⃗ − a3 (k)] ⃗ ∣Ψ⟩ = 0 [a0 (k)

(9.140)

⃗ The Gupta–Bleuler condition (9.140) is sufficient to ensure that there are for all k. no scalar and no longitudinal photons present in ∣Ψ⟩. An intuitive argument, valid for states with a finite number of photons, can be given as follows. The operators ⃗ and a3 (k) ⃗ annihilate different photon polarizations. Hence, the occupation a0 (k) ⃗ ⃗ numbers for scalar and longitudinal photons in the states a0 (k)∣Ψ⟩ and a3 (k)∣Ψ⟩ ⃗ ∣Ψ⟩ has to be equal to a3 (k) ⃗ ∣Ψ⟩, the only way to fulfill are different. Because a0 (k) Eq. (9.140) is for ∣Ψ⟩ not to contain any scalar or longitudinal photons in the first place. More formally, the proof is as follows. We first denote the allowed wave vectors of the electromagnetic field as k⃗1 , k⃗2 , . . . . Then, we decompose ∣Ψ⟩ in an occupation number representation of Fock space in terms of states of the form ∣Ψ⟩ = ∣n0 (k⃗1 ), n0 (k⃗2 ), . . . , n1 (k⃗1 ), n1 (k⃗2 ), . . . , n2 (k⃗1 ), n2 (k⃗2 ), . . . , n3 (k⃗1 ), n3 (k⃗2 ), . . .⟩, (9.141) which enumerates the occupation numbers of the photon states with polarizations λ = 0, 1, 2, 3 (subscript of the occupation number n) and with the allowed wave ⃗ is lifted from the vectors k⃗j (j ∈ N). Here, the state with occupation number nλ (k) vacuum using the photon creation operators, ⃗ n ∣0, . . . , 0, . . . ⟩ . ⃗ ...⟩ = √ 1 (a+λ (k)) ∣0, . . . , nλ (k), ⃗ nλ (k)!

(9.142)

⃗ − a3 (k) ⃗ (with an arbitrary allowable wave The application of the operator a0 (k) ⃗ leads to the equation vector labeled k) √ √ ⃗ ∣Ψ⟩ = n0 (k) ⃗ ∣ . . . , n0 (k) ⃗ − 1, . . . ⟩ = a3 (k) ⃗ ∣Ψ⟩ = n3 (k) ⃗ ∣ . . . , n3 (k) ⃗ − 1, . . . ⟩ . a0 (k) (9.143) We notice the reduction by one of the relevant occupation numbers. This equation involves two states with different occupation numbers for longitudinal and scalar ⃗ Hence, the only possibility for it to be fulfilled is if modes with wave vector k. ⃗ = n3 (k) ⃗ = 0. n0 (k)

(9.144)

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The Gupta–Bleuler condition (9.139) or alternatively (9.140) thus implies that there are no longitudinal or scalar photons present at all in the physical state ∣Ψ⟩. Let us figure an analogy to the reasoning just presented: One could say that (9.140), when applied to a box full of “matchbox” cars, says that after one “green” car (provided one is present) has been taken away, the resulting state is the same as the one obtained when one takes one “blue” car away (the color denotes the polarization). If there is only a finite number of cars, then the only way that could happen is if there were no blue and no green cars present in the box in the first place. And that is precisely the Gupta–Bleuler condition. 9.6

Wick Theorem and Applications

9.6.1

Time Ordering and Wick Theorem

The Wick theorem states, essentially, that the time-ordered product of field operators is equal to the normal-ordered product, plus all possible contractions. For the photon field, we can derive the Wick theorem by investigating a number of identities, which pertain to exponentiated field operators and Hamiltonians, where the classical current acts as a source for the quantized electromagnetic field. This relatively little-known set of identities is the subject of the current derivation. We begin by deriving an important identity which is valid for any time-ordered exponential. For a Hamiltonian H = H0 + H1 , the interaction Hamiltonian in the interaction picture (this is not redundant) acquires a time dependence according to Eq. (2.108), HI (t) = exp(−iH0 t) H1 exp(iH0 t) .

(9.145)

The time-ordered exponential of the interaction Hamiltonian is ∞

tf tf (−i)n dtn T [HI (t1 ) . . . HI (tn )] . dt1⋯ ∫ ∫ ti ti ti n=0 n! (9.146) Our purpose will be to rewrite the time-ordered exponential into a potentially easier form. We introduce a tiling of the time interval (ti , tf ), which for N → ∞ reads as

T exp [−i ∫

∆t =

tf

dt HI (t)] = ∑

tf − ti , N

tk = ti + (k − 12 ) ∆t ,

k ∈ (1, . . . , N ) ,

(9.147)

so that T exp [−i ∫

ti

tf

dt HI (t)] = e−i ∆t HI (tN ) e−i ∆t HI (tN −1 ) ⋯e−i ∆t HI (t1 ) .

(9.148)

The right-hand side of (9.148) contains products of exponentials which can be transformed using the Campbell–Baker–Hausdorff identity eA eB = eA+B+[A,B]/2 ,

(9.149)

which is valid if [A, [A, B]] = [B, [A, B]] = 0. We shall assume in the following that the commutator of two interaction Hamiltonians is a complex number, possibly a

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function of space and time, and commutes with the interaction Hamiltonian itself, which is supposed to contain no derivative operators. We begin by considering the product of the two leftmost factors in Eq. (9.148), 1

e−i ∆t HI (tN ) e−i ∆t HI (tN −1 ) = e−i ∆t [HI (tN )+HI (tN )] e 2 [−i ∆t HI (tN ),−i ∆t HI (tN −1 )] 1

= e−i ∆t [HI (tN )+HI (tN −1 )] e− 2 ∆t

2

[HI (tN ),HI (tN −1 )]

. (9.150)

This consideration generalizes to three temporal arguments as follows, e−i ∆t HI (tN ) e−i ∆t HI (tN −1 ) e−i ∆t HI (tN −2 ) 1

= e−i ∆t [HI (tN )+HI (tN −1 )] e−i ∆t HI (tN −2 ) e− 2 ∆t

2

[HI (tN ),HI (tN −1 ]

= e−i ∆t [HI (tN )+HI (tN −1 )+HI (tN −2 )] 1

× e− 2 ∆t

2

([HI (tN ),HI (tN −1 )]+[HI (tN −1 ),HI (tN −2 )]+[HI (tN ),HI (tN −2 )])

.

(9.151)

In the commutators on the right-hand side, the time argument in the left Hamiltonian is greater than the time in the right Hamiltonian. In the limit N → ∞, we thus have T exp [−i ∫

tf ti

dt HI (t)] = exp [−i ∫

tf ti

× exp [− 21 ∫

dt HI (t)]

ti

tf

dt ∫

tf ti

dt′ Θ(t − t′ )[HI (t), HI (t′ )]] . (9.152)

In the following, we use the quantum electrodynamic interaction Hamiltonian in the form HI (t) = ∫ d3 x Aµ (x) Jµ (x) = ∫ d3 x A(x) ⋅ J(x) ,

(9.153)

where Jµ (x) is assumed to be a classical current distribution, not a quantum mechanical operator [see also Eq. (2.108)]. We recall that in our formulation of the photon field operator, we have induced the time-dependence by going to the interaction picture. The S-matrix is given by the time-ordered exponential S = T exp [−i ∫ d4 x A(x) ⋅ J(x)] = T exp [−i ∫

∞ −∞

dt HI (t)] .

(9.154)

In the following, we are inspired by the derivation following Eq. (4-60) of Ref. [2], but we found it useful to add a number of explanations for intermediate steps which make the derivation much easier to follow and put the considerations into their proper context. The commutator of the interaction Hamiltonians HI (t) and HI (t′ ) is [see Eq. (9.116)] [HI (t), HI (t′ )] = ∫ d3 x ∫ d3 x′ Jµ (x) [Aµ (x), Aν (x′ )] Jν (x′ ) = i g µν ∫ d3 x ∫ d3 x′ Jµ (x) G− (x, x′ ) Jν (x′ ) = i ∫ d3 x ∫ d3 x′ Jµ (x) G− (x, x′ ) J µ (x′ ) ,

(9.155)

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where G− (x, x′ ) = GR (x, x′ ) − GA (x, x′ ) according to Eq. (9.38). We note that since HI (t) is a Hermitian operator, the commutator [HI (t), HI (t′ )] is antihermitian, and since we assume J µ (x) to be a classical current distribution, the commutator can be expressed as [HI (t), HI (t′ )] = i f (t, t′ ) where f (t, t′ ) is a real function. We conclude that S = exp [−i ∫

∞

−∞

dt HI (t)] exp [− 21 ∫

∞

−∞

dt ∫

∞

−∞

dt′ Θ(t − t′ )[HI (t), HI (t′ )]]

i = exp [−i ∫ d4 x A(x) ⋅ J(x)] exp [− ∫ d4 x ∫ d4 x′ Jµ (x)GR (x, x′ )J µ (x′ )] , 2 (9.156) where we have used the connection of the retarded Green function and the field commutator, namely GR (x, x′ ) = Θ(t − t′ ) G− (x, x′ ). The presence of the retarded Green function instead of the time-ordered Green function, or Feynman propagator, is disturbing. We can now proceed by carefully analyzing the term exp [−i ∫ d4 x A(x) ⋅ J(x)]. To this end, we first decompose the expression exp [−i ∫ d4 x A(x) ⋅ J(x)] into creation and annihilation parts, which are labeled as positive-frequency (annihilation) components [superscript (+)] and negative-frequency (creation) components [superscript (−)], exp [−i ∫ d4 x A(x) ⋅ J(x)] = exp [−i ∫ d4 x [A(+) (x) + A(−) (x)] ⋅ J(x)] = exp [−i ∫ d4 x A(−) (x) ⋅ J(x)] exp [−i ∫ d4 x′ A(+) (x′ ) ⋅ J(x′ )] × exp [ 21 ∫ d4 x ∫ d4 x′ Jµ (x) [A(−) µ (x), A(+) ν (x′ )] Jν (x)] .

(9.157)

Again, we have used the formula eA+B = eA eB e−[A,B]/2 . We can now identify the expression exp [−i ∫ d4 x A(−) (x) ⋅ J(x)] exp [−i ∫ d4 x A(+) (x) ⋅ J(x)] ≡ 1+ ∶ S ∶ ,

(9.158)

where ∶ S ∶ is the normal-ordered version of the S matrix defined in Eq. (9.154), because the annihilation operators contained in A(+) (x) stand to the right of the creation operators contained in A(−) (x). Note that the presence of the unit operator on the right-hand side is essential because the leading term in the expansion of the operator on the left-hand side of the above equation (in the limit J µ (x) → 0) is just unity, and the normal-ordered variant of the unit operator vanishes. Putting everything together, we thus have proven the identity S = (1+ ∶ S ∶) exp [ 21 ∫ d4 x ∫ d4 x′ Jµ (x) ([A(−) µ (x), A(+) ν (x′ )] − Θ(t − t′ ) [Aµ (x), Aν (x′ )]) Jν (x′ ) ] .

(9.159)

Here, we have rewritten the retarded Green function again as a field commutator, using the identity iJµ (x) GR (x − x′ )J µ (x′ ) = Θ(t − t′ ) Jµ (x) [Aµ (x), Aν (x′ )] Jν (x′ ) .

(9.160)

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The expression [A(−) µ (x), A(+) ν (x′ )]−Θ(t−t′ ) [Aµ (x), Aν (x′ )] is a field commutator and therefore equal to its vacuum expectation value. The time-ordered product emerges as follows, ⟨0 ∣[A(−) µ (x), A(+) ν (x′ )] − Θ(t − t′ ) [Aµ (x), Aν (x′ )]∣ 0⟩ = ⟨0 ∣−A(+) ν (x′ )A(−) µ (x) − Θ(t − t′ )Aµ (x)Aν (x′ ) + Θ(t − t′ )Aν (x′ )Aµ (x)∣ 0⟩ = ⟨0 ∣−Aν (x′ ) Aµ (x) + Θ(t − t′ ) Aν (x′ ) Aµ (x) − Θ(t − t′ ) Aµ (x) Aν (x′ )∣ 0⟩ = ⟨0 ∣−[1 − Θ(t − t′ )] Aν (x′ ) Aµ (x) − Θ(t − t′ ) Aµ (x) Aν (x′ )∣ 0⟩ = ⟨0 ∣−Θ(t′ − t) Aν (x′ ) Aµ (x) − Θ(t − t′ ) Aµ (x) Aν (x′ )∣ 0⟩ = − ⟨0 ∣T Aµ (x) Aν (x′ )∣ 0⟩ .

(9.161)

Our end result is S = (1+ ∶ S ∶) exp [− 12 ∫ d4 x ∫ d4 x′ Jµ (x) ⟨0 ∣T Aµ (x) Aν (x′ )∣ 0⟩ Jν (x′ )] , (9.162) or, written explicitly, T exp [−i ∫ d4 x A(x) ⋅ J(x)] = (1+ ∶ exp [−i ∫ d4 x A(x) ⋅ J(x)] ∶) × exp [− 21 ∫ d4 x ∫ d4 x′ Jµ (x) ⟨0 ∣T Aµ (x) Aν (x′ )∣ 0⟩ Jν (x′ )] .

(9.163)

This formula constitutes an “exponentiated” version of the Wick theorem. The classical current J µ (x) acts as a source for the quantized electromagnetic field. Upon expansion in powers of the vector potential, mediated by functional differentiation of Eq. (9.163) with respect to J µ (x), we recover the identities which make up the Wick theorem. When we act with δ 2 /(δJ µ (x)δJ ν (x′ )) on both sides of Eq. (9.163), we see that T Aµ (x) Aν (x′ ) = ∶ Aµ (x) Aν (x′ ) ∶ + ⟨0 ∣T Aµ (x) Aν (x′ )∣ 0⟩ .

(9.164)

The functional differentiation δ 3 /(δJ µ (x)δJ ν (x′ )δJ ρ (x′′ )) leads to the identity TAµ (x)Aν (x′ )Aρ (x′′ ) = ∶ Aµ (x)Aν (x′ )Aρ (x′′ ) ∶ + ∶ Aµ (x) ∶ ⟨0 ∣TAν (x′ )Aρ (x′′ )∣ 0⟩ + ∶ Aν (x′ ) ∶ ⟨0 ∣TAµ (x)Aρ (x′′ )∣ 0⟩ + ∶ Aρ (x′′ ) ∶ ⟨0 ∣TAµ (x)Aν (x′ )∣ 0⟩ .

(9.165)

In a general order, we have T Aµ1 (x1 )⋯Aµn (xn ) = ∶ Aµ1 (x1 )⋯Aµn (xn ) ∶ + ∑ ∶ Aµ1 (x1 )⋯Aµk (xk )⋯Aµl (x` )⋯Aµn (xn ) ∶ ⟨0 ∣TAµk (xk ) Aµl (x` )∣ 0⟩ k 0. which is completed to a four-vector by adding the timelike component k0 = k The mathematical momentum, required for momentum conservation at the vertex, is being denoted as p. The dot on one end of external lines indicates the presence of an internal vertex. The polarization vectors for incoming and outgoing photons are defined according to Eq. (9.91) [see also Eqs. (9.99) and (9.100)]. While the polarization vectors can be chosen as real rather than complex, we indicate the complex conjugate of the polarization vector for an outgoing photon, in order to account for possible manifestly complex choices of the photon polarization vectors.

Incoming electron

⃗ = Uσ (⃗ Uσ (k) p)

Outgoing electron

⃗ = U σ (⃗ U σ (k) p)

Incoming positron from the past

⃗ = V σ (−⃗ V σ (k) p)

Outgoing positron into the future

⃗ = Vσ (−⃗ Vσ (k) p)

Incoming photon

⃗ µ (k) λ

Outgoing photon

⃗ ∗ [µ (k)] λ

Photon line

µν iDF (p)

Fermion line

iSF (p)

Fermion-photon vertex

−i e γµ

Outgoing positron Interchanged fermion field operator Closed fermion loop

(−1) (−1) (−1)

p=k

p=k

p = −k p = −k k

k

p

p

√ energy component of this four-vector is k 0 = k⃗ 2 + m2 > 0. So, provided we draw the arrows for the momenta correctly in a Feynman diagram, we have to represent the position state by the bispinor V σ (⃗ p), where p⃗ = −k⃗ is the momentum in the diagram flow, which is opposite to the physical momentum k⃗ of the incoming positron.

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There are further overall phase factors to consider for Feynman diagram calculations. These are as follows, ● (−1) for an outgoing positron, ● (−1) for fermion field operator interchange, ● (−1) for a closed fermion loop. The (−1) for the outgoing positron line is understandable because the outgoing positron reverses the sequence of the field operators ψ(x) and ψ(x) in an internal line leading to the outgoing positron. That is to say, we observe that an outgoing positron is described by an expression ⟨0∣dσ (k) in a diagram; internal lines are generated by instances of the interaction Lagrangian −eψ(x) γ µ ψ(x) Aµ (x). However, the positron annihilation operator dσ (k) which pairs off with the positron creation operator d+σ (k), is to be found in ψ(x), not ψ(x). Because of the negative sign in the time-ordered product (7.145), the required “flipping” of the sequence of fermionic field operators implies the presence of an additional minus sign, which is incurred for an outgoing positron. Once the sign flipping is done, no additional minus sign is incurred for incoming positrons. For a closed fermion loop, we can understand the minus sign by the relative minus sign between the two terms in time-ordered product (7.145). Namely, working in space and time coordinates, we see that a loop can only be closed if the time ordering of the last pair in the sequence x1 → x2 → . . . → xn → x1 is inverted: (xn , x1 ) → (x1 , xn ). Again, considering the minus sign in Eq. (7.145), this leads to a factor (−1). The factor (−1) for fermion field operator interchange is sometimes not obvious just by inspecting Feynman diagrams. The presence of the factor is clear when, say, it pertains to the interchange of fermions in the out channel of, e.g., Møller scattering (e− e− → e− e− ), where it corresponds to an antisymmetrization of the wave function of the outgoing fermions. However, for Bhabha scattering (e+ e− → e+ e− ), there is also a relative minus sign between the so-called t channel (photon exchange) and the so-called s channel (annihilation diagram). The presence of this latter relative minus sign in Bhabha scattering is less obvious and requires a careful consideration of the assignment of incoming and outgoing momenta, which is left as an exercise to the reader. A coherent (amplitude) sum is formed over all diagrams which contribute to a given process in a given order of perturbation theory. This gives the S matrix element. The modulus square is then formed once the coherent sum over all amplitudes is taken. 10.3 10.3.1

Vertex Correction Vertex and Pauli–Villars Regularization

In principle, the Feynman rules provide us with the necessary tools to translate diagrams into formulas. The vertex correction is shown in Fig. 10.2. The Feynman

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k

p′

p

p′ − k

p−k

p′ − p

Fig. 10.2

Feynman diagram for the vertex correction with momentum flow.

rules, rather than being used in order to evaluate a cross section, here lead to a one-loop correction to the electron-photon vertex. One compares the simple factor −ieγ µ , which is usually associated to a vertex (see Table 10.1), to the one-loop correction, which depends on the momenta p and p′ . The replacement reads as follows, −ieγ ν → − ieγ ν − ieΛν (p, p′ ) , −ieΛν (p, p′ ) = ∫

(10.45a)

4

d k (iDFµρ (k)) (−ieγµ ) (2π)4

× (iSF (p′ − k)) (−ieγ ν ) (iSF (p − k)) (−ieγρ ) . µ

(10.45b)

µ

The correction Λ to the vertex factor Γ , which needs to be added to the Dirac matrix γ µ , is defined by Γν (p, p′ ) = γ ν + Λν (p, p′ ) .

(10.46)

Using the expression in Eq. (10.44), the correction can be expressed as 1 1 d4 k µρ D (k) γµ ′ γν γρ (2π)4 F p − k − m + i p − k −



m + i ′ ν γµ (p d4 k µρ
− k
+ m) γ (p
− k
+ m) γρ D = ie2 ∫ (k) . (10.47) F (2π)4 [(p′ − k)2 − m2 + i] [(p − k)2 − m2 + i]

Λν (p, p′ ) = ie2 ∫

In typical loop calculations, it is convenient to use the Feynman gauge, g µρ . (10.48) + i For ∣k∣ → ∞, the integral diverges logarithmically. We can establish this observation by simply counting the powers of k in the Euclidean region where we can replace DFµρ (k) = −

k2

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any k by its modulus ∣k∣, for power counting arguments. This means that the integral (10.47) diverges logarithmically at large momenta. The usual trick to still be able to evaluate such an integral is to introduce a regularization, i.e., we use an additional parameter (in this case denoted as Λ) which cuts off the divergence at large momenta. The integral no longer diverges, and we can take the limit Λ → ∞ at the end of the calculation (i.e., after regularization and renormalization, see Sec. 8.4 of Ref. [81]). In Pauli–Villars regularization, we subtract from the photon propagator a term describing a photon with a very large mass, g µρ g µρ DFµρ (k) → − 2 + 2 . (10.49) k + i k − Λ2 + i Anticipating potential infrared divergences, we also introduce an infrared regulator (infinitesimal photon mass) λ as follows, 2

Λ g µρ g µρ g µρ + 2 =∫ (10.50) dL 2 2 2 2 2 k − λ + i k − Λ + i (k − L + i)2 λ where the limits λ → 0 and Λ → ∞ are tacitly assumed to be taken at the end of the calculation. The loop correction to the vertex can be written as

DFµρ (k) → −

′ ν Λ γ µ (p d4 k dL
− k
+ m)γ (p
− k
+ m)γµ , Λ (p, p ) = 4πiα∫ ∫ (2π)4 λ2 (k 2 − L + i)2 [(p′ − k)2 − m2 + i][(p − k)2 − m2 + i] (10.51) where we have used the relation e2 = 4πα. For practical reasons, we focus on the matrix element of the vertex correction Λν (p, p′ ) between on-shell (OS) bispinors, which we write as ΛνOS (p, p′ ) = Λν (p, p′ )∣p=p′ =m , (10.52)

ν ′ ν where ΛOS (p, p ) is the on-shell vertex correction. In the evaluation of ΛOS (p, p′ ), we can replace any occurrence of a p2 or p′2 with the mass square, m2 . Also, because Uσ (⃗ p) fulfills the Dirac equation, we have p p) = m Uσ (⃗ p). In the same
Uσ (⃗ ′ way U σ′ (⃗ p′ ) has to fulfill the conjugate Dirac equation, U σ′ (⃗ p′ ) p
= U σ′ m. It is ν ′ understood that whenever p in the numerator of Λ (p, p ) is brought to the right, OS
′ or p is brought to the left, it can be replaced by the mass m, implementing the OS
(on shell) condition. Using the on-shell condition, we can simplify the denominator and write Λ2 d4 k dL ΛνOS (p, p′ ) = 4πiα ∫ ∫ 4 2 2 (2π) λ (k − L + i)2 µ ′ γ (p − k
+ m) γ ν (p
− k
+ m) γµ ∣ × 2
′ . (10.53) 2 [k − 2p ⋅ k + i] [k − 2p ⋅ k + i] p=p′ =m

As a next step, we consider the algebra of Dirac matrices and recall that the anticommutation relations read as {γ µ , γ ν } = 2g µν with a diagonal flat-space metric g µν = diag(1, −, 1, −1, −1). In view of the Dirac matrices in Eq. (10.53), we now have to practice a little Dirac algebra. The first, rather trivial, relation reads as 1 γ µ γµ = gµν γ µ γ ν = gµν {γ µ , γ ν } = gµν g µν = d . (10.54) 2 2

ν

′

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We here take the opportunity to supplement a generalization of this formula, valid for d space-time dimensions, which we evaluate, for later purposes, in d space-time dimensions. This result will become useful later, in the context of dimensional regularization, where one usually sets d = 4 − 2 , i.e., the number of dimensions is slightly lower than four. Using simple commutator relations, it is then easy to find formulas for slightly more complex Dirac structures, µ ν ν µ µν ν ν γ µ aγ
µ = γ γ γµ aν = −γ γ γµ aν + 2g γµ aν = −dγ aν + 2γ aν = (2 − d)a
. (10.55)

For two Feynman slashes sandwiched between Dirac γ µ matrices, we find µ ν κ ν κ µ ν µκ κ ν γµa

bγµ = γ γ γ γµ aν bκ = γ γ γ γµ aν bκ − 2 γ g γµ aν bκ + 2 γ γ aν bκ

= d γ ν γ κ aν bκ − 2 γ ν γ κ aν bκ − 2 γ ν γ κ aν bκ + 4 g κν aν bκ = (d − 4) γ ν γ κ aν bκ + 4 aκ bκ = (d − 4) a

b + 4a ⋅ b .

(10.56)

For three Feynman slashes sandwiched between Dirac γ µ and γµ matrices, we commute the leftmost γ µ sufficiently to the right before it “sees” a γµ , to obtain µ ν κ ρ ν µ κ ρ µν κ ρ γµa

b
cγµ = γ γ γ γ γµ aν bκ cρ = −γ γ γ γ γµ aν bκ cρ + 2 g γ γ γµ aν bκ cρ

= − γ ν γ κ γ ρ γ µ γµ aν bκ cρ + 2 γ ν γ κ g µρ γµ aν bκ cρ − 2 γ ν γ ρ γ κ aν bκ cρ − 2 γ ρ γ κ γ ν aν bκ cρ + 4 g κρ γ ν aν bκ cρ = − d γ ν γ κ γ ρ aν bκ cρ + 2 γ ν γ κ γ ρ aν bκ cρ + 2 γ ν γ κ γ ρ aν bκ cρ − 4 γ ν g ρκ aν bκ cρ − 2 γ ρ γ κ γ ν aν bκ cρ + 4 γ ν aν bρ cρ = (4 − d) a

b
c − 2
c
ba
.

(10.57)

We have used the relations γ µ γµ = d ,

γ ρ γ κ = 2g ρκ − γ κ γ ρ ,

(10.58)

′

repeatedly. Let use now assume that a
=p
and that
c = p.
The Feynman slashes are commuted to the outside, but to the wrong side. We would like to have expressions ′ ′ ′ like U σ′ (p′ ) p
and p
Uσ (p), not p
Uσ (p) and U σ′ (p ) p.
However, one can move the a to the left and the c to the right by writing identities like

ρ κ ν ρκ κ ρ ν
c
ba
= γ γ γ cρ bκ aν = (2g − γ γ ) γ bκ cρ aν = −a

b
c + 2 (a ⋅ b)
c − 2 (a ⋅ c)
b + 2 (b ⋅ c) a
.

(10.59)

After repeatedly commuting the Dirac γ matrices, one arrives at γµa

b
cγµ = (6 − d) a

b
c − 4 (a ⋅ b)
c + 4 (a ⋅ c)
b − 4 (b ⋅ c) a
,

(10.60)

where now all the slashed vectors keep their order. The numerator of the integrand in Eq. (10.53) can be rewritten as ′ ν N ν = γ µ (p
− k
+ m) γ (p
− k
+ m)γµ

′ ν µ ′ ν µ ν µ ν = γ µ (p
− k
)γ (p
− k
)γµ + γ (p
− k
)γ mγµ + γ mγ (p
− k
)γµ + γ mγ mγµ , (10.61)

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where we have separated the slashed momenta and the masses. We can rewrite this expression using Eq. (10.60) for the first term (observing that
b = γ ρ bρ = γ ν for bρ = g ν ρ ), Eq. (10.56) for the second and third, and finally Eq. (10.55) for the last term. So we find ′ ν ′ ν ν ′ N ν = (6 − d)(p
− k
)γ (p
− k
) − 4(p − k) (p
− k
) + 4γ (p − k) ⋅ (p − k)

′ ′ ν ′ ν ν − 4(p − k)ν (p
− k
) + (d − 4)m(p
− k
)γ + 4 m(p − k) + (d − 4)mγ (p
− k
)

+ 4 m(p − k)ν + (2 − d)m2 γ ν .

(10.62)

′ Using the on-shell condition, we now apply p
and p
directly onto their respective bispinors. After some additional manipulation, we obtain finally d N ν = −4m k ν +(d−6) γ ν k 2 +4k
[p′ν + pν − ( − 1) k ν ]+4 γ ν (p′ −k)⋅(p−k) . (10.63) 2 We shall now return to the case d = 4, where we have

N ν = −4m k ν − 2γ ν k 2 + 4k
[p′ν + pν − k ν ] + 4 γ ν (p′ − k) ⋅ (p − k) .

(10.64)

Now, let us investigate the denominator of the integrand of the vertex correction ΛµOS (p, p′ ). It contains the expressions [k 2 − 2p′ ⋅ k +i] and [k 2 − 2p ⋅ k +i] which are not “spherically” symmetric in the four-vector k. However, we can enforce this symmetry, by joining denominators using so-called Feynman parameters, based on formulas like ∫

1 0

2

1 1 ) = , dx ( A x + B (1 − x) AB

∫

0

1

dx

6 x (1 − x) 1 = , [A x + B (1 − x)]2 (A B)2

(10.65) −2

1

which can be generated by differentiation of the integral ∫0 dx (A x + B (1 − x)) with respect to parameters. After joining the denominators, one can simply shift the integration (loop) momentum, to obtain “spherically” symmetric integrals. We begin by combining the two denominators from the fermion propagators, writing (k 2

1 1 1 = ∫ dy , 2 2 ′ − 2p ⋅ k + i) (k − 2p ⋅ k + i) 0 (k 2 − 2py ⋅ k+ i)

(10.66)

where we have defined py = y p + (1 − y)p′ . The next step is to combine this denominator with the one from the photon propagator, using the second entry in Eq. (10.65), ∫

Λ2

dL

λ2

=∫

Λ2 λ2

1 1 1 ∫ dy 2 2 2 − L + i) 0 (k − 2py ⋅ k + i)

(k 2

dL ∫

0

1

dz ∫

1 0

dy

6z(1 − z) 4

(Q 2 − Lz − py2 (1 − z)2 + i)

,

(10.67)

where we have completed the square to shift the integration variable according to Q = k − py (1 − z) ,

(10.68)

using the identity k 2 − 2py ⋅ k(1 − z) = [k − py (1 − z)]2 − p2y (1 − z 2 ) = Q2 − p2y (1 − z 2 ). The integrand is now “spherically”’ symmetric in the new integration momentum

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Q. The shift leaves the momentum integration measure d4 k → d4 Q invariant, as we integrate over R4 . We now use the identity k = Q + py (1 − z) in Eq. (10.63) in order to reformulate the numerator N ν , very explicitly, as follows, N ν = − 4m Qν − 4m(1 − z)pνy − 2 γ ν Q 2 − 4 γ ν (1 − z) Q ⋅ py − 2 γ ν (1 − z)2 p2y ν Q Qν − 4(1 − z)p Q [p′ν + pν − (1 − z)pνy ] − 4  + 4
y Q

′ν ν ν ν ′ρ ν 2 + 4(1 − z)p
y [p + p − (1 − z)py ] − 4γ Q ⋅ [p + p − 2(1 − z)py ] + 4γ Q

+ 4 γ ν (p′ − (1 − z)py ) ⋅ (p − (1 − z)py ) .

(10.69)

We now distinguish terms quadratic in Q, terms linear in Q and terms without Q. Linear terms vanish after integration over Q because of the R4 symmetry (after the Wick rotation, to be discussed below). The only contributing terms are summarized in a redefined expression N ν , QQν − 4m(1 − z)pνy N ν = 2 γ ν Q 2 − 4

′ν ν ν − 2 γ ν (1 − z)2 p2y + 4(1 − z)p
y [p + p − (1 − z)py ]

+ 4 γ ν (p′ ⋅ p − (1 − z)p′ ⋅ py − (1 − z)p ⋅ py + (1 − z)2 p2y ) .

(10.70)

The two first expressions are quadratic in Q, and we replace them as follows, 1 2 γ ν Q 2 − 4 QQν = 2 γ ν Q 2 − 4γµ (Qµ Qν ) → 2 γ ν Q 2 − 4γµ ( g µν Q2 ) = γ ν Q 2 . (10.71) 4 We have carried out a replacement based on the formula g µν d 2 2 d µ ν 2 (10.72) ∫ d Q Q f (Q ) , ∫ d Q Q Q f (Q ) = d applied to the case d = 4. This relation follows because any general second-rank tensor B µν in four-dimensional space-time has a decomposition, much in analogy to Eq. (6.13), g µν B µν = T + Aµν + S µν , (10.73) d where T is its trace, A an antisymmetric tensor (Aµν = −Aνµ ), and S is a traceless, symmetric tensor (S µν = S νµ , and gµν S µν = 0). These tensors do not mix under Lorentz transformations. When integrated with a function symmetric in Q2 over all space-time, the anti-symmetric part of the tensor Qµ Qν vanishes due to symmetry. The same is true of the symmetric, traceless part. Only the scalar part (trace) yields a nonvanishing contribution to the integral, as expressed in Eq. (10.72). The part of N µ without Q can be further simplified with the help of the Dirac ′ equation, which allows the replacement p
y → m, because p
and p
act to the right ′ and left. Using this identity, expressing py = yp + (1 − y)p , and replacing p2 → m2 , p′2 → m2 , as well as p′ ⋅ p → m2 − q 2 /2, with q = p′ − p ,

(10.74)

we get the result that N can be equivalently replaced under the integral sign, by ν

N ν = γ ν Q 2 + 4m(1 − z)yq ν − 4m(1 − z)2 (1 − y) q ν − 2 γ ν m2 (1 − 2z − z 2 ) + 4z (1 − z) mpν + 2 γ ν q 2 [y 2 (1 − z)2 − y (1 − z)2 − z] .

(10.75)

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We now recall that in calculating the numerator structure N ν , we have always assumed that the numerator is “sandwiched” between bispinors U σ′ (p′ ) and Uσ (p) (see Sec. 7.3.1). We shall use the on-shell condition one more time, in order to get rid of the term proportional to pν in Eq. (10.75) and write it in terms a function of the momentum transfer q = p′ − p. To this end, we need the Gordon identity, 1 ′ ν ν U σ′ (⃗ U σ′ (⃗ p′ ) γ ν Uσ (⃗ p) = p′ ) [ p p)
γ + γ p]
Uσ (⃗ 2m 1 1 1 1 1 U σ′ (⃗ p) = p′ )[ {γ µ , γ ν } p′µ + {γ ν , γ µ } pµ + [γ µ , γ ν ] p′µ + [γ ν , γ µ ] pµ ] Uσ (⃗ 2m 2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ 2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ 2 ´¹¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¶ 2 ´¹¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¶ =2g νµ

= U σ′ (⃗ p′ ) [

p +p 2m ν

′ν

+

=2g νµ

=2iσ νµ

=−2iσ νµ

νµ

iσ qµ ] Uσ (⃗ p) . 2m

(10.76)

Using (again) the identity p′ = q + p, this leads to the relation q ν iσ νµ qµ − ] Uσ (⃗ p) , (10.77) 2 2 which allows to express the term proportional to pν in Eq. (10.75) by a matrix element of the Dirac γ function, and terms that depend on the momentum transfer q. We thus find that N ν can be equivalently replaced by U σ′ (⃗ p′ ) pν Uσ (⃗ p) = U σ′ (⃗ p′ ) [mγ ν −

N ν = γ ν Q 2 + 2mq ν (2y − 1)(z − 1)(z − 2) − 2 γ ν m2 (1 − 4z + z 2 ) − 2z(1 − z)miσ νµ qµ + 2 γ ν q 2 [y 2 (1 − z)2 − y(1 − z)2 − z] .

(10.78)

After all these simplifications to the numerator, we can finally proceed to the evaluation of the Q-integral. The integral over d4 Q is normally carried out in Minkowski space with a metric g µν = diag(1, −1, −1, −1). However, if we rotate the integral of the timelike component d`0 → i d`0E , then this so-called Wick rotation shifts the integration contour of the timelike component of the loop momentum integration by π/2, in the mathematically positive direction i.e. counterclockwise. Let us remember that the denominator D of the integrand in Eq. (10.67) is of the form D = Q 2 − X 2 + i ,

X 2 = Lz + py2 (1 − z)2 > 0 .

(10.79)

Here, we recall that the integration domain for z is the interval z ∈ (0, 1), and that py = yp + (1 − y)p′ , where y ∈ (0, 1). The next step is to combine the zeros of the denominator as given by the equation √ 2 + X 2 − i , Qµ = (ωQ , k⃗Q ) . (10.80) Q2 = X 2 − i , ωQ = ± k⃗Q Expanding in , and redefining , we find that the position of the poles in the ωQ integration contour is thus given by (see also Fig. 10.3) √ √ 2 + X 2 ∣ − i , 2 + X 2 ∣ + i , ωQ = ∣ k⃗Q ωQ = − ∣ k⃗Q (10.81) i.e., in the second and fourth quadrant of the complex plane. The integrand in Eq. (10.67) is an analytic function, which can be continued to complex momenta

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Im(ω) ✻

Wick–Rotated Contour

√ − −iǫ q

✻

q√

✲ Re(ω) −iǫ

✻

Fig. 10.3 The Wick-rotated integration contour for the vertex correction extends along the imaginary axis in the complex ω plane, where we understand that ω ≡ ωQ is the timelike component of the four-vector Q [see Eqs. (10.68) and √ (10.80)]. The denominator of the vertex function has poles ⃗2 − X 2 + i = 0, i.e., at ω = ± k ⃗2 + X 2 − i, where ω = ωQ and k ⃗=k ⃗Q . The minimum value at ω 2 − k of X 2 in the relevant range of integration parameters is zero, and so, as we integrate over d3 k, a √ 2 branch cut is generated, with branch points at ± X − i [see also Eq. (9.34)]. The curves in the figure display the geometric locus of the branch cuts for X = 0. We observe that the fermionic ⃗ ⋅ p⃗ + βm (see denominators, after multiplication with γ 0 = β, involve the free Dirac Hamiltonian α also Chap. 7). Let us include the cryptic remark that, for a free reference Dirac state, there is no further branch cut from the fermion propagator, because the reference state is an eigenstate of the momentum operator, and also, the Dirac Hamiltonian in the denominator only involves the momentum operator.

and falls off sufficiently fast for large ∣ωQ ∣. We can thus actively shift the ωQ integration contour from “horizontal” to “vertical” in the complex plane, changing the integration measure from Minkowskian to Euclidean, dωQ → idωQ,E , k⃗Q → k⃗Q,E , d4 Q → i d4 QE , (10.82) 2 2 Q2 → − Q2E = −ωQ,E − k⃗Q,E . (10.83) The integral over d4 QE also extends over the entire space R4 . As is illustrated in Fig. 10.3, the Wick rotation is compatible with the Feynman prescription for the poles of the denominators, because of the active shift by an angle +π/2 in the complex plane. This implies that we can drop the infinitesimal imaginary part i in the on-mass shell vertex function [see Eqs. (10.53), (10.67) and (10.78)], Λ2 1 1 d4 Q 6z(1 − z) N ν ΛνOS (p, p′ ) = 4πiα ∫ ∫ 2 dL ∫ dz ∫ dy 4 4 (2π) λ 0 0 (Q 2 − Lz − p 2 (1 − z)2 ) y

Λ 1 1 d4 QE 6z(1 − z) N ν = −4πα ∫ dL dz dy . ∫ ∫ ∫ 4 (2π)4 λ2 0 0 (−QE2 − Lz − py2 (1 − z)2 ) (10.84) 2

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We have already transformed the integral into a spherically symmetric form. Now, it remains to separate off the angular and radial parts. For the angular part, we need the (generalized) surface are Ωd of the (generalized) unit sphere embedded in d-dimensional space, in order to plug it into the formula d 2 d−1 2 ∫ d QE f (QE ) = ∫ dΩd ∫ dQE QE f (QE ) ,

(10.85)

which holds for any function f (Q2E ) of the square of the Euclidean momentum. The volume of the d-dimensional unit sphere is Ωd = ∫ dΩd . We here discuss the general result for non-integer dimension d, as it will become useful later. One possible way to derive a formula for Ωd is as follows. We consider the ddimensional Gaussian integral of exp(−k⃗2 ) and calculate it in two ways. First, we express k⃗2 in terms of its d components and integrate each of them separately, with the result √ d d ∞ dk π 1 π d/2 dd k −k⃗2 −k2 e = [ e ] = ( ) = d d/2 = . (10.86) ∫ ∫ d d 2π (2π)d −∞ 2π R (2π) 2 π On the other hand, if we single out the integral over the angular variable ∫ dΩd and perform only a radial integral, then we have ∞ 2 dd k −k⃗2 1 Γ(d/2) e = [∫ dΩd ] ∫ dk k d−1 e−k = [∫ dΩd ] , (10.87) ∫ d d d (2π) 2(2π)d R (2π) 0 ∞

where the Γ function is defined as Γ(x) = ∫0 tx−1 exp(−t) dt, for x > 0. So, we have Ωd =

2π d/2 , Γ ( d2 )

(10.88)

which yields Ω4 = 2π 2 in d = 4. This result could have been obtained by more elementary means for integer dimension, e.g., for the case d = 4, by integrating Eq. (4.196) over the angles. However, anticipating the discussion in Sec. 10.3.4, we need a more general expression for non-integer d. Putting together Eqs. (10.78) and (10.84), we have Λ 1 1 d4 Q 6z(1 − z) N ν dL dz dy ∫ ∫ ∫ 4 (2π)4 λ2 0 0 (−QE2 − Lz − py2 (1 − z)2 ) 2

ΛνOS (p, p′ ) = −4πα ∫ 1

1

= −4πα∫ dz ∫ dy[6z(1 − z)] [−I2 (y, z) γ ν + I0 (y, z) 0

0

× {2mq ν (2y − 1)(z − 1)(z − 2) + γ ν (2 q 2 [y 2 (1 − z)2 − y(1 − z)2 − z] − 2m2 (1 − 4z + z 2 )) − 2z(1 − z)miσ νµ qµ }] .

(10.89)

The d4 QE integrals have been absorbed in I2 (y, z) and I0 (y, z). The integral I2 (y, z) carries two powers of Q2E in the numerator, 2

I2 (y, z) = ∫ =

Q2E d4 QE Λ ∫ 2 dL 2 4 4 (2π) λ (QE + Lz + py2 (1 − z)2 )

p2y (1 − z)2 + zΛ2 1 1 1 zΛ2 ln [ ] = ln [ ] γµ . 8π 2 6z p2y (1 − z)2 + zλ2 48π 2 z p2y (1 − z)2

(10.90)

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We have used the identity ∫ d4 QE = 2π 2 ∫ dQE Q3E , which is valid for radially symmetric integrands. This expression is ultraviolet divergent in the limit Λ → ∞, while the infrared limit λ → 0 is safe. This confirms the result of naive power counting. By contrast, the integral I0 (y, z), by naive power counting, is infrared divergent but ultraviolet safe, Λ2

Λ2

∞

dQE Q3E dL d4 QE dL I0 (y, z) =∫ = ∫ ∫ ∫ 4 2 (2π)4 2 (Q 2 + Lz + p 2 (1 − z)2 )4 2 8π 2 2 2 y 0 (QE + Lz + py (1 − z) ) E λ λ =

1 Λ2 − λ2 1 . = 2 2 2 8π 12(py (1 − z) + zλ2 )(p2y (1 − z)2 + zΛ2 ) 96π 2 z(p2y (1 − z)2 + zλ2 ) (10.91)

In the last step, the limit Λ → ∞ is taken. With these results, we are now in a position to write for the full one-loop vertex correction [p2y = m2 − q 2 y(1 − y)] ΛνOS (p, p′ ) =

1 1 zΛ2 α ν ] ∫ dz ∫ dy {(1 − z) γ ln [ 2π 0 (1 − z)2 (m2 − q 2 y(1 − y)) 0 (1 − z) + (−mq ν (2y − 1)(z − 1)(z − 2) (1 − z)2 (m2 − q 2 y(1 − y)) + zλ2

+ γ ν (q 2 [y (1 − y) (1 − z)2 + z] + m2 (1 − 4z + z 2 )) + z(1 − z)miσ νµ qµ)}. (10.92) Based on symmetry properties, we can show that the term proportional to q ν vanishes after integration over y. Namely, the numerator of this term is directly proportional to 2y − 1, which is antisymmetric in the interval y ∈ [0, 1], whereas the denominator can be written in terms of the variable y (1 − y), which is symmetric in the interval y ∈ [0, 1]. So, ΛνOS (p, p′ ) simplifies to ΛνOS (p, p′ ) =

1 1 α zΛ2 ν ] ∫ dz ∫ dy {(1 − z) γ ln [ 2 2π 0 (1 − z) (m2 − q 2 y(1 − y)) 0 (1 − z) (γ ν m2 (1 − 4z + z 2 ) + (1 − z)2 (m2 − q 2 y(1 − y)) + zλ2

+ γ ν q 2 [y(1 − y)(1 − z)2 + z] + z(1 − z)miσ νµ qµ )} .

(10.93)

This concludes our discussion of the regularized vertex function. We now need to take care of the renormalization, and calculate the renormalized vertex function, as well as the form factors of the electron. 10.3.2

Vertex and Form Factors

Intuitively, one would assume that the vertex correction ΛνOS (p, p′ ) has to vanish for zero momentum transfer, i.e., for q = 0. This is because, according to Eq. (10.45), any correction to the “vertex factor” −i e γ ν at zero momentum transfer counts

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as a modification δe of the electron’s charge, −i e γ ν → −i (e + δe) γ ν . Since the e parameter in our theory parameterizes the physical electron charge, this implies that we should interpret the term δe as a perturbative, radiative contribution to the electron’s charge which should be naturally absorbed in the physical value of e. The original parameter e of the theory is then reinterpreted as a bare parameter, to which we have to add a radiative correction. Variously, if we wish to keep the interpretation of e as the physical, renormalized electron charge, then we need to subtract the radiative contribution δe because it is already contained in the physical value of e; we just did not know, when we wrote down the original QED Lagrangian, that the parameter e that appears in it also contained a radiative correction. The general motivation and implementation of the renormalization program of QED will be discussed in Sec. 10.3.3, where we shall introduce the charge renormalization constant Z1 ≡ Z1 (Λ, µ). It follows that we should subtract from the expression (10.93) the vertex function at zero momentum transfer (p = p′ ), ⎡ ⎤ 1 ⎢ ⎛ zΛ2 1 − 4z + z 2 ⎞⎥⎥ ν ν α ⎢ ΛOS (p, p) = γ ]+ ∫ dz ⎢(1 − z) ln [ 2 λ2 ⎠⎥ 2π 0 m (1 − z)2 ⎝ ⎢ ⎥ (1 − z)2 + z m 2 ⎣ ⎦ 2 2 2 α 1 Λ 1 m 9 λ = γ ν [ ln ( 2 ) − ln ( 2 ) + + O ( 2 )] . (10.94) π 4 m 2 λ 8 m This function is conveniently proportional to γ ν . At zero momentum transfer, the unrenormalized vertex function is ultraviolet and infrared divergent (limits Λ → ∞ and µ → 0). The on-shell, renormalized [subscript (R)] vertex function is obtained as ΛνR (p, p′ ) = ΛνOS (p, p′ ) − ΛνOS (p, p) =

1 1 α ν ∫ dz ∫ dy{γ (1 − z) 2π 0 0

⎡ −1 ⎤ ⎥ ⎢ (1 − z) q2 ⎢ × ln ⎢(1 − 2 y(1 − y)) ⎥⎥ + (γ ν (1 − 4z + z 2 ) m ⎢ ⎥ (1 − z)2 [1 − q22 y(1 − y)] + z λ22 ⎦ ⎣ m m 2 q 1 − 4z + z 2 q µ }. + γ ν 2 [y(1 − y)(1 − z)2 + z] + z(1 − z)iσ νµ ) − γ ν (1 − z) λ2 m m (1 − z)2 + z m 2 (10.95) The renormalized vertex function is still infrared divergent. Let us define the renormalized on-shell vertex function ΓνR (q) to be the sum of the Dirac γ ν matrix and the renormalized one-loop correction ΛνR (p, p′ ). An inspection of Eq. (10.95) reveals that we can write Γν (q) in terms of two form factors F1 and F2 , i νµ σ qµ . (10.96) 2m We assume on-shell incoming and outgoing bispinors. A more general analysis of the structure of possible interactions between a fermion and gauge fields would ΓνR (q) = γ ν + ΛνR (p, p′ ) = F1 (q 2 ) γ ν + F2 (q 2 )

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otherwise lead to a form [see Eq. (1.2) of Ref. [308]] i νρ 1 νρ 5 ΓνR (q) = F1 (q 2 ) γ ν + F2 (q 2 ) σ qρ + F3 (q 2 ) σ γ qρ 2m 2m GF a + FA (q 2 ) √ (γ ν γ 5 q 2 − 2m γ 5 q ν ) . (10.97) 8π 2 Here, σ νρ = i [γ ν , γ ρ ]/2, and a is the contribution of the anapole moment [309]. A nonvanishing value of F3 (0) would otherwise describe an electric dipole moment of the electron [see Eq. (1.3) of Ref. [308]]. Within QED, one obtains nonvanishing contributions for the F1 and F2 form factors (which are also called the Dirac and the Pauli form factors, respectively). By inspection of Eq. (10.95), one infers the integral representations ⎡ −1 ⎤ ⎧ 1 1 ⎪ ⎥ q2 α ⎪ ⎢⎢ ⎥ dy (1 − z) ⎨ ln dz (1 − F1 (q 2 ) = 1 + y(1 − y)) ∫ ∫ ⎢ ⎥ 2 ⎪ 2π 0 m 0 ⎢ ⎥ ⎪ ⎩ ⎣ ⎦ + F2 (q 2 ) =

q2 [y(1 − y)(1 − z)2 + z] m2 2 q λ2 (1 − z)2 [1 − m 2 y(1 − y)] + z m2

(1 − 4z + z 2 ) +

⎫ ⎪ ⎪ − ⎬, 2 µ ⎪ 2 (1 − z) + z m2 ⎪ ⎭ 1 − 4z + z 2

1 α 1 (1 − z)2 z . ∫ dz ∫ dy q2 λ2 π 0 0 (1 − z)2 [1 − m 2 y(1 − y)] + z m2

(10.98)

(10.99)

For the infrared finite form factor F2 , we can set the photon mass λ = 0, F2 (q 2 ) =

1 α 1 ∫ dz z ∫ dy π 0 0 1− ´¹¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¶

1 q2 y(1 − y) m2

=

1 q2 α 1 q4 q6 [1 + + + O ( )] . 2π 6 m2 30 m4 m6

=1/2

The integrals for F1 (q ) = part is 2

[1]

F1 (q 2 ) =

[1] F1 (q 2 )

+

[2] F1 (q 2 )

(10.100) are more involved. The logarithmic

⎡ −1 ⎤ 1 1 ⎢ ⎥ α q2 ⎢ ⎥ (1 − dz(1 − z) dy ln y(1 − y)) ∫ ∫ ⎢ ⎥ 2 2π 0 m 0 ⎢ ⎥ ⎣ ⎦ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ =1/2

=

2

α q 1 q4 1 q6 [ 2 + 4 + O ( 6 )] . 2π m 12 m 120 m

(10.101)

The nonlogarithmic term is expanded as follows, ⎧ q2 2 1 1 ⎪ (1 − 4z + z 2 ) + m α 2 [y(1 − y)(1 − z) + z] ⎪ [2] 2 F1 (q ) = ∫ dz ∫ dy (1 − z)⎨ q2 λ2 ⎪ 2π 0 0 ⎪ (1 − z)2 [1 − m 2 y(1 − y)] + z m2 ⎩ ⎫ 1 − 4z + z 2 ⎪ ⎪ − ⎬ λ2 ⎪ (1 − z)2 + z m 2 ⎪ ⎭ 2 α q 2 m 1 q4 1 m 1 q6 = { 2 [ ln ( ) − ] + 4 [ ln ( ) − ] + O ( 6 )} . (10.102) 2π m 3 λ 3 m 10 λ 10 m

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The integration over the Feynman parameters is greatly simplified if one observes that the infrared regulator is only relevant in the region z ≈ 1; hence, one can replace (twice) zλ2 /m2 → λ2 /m2 in the integrand of Eq. (10.102). In the final result, we only keep terms which are divergent or finite in the limit λ → 0. The F1 form factor [2] [1] in one-loop order is the sum of F1 (q 2 ) and F1 (q 2 ), and of the tree-level leading term (unity) from Eq. (10.98), F1 (q 2 ) = 1 +

α q2 1 m 1 α q4 1 m 11 q6 [ [ ) . (10.103) ln ( ) − ] + ln ( ) − ] + O ( π m2 3 λ 8 π m4 20 λ 240 m6

One cannot stress enough that this result is still infrared divergent; it contains the photon mass λ. That is to say, despite all our heroic efforts to remove the (ultraviolet) divergences by means of renormalization, there are still divergences (of infrared nature) left in the final result. This observation finds a natural explanation if one considers a compensating divergence in the soft bremsstrahlung correction. Roughly speaking, the explanation is as follows. The infrared divergence in the result (10.103) describes the physical fact that it is infinitely easy for the electron to emit, and reabsorb, a virtual photon before and after the interaction with the external field (at the vertex). Hence, this infrared divergence needs to be cut off, at an energy scale commensurate with the photon mass λ. If one considers the high-energy part of the bound-electron self-energy for the Lamb shift, then the infrared divergence finds a natural explanation: simply, this infrared divergence is cut off at the scale of the atomic binding energy, where the energy could not be radiated and absorbed any more in the form of virtual infrared photons of arbitrarily small energy, but rather, the bound-state spectrum should be taken into account. We anticipate that the infrared divergence enters the expressions for the high-energy part of the bound-electron self-energy in photon mass regularization [see Eq. (11.148)] as well as in photon energy regularization [see Eq. (11.150)], with the matching given in Eq. (11.151). The infrared divergence of the high-energy part is compensated when the low-energy part is added. The low-energy part is given in Eq. (11.147) (photon mass regularization) and in Eq. (4.325) (photon energy regularization). The total result for the self-energy shift of an atomic level is independent of the infrared regularization procedure. One might now ask how all of this plays out when one considers bremsstrahlung corrections to a scattering process. In this case, the infrared divergence of the vertex correction is compensated by a corresponding logarithmic infrared divergence of the infrared bremsstrahlung correction, which describes the (almost collinear) emission of infrared photons by the outgoing electron. Infrared divergences of the vertex correction and of the bremsstrahlung correction to the total cross-section can be shown to cancel, and the so-called infrared catastrophe is avoided (see Chap. 7 of Ref. [2]). In this case, the infrared photon energy cutoff  [see Eq. (11.150)] takes the role of the detector resolution ∆E = , where ∆E =  is the smallest photon energy detectable by the apparatus. The matching between the photon energy and photon mass regularizations is given in Eq. (11.151).

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Incidentally, one further aspect is of utmost interest. While, at every order of perturbation theory, infrared divergences of vertex and bremsstrahlung corrections cancel, one might ask about the total contribution of the vertex-only corrections, i.e., of the processes in which manifestly no extra bremsstrahlung photons are emitted. This question is answered in the works of Yennie,2 Frautschi3 and Suura,4 and Sudakov.5 Roughly speaking, one can ascertain that the corrections to the cross section, once all orders of perturbation theory are added, behave, to logarithmic accuracy, just like the following infinite series, ∞

α m n α m  α/π 1 [− ln ( )] = exp [− ln ( )] = ( ) → 0, π  π  m n=0 n! ∑

(10.104)

where  is the infrared cutoff. The summation of all orders of perturbation theory leads to a vanishing of the infrared divergence. Therefore, in the limit  → 0, this leaves only a finite, infrared safe correction to the entire process. The process of the cancellations is similar to the calculation of the exponential function of large negative argument, 10002 + ⋯. (10.105) 2! Each single term on the right-hand side is very large, but their sum approaches exp(−1000), which is very close to zero. exp(−1000) = 1 − 1000 +

10.3.3

Vertex and Renormalization

As our considerations regarding the vertex charge renormalization show [see the beginning of Sec. 10.3.2], the renormalization establishes a relation between the bare quantities, which are present in the “original” QED Lagrangian, and the renormalized, physical quantities, such as the renormalized, physical charge. In the context of the charge renormalization, we recall Eq. (10.45), which at p′ = p, or q = 0, and between on-shell bispinors, reads as − ieγ ν → −ieγ ν − ieΛνOS (p, p) .

(10.106)

Furthermore, we have Eq. (10.94), which we write as follows, ΛνOS (p, p) = γ ν

Λ2 1 m2 9 λ2 e2 1 [ ln ( ) − ln ( ) + + O ( )] . 16π 2 4 m2 2 λ2 8 m2

(10.107)

The parameter e in the original QED Lagrangian is thus seen to be the bare charge e0 , which receives a perturbative correction δe = 2 Donald

e3 1 Λ2 1 m2 9 [ ln ( ) − ln ( )+ ] . 16π 2 4 m2 2 λ2 8

Robert Yennie (1924–1993). Clark Frautschi (b. 1933). 4 Hiroshi Suura (1925–1998). 5 Vladimir Vasilievich Sudakov (1925–1995). 3 Stephen

(10.108)

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This correction is perturbatively small (of higher order in the coupling parameter, √ which is the electron charge e = 4πα) and yet, divergent in the limits Λ → ∞ and λ → 0. Upon re-identifying the parameter e in the relations above as the bare electron charge, we can write the following relation between the physical, renormalized charge ephys = eR , and the bare charge e0 , ephys = eR = e0 + δe = e0 {1 +

e20 1 Λ2 1 m2 9 [ ln ( ) − ln ( ) + ]} . 2 2 2 16π 4 m 2 λ 8

(10.109)

This equation suggests that we can express, order by order in perturbation theory, the physical charge of the electron, in terms of the bare charge. The renormalization program is characterized by the following steps. (i) The original Lagrangian is identified as the bare Lagrangian, expressed in terms of the bare quantities. (ii) The perturbative loop corrections identify counterterms, i.e., terms which are naturally added to the Lagrangian. These are proportional to terms of the original Lagrangian. These so-called counterterms shift the bare quantities so that, upon addition of radiative corrections, finite results are obtained from the theory. The postulate of renormalizability then states that the physical, renormalized Lagrangian, expressed in terms of the physical quantities, should have the same functional form as the bare Lagrangian, expressed in terms of the bare quantities, up to the addition of the counterterms. It will turn out that the bare and physical quantities are related by multiplicative renormalization constants. (iii) One then rewrites the bare Lagrangian, originally expressed in terms of the bare quantities, in terms of the physical quantities. This operation is equivalent to the combined operation of re-interpreting the parameters in the original Lagrangian in terms of the physical, renormalized parameters, and adding the counterterms (in the sense of the relation e0 = eR − δe). Here, δe is the radiative contribution to the electron charge. Added to the bare charge, it gives the renormalized, physical charge. On the other hand, if we add, to the original Lagrangian, the counter term −δe, and re-interpret the charge parameter in the original Lagrangian as the renormalized charge, then, upon switching the radiative corrections “back on”, finite physical predictions can be obtained from the theory. Due to the multiplicative nature of renormalization, the addition of the radiative corrections (subtraction of the counter terms) amounts to a multiplication of the bare parameters by renormalization constants. The final result of all these operations is that the bare Lagrangian, written in terms of the bare quantities, is equal to the renormalized Lagrangian, written in terms of the renormalized (physical) quantities. As we carry out the entire renormalization procedure on the example of quantum electrodynamics, we will soon realize that the charge renormalization considered above, in fact, is a so-called Z1 renormalization at the vertex, which cancels with the Z2 wave function renormalization by virtue of the Ward6 –Takahashi7 identity. But let us not anticipate details of the second step before completing the first. 6 John

Clive Ward (1924–2000). Takahashi (1924–2013).

7 Yasushi

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Rather, let us start by realizing that the quantum field operators in Eq. (10.1) constitute the unrenormalized, bare quantities that are obtained when we neglect the radiative corrections that lead to the necessity of renormalization. The interaction Hamiltonian originally given in Eq. (10.3) is renormalized as follows, LI = −e0 ψ 0 A0 ψ0 = −Z1 eR ψ R AR ψR .

(10.110)

For full clarity, all renormalized quantities will be indicated with the subscript R in the following discussion. Surprisingly, the bare Lagrangian, once all renormalizations are taken into account (charge, wave function and four-vector potential), is formally equal to the renormalized Lagrangian, but with the bare parameters replaced by their renormalized counterparts and with counterterms being added. We note that the first form of the Lagrangian (10.110) is the QED interaction Lagrangian in the interpretation (i) given above, whereas the second form actually corresponds to the interpretation (iii). Indeed, the full renormalization program of QED, including the Ward–Takahashi identity, implies that the bare quantities are related to the physical quantities as follows, Z1 1/2 1/2 e , A 0 = Z 3 AR , Z 1 = Z2 . (10.111) ψ0 = Z 2 ψR , e0 = 1/2 R Z2 Z3 Concerning the multiplicative renormalizations of the electron in Eq. (10.111), we will soon realize that, for the electron charge, the Z2 renormalizations of the wave function and the Z1 vertex renormalization of the charge cancel, and the only surviving renormalization is the Z3 renormalization of the charge due to a phenomenon known as vacuum polarization. For the entire interaction Lagrangian given in Eq. (10.110), the only surviving renormalization is the overall Z1 renormalization of the charge at the vertex. The renormalization constants Z1 , Z2 and Z3 relate the bare to the physical quantities. The physical interaction Lagrangian is LI,R = −eR ψ R AR ψR ,

(10.112)

where all quantities correspond to the physical ones, and LI,R is renormalized. The bare Lagrangian is LI,0 = −e0 ψ 0 A0 ψ0 .

(10.113)

As a result of our calculations, we obtain a relation of the form LI,0 + LRAD = LI,R ,

LI,0 = LI,R − LCT ,

LCT = LRAD ,

(10.114)

where LRAD = LCT contains the radiative corrections to the physical quantities of the theory (only those radiative corrections that need to be absorbed into physical parameters). In order to obtain finite results, we need to subtract, from the Lagrangian LI,R formulated in terms of finite, physical quantities, the term LCT which contains the counter terms that we still have to identify. From Eqs. (10.110) and (10.111), we see that LCT = (Z1 − 1) eR ψ R AR ψR = (1 − Z1−1 ) e0 ψ 0 A0 ψ0 = − (Z1−1 − 1) e0 ψ 0 A0 ψ0 . (10.115)

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In the framework of the renormalization program, we thus have to identify the counter term from the vertex correction as follows, LCT = − (Z1−1 − 1) e0 ψ 0 A0 ψ0 .

(10.116)

Now, we add this term to the Lagrangian and convince ourselves one more time that the physical Lagrangian is obtained, as a function of the physical parameters, LI,0 + LCT = − e0 ψ 0 A0 ψ0 − e0 (Z1−1 − 1) ψ 0 A0 ψ0 = − e0 Z1−1 ψ 0 A0 ψ0 = −eR ψ R AR ψR ,

(10.117)

where we carefully distinguish between bare quantities (with a subscript zero) and physical quantities (with a subscript R), and all quantities have been replaced by their renormalized counterparts [see also Eq. (10.111)]. We should mention that, in the sense of Eq. (10.111), we have only taken into account the charge renormalization at the vertex, which is given by the Z1 renormalization constant, so that the simplified relation of the renormalized and the bare charge reads as eR = Z1−1 e0 . In leading order in the QED perturbative expansion, the renormalization constants Z1 , Z2 , and Z3 all have the expansion 1 + O(α). In our calculation of the vertex correction, we had found the decomposition [see Eqs. (10.45), (10.46), and (10.52)] ΓνOS (p, p′ ) = γ ν + ΛνOS (p, p′ ) ,

ΛνOS (p, p′ ) = Λν (p, p′ )∣p=p′ =m . (10.118)

ν We thus have to write the unrenormalized on-shell vertex function ΛOS (p, p′ ) as follows, ΛνOS (p, p′ ) = γ ν (Z1−1 − 1) + Z1−1 ΛνR (p, p′ ) ,

ΓνOS (p, p′ ) = Z1−1 ΓνR (p, p′ ) . (10.119)

We might note that this definition of the renormalized vertex function only works for on-shell momenta. The renormalized vertex function also exists for more general values of p and p′ , and has been investigated intensively in the latter kinematic region, in part as an input in multiloop calculations in QED bound-state problems [310, 311]. The renormalized on-shell vertex function ΛνR (p, p′ ) vanishes at zero momentum transfer [see Eqs. (10.94) and (10.95)]. In this case, the unrenormalized vertex factor −ie0 γ ν , plus the radiative correction, −ie0 ΛνOS (p, p), at zero momentum transfer, finds the representation − ie0 γ ν − ie0 ΛνOS (p, p) = −ie0 γ ν − ie0 γ ν (Z1−1 − 1) = −iZ1−1 e0 γ ν = −i eR γ ν . (10.120) Now, the question is how the separation into the renormalization (counter-)term γ ν (Z1−1 − 1) and the renormalized vertex function has to be accomplished. The answer to this question can be found as follows. One has to formulate renormalization conditions, which determine how the bare quantities are related to the physical ones. For the renormalized vertex function, the on-shell renormalization condition reads as follows, ΓνR (p, p)∣p=m = γ ν , ΛνR (p, p) = 0 ,
ΛνOS (p, p) = γ ν (Z1−1 − 1) .

(10.121a) (10.121b)

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This condition forces us to subtract the vertex at zero momentum transfer, i.e., at q = p − p = 0. Furthermore, because Z1 = 1 + O(α) is equal to unity in leading order in α, we have ΛνOS (p, p) = γ ν (Z1−1 − 1) ≈ γ ν (1 − Z1 ) , ΛνR (p, p′ )

=

Z1 ΛνOS (p, p′ ) − γ ν (1 − Z1 )

(10.122a) ≈

ΛνOS (p, p′ ) − ΛνOS (p, p) .

(10.122b)

This consideration finally justifies our subtraction, carried out in Eq. (10.95) to one-loop order, i.e., in the first order in the fine-structure constant α. If we keep Z1 in the intermediate steps (no expansion in α), then we need to write the equations Z1 ΛνOS (p, p) = γ ν (1 − Z1 ) , ΛνR (p, p′ )

=

(10.123a)

Z1 [ΛνOS (p, p′ ) − ΛνOS (p, p)]

,

(10.123b)

again with a single subtraction at zero momentum transfer. We can now also understand why there is an additional factor Z1−1 in front of the renormalized terms in Eq. (10.119). Namely, after the absorption of the first term, γ µ (Z1−1 − 1), into the physical electron charge, there is still a term left, which multiplies the unrenormalized charge e0 in the bare Lagrangian, and which must be multiplied with Z1−1 , to obtain the physical charge which multiplies the renormalized vertex function. Let us now extend this program to the complete QED Lagrangian. The original (0) QED Lagrangian L ≡ LQED , given in Eq. (10.1), is identified as the bare Lagrangian. We add a gauge-fixing term as well as a small photon mass, which as we saw comes in useful for the calculation of vertex corrections, λ0 1 2 (0) (∂µ Aµ0 ) LQED = − (∂ µ Aν0 − ∂ ν Aµ0 ) (∂µ A0ν − ∂ν A0µ ) − 4 2 λ2 + 0 A20 + ψ 0 (i∂
− e0 A0 − m0 ) ψ0 . (10.124) 2 This is the QED Lagrangian in the interpretation (i) given above: All terms are expressed in terms of bare quantities. The term proportional to λ20 A20 describes the infinitesimal photon mass. We apologize to the reader for the similarity of the fonts used for the (bare) photon mass λ0 and the (bare) gauge fixing parameter λ0 . In view of the presence of a number of variations of µ in this treatise, among them, µ for the reduced mass of a two-body bound system, and µ for the renormalization scale, we prefer λ for the photon mass. (0) Now, we can write the original Lagrangian L ≡ LQED , given in Eq. (10.1), in view of the relations (10.111), as follows, λ 0 Z3 Z3 2 (0) (∂ µ AνR − ∂ ν AµR ) (∂µ AR ν − ∂ν AR µ ) − (∂µ AµR ) LQED = − 4 2 λ2 Z 3 + 0 A2R + Z2 ψ R (i∂
− m0 ) ψR − Z1 eR ψ R AR ψR . (10.125) 2 This is the QED Lagrangian in the interpretation (iii) as discussed above. Moreover, Ward–Takahashi identities imply that we can consistently renormalize QED, Z1 = Z2 ,

λ R = Z3 λ 0 ,

λ2R = Z3 λ20 .

(10.126)

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The renormalization of the gauge-fixing parameter λ, and of the photon mass λ, is physically irrelevant. This is because the gauge-fixing parameter constitutes an auxiliary, variational parameter, and because the photon mass is infinitesimally small (and remains so after renormalization). The identity Z1 = Z2 is especially important because both renormalization constants Z1 and Z2 are gauge dependent: If they were not equal, then, by choosing a different gauge, we could assign a (R) different multiplicative correction to the renormalized interaction Lagrangian LI as opposed to the free fermionic term ψ 0 (i∂
− m0 ) ψ0 . This would destroy the gauge invariance of the theory on the level of loop corrections. Finally, the Ward– Takahashi identities imply that, after the specification of the counterterms, the QED Lagrangian reads as λ 1 µ ν 2 (∂ AR − ∂ ν AµR ) (∂µ ARν − ∂ν ARµ ) − (∂µ AµR ) 4 2 λ + A2R + ψ R (i∂
− e AR − mR ) ψR + LCT , (10.127) 2 where all quantities are equal to the renormalized, physical entities. All counterterms in LCT will be specified in the following. The bare mass m0 and the renormalized mass mR are related as m0 = mR − δm, where δm is the mass counter term. In Eq. (10.124), we had expressed the unrenormalized (bare) Lagrangian in terms of the bare quantities. By contrast, in Eq. (10.127), the renormalized Lagrangian is expressed in terms of the physical fields. The observation is that the unrenormalized (bare) Lagrangian in terms of the bare quantities, has the same functional form as the renormalized Lagrangian, expressed in terms of the physical fields. Let us therefore define a mapping of the vector of renormalization constants and physical parameters onto the Lagrangian, which read as follows, (0)

LQED = −

(0) ⃗ δm, v⃗) . (Z⃗ = (Z1 , Z2 , Z3 ), δm, v⃗ = (e, m, ψ, A, λ, λ)) → LQED (Z,

(10.128)

Defining 1⃗ = (1, 1, 1), one can write the relation (0) (0) ⃗ δm, v⃗R ) = L(0) (1, ⃗ δm, v⃗R ) , ⃗ 0, v⃗R ) + LCT (Z, LQED (⃗1, 0, v⃗0 ) = LQED (Z, QED

(10.129)

where v⃗0 is the input “vector” of bare parameters, and v⃗R is the input “vector” of renormalized parameters, v⃗0 = (e0 , µ0 , m0 , ψ0 , A0 , λ0 , λ0 ),

v⃗R = (eR , µR , mR , ψR , AR , λR , λR ).

(10.130)

This relation describes the renormalizability of the theory: The physical, renor(0) ⃗ δm, v⃗R ) expressed malized Lagrangian LR of the theory, given as LR = LQED (Z, in terms of the physical, renormalized quantities, has the same functional form as the bare Lagrangian, expressed in terms of the bare quantities. It is often stated that QED actually is defined at some ultraviolet scale and then evolved, using renormalization-group equations, into the infrared (low-energy) region. This statement finds an easy explanation: Namely, looking at a charged electron at very high momenta (very small distances), we can say that the vacuum-polarization charge

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cloud is gradually “taken away”, and in fact we see the more of the “bare” charge of the electron, which is larger than the charge seen at low momenta (large distance). Bare parameters are relevant to high energy scales. This explains why we say that the theory is defined at high energy, while the renormalization conditions used here actually describe low-energy physics. In order to derive the Ward–Takahashi identities, it is in fact very useful to take a step back and recall the vertex function at q = 0, i.e. p′ = p, initially without assuming that the incoming and outgoing bispinors are on the mass shell, Λµ (p, p) = i e2 ∫

1 d4 k νρ 1 D (k) γν γµ γρ . (2π)4 F p
− k
− m + i p
− k
− m + i

(10.131)

Using the operator identity ∂ 1 1 1 γµ =− , ∂p p − k − m + i p − k − m + i p − k − µ

m + i





(10.132)

where the sequence of terms matters, we can rewrite the one-loop vertex function at zero momentum transfer as follows, d4 k νρ 1 γρ , D (k) γν (2π)4 F p − k −

m + i (10.133) where Σ(p) is the one-loop self-energy. We can even ascertain that the first equality in the relation (10.133) is general and holds to all loop orders. A handwaving argument is as follows: One considers an arbitrary multi-loop self-energy diagram with the derivative ∂/∂pµ acting on all the different internal fermion propagators due to the product rule. This differentiation accounts for all possibilities to insert a γ µ matrix. The Ward–Takahashi identity is appropriately named because it was Takahashi who generalized this for arbitrary momentum transfer q = p′ − p to Λµ (p, p) = −

∂ Σ(p) , ∂pµ

Σ(p) = ie2 ∫

(p′µ − pµ )Γµ (p, p′ ) = SF−1 (p′ ) − SF−1 (p) ,

(10.134)

where SF−1 (p) is the inverse of the full (up to all orders) Feynman propagator (including self-energy insertions), and we refer to Eq. (10.118) for a definition of the vertex function, which we use in an extended sense here, writing Γµ (p, p′ ) = γ µ + Λµ (p, p′ ), where Λµ (p, p′ ) is an all-order vertex correction. In full analogy with Eq. (10.119), we can write an expansion for the self-energy, −1 Σ(p) = δm − [Z2−1 − 1] (p
− m) + Z2 ΣR (p) ,

(10.135)

where δm is the mass counterterm, i.e., m = m0 + δm. With the help of the Ward– Takahashi identity, we can then write for zero momentum transfer and on mass shell (p
= m), in view of the relation (10.135), ∂ Σ(p)∣p=m = γ µ (Z2−1 − 1) , Λµ (p, p)∣p=m = ΛµOS (p, p) = −

∂pµ

Z1 = Z2 . (10.136)

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Indeed, a quick look at Eq. (10.121a) then reveals that Z1 = Z2 . According to Eqs. (10.94) and (10.107), the renormalization constants can be calculated as follows, ΛµOS (p, p) = γ ν [Z1−1 − 1] = γ ν [Z2−1 − 1] = γν

α 1 Λ2 1 m 9 λ2 [ ln ( 2 ) − ln ( 2 ) + + O ( 2 )] . π 4 m 2 λ 8 m

(10.137)

For the renormalized vertex function we thus have the integral given in Eq. (10.95). 10.3.4

Detour on Dimensional Regularization

A very brief detour on dimensional regularization is in order. We follow Sec. 8.2.5 of Ref. [81]. Dimensional regularization proceeds by artificially modifying the dimension of space-time into non-integer values, the notion being that typical (logarithmic) ultraviolet as well as infrared divergences can be eliminated once the quantum-field theoretical integrals are continued analytically into a fictitious space with a non-integer value of dimensions. The analytic continuation is inspired by mathematical considerations which lead to an analytic continuation of the solid angle into an arbitrary number of dimensions. Pauli–Villars regularization respects radial symmetries of the problem; dimensional regularization does so automatically, and in addition, it also respects gauge invariance. Strictly speaking, the introduction of the photon mass in dimensional regularization breaks gauge invariance, albeit only in terms of a mathematical computation device. In compensation, in parity violating weak interactions, which involve the fifth current matrix γ 5 , there are problems associated with dimensional regularization: By construction of the electroweak theory, the γ 5 matrix is required to anticommute with all other γ µ matrices, a property required in order to consistently define chirality. However, in an arbitrary number of dimensions d, this concept is problematic; we shall not dwell into this matter here in too much detail but point out that remedies (including the so-called naive γ 5 scheme) exist (see also Sec. 10.7). In order to proceed, we need the volume surface area ΩD of the unit sphere embedded in D-dimensional space. The matching of the result for integer dimensions to the result for non-integer dimensions is recalled from Eq. (10.88) as follows, ΩD = ∫ dΩD =

2 π D/2 . Γ(D/2)

This result takes the following values for integer dimension, ⎧ 2 (D = 1) ⎪ ⎪ ⎪ ⎪ ⎪ (D = 2) ⎪ 2π ΩD = ⎨ . ⎪ 4π (D = 3) ⎪ ⎪ ⎪ 2 ⎪ ⎪ (D = 4) ⎩ 2π

(10.138)

(10.139)

One may ask about the origin of the result ΩD=1 = 2. The reason is that we attempt to replace an integral over the entire D-dimensional space ∫RD dD r with an integral

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∞

of the form proportional to ∫0 dr rD−1 over a radially symmetric function. Then, in one dimension, for a function f (x) = f (−x), with x ∈ R, we have a factor two because the “solid angle” in one dimension simply counts the number of endpoints of the line segment ∣x∣ ≤ 1, which is the one-dimensional analog of a unit sphere. Regimes x ∈ (−∞, 0) and x ∈ (0, ∞) have to be treated separately. For a radially symmetric integrand f (⃗ r) = f (r), one has ∫

R

dD r f (⃗ r ) = ΩD ∫ D

∞ 0

dr rD−1 f (r) ,

(10.140)

in a general dimension D (where r = ∣⃗ r∣ and the function f is defined accordingly, depending on the scalar or vector character of its argument). In the following, we will assume a non-integer dimension D, slightly displaced from the n = 3 by 2ε, D = 3 − 2ε ,

(10.141)

and consider the continuation of the Coulomb potential to a non-integer number of dimensions. Note that we here denote the deviation from an integer number of dimensions by the letter ε, while energy cutoffs are denoted as  (see Chap. 4). We also use an infinitesimal parameter  > 0 in propagator denominators. Our considerations here supplement other works (see Chap. 8 of Ref. [81] and Ref. [307]) where, within classical as opposed to quantum electrodynamics, electrostatic potentials have been evaluated using dimensional regularization. In the following, we typically use the symbol D = 3 − 2ε for the dimensionality of a regularized three-dimensional space, and d = 4 − 2ε

(10.142)

for the dimensionality of a regularized four-dimensional space-time. Per Eq. (10.138), the surface of the unit sphere in D = 3 − 2ε dimensions is Ω3−2ε = 2π 3−2ε /Γ[(3 − 2ε)/2]. The radial part of the Laplacian is ⃗2 ∼ ∇

D−1 ∂ ∂2 + . ∂r2 r ∂r

(10.143)

Let us define the Coulomb potential (strictly speaking, the potential energy experienced by a bound particle of charge number Z) in D dimensions as the radially symmetric potential V that solves the equation (

∂2 2 − 2ε ∂ + ) V (r) = 4πZα µ3−D δ (D) (⃗ r) . ∂r2 r ∂r

(10.144)

Here, the mass scaling is as follows. The potential V has mass dimension one, the Laplacian has mass dimension two. The Dirac-δ in D dimensions, δ (D) (⃗ r), has mass dimension D, because the dimension of the D-dimensional Dirac-δ is the inverse of the dimension of its argument. In order for the right-hand side of Eq. (10.144) to have the same mass dimension as the left-hand side, we need to insert a factor µ3−D = µ2ε , which is typical for calculations that involve dimensional regularization.

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The right-hand side of Eq. (10.144) is zero almost everywhere, namely, except at the origin of D-dimensional space. Hence, we use an ansatz of the form C C = rD−2 r1−2ε for the potential. Let us now return to the defining equation V =

(10.145)

⃗ 2 V (r) = 4πZα µ2ε δ (D) (⃗ ∇ r) ,

(10.146)

and try to determine the coefficient C in the ansatz given in Eq. (10.145). Certainly, the integral over ∫ dD r of the right-hand side of Eq. (10.146) is equal to 4πZα µ2ε , by a fundamental property of the D-dimensional Dirac-δ. For the left-hand side of Eq. (10.146), one can integrate over the D-dimensional sphere SR of radius R and obtain ∫

SR

⃗ 2 V (r) = ∫ dD r ∇

∂SR

⃗ (r) = RD−1 ΩD ( dA⃗ ⋅ ∇V

∂ V (r))∣ , ∂r r=R

D = 3 − 2ε .

(10.147) Here, Gauss’s8 theorem has been used in D dimensions, and we have taken advantage of the radial symmetry of the potential. The surface of the D-dimensional sphere of radius R is denoted as ∂SR . In view of the identity [see Eq. (10.145)] (

∂ V (r))∣ = −C (D − 2) R1−D , ∂r r=R

(10.148)

one can derive the result 1 C = −Zα µ2ε π −1/2+ε Γ ( − ε) . 2 So, the Coulomb potential in D = 3 − 2ε dimensions is V (r) = −

Zα 2ε −1/2+ε 1 µ π Γ ( − ε) . 1−2ε r 2

(10.149)

(10.150)

This relation confirms results given in Refs. [312] and [313]. 10.3.5

Detour on Feynman Parameterization

Feynman parameters have been outlined in Eq. (10.65). We will need the appropriate generalization to m denominators, which reads as follows, 1 ⎛m ⎞ 1 δ(1 − u1 − u2 − ⋯ − um ) = (m − 1)! ∏ ∫ duj . A1 A2 ⋯Am ⎝j=1 0 ⎠ [A1 u1 + A2 u2 + ⋯ + Am um ]m

(10.151)

For m = 2, we recover Eq. (10.65). There are two general identities which are useful for integrating out Feynman parameters. The first of these is given as ∫

1 0

dum δ(1 − u1 − u2 − ⋯ − um ) f (u1 , u2 , . . . , um )

= Θ(1 − u1 − u2 − ⋯ − um−1 ) f (u1 , u2 , . . . , 1 − u1 − u2 − ⋯ − um−1 ) . 8 Johann

Carl Friedrich Gauss (1777–1855).

(10.152)

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This identity holds because, if the expression u1 + u2 + ⋯ + um−1 turns out to be negative or greater than one, then the peak of the Dirac-δ function will be outside of the interval um ∈ (0, 1). If we did not assume that all the ui with i = 1, . . . , m are positive, then we would have to indicate an extra Heaviside step function Θ(u1 + u2 + ⋯ + um−1 ) on the right-hand side. The second identity is ∫

1

dum−1 Θ(1 − u1 − u2 − ⋯ − um−1 ) f (u1 , u2 , . . . , 1 − u1 − u2 − ⋯ − um−1 )

0

= Θ(1 − u1 − u2 − ⋯ − um−2 ) ×∫

1−u1 −⋯−um−2 0

dum−1 f (u1 , u2 , . . . , 1 − u1 − u2 − ⋯ − um−1 ) .

(10.153)

The upper integration limit on the right-hand side is 1 − u1 − ⋅ ⋅ ⋅ − um−2 . We need to indicate an extra Heaviside step function Θ(1 − u1 − u2 − ⋯ − um−2 ) on the righthand side because, if the expression u1 + u2 + ⋯ + um−2 is greater than one, then the integrand vanishes on the entire integration interval um−1 ∈ (0, 1). With the help of the identities (10.152) and (10.153), one can subsequently integrate out the Feynman parameters. For example, one has ∫

1 0

=∫

du1 ∫ 1

0

1 0

du1 ∫

du2 ∫ 1−u1

0

1 0

du3 δ(1 − u1 − u2 − u3 ) f (u1 , u2 , u3 )

du2 f (u1 , u2 , 1 − u1 − u2 ) ,

(10.154)

where we note the integration limits, and ∫

1 0

=∫

du1 ∫ 1

1 0

du2 ∫ 1−u1

1 0

1

du4 δ(1 − u1 0 1−u1 −u2

du3 ∫

− u2 − u3 − u4 ) f (u1 , u2 , u3 , u4 )

du2 ∫ du3 f (u1 , u2 , u3 , 1 − u1 − u2 − u3 ) . (10.155) 0 We can thus rewrite Eq. (10.151) as follows, 1 1−u1 −u2 −⋯−um−2 1−u1 1 dum−1 = (m − 1)! ∫ du1 ∫ du2 ⋯ ∫ A1 A2 ⋯Am 0 0 0 1 × . [A1 u1 + A2 u2 + ⋯ + Am−1 um−1 + Am (1 − u1 − u2 − ⋯ − um−1 )]m (10.156) Using the substitution (with unit Jacobian) v1 = 1 − u1 , v2 = 1 − u1 − u2 , (10.157) 0

du1 ∫

0

v3 = 1 − u1 − u2 − u3 , . . . , (10.158) one can show the validity of an alternative representation 1 v1 vm−2 1 = (m − 1)! ∫ dv1 ∫ dv2 ⋯ ∫ dvm−1 A1 A2 ⋯Am 0 0 0 1 × , (10.159) [A1 + v1 (A2 − A1 ) + ⋅ ⋅ ⋅ + vm−1 (Am − Am−1 )]m which is sometimes useful. By a suitable differentiation under the integral sign, one can show the more general formula α1 −1 m −1 1 1 δ(1 − ∑m ⋯uα Γ(∑m 1 k=1 uk )u1 m k=1 αk ) = du . . . , (10.160) du m 1 m ∫ ∫ m αm 1 ∑k=1 αk Aα 0 ∏k=1 Γ(αk ) 0 ( ∑m 1 ⋯Am k=1 uk Ak ) which is valid for general exponents α1 , . . . , αm .

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10.3.6

Vertex and Dimensional Regularization

Let us now delve into a calculation to illustrate the method. Eq. (10.47), generalized to d dimensions, Λν (p, p′ ) = ie2 ∫

We start from

′ ν γµ ( p dd k µρ
− k
+ m)γρ
− k
+ m)γ (p DF (k) . (10.161) d ′ 2 2 (2π) [(p − k) − m + i][(p − k)2 − m2 + i]

It is important to mention here that also in d = 4 − 2ε dimensions [see Eq. (10.142)], we still have one time-like dimension and thus d − 1 space-like dimension. Consequently, the dimension of the Wick rotation, which affects only the time-like component of the photon four-momentum, remains unchanged. One immediately visible advantage of dimensional regularization, in comparison to Pauli–Villars regularization (see Sec. 10.3.1), is that we do not need to introduce a separate infrared regulator (photon mass λ). The integration over the L parameter, per Eq. (10.50), is thus unnecessary, and we can directly proceed to the combination of the propagator denominators via Feynman parameterization. Evaluating Eq. (10.161) on mass shell, we find ′ ν dd k g µρ γµ (p
− k
+ m) γ (p
− k
+ m)γρ ∣ . d 2 2 ′ 2 (2π) k + i [k − 2p ⋅ k + i][k − 2p ⋅ k + i] p=p′ =m

(10.162) Here, we use the same definition of the on-mass-shell condition as in Eq. (10.53): ν ′ ′ whenever p
in the numerator of ΛOS (p, p ) is brought to the right, or p
is brought to the left, it will be replaced by the mass m. We can use our result from Eq. (10.63), where we already calculated the Dirac algebra in d dimensions, to write

ΛνOS (p, p′ ) = −i e2 ∫

Nν dd k , (10.163a) d 2 2 ′ (2π) [k + i][k − 2p ⋅ k + i][k 2 − 2p ⋅ k + i] d N ν = −4 mk ν + (d − 6)γ ν k 2 + 4k
[p′ν + pν −( − 1) k ν]+ 4γ ν (p′ − k) ⋅ (p − k) . 2 (10.163b)

ΛνOS (p, p′ ) = −ie2 ∫

In combining the denominators, we proceed as outlined in Sec. 10.3.4, use Eq. (10.156) for m = 3, to combine the denominators in Eq. (10.163), and find ΛνOS (p, p′ ) = −ie2 (2!) ∫ ×∫

CF

CF

0

du1 ∫

1 0

du2 ∫

0

1

du3 δ(1 − u1 − u2 − u3 )

Nν dd k , (2π)d [u1 k 2 + u2 k 2 − 2u2 p′ ⋅ k + u3 k 2 − 2u3 p ⋅ k]3

= −2ie2 ∫ ×∫

1

1 0

du1 ∫

1 0

du2 ∫

1 0

(10.164)

du3 δ(1 − u1 − u2 − u3 )

Nν dd k , (2π)d [k 2 − 2k ⋅ (u2 p′ + u3 p)]3

where we have suppressed the infinitesimal imaginary part i in view of the specification of the Feynman contour CF . The integrand could be rearranged with the

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help of the identity u1 + u2 + u3 = 1. The square in the denominator is completed according to k = ` + u2 p′ + u3 p ,

(10.165a)

′

2

D = k − 2k ⋅ (u2 p + u3 p) = ` − ∆ ,

(10.165b)

∆ = m (u2 + u3 ) − q u2 u3 .

(10.165c)

2

2

2

2

We rewrite the loop momentum k in terms of `, with unit Jacobian, and use the replacement 1 (10.166) `µ `ν → g µν `2 , d in the numerator in Eq. (10.164), motivated by the angular structure in d dimensions. Terms linear in ` in the numerator vanish after loop momentum integration, due to symmetry. We can thus replace the numerator structure in Eq. (10.164), originally defined in Eq. (10.63), as follows, d (d − 2)2 ν 2 γ ` + 2γ ν m2 {2 − 2u2 + (1 − ) u22 − 2u3 + (2 − d) u2 u3 d 2 d d 2 ν 2 + (1 − ) u3 } + 2γ q [u2 + u3 − 1 + (1 − ) u2 u3 ] 2 2 d d d − 4m i σ νρ qρ [u2 + (1 − ) u22 + u3 + 2 (1 − ) u2 u3 + (1 − ) u23 ] . (10.167) 2 2 2 Here, we have left out a term proportional to q ν on the right-hand side, which, per our discussion following Eq. (10.92), vanishes after integration. Another argument to show this is as follows. From the Ward–Takahashi identity, given in Eq. (10.134), which we use in the form (p′ν − pν ) Λν (p, p′ ) = SF−1 (p′ ) − SF−1 (p), one can infer that any conceivable contribution to Λµ (p, p′ ), proportional to q ν , needs to vanish. This is because it would lead to a term (p′ν − pµ )Λν (p, p′ ) → qν q ν = q 2 , which cannot be reproduced by any on-mass-shell self-energy insertion on the right-hand side SF−1 (p′ ) − SF−1 (p), in view of the decomposition (10.135). We are interested in the numerator N ν for d = 4 − 2ε dimension, i.e., close to the ordinary four dimensions of space-time. As far as N ν is concerned, it turns out to be advantageous to set d = 4 − 2ε everywhere except the first term, and we write (d − 2)2 ν 2 γ ` + γ ν {2m2 [2 − 2u2 + (ε − 1) u22 − 2u3 + 2 (ε − 1) u2 u3 + (ε − 1) u23 ] Nν = d + 2q 2 [u2 + u3 − 1 + (ε − 1) u2 u3 ]} Nν →

− 2mi σ νρ qρ [u2 + (ε − 1) u22 + u3 + 2 (ε − 1) u2 u3 + (ε − 1) u23 ] =

(d − 2)2 ν 2 γ ` + γ ν N1 + iσ νρ qρ N2 . d

(10.168)

Here, N1 = 2m2 [2 − 2u2 + (ε − 1) u22 − 2u3 + 2 (ε − 1) u2 u3 + (ε − 1) u23 ] + 2q 2 [u2 + u3 − 1 + (ε − 1) u2 u3 ] , N2 =

− 2m [u2 + (ε − 1) u22

+ u3 + 2 (ε − 1) u2 u3 + (ε − 1) u23 ]

(10.169a) .

(10.169b)

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We can now put these simplified expressions for the numerator and denominator back into ΛνOS . However, there is an obstacle. Namely, the charge e in d dimensions cannot be dimensionless. In order to see this, let us consider the QED interaction Lagrangian in d dimensions [compare with Eq. (10.3) for the Lagrangian density], LI = −e ∫ dd x ψ(x) γ µ ψ(x) Aµ (x) .

(10.170)

Let M temporarily denote a mass scale, which, in natural units, is equal to the inverse of a length scale. Then, if we denote the mass dimension of a physical quantity X by [X], [dd x] = M −d ,

[ψ] = M (d−1)/2 ,

[Aµ ] = M (d−2)/2 .

(10.171)

The latter mass scaling can be derived based on the requirement that the free electromagnetic part of the Lagrangian (the Maxwell Lagrangian in d dimensions) be dimensionless. Namely, we have L = −(1/4) ∫ dd x Fµν F µν , where Fµν = ∂µ Aν − ∂ν Aµ . The mass scaling of ψ can be derived from the requirement that the integral of the spatial probability density operator (absolute square of the field operator), given as ∫ dd−1 x ψ + (x) ψ(x), must be dimensionless. Now, for LI in Eq. (10.170) to be dimensionless, in view of the mass scalings in Eq. (10.171), LI can be made dimensionless provided e acquires a mass dimension of [e] = M 2−d/2 = M ε ,

d = 4 − 2ε .

(10.172)

We verify that if d = 4, the charge e is dimensionless. In a general number of d = 4 − 2ε dimensions, it turns out that a favorable ansatz (known as “modified minimal subtraction”, or MS) for the square of the charge is e2 = (4π)d/2−1 α µ4−d eγE (2−d/2) = (4π)1−ε α µ2ε eγE ε ,

(10.173)

where γE = 0.57721 56649 . . . is the Euler–Mascheroni constant, which we had already encountered in Eq. (4.272). Notice that we were forced to introduce an arbitrary mass parameter µ in order to keep the dimensions consistent. The introduction of µ is characteristic of dimensional regularization and has already appeared in Eq. (10.144). The parameter µ finds a natural interpretation as a renormalization scale. With the help of Eqs. (10.164) and (10.173), as well as Eq. (10.168), we find for ΛOS (p, p′ ), ΛνOS (p, p′ ) = − 8πiαµ2ε eγE ε (4π)−ε ∫ × δ(1 − u1 − u2 − u3 ) ∫

1 0

du1 ∫

dd ` (2π)d

1

du2 ∫

1

0 0 (d−2)2 ν 2 γ ` d

du3

+ γ ν N1 + iσ νρ qρ N2 . (`2 − ∆ + i)3 (10.174)

Following the discussion in Sec. 10.3.1 [see the text surrounding Eq. (10.84)], we actively shift the integration contour along the time-like dimension to the imaginary

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axis. We proceed to do this in one time-like (Wick-rotated) dimension, and 3 − 2ε spatial dimensions, `⃗ = `⃗E ,

`0 → i`0E ,

`2 → −`2E ,

d d ∫ d ` → i ∫ d `E .

(10.175)

This results in ΛνOS (p, p′ ) = 8παµ2ε eγE ε (4π)−ε ∫

1

0

du1 ∫

1 0

du1 ∫

1 0

du2 ∫

1 0

0

dd `E (2π)d

× δ(1 − u1 − u2 − u3 ) ∫ = 8πα ∫

1

du2 ∫ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

1

du3 0 2 (d−2) γ ν `2E d (`2E + ∆)3

−

⎤ iσ νρ qρ N2 ⎥⎥ γ ν N1 − (`2E + ∆)3 (`2E + ∆)3 ⎥⎥ ⎦

du3 δ(1 − u1 − u2 − u3 ) [T1ν (q 2 ) + T2ν (q 2 ) + T3ν (q 2 )]

= I1ν (q 2 ) + I2ν (q 2 ) + I3ν (q 2 ) ,

(10.176)

where q = p′ − p and the terms Tjν (q 2 ) and Ijν (q 2 ) (k = 1, 2, 3) are defined in the obvious way; they will be used in the following. We shall make use of the fact that the vertex correction can be expressed in terms of the momentum transfer q. We note the overall sign change in the denominator term (`2 − ∆)3 → −(`2E + ∆)3 . The term T1ν (q 2 ) with the Euclidean loop momentum square in the numerator is, very explicitly, T1ν (q 2 ) = γ ν µ2ε eγE ε (4π)−ε = γ ν µ2ε eγE ε (4π)−ε = γ ν µ2ε eγE ε

`2E dd `E (d − 2)2 ∫ 2 d d (2π) (`E + ∆)3 ∞ `d+1 (d − 2)2 2π d/2 E d` E ∫ d (`2E + ∆)3 (2π)d Γ( d2 ) 0

(1 − ε)2 1 1 , 16 π Γ(1 − ε) sin(επ) ∆ε

(10.177)

where we set d = 4−2ε in the result, and we have used the result for the d-dimensional solid angle, Ωd = (2π d/2 )/Γ( d2 ). We also note that the dependence on the momentum transfer enters via the implicit dependence of ∆ on q 2 . We must restrict d to the range −2 < d < 4 in order for the integral over `E to converge. For the second and third integrals, using a similar procedure, we get dd `E γ ν N1ν + iσ νρ q ρ N2 (2π)d (`2E + ∆)3 ν 1 1 γ N1 + iσ νρ q ρ N2 . 32π Γ(−ε) sin(επ) ∆1+ε

T2ν (q 2 ) + T3ν (q 2 ) = − µ2ε eγE ε (4π)d/2−2 ∫ = µ2ε eγE ε

(10.178) (10.179)

For I1ν (q 2 ), we find I1ν (q 2 ) = 8πα ∫ = γν

1 0

du1 ∫

1

du2 ∫

1

du3 δ(1 − u1 − u2 − u3 ) T1ν (q 2 )

0 0 1 (1 − ε)2 µ2ε eγE ε

(10.180)

1−u2 α 1 . du3 ∫ du2 ∫ 2 Γ(1 − ε) sin(επ) 0 (m2 (u2 + u3 )2 − q 2 u2 u3 )ε 0

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We recall the definition of ∆ in Eq. (10.165c). Because the calculation is already complicated as it stands, we elect to expand the integrand of I1 in q 2 and obtain α (1 − ε)2 µ2ε eγE ε 1 I1ν (q 2 ) = γ ν 2 Γ(1 − ε) sin(επ) [m(u2 + u3 )]2ε q4 q 2 u2 u3 ε u22 u23 ε + ) , (10.181) m2 (u2 + u3 )2 2m4 (u2 + u3 )4 0 0 where we ignore terms of order ε2 in the integrand (it would otherwise be incurred in the term proportional to q 4 ). This happens as we anticipate that these vanish in the limit ε → 0 in the final result for the form factors. We also note that ε acts as a suitable ultraviolet as well as infrared regulator: Simply, it regularizes the dimensionality of space-time, regularizing, simultaneously, infrared as well as ultraviolet divergences. We change variables as follows, ×∫

1

du2 ∫

1−u2

du3 (1 +

u2 = ρ ω ,

u3 = ρ (1 − ω) ,

(10.182)

which allows to convert the integration limits in both variables to the interval (0, 1), the Jacobian being equal to ρ. Hence, we can write I1ν (q 2 ) as α (1 − ε)2 µ 2ε eγE ε I1ν (q 2 ) = γ ν ( ) 2 Γ(1 − ε) m sin(επ) ×∫

1 0

dρ ∫

1 0

dω ρ1−2ε (1 +

q2 ε q4 ε 2 ω(1 − ω) + ω (1 − ω)2 ) m2 2m4

1 q2 1 q4 q6 α 1 µ2 [ − 1 + ln ( 2 ) + + + O ( 6 )] . (10.183) 2 4 4π ε m 6m 60 m m We now turn our attention to the infrared divergent integral I2ν (q 2 ), which contributes to the form factor F1 (q 2 ), in contrast to the term T3ν (q 2 ), which contributes to the magnetic F2 (q 2 ) form factor. One finds, after an expansion in powers of the momentum transfer, = γν

I2ν (q 2 ) = 8πα ∫

1 0

du1 ∫

1

du2 ∫

0 2ε γE ε

1 0 1

du3 δ(1 − u1 − u2 − u3 ) T2ν (q 2 )

= γν

1−u2 N1 α 1 µ e du3 ∫ du2 ∫ 4 Γ(−ε) sin(επ) 0 (m2 (u2 + u3 )2 − q 2 u2 u3 )1+ε 0

= γν

α 1 5 µ2 1 q2 1 µ2 { + + ln ( 2 ) − [1 + + ln ( )] 2π ε 2 m 3 m2 ε m2

1 q4 1 µ2 q6 [2 + + ln ( 2 )] + O ( 6 )} . 4 20 m ε m m ν 2 Similarly, we carry out the calculation of I3 (q ), with the result −

I3ν (q 2 ) = 8πα ∫

1 0

= iσ νρ qρ =

du1 ∫

1 0

du2 ∫

0 2ε γE ε

1

(10.184)

du3 δ(1 − u1 − u2 − u3 ) T3ν (q 2 )

1 1−u2 N2 α 1 µ e du3 ∫ du2 ∫ 2 4 Γ(−ε) sin(επ) 0 (m (u2 + u3 )2 − q 2 u2 u3 )1+ε 0

iσ νρ qρ α 1 q2 1 q4 q6 [1 + + + O ( 6 )] . 2 4 2m 2π 6m 30 m m

(10.185)

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For the anomalous magnetic moment term, one can actually set ε = 0 in all intermediate steps, since neither infrared nor ultraviolet divergences are being encountered. For amusement, it is actually quite entertaining to renounce on the expansion in q 2 , do the same substitutions as for the integral I2ν (q 2 ), to obtain −1 1 1 iσ νρ qρ α q2 dω [1 − ω (1 − ω)] I3ν (q 2 ) = ∫ ∫ 2m 2π 0 0 m2 ¿ ⎛Á q 2 /m2 ⎞ iσ νρ qρ α 4 À ⎟, √ √ = arctan ⎜Á (10.186) 2 2 2m 2π q 2 /m2 4 − q 2 /m2 ⎝ 4 − q /m ⎠ which confirms the corresponding terms in Eqs. (10.95), (10.99), and (10.100). Based on Eqs. (10.176), (10.183), (10.184), and (10.185), we can write down the result for the unrenormalized, on-shell one-loop vertex correction ΛνOS (p, p′ ), in dimensional regularization, as follows, ΛνOS (p, p′ ) = I1ν (q 2 ) + I2ν (q 2 ) + I3ν (q 2 ) = γν +

3 3 µ2 α q2 1 1 1 m2 α [1 + + ln ( 2 )] + γ ν [ 2 (− − + ln ( 2 )) π 4ε 4 m π m 8 6ε 6 µ 11 1 1 m2 q6 q4 (− − + ln ( 2 )) + O ( 6 )] 4 m 240 40ε 40 µ m

iσ νρ qρ α 1 q2 1 q4 q6 [1 + + + O ( )] . (10.187) 2m 2π 6 m2 30 m4 m6 The on-mass-shell renormalization condition can be implemented by subtracting the q 2 -independent term, as outlined in Sec. 10.3.1 [see Eq. (10.95)], namely ΛνR (p, p′ ) = ΛνOS (p, p′ ) − ΛνOS (p, p) . (10.188) Hence, ΛνR (p, p′ ) differs from Eq. (10.163a) by the subtraction of the q 2 -independent term, which also contains the logarithm ln (µ2 /m2 ). Following Eq. (10.96), we use the form factors F1 and F2 , i νµ ΓνR (q) = γ ν + ΛνR (p, p′ ) = F1 (q 2 ) γ ν + F2 (q 2 ) σ qµ . (10.189) 2m From Eq. (10.163a), we read off the results α q2 1 1 1 m2 + ln ( 2 )) F1 (q 2 ) = 1 + [ 2 (− − π m 8 6ε 6 µ +

+

11 1 1 m2 q6 q4 (− − + ln ( )) + O ( )] , m4 240 40ε 40 µ2 m6

(10.190a)

α 1 q2 1 q4 q6 [1 + + + O ( )] . (10.190b) 2π 6 m2 30 m4 m6 If we set the renormalization mass scale equal to the electron mass, µ = m, then the results specialize to α q2 1 1 q4 11 1 q6 F1 (q 2 ) = 1 + [ 2 (− − ) + 4 (− − ) + O ( 6 )] , (10.191a) π m 8 6ε m 240 40ε m F2 (q 2 ) =

F2 (q 2 ) =

α 1 q2 1 q4 q6 [1 + + + O ( 6 )] . 2 4 2π 6m 30 m m

(10.191b)

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k+q

k

k

q Fig. 10.4

The Feynman diagram for the vacuum polarization involves a closed fermion loop.

The result given in Eq. (10.191a) constitutes the conversion of the one given in Eq. (10.103) into dimensional regularization, while the result in Eq. (10.191b) confirms Eq. (10.100). In Eq. (10.191a), ε acts as an infrared regulator. Note that, in contrast to the result (10.103), the photon mass λ has been replaced by a dimensional infrared regulator. We anticipate that, as we will show in Sec. 11.4.4, the infrared divergence of the high-energy part of the bound-state self-energy (in dimensional regularization) will be compensated by an opposite divergent term (in 1/ε) coming from the low-energy part of the Lamb shift. At this point, it is useful to remember that the on-shell condition states that F1 (q 2 = 0) = 1 (because of the charge renormalization), while the value F2 (q 2 = 0) does not require any additional normalization. In fact, the result F2 (q 2 = 0) = α/(2π) reproduces Schwinger’s expression [53] for the anomalous magnetic moment of the electron. 10.4 10.4.1

Vacuum Polarization Initial Considerations

We consider the Feynman diagram in Fig. 10.4. The photon four-momentum k enters the loop, while the fermion momentum circulating inside the loop is q. According to Table 10.1, we need to assign the following expressions: −i e γ µ and −i e γ ν for the fermion-photon vertices, iSF (q) and iSF (k + q) for the fermion propagators, and iDρµ as well as iDνσ for the photon propagators. A further factor (−1) is incurred for the closed fermion loop. Hence, the vacuum-polarization insertion into the photon propagator leads to the following expression, iDρσ (k) → (−1) ∫

dd q iDρµ (k)(−ieγ µ )αβ i [SF (q + k)]βγ (2π)d

× (−ieγ ν )γδ i [SF (q)]δα iDνσ (k) .

(10.192)

Here, α, β, γ and δ are spinor indices, which are summed over. Because the first and the last spinor index are paired off, we encounter the trace in spinor space. By contrast, ρ, µ, ν and σ are Lorentz indices; repeated Lorentz indices are also summed over.

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This can be reformulated as dd q Tr [γ µ SF (q + k) γ ν SF (q)]} iDνσ (k) . (2π)d ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

iDρσ (k) → iDρµ (k) {(−1) e2 ∫

= i Πµν (k)

(10.193) We write the sum of the tree-level photon propagator and the one-loop correction in terms of the vacuum-polarization tensor Πµν (k), iDρσ (k) = iDρσ (k) + iDρµ (k) [i Πµν (k)] iDνσ (k) + ⋯ ,

(10.194)

where i Πµν (k) = (−1) e2 ∫

dd q Tr [γ µ SF (q + k) γ ν SF (q)] . (2π)d

(10.195)

The following considerations are easiest when we consider the original photon propagator to be formulated in Landau gauge, 1 kµ kν Dµν (k) = − 2 P µν (k) , P µν (k) = g µν − 2 . (10.196) k + i k The projector P µν (k) has the property P µν (k) Pνρ (k) = P µ ρ (k) .

(10.197)

Let us have a look at the exact photon propagator and how it is affected by vacuum polarization. From Eq. (10.194), we have Dρσ (k) = Dρσ (k) − Dρµ (k) Πµν (k) Dνσ (k) + ⋯.

(10.198)

One can show (see also Sec. 10.7) that the vacuum-polarization tensor Πµν (k) has the following structure, Πµν (k) = k 2 P µν (k) Π(k 2 ) ,

(10.199)

with a scalar vacuum polarization correction Π(k ). The modified photon propagator becomes 1 Pρσ (k) Dρσ (k 2 ) = − 2 k + i 1 1 − (− 2 Pρµ (k)) k 2 P µν (k) Π(k 2 ) (− 2 Pνσ (k)) + ⋯ k + i k + i 2 1 1 = − 2 Pρσ (k) − ( 2 ) [k 2 Π(k 2 )] Pρσ (k) + ⋯ k + i k + i 1 = − 2 Pρσ (k) . (10.200) k [1 − Π(k 2 )] + i 2

The residue of the photon propagator at k 2 = 0 is thus shifted from unity, to a value 1 ≈ 1 + Π(0) = Z3 , Z3 ≡ 1 + Π(0) . (10.201) 1 − Π(0) In Eq. (10.295), we shall realize that the proper identification is Z3 ≡ 1 + ∏(0), even if, in one-loop order, the difference does not matter. If we wish to retain a unit

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residue, then we must add to the QED Lagrangian a counter term, to be specified in the following, so that the summation of diagrams due to the counter term results in the subtraction of the polarization operator insertion at k 2 = 0, to yield the renormalized vacuum-polarization insertion ΠR (k 2 ) = Π(k 2 ) − Π(k 2 = 0) .

(10.202)

A closer inspection reveals that, somewhat surprisingly, this can be achieved if one adds, to the QED Lagrangian, a counter term [see also Eq. (10.270c)] 1 (10.203) LCT = − (Z3 − 1)F µν Fµν , 4 where Z3 is defined in Eq. (10.201), and F µν = ∂ µ Aν − ∂ ν Aµ is the field-strength tensor of QED. This is most easily seen by going into momentum space, where Fµν (k) = −i (kµ Aν (k) − kν Aµ (k)), and considering all possible contractions of the field operators (see also Sec. 10.6.3). One treats the counter term given in Eq. (10.203) perturbatively. A summation of diagrams involving, instead of the vacuum-polarization insertion given in Fig. 10.4, the counter term LCT , then yields a modified photon propagator of the form 1 Pµν (k) (10.204) DR,ρσ (k 2 ) = − 2 k Z3 [1 − Π(k 2 )] + i where Z3 [1 − Π(k 2 )] ≈ [1 + Π(0)] [1 − Π(k 2 )] ≈ 1−[Π(k 2 ) − Π(0)] = 1−ΠR (k 2 ) . (10.205) Alternatively, we observe that a change in the residue of the photon propagator at k 2 = 0 is equivalent to a renormalization of the charge. Specifically, the physical √ charge eR is obtained as eR = Z3 e0 , where e0 is the bare charge, i.e. the charge parameter that appears in the bare Lagrangian (see also Sec. 10.6.1). 10.4.2

Vacuum Polarization and Dimensional Regularization

After this initial discussion, let us turn to the actual calculation of the vacuum polarization diagram in 1-loop order. Using the well-known expressions for the propagator SF [see Eq. (7.154)] in Eq. (10.195), we obtain i Πµν (k 2 ) = −e2 ∫

γ µ (
q + k
+ m)γ ν (
q + m) dd q Tr { }, (2π)d (q 2 − m2 + i) [(q + k)2 − m2 + i]

(10.206)

where d = 4 − 2ε is the dimension of space-time. By naive power counting, the integral of the vertex correction, given in Eq. (10.47), is logarithmically divergent at large loop momentum k. This is because the photon propagator goes as ∝ 1/k 2 , while both fermionic propagators are proportional to 1/k. Together with the integration measure ∫ d4 k, logarithmic divergence is observed. For the integral (10.206), naive power counting results in a quadratic divergence for large q. We should note that the integration over ∫ dd q in Eq. (10.206) is

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an integration over the four-momentum of a fermionic particle, so that the Pauli– Villars prescription of subtracting an auxiliary heavy photon propagator, as done in Sec. 10.3.1, requires a modification. Namely, one has to subtract the contribution from auxiliary heavy fermions (possibly, more than one) inside the fermion loop. Naively, one might think that it will be required to subtract the contribution of at least three auxiliary fermions in order to reduce the degree of divergence of the integral from quadratic to linear, and then, to logarithmic. Finally, with the third subtraction, convergence will be reached. However, a more careful analysis, to be completed below, reveals that Πµν (k 2 ) has the form Πµν (k 2 ) = k 2 (g µν − k µ k ν /k 2 ) Π(k 2 ), where Π(k 2 ) has a finite limit as k 2 → 0. The additional factor k 2 implies that the divergence of the integral given in Eq. (10.206) is only logarithmic, and the subtraction of the contribution of only one auxiliary heavy fermion is sufficient to ensure the convergence of the loop integral. Still, the calculation of the integral in Eq. (10.206) is much easier in dimensional as opposed to Pauli–Villars regularization. One avoids any problems involved with the introduction of “auxiliary heavy fermions”, and the subtraction procedure becomes much more clear. In addition, one can point out that the Pauli–Villars subtraction procedure only works if the mass M of the auxiliary fermion is much larger than the incoming photon momentum scale k. This issue is circumvented when using dimensional regularization. In evaluating the Dirac trace in Eq. (10.206), we naturally encounter the problem of calculating traces of Dirac γ matrices in d dimensions. Some introductory notes on this topic are therefore required. The relations we are going to find have a strong connection to those that we encountered in Sec. 10.3.1. The trace of the product of two γ matrices can be calculated with the help of the fundamental anticommutator relation, Tr [γ µ γ ν ] = Tr [2g µν 1 − γ ν γ µ ] = 8g µν − Tr [γ ν γ µ ] .

(10.207)

It follows that Tr [γ µ γ ν ] = 4g µν ,

Tr [γµ γ µ ] = 4g µν gµν = 4d .

(10.208)

Despite working in d dimensions, we set Tr 1 = 4 ,

(10.209)

because the trace of the unit matrix is obtained as a universal prefactor in all traces and can be safely set equal to four right at the start of the calculation. The reason is that d = 4 − 2ε differs from four only by a term of order ε. Because the term of order 1/ε of the vacuum-polarization function Π(k 2 ) is momentum-independent (just a constant). An alternative convention like Tr 1 = d would thus only add a constant term to Π(k 2 ), which is absorbed in the renormalization anyway. It is possible to show that the trace of an odd number of Dirac γ matrices vanishes, Tr [γ µ1 γ µ2 ⋯ γ µ2n+1 ] = 0 .

(10.210)

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For the trace of four γ matrices, one has, based on the repeated application of relations akin to γ ν γ κ = {γ ν , γ κ } − γ κ γ ν , Tr [γ µ γ ν γ κ γ λ ] = Tr [2g µν γ κ γ λ ] − Tr [2g µκ γ ν γ λ ] + Tr [2g µλ γ ν γ κ ] − Tr [γ ν γ κ γ λ γ µ ] . (10.211) Based on a cyclic permutation of the matrix product (argument of the last trace), and the property Tr [γ κ γ λ ] = 4g κλ [see Eq. (10.207)], we have Tr [γ µ γ ν γ κ γ λ ] = 4g µν g κλ − 4g µκ g νλ + 4g µλ g νκ .

(10.212)

Further considerations on traces of Dirac γ matrices can be found in Sec. 11.6. Using the derived formulas for the traces in Eqs. (10.207) and (10.212) we find for the numerator of Eq. (10.206), N µν = Tr [γ µ (
q + k
+ m)γ ν (
q + m)] = Tr [γ µ (
q + k
)γ ν
q] + m2 Tr [γ µ γ ν ] = g µν [4m2 − 4q ⋅ (q + k)] + 4q µ (q + k)ν + 4q ν (q + k)µ ,

(10.213)

where in going from the first to the second line, we have used the fact that the trace of an odd number of γ-matrices vanishes. The two denominators in Eq. (10.206) are combined using a Feynman parameter, 1 1 1 = ∫ dx , AB [A x + B (1 − x)]2 0

(10.214)

which together with the above result for the trace leads to 1 dd q g µν [m2 − q ⋅ (q + k)] + q µ (q + k)ν + q ν (q + k)µ dx . ∫ (2π)d 0 [q 2 + 2q ⋅ k(1 − x) + k 2 (1 − x) − m2 ]2 (10.215) In order to complete the square in the denominator, we shift the integration variable according to

iΠµν (k 2 ) = −4e2 ∫

` = q + k (1 − x) ,

(10.216)

which yields for Π (k ) the formula µν

2

iΠµν (k 2 ) = −4e2∫

1 N µν dd ` dx , ∫ (2π)d 0 [`2 + k 2 x(1 − x) − m2 ]2

(10.217)

where the numerator is N µν = g µν (m2 − `2 + k ⋅ `(1 − 2x) + k 2 x(1 − x)) + 2`µ `ν − (1 − 2x)(k µ `ν + k ν `µ ) − 2k µ k ν x(1 − x) .

(10.218)

Due to Lorentz invariance, all terms linear in ` vanish when integrated over the d-dimensional space. Furthermore, as described for the vertex correction, we have `µ `ν → d1 g µν `2 . The expression for Πµν (k 2 ) thus simplifies to iΠµν (k 2 ) = −4e2 ∫

1 ) `2 + m2 + k 2 x(1 − x)] − 2k µ k ν x(1 − x) g µν [( 2−d dd ` d , dx ∫ (2π)d 0 (`2 − ∆)2 (10.219)

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where ∆ = m2 − k 2 x(1 − x) .

(10.220)

A suitable Wick rotation of the integration momentum, following `0 → i`0E and `2 → −`2E , results in d−2 µν 2 2 2 µ ν dd `E 1 g [( d ) `E + m + k x(1 − x)] − 2k k x(1 − x) dx , ∫ (2π)d 0 [`2E + ∆]2 (10.221) where e2 in d dimensions is, according to Eq. (10.173),

iΠµν (k 2 ) = −4ie2 ∫

e2 = 4πα µ2ε eγE ε (4π)−ε .

(10.222)

It is now possible to carry out the d-dimensional integration over the loop momentum, in terms of integrals K1 and K2 , to be defined according to Πµν (k 2 ) = − 16πα µ2ε eγE ε (4π)−ε ∫ × [g µν (

1 0

dx ∫

dd `E 1 (2π)d [`2E + ∆]2

d−2 2 ) `E + g µν (m2 + k 2 x(1 − x)) − 2k µ k ν x(1 − x)] d 1

= − 16πα ∫ dx {g µν K1 + [g µν (m2 + k 2 x(1 − x)) − 2k µ k ν x(1 − x)] K2 } 0

= J1µν (k 2 ) + J2µν (k 2 ) .

(10.223)

The terms J1µν (k 2 ) and J2µν (k 2 ) are generated by the terms K1 = K1 (k 2 ) and K2 = K2 (k 2 ) in the integrand. The K1 term reads as follows, `2E d−2 dd `E ∫ d (2π)d (`2E + ∆)2 1 1 = − µ2ε eγE ε ∆1−ε , 16 π Γ(1 − ε) sin(επ)

K1 (k 2 ) = µ2ε eγE ε (4π)−ε

(10.224)

while K2 is given as dd `E 1 2 d (2π) (`E + ∆)2 1 1 1 = µ2ε eγE ε . 16 π Γ(1 − ε) sin(επ) ∆ε

K2 (k 2 ) = µ2ε eγE ε (4π)−ε ∫

(10.225)

The dependence on k 2 and x is due to the denominator structure ∆ defined in Eq. (10.220). We have 1 µ2ε eγE ε 1−ε (10.226) ∫ dx ∆ , Γ(1 − ε) sin(επ) 0 1 µ2ε eγE ε −ε µν 2 2 µ ν J2µν (q 2 ) = −α ∫ dx ∆ [g [m + k x(1 − x)] − 2k k x(1 − x)] , Γ(1 − ε) sin(επ) 0 (10.227)

J1µν (q 2 ) = α g µν

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(overall minus sign for J2µν (q 2 )). In the sum of J1µν (q 2 ) and J2µν (q 2 ), using Eq. (10.220), we can find the tensor structure Πµν (k 2 ) = (k 2 g µν − k µ k ν ) Π(k 2 ) .

(10.228)

The scalar function Π(k 2 ) is calculated as Π(k 2 ) = − 2α = −

1 µ2ε eγE ε −ε ∫ dx x(1 − x) ∆ Γ(1 − ε) sin(επ) 0

1 α 1 µ2 2α k2 [ + ln ( 2 )] + ∫ dx x(1 − x) ln (1 − x(1 − x) 2 ) 3π ε m π 0 m

µ2 α 1 [ + ln ( 2 )] + ΠR (k 2 ) , 3π ε m ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

=−

(10.229)

=Π(k2 =0)

where 1 2α k2 ∫ dx x(1 − x) ln (1 − x(1 − x) 2 ) π 0 m ¿ ¿ ⎧ ⎫ ⎛Á k 2 /m2 ⎞⎪ ⎪ ⎪ 4 − k 2 /m2 α ⎪ ⎪ 12 + 5k 2 /m2 2(2 + k 2 /m2 ) Á ⎪ Á Á À À ⎟⎬ . + arctan ⎜ = ⎨− 2 2 2 2 2 2 2 2 π ⎪ 9k /m 3(k /m ) k /m 4 − k /m ⎪ ⎪ ⎝ ⎠⎪ ⎪ ⎪ ⎩ ⎭ (10.230)

ΠR (k 2 ) =

We encounter the same variable

¿ ⎛Á k 2 /m2 ⎞ À ⎟ ξ(k 2 ) = arctan ⎜Á 2 2 ⎝ 4 − k /m ⎠

as in Eq. (10.186). The asymptotic limits are as follows, ⎡ ⎤ 2 3 ⎧ ⎥ ⎪ 1 k2 1 k2 α ⎢⎢ 1 k 2 ⎪ ⎥ ⎪ + ( ) + ( ) + ⋯ − ⎪ ⎢ ⎥ ⎪ 2 2 2 ⎪ π ⎢ 15 m 140 m 945 m ⎥ ⎪ ⎪ 2 ⎣ ⎦ ΠR (k ) = ⎨ ⎪ ⎪ 2 2 ⎪ α 1 k 5 2m ⎪ ⎪ ⎪ [ ln (− 2 ) − − 2 + ⋯] ⎪ ⎪ π 3 m 9 k ⎩

(10.231)

k 2 ≪ m2 . k 2 ≫ m2

(10.232) For k 2 /m2 ≫ 1, the incoming virtual photon can produce a pair, which is manifest in the imaginary part due to the logarithm. 10.4.3

Vacuum Polarization and Coulomb Potential

The Coulomb potential is due to the exchange of space-like virtual photons with ⃗ four-momenta k µ = (0, k), ⃗ = − 4πZα . V (k) (10.233) k⃗2

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Here, Z is the nuclear charge number, while α is the fine-structure constant. Vacuum polarization modifies the Coulomb potential to ⃗ =− V (k)

4πα

k⃗ 2 [1 − ΠR (−k⃗2 )]

≈−

4πα [1 + ΠR (−k⃗2 )] . k⃗ 2

(10.234)

The result is a small modification of the Coulomb potential, which is especially ⃗ 2 , i.e. small distances. Indeed, in coordinate space, vacuum prevalent for large ∣k∣ polarization yields a Dirac-δ modification. Let us evaluate the leading correction to the Coulomb potential, which reads as V (⃗ r) ≈ − 4πZα ∫

1 d3 k ik⋅⃗ α ⃗ e r( ) + ⃗ 2 15πm2 (2π)3 ∣k∣

Zα 4 α (Zα) (3) − δ (⃗ r) . (10.235) r 15 m2 In hydrogen atoms, the consequence is that, in leading order, only S states receive a correction through vacuum polarization because only these states have an overlap with the core (for a point-like core at least). It is very instructive, and a good exercise in bound-state calculations, to derive a concise integral representation for V (⃗ r), including the vacuum-polarization correction, based on complex contour integration. The Cauchy residue theorem will ⃗ be of importance. We start from (k ≡ ∣k∣) = −

V (⃗ r) = − ∫ = − ≈

4πZα d3 k ik⋅⃗ ⃗ e r . 2 ⃗ ⃗ 2 )) − i (2π)3 ∣k∣ (1 − ΠR (−∣k∣

ikr 1 − e−ikr 4πZα ∞ 2e dk k ∫ (2π)2 0 ikr k 2 (1 − ΠR (−k 2 )) − i

iZα ∞ eikr (1 + ΠR (−k 2 )) . dk k 2 ∫ πr −∞ k − i

(10.236)

We have used the fact that, due to radial symmetry, we can set k⃗ ⋅ r⃗ = kr cos(θ), carry out the integration over the azimuth angle, and then, do the polar angle integration. In the second and third lines of Eq. (10.236), we have changed the ⃗ notation to k ≡ ∣k∣. We must close the contour in Eq. (10.236) in the positive half-plane. Let us set √ k = iq. The expression in Eq. (10.236) has a pole at k ≈ ± i. However, because the argument of the logarithm in ΠR (−k 2 ) becomes negative for −k 2 = q 2 > 4m2 , the integrand in Eq. (10.236) has a branch cut starting at k = ±2mi, extending along the imaginary axes in the complex k-plane. We now write ∞ iZα eikr iZα eikr (1+ΠR (−k 2 )) = (1+ΠR (−k 2 )) (∫ +∫ +∫ )dk k 2 ∮ dk k 2 πr C k − i πr −∞ CH CB k − i iZα eikr (1 + ΠR (−k 2 ))} . = (2πi) Res √ {k 2 πr k − i k= i (10.237)

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Im(k)

✙

✻ ❨ CH

CH

CB ✻ CB ❄ s √ − iǫ ×

×

2i m √

iǫ

✲ Re(k)

Fig. 10.5 The integration contours used in the representation (10.237) of the vacuum polarization ⃗ = involve an integration around the branch cut of the vacuum-polarization insertion from k = ∣k∣ 2im.

The integration contours are shown in Fig. 10.5. The integral ∫CH vanishes because of the exponential suppression of the integrand. Then, we have 2Zα eikr iZα eikr (1 + ΠR (−k 2 ))} − (1 + ΠR (−k 2 )), Res ∫ dk k 2 √ {k 2 r k= i k − i πr CB k − i ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

V (⃗ r) = −

=VC (r)

=VR (r)

(10.238) where we define the Coulomb term VC (r) and the radiative correction VR (r). In order to evaluate the pole term, one writes the identity √ √ (10.239) k 2 − i = (k + i) (k − i) . √ From the residue term at k = i, we have eikr 2Zα (1 + ΠR (−k 2 ))} {k Res r k=√i k 2 − i √ 2Zα 1 Zα = − × × ei ir (1 + ΠR (0)) = − . (10.240) r 2 r This term reproduces the Coulomb potential. The second term contributing to the potential is due to the branch cut, taken along the contour CB , with the branch cut extending from pair-production threshold to infinity, VC (⃗ r) = −

iZα dk k eikr iZα i∞ dk k eikr 2 (1+Π {i ImΠR (−k 2 )} (−k )) = +2 R ∫ ∫ πr CB k 2 − i πr k=2mi k 2 − i ∞ Zα 2 e−qr =− dq Im[ΠR (q 2 − i)] . (10.241) ∫ r π 2m q

VR (r) = −

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The form of the integral now forces us to find the imaginary part of ΠR if we want to find the one loop result for the correction to the Coulomb potential. According to Eq. (10.230), we have Im ΠR (q 2 − i) =

q2 2α 1 ∫ dx x(1 − x) Im [ln (1 − x(1 − x) 2 + i)] . π 0 m

(10.242)

We thus have to evaluate, as a function of q 2 , the imaginary part of the integral. The imaginary part of the logarithm is +π when the real part of the argument of the logarithm is negative. The imaginary part of the argument of the logarithm is infinitesimally positive. We thus need to evaluate the branch points, as solutions of the equation √ q2 1 1 4m2 1 − 2 x(1 − x) = 0 , x = x± ≡ ± (10.243) 1− 2 . m 2 2 q Hence, x+ α 2α dx x (1 − x) = Im[ΠR (q − i)] = (+π) ∫ π 3 x− 2

√ 1−

4m2 2m2 (1 + ) . (10.244) q2 q2

The radiative correction to the Coulomb potential is finally evaluated as √ 2α(Zα) ∞ e−qr 4m2 2m2 VR (r) = − dq 1 − 2 (1 + 2 ) . (10.245) ∫ 3πr q q q 2m We have a found a compact integral representation of the one loop vacuum polarization correction to the Coulomb potential, that is, the so-called Uehling9 potential. 10.4.4

Vacuum Polarization and Asymptotics

It is extremely useful to explore the asymptotics of the Uehling potential, in the limits r ≫ 1/m and r ≪ 1/m. For r ≫ 1/m, the exponential suppression in Eq. (10.245) is dominant. We substitute t = q − 2m, which leads to ¿ e−(2m+t)r Á 2α(Zα) ∞ Á À t(4m + t) (t2 + 4mt + 6m2 ) dt VR (r) = − ∫ 3 3πr (2m + t) (2m + t)2 0 √ α (Zα) ∞ e−(2m+t)r t 29 t 445 t2 3037 t3 = − dt (1 − + − + ⋯) ∫ 2 2πr m m 24 m 384 m 3072 m3 0 = −

α Zα e−2mr 29 1 2225 1 106295 1 √ (1 − + − + ⋯) . r 4 π (mr)3/2 16 mr 512 (mr)2 8192 (mr)3 (10.246)

For hydrogen, we can use the above formula as an improvement over Eq. (10.235). The determination of the asymptotic behavior for r → 0 requires the use of another trick, which is an application of the method of the overlapping parameter 9 Edwin

Albrecht Uehling (1901–1985).

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(see Sec. 4.5.2) or, a special case for the method of regions outlined in Sec. 16.3.2. A first substitution q = 2mu brings the integral √ (10.245) 2into the form ∞ α Zα u2 − 1 (2u + 1) du e−2mur VR (r) = [− ]∫ . (10.247) π r 3u4 1 2 −1/2 A second substitution u = (1 − v ) gives an alternative and perhaps even more economical representation [314], 2 v 2 (1 − v3 ) 1 α Zα 2mr ] VR (r) = [− ] ∫ dv exp [− √ . (10.248) π r 1 − v2 0 1 − v2 The advantage of the representation in Eq. (10.248) above is that we have projected the infinite integration regime into a finite and very simple interval [0, 1]. A naive expansion of the integrand in Eq. √ (10.248) in powers r leads to problems in the range v ≈ 1, due to the expression 1 − v 2 in the denominator. We thus split the integral into VR (⃗ r) = lim [VL (r) + VH (r)] , (10.249a) →0

2

v 2 (1 − v3 ) 1− α Zα 2mr VL (r, ) = [− ]∫ dv exp [− √ ] , π r 1 − v2 0 1 − v2 2

(10.249b)

v 2 (1 − v3 ) 1 2mr α Zα ] ∫ dv exp [− √ ] VH (r, ) = [− . (10.249c) π r 1 − v2 1− 1 − v2 In VL (r), we can expand the integrand (including the exponential) in r, without problems, because we avoid the problematic region v ≈ 1. A subsequent expansion in  leads to the result √ 5 Zα Zα  α 2 2Zαm π √ − Zαm + + ln ( ) + O(r)] . (10.250) VL (r, ) = [ π 2 9 r 3r 2 3  For the high-v part VH of the vacuum polarization correction, we set v 2 = 1 − h, so that the integration variable is small. Hence, the integral for the upper part takes the form 2−2 α Zα 2mr 2 − h − h2 √ . (10.251) VH (r, ) = [− ]∫ dh exp[− √ ] π r 0 h 6h 1 − h Because of √ the exponential suppression factor, the relevant r-range is of the same order as h, and therefore, an expansion in h of the non-exponential terms corresponds to an expansion in √ r. The result is α 2 2Zαm Zα 2mr √ + [2γE + ln ( )] + O(r)} , (10.252) VH (r, ) = {− π 3r  3  where γE , again, is the Euler gamma constant. The addition of VL and VH leads to the cancelation of the  parameter, and gives the short-range asymptotics of VR (r), α Zα 1 π 5 VR (r) = [ (ln(mr) + 2γE ) − mr + ] + O(r) . (10.253) π r 3 2 9 It is important to mention that the argument of the logarithm is, in fact, dimensionless: We recall that in the natural unit system, the mass is given in inverse units of length. Moreover, the leading term ∼ ln(mr)/r constitutes an attractive potential for distances r < 1/m since the logarithm is negative if the argument is smaller than unity.

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k

p

p−k

p

Fig. 10.6 The four-vector p of the incoming fermion splits into the photon loop momentum k, and the fermion loop momentum p − k, in the one-loop self-energy diagram.

10.5 10.5.1

Self–Energy Operator Self–Energy and Pauli–Villars Regularization

We have discussed, in great detail, the evaluation of the vertex correction (Sec. 10.3) as well as the vacuum polarization correction to the photon propagator (Sec. 10.4). In the evaluation of the quantum electrodynamic self-energy (see Fig. 10.6), we will thus concentrate on the results. In Eq. (10.45), we had derived the vertex correction based on the addition to the vertex factor −ieγ ν . For the vacuum-polarization diagram, we had considered the insertion of the fermionic loop into the photon propagator in Eq. (10.192). Based on the Feynman rules of QED, the self-energy diagram (see Fig. 10.6) leads to the following insertion into the fermionic propagator, iSF (p) → iSF (p) +∫

d4 k iSF (p)(iDµν (k))(−ieγ µ )(iSF (p − k))(−ieγ ν )iSF (p) + ⋯. (2π)4 (10.254)

On the other hand, if we can show that the self-energy insertion can be formulated in terms of an operator Σ(p) as iSF (p) → iSF (p) + iSF (p)Σ(p) SF (p) + ⋯ ,

(10.255)

then the infinite series of insertions can be summed into the propagator denominator, as follows, i

1 1 1 1 →i [ + Σ(p) + ⋯] p − m + i p − m + i p − m + i p − m + i



1 =i [ ]. p − m − Σ(p) + i


(10.256)

We appeal to Eq. (7.154) for the functional form of the Feynman propagator in momentum space. A comparison of Eqs. (10.254) and (10.256) reveals that Σ(p) = ie2 ∫

d4 k Dµν (k) γ µ SF (p − k)γ ν . (2π)4

(10.257)

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In Feynman gauge [see Eq. (9.121)], one can write this as follows, gµν d4 k 1 γν Σ(p) = e2 ∫ γµ (2π)4 i k 2 + i p − k −

m + i

1 1 1 d4 k ( 2 − 2 ) γµ γµ . (10.258) 4 2 2 (2π) i k − Λ + i k − λ + i p
− k
− m + i In the second step, we have introduced a photon mass and a Pauli–Villars regularization term Sec. 10.3.1. As we can see in Eq. (10.256), the self-energy insertion Σ(p) is an additive insertion in the propagator denominator, to be added to the particle’s mass. Because the mass of a particle is fixed by experiment, the term δm in the expansion = e2 ∫

−1 Σ(p) = δm − (Z2−1 − 1) (p
− m − δm) + Z2 ΣR (p) −1 = Z2−1 δm − (Z2−1 − 1) (p
− m) + Z2 ΣR (p)

(10.259)

constitutes an electromagnetic radiative contribution to the particle’s mass, which, according to modern understanding, should be added to the mass term which is otherwise generated by the Higgs10 mechanism. In the spirit of Sec. 10.3.3, we thus have to interpret δm as a correction to the mass which has to absorbed in the physical parameters of the theory. The summed and self-energy corrected propagator reads as follows, Z2 1 = . (10.260) p − m − Σ(p) + i p − m − δm − ΣR (p) + i

This will be discussed in further detail in Sec. 10.6.2; the physical mass is m + δm, so that the initial parameter m in the theory must be interpreted as the bare mass. Alternatively, if we wish to keep the interpretation of m as the physical mass of the electron, then we need to subtract a so-called mass counter term, proportional to δm, from the Lagrangian. A calculation using Pauli–Villars regularization reveals that Λ2 1 3α [ln ( 2 ) + ] . (10.261) δm = m 4π m 2 Note that, in leading order in α, we have Z2 = 1+O(α), so that Z2 δm = δm+O(α2 ). The result for the renormalization constant Z2 = Z1 can be found in Eq. (10.137), per Eq. (10.134). In Pauli–Villars regularization, just as for the vertex correction, we use a photon mass λ in order to regularize infrared divergences, which can persist even in the renormalized expressions (see Sec. 10.3.2). Indeed, the renormalized self-energy function is found as (p2 < m2 ) αm 2(m2 − p2 ) p2 ln (1 − )] ΣR (p) = [−1 + 2π p2 m2 αp 3 m2 m4 − (p2 )2 p2 α λ2 +
[ − 2 − ln (1 − )] + ( p − m) ln ( ). 2π 2 2 p 2 (p2 )2 m2 2π
m2 (10.262) Here, λ is the photon mass. 10 Peter

Ware Higgs CH FRS (b. 1929).

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Self–Energy and Dimensional Regularization

We use the dimensionally regularized Dirac algebra outlined in Sec. 10.3.1, and the dimensionally regularized electron charge as given in Eq. (10.173) (inspired by the MS scheme). The renormalization procedure is the same as in Sec. 10.5.1, and the renormalization constants in the dimensional regularization are listed in Table 10.2. Using a judicious choice of the integration methods, it is possible to derive the following closed-form result for the self-energy insertion, ΣR (p) =

2(m2 − p2 ) p2 αm [−1 + ln (1 − )] 2π p2 m2 p2 α αp 3 m2 m4 − (p2 )2 1 µ2 ln (1 − )] + )] . ×
[ − 2 − ( p − m) [ + ln ( 2π 2 2 p 2 (p2 )2 m2 2π
 m2 (10.263)

Note that the above result is obtained from the result in Pauli–Villars regularization by the substitution ln (

λ2 1 µ2 ) → + ln ( ). m2  m2

(10.264)

The proof is left as an exercise to the reader. 10.6 10.6.1

Renormalization of QED Bare and Renormalized Lagrangian

We are now in the position to expand on the arguments presented in Sec. 10.3.3. For simplicity and clarity, we shall exclude the gauge-fixing term, and the photon mass, from the discussion. The bare QED Lagrangian, which is identified as the “original” Lagrangian, i.e., the one which we use in order to start constructing the quantum field theory, is 1 L0 = − F02 + ψ 0 (i∂
− m0 ) ψ0 − e0 ψ 0 A0 ψ0 , 4

(10.265)

F02 = F0,µν F0µν = (∂µ A0,ν − ∂ν A0,ν ) (∂ µ Aν0 − ∂ ν Aν0 ) .

(10.266)

where

The unrenormalized, bare physical quantities carry a subscript zero. This Lagrangian can describe all physical processes at tree level. Furthermore, at tree level, the unrenormalized quantities are equal to the physical ones (there is no renormalization necessary at tree level). The “renormalized QED Lagrangian”, expressed in terms of renormalized quantities (denoted by the subscript R), is 1 L0 = LR = − Z3 FR2 + Z2 ψ R (i∂
− mR ) ψR + Z2 ψ R δmR ψR − Z1 e ψ R AR ψR , 4 (10.267)

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where FR2 = FR,µν FRµν = (∂µ AR,ν − ∂ν AR,ν ) (∂ µ AνR − ∂ ν AνR ) .

(10.268)

The renormalized (physical) and bare parameters are related as follows, √ m0 = mR − δm , (10.269a) ψ0 = Z2 ψR , √ −1 −1/2 A0 = Z 3 AR , (10.269b) e 0 = Z1 Z2 Z3 eR . The renormalized Lagrangian can be written as follows [see also Eq. (10.129)], LR = L1 + L2 , (10.270a) 1 2 L1 = − FR + ψ R (i∂
− mR ) ψR − eR ψ R AR ψR , (10.270b) 4 1 L2 = − (Z3 − 1) FR2 + (Z2 − 1) ψ R (i∂
− mR ) ψR − (Z1 − 1) eR ψ R AR ψR 4 + Z2 ψ R δmR ψR . (10.270c) Here, the functional form of L1 is equal to that of the original Lagrangian (10.265), with the bare parameters replaced by the renormalized ones, while L2 contains the counter terms which render the physical Green functions finite. We may add the following two general summary statements: (i) The unrenormalized Lagrangian contains bare quantities but is void of counterterms. (ii) The renormalized Lagrangian contains physical quantities and all counter terms. The renormalization program will be illustrated in the following. 10.6.2

Renormalization of Vertex and Self–Energy

The renormalization program sketched above will now be illustrated. From Sec. 10.5, we recall that the self-energy has the expansion −1 Σ = Z2−1 δm − (Z2−1 − 1) (p
− m0 ) + Z2 ΣR ,

(10.271)

this time, suppressing the argument of Σ, in order to simplify the notation. Unlike in Eq. (10.259), we denote the initial mass parameter of the theory, which has been identified as the bare mass m0 , with the corresponding subscript zero. Through an iterative insertion of self-energy diagrams, the unrenormalized self-energy insertion Σ is summed into the propagator denominator as follows, 1 1 1 1 1 1 1 + Σ + Σ Σ +⋯= . (10.272) p − m p − m p − m p − m p − m p − m p − m 0 0 0 0 0 0 0−Σ






In view of the relation of the unrenormalized and renormalized quantities (we carefully distinguish m0 and mR ), one can write the relation −1 −1 p
− m0 − Σ = p
− m0 − δm + (Z2 − 1) (p
− mR ) − Z2 ΣR −1 −1 =p
− mR + (Z2 − 1) (p
− mR ) − Z2 ΣR

= Z2−1 (p
− m R − ΣR ) ,

(10.273)

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where mR = m0 + δm. The summation of self-energy insertions thus leads to the following modification of the fermionic propagator, 1 Z2 (0) SF = = . (10.274) p
− m0 − Σ p
− mR − ΣR The unrenormalized theory, after the identification of the mass counter term, leads to a propagator whose residue at p
= mR is not equal to unity any more. Now, let us consider the corresponding terms in the renormalized Lagrangian (10.267), and check whether they produce finite results, L ∼ Z2 ψ(i∂
− mR )ψ + Z2 δm ψ ψ .

(10.275)

Upon quantization, these terms create a propagator (indeed) of the functional form SF′ =

1 1 , Z2 p −
mR

(10.276)

where the prime only denotes the modified form (no differentiation). We recall that the propagator, in the momentum representation, has to be equal to the inverse of the kinetic operator, in this case, the inverse of the operator sandwiched in between ψ and ψ. Let us also consider the one-loop self-energy insertion derived from the renormalized Lagrangian. The self-energy has two vertices and one propagator. The vertices acquire a factor Z1 , while the propagator acquires a factor 1/Z2 . Altogether, one has the factor 1 Z1 = Z2 , Z1 = Z2 , (10.277) Z1 Z2 where advantage has been taken of the Ward–Takahashi identity [see Eq. (10.136)]. So, using the renormalized Lagrangian, taking into account the mass counter term perturbatively in each order of perturbation theory, the summation of selfenergy insertions gives 1 1 1 + (Z2 Σ − Z2 δm) +⋯ SF′′ = Z2 (p − m ) Z ( p − m ) Z ( p R 2 R 2


− mR ) 1 = , (10.278) Z2 (p
− mR + δm − Σ) the mass counter term ending up being subtracted from mR . Again, we emphasize that the double prime on SF′′ denotes the modified form of the fermionic Feynman propagator obtained from the renormalized Lagrangian (no differentiation). But then, the expression in the denominator of SF′′ can be reformulated as follows, −1 −1 Z2 (p
− mR + δm − Σ) = Z2 [p
− mR + δm − δm + (Z2 − 1) (p
− mR ) − Z2 ΣR ] −1 −1 = Z 2 [p
− mR + (Z2 − 1) (p
− mR ) − Z2 ΣR ]

= (p
− mR − ΣR ) .

(10.279)

We are using Eq. (10.259) in the form −1 Σ(p) = Z2−1 δm − [Z2−1 − 1] (p
− mR ) + Z2 ΣR (p) .

(10.280)

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So, we finally obtain the result SF′′ =

1

, (10.281) p
− mR − ΣR which is finite and has unit residue at p
= mR . Consistent renormalized results have been obtained from the renormalized Lagrangian (10.270). A further remark is in order. Based on a general consideration, we have justified, in the text following Eq. (10.276), that the Lagrangian L = Z2 ψ R (i∂
− m) ψR + ⋯ −1 generates a term Z2−1 (p
− m) in the propagator. This can be further justified as follows. First, one quantizes the theory by postulating the following equal-time anticommutation relation of the field ψR and its conjugate momentum πR , {ψR (t, r⃗), πR (t, r⃗′ )} = i δ (3) (⃗ r − r⃗′ ) .

(10.282)

From L, the canonical momentum is πR =

∂L + . = i Z 2 ψ R γ 0 = i Z 2 ψR ∂(∂t ψ)

(10.283)

This expression differs from the usual field momentum given in Sec. 7.4.1 [see Eq. (7.160)] only in the multiplicative factor Z2 . If we impose the anticommutator condition + {ψR (t, r⃗), πR (t, r⃗′ )} = i Z2 {ψR (t, r⃗), ψR (t, r⃗′ )} = i δ (3) (⃗ r − r⃗′ ) ,

(10.284)

then it become clear that this condition can be fulfilled if we choose for ψR and + ψR field operators which formally differ from the usual definitions only in a multi√ plicative factor 1/ Z2 . This observation finally justifies the form of the propagator as derived from the renormalized Lagrangian. Because the Feynman propagator is obtained as the time-ordered product of two field operators, the factor Z2 in Eq. (10.276) is consistently obtained. 10.6.3

Renormalization of Vacuum Polarization

Let us supplement the considerations outlined in Sec. 10.4.1, regarding the renormalization of the vacuum-polarization insertions, by additional considerations, which are also connected to the renormalized Lagrangian given in Eqs. (10.267) and (10.270). Let us first treat the counter terms in Eq. (10.270c) perturbatively. A quantization of the electromagnetic field based on the Lagrangian (10.270b) leads to the photon propagator [in Landau gauge, see Eq. (10.196)]. Let us also remember Eq. (10.41), which in Landau gauge reads as d4 k Dµν (k) e−ik⋅(x−z) = iDFµν (x − z) , (2π)4 F i kµ kν iDFµν (k) = − 2 P µν (k) , P µν (k) = g µν − 2 . k + i k

⟨0∣T AµR (x) AνR (z)∣0⟩ = i ∫

(10.285a) (10.285b)

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The counter term is 1 (10.286) L2 ∼ (Z3 − 1) {− FR,µν FRµν } 4 from Eq. (10.270c). We treat it perturbatively. One insertion of the perturbative counter term leads to 1 i∆Dµν = (Z3 − 1) i ∫ d4 y {− ⟨0∣T AµR (x) [∂ρ AR,σ (y) − ∂σ AR,ρ (y)] 4 × [∂ ρ AσR (y) − ∂ σ AρR (y)] AνR (z)∣0⟩} ∼ −

Z3 − 1 i 4

× ∫ d4 y ⟨0∣T AµR (x) [∂ρ AR,σ (y) ∂ ρ AσR (y) + ∂σ AR,ρ (y) ∂ σ AρR (y)] AνR (z)∣0⟩ , (10.287) where we have selected, in the second step, after the ∼ sign, the terms with pairedoff Lorentz indices in the derivative operators. The other terms would generate terms proportional to ∂ρ DFρσ (x − y) ∼ kρ DFρσ (k) = 0 (in momentum space). We are considering only those contractions in Eq. (10.287) which lead to connected diagrams, i.e., the operators AµR (x) and AνR (z) both get contracted with four-vector potentials at y. One finally gets four equivalent terms, from the contractions of each of AµR (x) and AνR (z) with one of the two AσR (y) [and AρR (y)], resulting in i∆Dµν ∼ − +

∂ ∂ Z3 − 1 i ∫ d4 y {[ ρ ⟨0∣T AµR (x) AR,σ (y)∣0⟩ ⟨0∣T AσR (y) AνR (z)∣0⟩ 2 ∂y ∂yρ ∂ ∂ ⟨0∣T AµR (x) AR,ρ (y)∣0⟩ ⟨0∣T AρR (y) AνR (z)∣0⟩]} ∂y σ ∂yσ

∂ ∂ (i DF µ σ (x − y)) (i DFσν (y − z))] , (10.288) ∂y ρ ∂yρ where in the last step, we have performed a partial integration. In momentum space, we can replace convolution by multiplication, and so, ∂ ∂ − ρ → −(−ikρ )(−ik ρ ) = k 2 . (10.289) ∂y ∂yρ So, the one-loop correction becomes = − (Z3 − 1) i ∫ d4 y [−

i∆Dµν ∼ − (Z3 − 1) ik 2 (iDF µ σ (k)) (iDFσν (k)) = (Z3 − 1) ik 2 (−

1 1 P µ σ (k))(− 2 P σν (k)) k 2 + i k + i

1 P µν (k) . (10.290) k 2 + i Hence, the series of perturbative insertions of the counter term given in Eq. (10.286) leads to i i iDµν (k) + i∆Dµν (k) + ⋯ = − 2 P µν (k) + (Z3 − 1) 2 P µν (k) + ⋯ . k + i k + i (10.291) = (Z3 − 1) i

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The resulting series can be summed, 1 − (Z3 − 1) + (Z3 − 1)2 + ⋯ =

1 1 = . 1 + (Z3 − 1) Z3

(10.292)

The same result would be obtained if we take the multiplicative character of the counter term in to account, and write the Lagrangian of the free photon field as 1 L ∼ − Z3 FR,µν F R,µν , 4

(10.293)

by arguments analogous to those used in the discussion surrounding the passage between Eqs. (10.282) and (10.284). We then sum all insertions of the counter term (10.286) and of the vacuumpolarization diagrams into the photon propagator, over all iterative insertions (including the mixed ones). This is in full analogy to the fermionic self-energy (see Sec. 10.5). This procedure leads to a modified propagator denominator, Dµν (k 2 ) = −

1 Pµν (k) . k 2 [1 + (Z3 − 1) − Π(k 2 )] + i

(10.294)

Writing Π(k 2 ) = Π(0) + ΠR (k 2 ) ,

Z3 = 1 + Π(0) ,

(10.295)

one obtains the result for Z3 . We conclude with an illustrative remark. Let us return to Eq. (10.269), and consider the relation of the bare charge, and bare four-vector potential, to the renormalized counterparts. Strictly speaking, the counter term (10.286) is due to a renormalization of the four-vector potential (and consequently, of the electromagnetic field). Of course, it enters the field-strength tensor, √ √ A0,µ = Z3 AR,µ . (10.296) F0,µν = Z3 FR,µν , This constitutes a renormalization of the photon field. However, per Eq. (10.267), the only renormalization which persists in the QED interaction Lagrangian is the Z1 renormalization at the vertex, L ∼ −Z1 e ψ R AR ψR .

(10.297) √ This implies that, in view of the renormalization of the Dirac field ψ0 = Z2 ψR , √ and of the photon field A0,µ = Z3 AR,µ , the electron charge has to be renormalized as −1/2

e0 = Z1 Z2−1 Z3

−1/2

eR = Z3

eR ,

(10.298)

where we make use of the Ward–Takahashi identity (10.136), which implies that Z1 = Z2 . In other words, the renormalization of the photon field, per Eq. (10.296), implies a renormalization of the electron charge, according to Eq. (10.298).

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419

Table 10.2 Summary of the one-loop renormalization constants using Pauli–Villars and dimensional regularization. Feynman gauge has been used throughout, although only Z1 = Z2 is gauge dependent. For the Pauli–Villars regularization scheme, we have included an additional artificial photon with a large mass Λ, and have also included a photon mass term λ to regulate infrared divergences [see Eq. (10.50)]. We see that Z1 = Z2 has both ultraviolet and infrared divergences. For dimensional regularization, we calculate in d = 4−2 dimensions and see that  regulates both ultraviolet and infrared divergences. The mass parameter µ is included to maintain dimensional consistency [see Eq. (10.173)]. One-Loop Renormalization Constants

Z1 = Z2

Z3

δm

10.6.4

1−

Pauli–Villars Regularization

Dimensional Regularization

(λ is the photon mass)

(µ is the renormalization scale)

α 1 Λ2 1 m2 9 [ ln ( 2 ) − ln ( 2 ) + ] π 4 m 2 λ 8

1−

α 1 Λ2 [ ln ( 2 )] π 3 m

αm 3 Λ2 3 [ ln ( 2 ) + ] π 4 m 8

1−

α 3 3 µ2 [ + ln ( 2 ) + 1] π 4 4 m

1−

α 1 1 µ2 [ + ln ( 2 )] π 3 3 m

αm 3 3 µ2 [ + ln ( 2 ) + 1] π 4 4 m

Compilation of Renormalization Constants

We would like to summarize the one-loop renormalization constants we have obtained in this chapter. We have employed two regularization schemes: Pauli–Villars and dimensional, and have used them to calculate the set of constants Z1 = Z2 , Z3 , and δm. Our results are presented in Table 10.2. The theory of QED is defined by the simple Lagrangian (10.1). However, for actual calculations a number of choices must be made, and these choices affect the character of a QED calculation in a profound way. The initial and unavoidable choice concerns the gauge for the quantization procedure. We have restricted ourselves to covariant gauges, and Feynman gauge in particular, in this chapter. In general, for calculations of cross sections and for renormalization related considerations, the covariant gauges have the great advantage of preserving Lorentz symmetry and allowing us to use arguments based on covariance to simplify calculations. On the other hand, the non-covariant Coulomb gauge is advantageous for the study of bound systems in QED. Once a gauge has been established, calculations beyond tree-level bring the need to evaluate loop integrals. As we have seen, these integrals often contain divergences: ultraviolet and infrared. Regularization procedures must be adopted in order to deal with these divergences. In this chapter, we have studied Pauli–Villars regularization, which involves the introduction of

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fictitious particles with large masses (that will eventually be taken to be infinitely large) coupled in such a way as to cancel the ultraviolet divergences that would otherwise be present. When using Pauli–Villars regularization, we made the choice to regulate the infrared by inserting an artificial small photon mass. The resulting photon propagator with Pauli–Villars regularization was given in Eq. (10.50): g µν g µν + , (10.299) k 2 − λ2 + i k 2 − Λ2 + i where λ is the small photon mass and Λ is the large mass of the unphysical (note the sign of its propagator) additional “photon”. We also illustrated the use of dimensional regularization, in which both ultraviolet and infrared divergences are regulated by working in a number of space-time dimensions different from four. The algebraic structure of QED is preserved with dimensional regularization since the propagators and vertices are unchanged, but tensor and spinor calculations and integrations must be done in d dimensions. When working in d = 4 − 2 dimensions, the elementary charge acquires a dimension, and we write e2 = 4πα(µ2 eγE /(4π)) where µ is a mass parameter that we are forced to introduce in order to keep dimensions consistent. It is convenient to introduce the terms involving the Euler– Mascheroni constant γE and π along with µ in order to minimize factors of γE and ln(4π) that would appear in our results, but this itself is a choice that is inspired by modified minimal subtraction (“MS-bar”, MS) [see also Eqs. (10.173) and (16.145)]. Other choices of factors in the relation between e2 and the dimensionless α are also possible. Moreover, other regularization schemes could have been chosen. Among them are “analytic regularization”, in which propagator denominators are generalized from k 2 − m2 + i to (k 2 − m2 + i)α . The diagrams become analytic functions of α in some region of the complex plane. Adjustment of α can be used to regulate ultraviolet and infrared divergences. Another regularization method involves the imposition of an explicit momentum-space cut-off. The cut-off is usually chosen to limit the Euclidean-space momentum magnitude to a finite value. These regularization procedures break gauge invariance and the equality Z1 = Z2 . For that reason, they are seldom used in serious calculations. Dimensional regularization is the usual choice due to its favorable mathematical qualities and the fact that it minimizes the structural disruption suffered by the theory upon regularization. When dealing with Coulombic bound states such as hydrogen, muonium, positronium, etc., choices about the procedure for dealing with the binding must also be made. Perhaps the most fundamental is whether to stick with traditional QED or move to an effective low-energy quantum field theory such as NRQED. An introduction to NRQED will be given in Chap. 17. DFµν (k) → −

10.6.5

Forest Formula

In diagrams with more than one nested loop, one may encounter so-called nested divergences. The renormalization, in this case, becomes a little more complicated. One first has to subtract all divergences associated with the inner loops; i.e., in a

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k1 k2

p

p − k1 p − k1 − k2 p − k1

p

Fig. 10.7 Two consecutive photon emissions and absorptions constitute the “rainbow” nestedloop two-loop self-energy Feynman diagram.

two-loop self-energy diagram like the one shown in Fig. 10.7, one would subtract the mass counter term, and the wave function renormalization term, from the inner selfenergy loop. The subtraction operations are done recursively. If γa is a divergent subgraph of a larger graph G, and Tγa is the appropriate subtraction information for the graph γa , IG is the integral defining the graph G, and IG/{γ1 ,...,γn } is the integral with the divergent subgraphs excluded, then the so-called forest formula reads as RG = IG +

∑

{γ1 , . . . , γn } γa ∩ γb = ⊘

n

IG/{γ1 ,...,γn } ∏ (−Tγa Rγa ) ,

(10.300)

a=1

RG = (1 − TG ) RG .

(10.301)

Here, RG is the renormalized integral corresponding to the graph G, and it requires a further, overall subtraction TG , while RG is a partially subtracted graph, where all subgraphs have been renormalized, but an overall remaining subtraction has not yet been implemented. For a two-loop vertex with the external field acting in the middle of the rainbow (see Fig. 10.7), with G being the whole diagram, and γ being the inner loop, this means that we can carry out the renormalization recursively. For the inner loop, we have RG = IG + IG/γ (−Tγ Rγ ) = (1 − Tγ ) IG .

(10.302)

Then, for the outer loop, there is a further subtraction, RG = (1 − TG ) RG = (1 − TG ) (1 − Tγ ) IG .

(10.303)

The leftmost subtraction is the outer subtraction, the second one is the inner subtraction.

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Further Thoughts

Here are some suggestions for further thought. (1) Form Factors and QED. Consider Eq. (10.97). Show that nonvanishing contributions to F3 and FA would be incompatible with the discrete symmetries of QED. (2) An Integral. Do the analytic evaluation of the integral given in the Eq. (10.94). Why can you replace zλ2 /m2 → λ2 /m2 in the integrand, in the physically relevant limit λ → 0? (3) Feynman Parameters. Verify the derivations of relevant formulas concerning Feynman parameters, given in Eqs. (10.151), (10.156), (10.159), and (10.160). (4) Magnetic Form Factor and Anomalous Magnetic Moment. Find the connection of the result for the magnetic (Pauli) form factor of the electron, given in Eqs. (10.100) and (10.191b), to the anomalous magnetic moment of the electron. (5) Two-Loop Form Factors. Study the two-loop generalizations of the results given in Eq. (10.191) and their role in bound-state calculations. Consult Refs. [199, 312, 315]. (6) Tensor Structure of the Vacuum-Polarization Insertion. Consider Eq. (10.195): dd q Tr [γ µ SF (q + k) γ ν SF (q)] i Πµν (k) = (−1) e2 ∫ (2π)d dd q 1 1 γν ] . (10.304) Tr [γ µ q − m + i (2π)d

q − k
− m + i In order to show that the vacuum-polarization has the tensor structure (10.199), one does not necessarily have to go through the derivation outlined in Sec. 10.4.2, which leads to the explicit result (10.228). it suffices to show that kµ Πµν (k) = 0. Show that kµ Πµν (k) = 0 , (10.305) µ by writing kµ γ = k
= −(
q − k
− m + i) + (
q − m + i), canceling denominators in Eq. (10.304), and using the cyclic invariance of the trace. (7) Self-Energy Operator. Repeat the calculation of the self-energy operator, discussed in Sec. 10.5 in Pauli–Villars regularization, using dimensional regularization. Be inspired by Secs. 10.3.6 and 10.4.2. In particular, find the one-loop mass counter term in dimensional regularization [see Eq. (10.261)]. (8) Two-Loop Renormalization. Consider Ref. [311]. Convince yourself that the renormalized fermion propagator given in Eq. (2.14) of Ref. [311] is consistent with Eq. (10.281). Then, continue to study the two-loop renormalization constants on the basis of Ref. [311], possibly, extending the treatment to a unified regularization of both infrared as well as ultraviolet divergences through dimensional regularization. = (−1) e2 ∫

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(9) Trace Technology for an Odd Number of Dirac Matrices. Investigate why the trace of an odd number of Dirac γ µ matrices vanishes. Consider the case of four space-time dimensions, and consider the definition of γ 5 given in Eq. (7.15). Then, one considers the series of transformations (2n + 1 is odd, where n is an integer), 2

2

2

Tr [γ µ1 γ µ2 ⋯γ µ2n+1 ] = Tr [(γ 5 ) γ µ1 (γ 5 ) γ µ2 ⋯ (γ 5 ) γ µ2n+1 ] = Tr [(γ 5 γ µ1 γ 5 ) (γ 5 γ µ2 γ 5 ) ⋯ (γ 5 γ µ2n+1 γ 5 )] = Tr [(−γ µ1 ) (−γ µ2 ) ⋯ (−γ µ2n+1 )] = (−1)2n+1 Tr [γ µ1 γ µ2 ⋯γ µn+1 ] = − Tr [γ µ1 γ µ2 ⋯γ µ2n+1 ] ,

(10.306) 2

from which the theorem follows. Clarify where the properties (γ 5 ) = 1, {γ µ , γ 5 } = 0, and the invariance of the trace under a cyclic permutation, have been used in the above transformations. Now, consider what happens in d dimensions. We cannot define γ 5 consistently in d dimensions, as we convinced ourselves. However, in the case of three γ µ matrices, we can use the fourth current, not contained in any of the three matrices, in place of the γ 5 . Also, because the case d = 4 has to be approached smoothly, any deviation from the result for d = 4 could only be of order . (10) Fifth Current and Dimensional Regularization. Investigate the possibility of defining a γ 5 matrix in d dimensions. Using the equality γ µ γµ = d 1, one can derive the following relation, under repeated use of the invariance of the trace under cyclic permutations of the matrices in the argument of the trace, as well as the assumed anticommutativity of the γ 5 matrix with all γ µ matrices, d Tr [γ 5 γ ρ γ σ γ κ γ λ ] = Tr [γ 5 γµ γ µ γ ρ γ σ γ κ γ λ ] = Tr [γ µ γ ρ γ σ γ κ γ λ γ 5 γµ ] = −Tr [γ µ γ ρ γ σ γ κ γ λ γµ γ 5 ] = − Tr [γ 5 γ µ γ ρ γ σ γ κ γ λ γµ ] = − Tr [γ 5 γ µ γ ρ γ σ γ κ {γ λ , γµ }] + Tr [γ 5 γ µ γ ρ γ σ γ κ γµ γ λ ] = − 2 Tr [γ 5 γ λ γ ρ γ σ γ κ ] + Tr [γ 5 γ µ γ ρ γ σ γ κ γµ γ λ ] = ... = − (d − 8) Tr [γ 5 γ ρ γ σ γ κ γ λ ] .

(10.307)

Here, we denote missing intermediate steps using dots; the corresponding transformations start with the use of the equality γ κ γµ = {γ κ , γµ } − γµ γ κ . One still has to permute the arguments cyclically a few times, and use the anticommutativity of the γ 5 matrix a few times. One ends up with the relation dTr [γ 5 γ ρ γ σ γ κ γ λ ] = −(d − 8) Tr [γ 5 γ ρ γ σ γ κ γ λ ] .

(10.308)

Alternatively, one can formulate this as follows, (d − 4) Tr [γ 5 γ ρ γ σ γ κ γ λ ] = 0 .

(10.309)

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If d ≠ 4, then this implies that the trace Tr [γ 5 γ ρ γ σ γ κ γ λ ] needs to vanish, and cannot be proportional to ρσκλ , as we would otherwise wish. (11) Investigate further properties of the fifth current, in particular, the trace of the γ 5 matrix in d dimensions. Convince yourself of the validity of all intermediate steps in the transformation dTr [γ 5 ] = Tr [γ 5 γµ γ µ ] = Tr [γ µ γ 5 γµ ] = −Tr [γ µ γµ γ 5 ] = −d Tr [γ 5 ] , (10.310) from which it follows that d Tr [γ 5 ] = 0 .

(10.311)

Then, consider the next more complicated trace containing γ 5 , which is dTr [γ 5 γ ρ γ σ ] = Tr [γ 5 γµ γ µ γ ρ γ σ ] = Tr [γ µ γ ρ γ σ γ 5 γµ ] = −Tr [γ µ γ ρ γ σ γµ γ 5 ] = − Tr [γ 5 γ µ γ ρ γ σ γµ ] = − Tr [γ 5 γ µ γ ρ {γ σ , γµ }] + Tr [γ 5 γ µ γ ρ γµ γ σ ] = ... = − (d − 4) Tr [γ 5 γ ρ γ σ ] .

(10.312)

Here, the same transformation has been used as for the trace of a γ 5 matrix and four Dirac γ matrices, in Eq. (10.307). From the relation (d−2)Tr [γ 5 γ ρ γ σ ] = 0, near d = 4 dimensions, one finds that the trace Tr [γ 5 γ ρ γ σ ] vanishes. (12) Fifth Current and Traces. In four space-time dimensions, the trace of a γ 5 matrix and four Dirac γ matrices has to be proportional to the LeviCivit` a symbol ρσκλ (where 0123 = +1). The constant can be determined by 2 2 recalling that (γ 0 ) = 1 and (γ i ) = −1, for i = 1, 2, 3 (no summation over i). Considering the definition (7.15) and the anticommutativity of the Dirac γ matrices, show that Tr [γ 5 γ ρ γ σ γ κ γ λ ] = −4 i ρσκλ .

(10.313)

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Chapter 11

Foldy–Wouthuysen Transformation and Lamb Shift

11.1

Overview

The Foldy–Wouthuysen transformation is an essential tool in the analysis of bound states, and interacting particles in general. The idea is the following: The free Dirac Hamiltonian, given in Eq. (7.18), HFD = (

m12×2 ⃗ ⋅ p⃗ σ

⃗ ⋅ p⃗ σ ), −m 12×2

(11.1)

couples upper and lower components of the Dirac bispinor. For zero momentum, we can ignore the kinetic term, and identify the upper Dirac spinor (in the bispinor) as the particle wave function, while the lower spinor is identified as an antiparticle wave function, but this is impossible for nonvanishing momenta. The aim of the (unitary) Foldy–Wouthuysen transformation is to separate the upper (particle) from the lower (antiparticle) degrees of freedom in higher orders in the momentum expansion. It is rather common wisdom that the basic relativistic corrections to the hydrogen spectrum, induced by the relativistic correction to the free-particle dispersion relation, the zitterbewegung term and the spin-orbit coupling, can be derived from the Dirac–Coulomb Hamiltonian, by means of a Foldy–Wouthuysen transformation (a particularly instructive derivation is presented in Ref. [316]). Yet, it is our goal here to go further, and to derive higher-order correction terms. In considering the Foldy–Wouthuysen transformation, we should remember that we apply it to the Dirac equation, which in principle is a one-particle equation. This equation is valid, a priori, only in the limit of infinite nuclear mass. The momentum of a bound particle in the Coulomb field of a heavy nucleus is of order Zαm, where Z is the nuclear charge number, α is the fine-structure constant, and m is the mass of the orbiting particle. Hence, the expansion in higher powers of the momenta is, in fact, an expansion in powers of Zα. We proceed as follows. In Sec. 11.2, we derive the leading relativistic corrections to the Foldy–Wouthuysen transformed Dirac Hamiltonian, for general electric and magnetic background fields. We include the anomalous magnetic moment into the transformation. Applications to the Coulomb field, and to magnetic-field coupling, and to the fermionic transition current, are discussed in Sec. 11.3. All of these 425

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calculations take into account the anomalous-magnetic-moment correction, given by the Pauli form factor, but not the Dirac form factor corrections. The latter aspect is treated in Sec. 11.4, including the high-energy part of the Lamb shift which can be derived on the basis of the transformed Hamiltonian. Further applications are discussed in Sec. 11.5. 11.2 11.2.1

Leading Relativistic Corrections Unitary Transformation and Hamiltonian

Let us study the transformation of the Hamiltonian operator under a general unitary transformation U = exp(iS) ,

S = S+ ,

(11.2)

where U may be time-dependent and S is Hermitian (not to be confused with the S matrix, of course). The transformed Hamiltonian acts on the transformed wave function ψ ′ = exp(iS) ψ, for which i∂t ψ ′ = i∂t exp(iS) ψ = exp(iS)(H − ∂t S) ψ = [exp(iS)[H − i∂t ] exp(−iS)] ψ ′ .

(11.3)

We thus write the Foldy–Wouthuysen (FW) transformed Hamiltonian as follows, HFW = exp(iS) (H − i ∂t ) exp(−iS) .

(11.4)

In typical cases, the operator S carries some power of the momenta, which is typically small in atomic systems, and it is thus possible to expand in nested commutators as follows, 1 [iS, [i S, H − i ∂t ]] HFW = H + [i S, H − i ∂t ] + 2! 1 1 + [iS, [iS, [i S, H − i ∂t ]]] + [iS, [iS, [iS, [i S, H − i ∂t ]]]] + ⋯ . (11.5) 3! 4! The order required for the latter transformation depends on the problem under study. However, in general, we can say that S will be of the same order as the momenta in the problem. In many cases, the momenta define a convenient coupling parameter for a bound-state problem. In the case of an explicit time dependence of the fields, we can formulate the first-order correction as follows, δH (1) = [iS, H − i∂t ] = [iS, H] − [−i ∂t (i S)] = i[S, H] − ∂t S .

(11.6)

The next higher-order corrections can be calculated iteratively as follows, 1 1 1 δH (2) = [iS, δH (1) ] , δH (3) = [iS, δH (2) ] , δH (n+1) = [iS, δH (n) ] . 2 3 n+1 (11.7) It is only in special cases that the Foldy–Wouthuysen transformation can be carried out to all orders in the coupling parameter.

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Free Dirac Particle

It is natural to apply the idea of the Foldy–Wouthuysen transformation first to the example of the free Dirac Hamiltonian. Indeed, in Chaps. 7 and 8, we have studied the relativistic theory of the electron, on the basis of the Dirac equation. The particle wave function (positive energy) and antiparticle wave functions (negative energy) were obtained within the fully relativistic theory, without approximations. The solutions of the free Dirac equation were given in Chap. 7, and those of the Dirac–Coulomb problem were discussed in Chap. 8. All of these solutions have nonvanishing entries for the upper and lower spinors, i.e., all four entries of the ⃗ ⋅ p⃗ in the Dirac Hamiltonian bispinor wave functions are populated. The operator α mixes the upper and lower components. Indeed, in the free Dirac Hamiltonian ⃗ ⋅ p⃗ + β m given in Eq. (7.18), the kinetic term α ⃗ ⋅ p⃗ mixes the upper and HFD = α lower spinor components. The Hamiltonian for a particle at rest, which reads as β m, is diagonal in bispinor space, with an energy eigenvalue +m for the particle solutions, and energy −m for the antiparticle solutions. The Foldy–Wouthuysen transformation serves to accomplish a separation of the total Hamiltonian into an effective upper-component Hamiltonian, and an effective lower-component Hamiltonian, perturbatively, in successively higher orders of the momentum operator. The Foldy–Wouthuysen transformation is best explained by way of example [317]. For illustrative purposes, let us therefore consider the free Dirac Hamiltonian defined in Eq. (7.18), ⃗ ⋅ p⃗ + βm ≡ H1 , HFD = α

(11.8)

where the subscript “1” denotes the first iteration of the Foldy–Wouthuysen transformation. For the first Foldy–Wouthuysen transformation, we choose S1 , a Hermitian operator, to be proportional to −iβ O1 , where O1 is the odd part (in bispinor space) of the Hamiltonian H1 . Specifically, we have i ⃗ ⋅ p⃗ . β O1 , O1 = α (11.9) 2m In view of the lack of an explicit time dependence of S1 [one compares with Eq. (11.4)], the transformation toward the Hamiltonian H2 is calculated in terms of commutators as follows, S1 = −

H2 = eiS1 H1 e−iS1 = H1 + i [S1 , H1 ] +

i2 [S1 , [S1 , H1 ]] 2!

i3 i4 [S1 , [S1 , [S1 , H1 ]]] + [S1 , [S1 , [S1 , [S1 , H1 ]]]] + ⋯ . (11.10) 3! 4! For a calculation in fourth order in the momenta, one needs all terms given in Eq. (11.10). The result reads as follows, +

H2 = β (m +

p⃗ 2 p⃗ 4 (⃗ α ⋅ p⃗)3 )− + O(p5 ) , − 3 2m 8m 3m2

(11.11)

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where O(p5 ) denotes a generic term, possibly involving Dirac matrices, of fifth power in momentum operator p⃗. For the second Foldy–Wouthuysen transformation, we choose the odd operator to be the operator derived from the Hamiltonian H2 , i (⃗ α ⋅ p⃗)3 . (11.12) S2 = − β O2 , O2 = − 2m 3m2 The second transformation HFW = eiS2 H2 e−iS2 (11.13) 3 eliminates the term proportional to (⃗ α ⋅ p⃗) and leads to the final form, p⃗ 2 p⃗ 4 H3 = β (m + − + O(p6 )) . (11.14) 2m 8m3 Eventually, extending the method to arbitrary order in the momenta, one arrives at the Hamiltonian (see also Sec. 11.6) √ √ p⃗2 + m2 ⋅ 12×2 √ 0 2 2 ), (11.15) H∞ = β p⃗ + m = ( 0 − p⃗2 + m2 ⋅ 12×2 where (2 × 2)-matrix notation has been used. The particle degrees of freedom (spinup and spin-down, upper components) have been separated from the antiparticle degrees of freedom (lower components). Furthermore, because an antiparticle solution with eigenvalue −E of the Hamiltonian is interpreted in terms of a physical energy +E, one verifies the dispersion relation √ E = p⃗2 + m2 (11.16) for electrons and positrons. Generically, it is customary in the literature to refer to the Foldy–Wouthuysen transformed Hamiltonian as the particle (not antiparticle) term, i.e., as the (2 × 2)-upper component of H3 . For the case of a free particle, one obtains √ p⃗ 2 p⃗ 4 − + O(p6 )) . (11.17) HFW = p⃗2 + m2 ⋅ 12×2 = 12×2 (m + 2m 8m3 The free Dirac Hamiltonian (11.8) is invariant under parity and so is the transformed Hamiltonian (11.17). It is instructive to observe that parity invariance is maintained in every step of the transformation. In view of Eq. (7.54), the first transformation S1 given in Eq. (11.9) transforms as follows, i i i P ⃗ ⋅ p⃗ Ð→ β [− ⃗ ⋅ (−⃗ ⃗ ⋅ p⃗ β = S1 . S1 = − βα βα p)] β = α (11.18) 2m 2m 2m In fact, it is known that spurious terms can be generated in variants of the Foldy– Wouthuysen program when parity invariance is broken in intermediate steps of the calculation [286]. In particular, attempts to short-cut the procedure are prone to generating spurious terms. For example, it is known that parity-breaking, still unitary transformation give rise to a deceptively simple diagonal form (in bispinor space), but feature spurious terms [255, 286, 287, 318–321]. The main difficulty concerned with the separation of the upper and lower components cannot be avoided in general cases. The Foldy–Wouthuysen transformation eliminates odd terms one after the other, perturbatively in the momenta. It is a tedious but unavoidable endeavor.

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Transformation in the General Case

Let us dwell a little more on the central idea the Foldy–Wouthuysen transformation, and anticipate a few aspects which will be important in more complex applications. The generalized Dirac Hamiltonians considered in Chaps. 7 and 8 have the structure H = E +O, E= O=

1 2 1 2

(11.19a)

(H + β H β) = βm + ⋯ ,

(11.19b)

⃗ ⋅ p⃗ + ⋯ . (H − β H β) = α

(11.19c)

Here, E is the even part in bispinor space, whereas O is the odd part, where even and odd are understood with respect to the separation of the Dirac matrix into 2×2 ⃗ ⋅ p⃗ is odd, submatrices. For example, β m is even, while α βm=(

m 12×2 0 ), 0 −m 12×2

⃗ ⋅ p⃗ = ( α

⃗ ⋅ p⃗ 0 σ ). ⃗ ⋅ p⃗ 0 σ

(11.20)

In typical cases, the dominant term in the even part consists of the term β m, which has eigenvalues ± m and describes a Dirac particle at rest. The dominant term in the ⃗ ⋅ p⃗. The central idea of the Foldy–Wouthuysen odd contribution is the kinetic term α transformation is based on the identity [β O, βm] = −2m O ,

(11.21)

which holds for a general odd term O, because β m is a matrix with constant entries in space and time. It is for this reason that one defines the Hermitian operator S and the unitary operator U = exp(iS) as follows, O S = −i β , U = eiS . (11.22) 2m One defines a perturbative expansion parameter, usually, a parameter connected with the average velocity of the quantum-mechanical wave packet, or with the spatial dimension of the system, and calculates all the terms of the Foldy–Wouthuysen transformation, to be defined in detail below, perturbatively, in the respective parameters. The calculation of multiple nested commutators of S and H, to describe the unitary transformation induced by U on H, then proceeds until further nested commutators yield higher-order terms beyond the order of interest. In many cases, the Foldy–Wouthuysen transformation needs to be iterated in order to yield the desired result. 11.2.4

Radiatively Corrected Dirac Hamiltonian

Before we discuss the Foldy–Wouthuysen transformation of a general electromagnetically coupled Dirac Hamiltonian, we should dwell on the radiatively corrected Dirac equation, because it is very useful to incorporate at least some radiative effects directly into the Foldy–Wouthuysen transformation. Our considerations rely on an understanding of the free Dirac Hamiltonian [see Eq. (7.18)] and of the Dirac– Coulomb Hamiltonian [see Eq. (8.9)]. Furthermore, in Eq. (10.96), we had already

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investigated the vertex function Λµ . Results for the Dirac form factor F1 , and for the Pauli form factor F2 , were given in Eqs. (10.103) and (10.100), respectively. In a general four-vector potential Aµ (x), the Dirac Hamiltonian is ⃗⋅π ⃗ + β m + e A0 , HD = α

(11.23)

⃗ is the canonical momentum, where e is the electron charge, and π ⃗ = p⃗ − e A⃗ . π

(11.24)

The Dirac form factor F1 , and the Pauli form factor F2 , are usually formulated as functions of the momentum transfer q 2 . However, the general Dirac Hamiltonian (11.23) is formulated in terms of differential operators, and multiplication operators, expressed in space-time coordinates. Let us assume static fields and define ⃗ r)] + β m + e A0 (⃗ ⃗ − e A(⃗ ⃗ ⋅ [−i∇ HD (⃗ r) = α r) .

(11.25)

Following the paradigm that multiplication in coordinate space is equivalent to convolution in momentum space, and differentiation in coordinate space is equivalent to multiplication in momentum space, we can formulate the Dirac Hamiltonian in momentum space as ⃗ p′ − p⃗)] + δ (3) (⃗ ⃗ ⋅ [δ (3) (⃗ HD (⃗ p′ , p⃗) = α p′ − p⃗)⃗ p − e A(⃗ p′ − p⃗) β m + e A0 (⃗ p′ − p⃗) . (11.26) This is consistent with our earlier considerations for the Schr¨odinger–Coulomb problem [Eqs. (4.171)–(4.173)]. One finally has for the action of the Hamiltonian HD in momentum space, [HD ψ] (⃗ p′ ) = ∫

d3 p HD (⃗ p′ , p⃗) ψ(⃗ p). (2π)3

(11.27)

An example is the Coulomb potential e A0 (⃗ r) = V (⃗ r) ,

V (⃗ q) = −

4πZα , q⃗2

q⃗ = p⃗′ − p⃗ .

(11.28)

A hypothetical multiplication of V (⃗ q ) by a function F (⃗ q ) of q⃗ would modify the potential. By Fourier transformation, one sees that the action of F (−⃗ q 2 ) V (⃗ q ) in 2 ⃗ momentum space is thus equivalent to the action of F (∇ ) V (⃗ r) in coordinate space, where the gradient operator only acts on the potential, but not on the following wave function. Thus, we replace, according to Eqs. (10.45), (10.118) and (10.189), i σ µν qν F2 (q 2 )Aµ (⃗ q) 2m µν i σ qν → γ µ F1 (−⃗ q 2 )Aµ (⃗ q) + F2 (−⃗ q 2 )Aµ (⃗ q) 2m µν iσ ⃗ 2 )Aµ (⃗ ⃗ 2 ) {i ∂ν Aµ (⃗ → γ µ F1 (∇ r) + F2 (∇ r)} . 2m

γ µ Aµ (⃗ q ) → γ µ F1 (q 2 )Aµ (⃗ q) +

(11.29)

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The Laplacian operators in the arguments of the form factors act only on the argument of the four-vector potentials. We use the identity 1 r) σ µν ∂ν Aµ (⃗ r) = − σ µν ∂µ Aν (⃗ r) = − (σ µν − σ νµ ) ∂µ Aν (⃗ 2 1 1 = − σ µν [∂µ Aν (⃗ r) − ∂ν Aµ (⃗ r)] = − σ µν Fµν . (11.30) 2 2 Hence, the replacement rule is ⃗ 2 ) µν F2 (∇ ⃗ 2 )Aµ (⃗ γ µ Aµ (⃗ q ) → γ µ F1 (∇ r) + σ Fµν . (11.31) 4m We recall that the covariant form of the Dirac equation (11.23) is [γ µ πµ − m] Ψ = [γ µ pµ − e γ µ Aµ − m] Ψ = 0 .

(11.32)

We can thus reformulate the covariant form of the Dirac equation, with the interaction term given in Eq. (11.31), in the static case, as e ⃗ 2 ) γ µ Aµ (⃗ ⃗ 2 ) σ µν Fµν − m] Ψ = 0 . [γ µ pµ − e F1 (∇ r) − F2 ( ∇ (11.33) 4m We now use the identity i ⃗ 0 σ ⃗, σ 0i F0i = [γ 0 , γ i ] E i = i β γ⃗ ⋅ E γ⃗ = ( ). (11.34) −⃗ σ0 2 The term σ i0 Fi0 leads to an equal term which has to be added. The magnetic interaction is written as ⃗ 0 ⃗ ⋅B ⃗, ⃗ = (σ σ ij Fij = ijk Σk (−ij` B ` ) = −2 Σ Σ ). (11.35) ⃗ 0σ Finally, we have e e µν ⃗ −Σ ⃗ ⋅ B) ⃗ . ⃗ 2) ⃗ 2) (i β γ⃗ ⋅ E (11.36) σ Fµν = F2 (∇ F2 ( ∇ 4m 2m Converting Eq. (11.33) to Hamiltonian form, we thus have the radiatively corrected Dirac Hamiltonian HR as ⃗ r)] + β m + F1 (∇ ⃗ 2 ) A(⃗ ⃗ 2 ) eA0 (⃗ ⃗ ⋅ [⃗ HR (⃗ r) = α p − e F 1 (∇ r) e ⃗ r) − β Σ ⃗ ⋅ B(⃗ ⃗ r)] . ⃗ 2) [i γ⃗ ⋅ E(⃗ + F2 (∇ (11.37) 2m In the following, we will use the approximations [see Eqs. (10.100) and (10.103)], α ⃗ 2 ) ≈ F1 (0) = 1 , ⃗ 2 ) ≈ F2 (0) ≡ κ ≈ F1 (∇ F2 (∇ , (11.38) 2π where the g factor of the electron is expressed as g = 2(1 + κ). Furthermore, under the approximations given in Eq. (11.38), the form factors become independent of time. In this case, we can relax again the assumptions on the time-independence of the vector and scalar potentials and write the radiatively corrected Dirac Hamiltonian as ⃗ r, t)] + β m + eA0 (⃗ ⃗ ⋅ [⃗ HR (⃗ r, t) = α p − e A(⃗ r, t) eκ ⃗ r, t) − β Σ ⃗ ⋅ B(⃗ ⃗ r, t)] . [i γ⃗ ⋅ E(⃗ (11.39) 2m The possible explicit time dependence will need to be taken into account as we discuss the Foldy–Wouthuysen transformation. +

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11.2.5

General Electromagnetic Coupling

We need to generalize the Foldy–Wouthuysen transformation, carried out at this point only for a free Dirac particle, to the case of an electromagnetic coupling. It is useful to also include leading radiative corrections, corresponding to the anomalous magnetic moment, as given in the effective Dirac Hamiltonian (11.37). We shall consider only terms linear in the electron anomaly κ defined in Eq. (1.9). With reference to Eqs. (11.24) and (11.39), we investigate the radiatively corrected Hamiltonian eκ ⃗ −βΣ ⃗ ⋅ B) ⃗ , (iβ α ⃗⋅E ⃗⋅π ⃗ + βm + e A0 + (11.40) HR = α 2m where we take into account the possibility of an explicit time dependence of the vector and scalar potentials, and the fields. We shall carry out the transformation of the Hamiltonian (11.40) up to eighth order in the momenta, under the premise of the following scaling, ⃗ ∼ α, π

eA⃗ ∼ α ,

⃗ ∼ α2 , eB

⃗ = −∇V ⃗ ∼ α3 , eE

⃗ ∼ α5 . e ∂t E

(11.41)

In the context of the general method outlined in Sec. 11.2.1, we find that the time derivative term in Eq. (11.4) is decisive in ensuring that the Foldy–Wouthuysen transformed Hamiltonian contains the complete electric field, and not only the transverse or longitudinal components separately. In Coulomb gauge, these components are naturally separated as follows, ⃗∥ = −∇A ⃗ 0, E

⃗⊥ = −∂t A⃗ . E

(11.42)

The result of the iterative eighth-order standard Foldy–Wouthuysen transformation [286], carried out up to order α8 , can be written in terms of HFW = H[0] + H[2] + H[3] + H[4] + H[5] + H[6] + H[7] + H[8] .

(11.43)

The superscript denotes the power of the coupling parameter at which the term becomes relevant. The coupling parameter is usually denoted as α. From the zeroth to third order in α, the terms read as follows, H[0] = βm , (11.44) 1 ⃗ ⃗ )2 + V , (Σ ⋅ π (11.45) H[2] = β 2m eκ ⃗ ⃗ H[3] = − βΣ⋅B. (11.46) 2m The α4 and α5 terms can be expressed very compactly, and are found in agreement with the literature [316, 322], 1 ⃗ ie ⃗ ⃗ ⋅E ⃗ ], ⃗ )4 − ⃗, Σ (Σ ⋅ π [Σ ⋅ π 8m3 8m2 ieκ ⃗ ⃗ ⋅E ⃗ ] + β eκ {Σ ⃗ ⋅π ⃗ ⋅π ⃗ ⋅B ⃗ }} . ⃗, Σ ⃗, { Σ ⃗, Σ H[5] = − [Σ ⋅ π 4m2 16m3 Here, {A, B} = A B + B A denotes the anticommutator. H[4] = − β

(11.47) (11.48)

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For the α6 terms, we indicate two alternative representations. The straightforward application of the method described in Sec. (11.2.1) leads to the result 1 5ie ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅E ⃗ ]]] ⃗ )6 − ⃗, [ Σ ⃗, [ Σ ⃗, Σ (Σ [Σ 5 16m 128m4 2 ⃗2 ie 2 ⃗ ⋅π ⃗ ⋅π ⃗ ⋅E ⃗ ]} + β e E . ⃗ ⃗ { ( Σ ) , [ Σ , Σ + 8m4 8m3

H[6] = β

(11.49)

Using the identity ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅ E]]] ⃗ ⃗ ⋅π ⃗ ⋅π ⃗ ⋅ E]} ⃗ − [(Σ ⃗ ⋅π ⃗ ⋅ E, ⃗ Σ ⃗ ⋅π [Σ ⃗ , [Σ ⃗ , [Σ ⃗, Σ ⃗ ) , [Σ ⃗, Σ ⃗ ) , {Σ ⃗ }] , = 2 {(Σ 2

2

(11.50) the result for the sixth-order terms can alternatively be written as 2 5ie 1 ⃗ ⋅π ⃗ ⋅π ⃗ ⋅ E, ⃗ Σ ⃗ ⋅π ⃗ )6 + ⃗ ) , {Σ ⃗ }] (Σ [(Σ 16m5 128m4 2 ⃗2 3i e ⃗ ⋅π ⃗ ⋅π ⃗ ⋅E ⃗ ]} + β e E . ⃗ )2 , [ Σ ⃗, Σ { (Σ + 4 64m 8m3

H[6] = β

(11.51)

The α7 terms contain the anomalous magnetic moment, proportional to κ. As we recall, the electron g factor is related to κ by the relations g = 2(1 + κ). The result, obtained using the method outlined in Sec. 11.2.1, is e2 κ ⃗ 2 eκ ⃗ ⋅π ⃗ ⋅ ∂t E ⃗ ⋅π ⃗ ⋅π ⃗ ⋅E ⃗ ]} ⃗ } + i e κ { (Σ ⃗, Σ ⃗ )2 , [ Σ ⃗, Σ E − β {Σ 8m3 16m3 16m4 eκ ⃗ ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅B ⃗ }} ]] ⃗ , [Σ ⃗, { Σ ⃗, { Σ ⃗, Σ −β [Σ ⋅ π 32m5 3e κ ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅B ⃗ }} }} ⃗ , {Σ ⃗, { Σ ⃗, { Σ ⃗, Σ −β {Σ 256m5 i e2 κ ⃗ ⃗ ⃗ ⃗ ⋅ B} ⃗ ]. ⃗, Σ + [ Σ ⋅ E, {Σ ⋅ π (11.52) 16 m4

H[7] = β

We here supplement a few terms which are relevant for strong magnetic fields, and which have not been taken into account in the literature. One notes that, in the seventh order in α, one cannot avoid the emergence of terms proportional to the time derivative of electric fields. In the somewhat non-standard approach chosen in Ref. [322], the time derivative terms already appear in the order α6 . Specifically, according to Eq. (18) of Ref. [322], ⃗˙ ⃗ , σ ⋅ E}. one encounters a term −[e/(16m3 )] {σ ⋅ π The terms involving the timederivative of the electric field were advocated to be eliminated by an additional unitary transformation given in Eq. (19) of Ref. [323]. In the standard Foldy– Wouthuysen approach, as we show here, the time derivative of the electric field appears only in the order α7 . In the order α8 , it is not necessary to consider the anomalous magnetic moment, provided we neglect terms proportional to κ2 . After a somewhat elaborate calculation, using computer algebra techniques [151] in order to keep the complexity of

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intermediate expressions under control, one arrives at the following terms, 2 i e2 ⃗ ⃗ ⃗ 5 8 ⃗ ⋅π ⃗ + 7 e β[Σ ⃗ ⋅π ⃗ ⋅ E][ ⃗ Σ ⃗ ⋅π ⃗ ⋅ E] ⃗ ⃗ ⃗, Σ ⃗, Σ ( Σ ) − [ Σ ⋅ E, Σ ⋅ ∂ E] H[8] = −β t 128m7 32m4 192m5 2 3 e2 ⃗ ⋅π ⃗ ⋅E ⃗ } {Σ ⃗ ⋅π ⃗ ⋅E ⃗ } − e β [Σ ⃗ ⋅π ⃗ ⋅π ⃗ ⋅ E) ⃗ 2 ]] ⃗ ⃗ ⃗, [ Σ ⃗ , (Σ − β { Σ , Σ , Σ 64m5 24m5 e ⃗ ⋅π ⃗ ⋅ ∂t E ⃗} ⃗ )3 , Σ β { (Σ + 48m5 5ie ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅E ⃗ ]]]]] ⃗, [ Σ ⃗, [ Σ ⃗, [ Σ ⃗, [ Σ ⃗, Σ [Σ − 1024m6 ie ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅E ⃗ ]]]}} ⃗, { Σ ⃗, [ Σ ⃗, [ Σ ⃗, [ Σ ⃗, Σ − {Σ 32m6 ie ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅π ⃗ ⋅E ⃗ ]}}}}. ⃗, { Σ ⃗, { Σ ⃗, { Σ ⃗, [ Σ ⃗, Σ − {Σ (11.53) 48m6 The α8 terms have been obtained by mapping the algebra of the commutators onto a computer symbolic program [151]. One can check that the terms given in Eqs. (11.44)–(11.53) fulfill the particleantiparticle symmetry relations, i.e., they are invariant under the application of the following operations: (i) multiplication by an overall factor −1, (ii) replacement ⃗ → −Σ, ⃗ and ⃗ → −⃗ β → −β, (iii) replacement π π , and ∂t → −∂t , (iv) replacement Σ ⃗ → −E. ⃗ All of these replacements are (v) replacements e → −e, V → −V and E dictated by the reinterpretation principle of quantum field theory. 11.2.6

General Particle Hamiltonians

Let us concentrate on the upper left 2 × 2 submatrix of HFW , which is the particle Hamiltonian. It is well known that the Dirac Hamiltonian describes particle and antiparticle states simultaneously, and that the lower right 2 × 2 submatrix of HFW describes the corresponding antiparticle. In principle, the particle Hamiltonian can be obtained from the results given in Eqs. (11.45)–(11.53) by simply replacing ⃗ →σ ⃗ , and β → 12×2 , but it is still instructive to give the results separately. Σ Because the rest mass term β m given in Eq. (11.44) is a physically irrelevant constant, we write the general particle Hamiltonian H under the presence of the external electric and magnetic fields as HFW = H [2] + H [3] + H [4] + H [5] + H [6] + H [7] + H [8] , (11.54) where we take into account up to eighth-order terms. One finds eκ 1 ⃗. ⃗ )2 + V , ⃗⋅B (⃗ σ⋅π H [3] = − σ (11.55) H [2] = 2m 2m In regard to Eq. (11.55), the following relation is of extreme importance, which is why we present the derivation in great detail, ⃗ )2 = σ i σ j π i π j = (δ ij + i ijk σ k ) π i π j (⃗ σ⋅π 1 1 ⃗ 2 + i ijk σ k [π i , π j ] = π ⃗ 2 + i ijk σ k [−i∇i (−e Aj ) + i∇j (−e Ai )] =π 2 2 1 ijk k 2 i j ⃗ . (11.56) ⃗ 2 − e kij σ k ∇i Aj = π ⃗2 − eσ ⃗⋅B ⃗ +  σ {2∇ (−e A )} = π =π 2

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The fourth-order terms find the compact representation 1 ie ⃗ ], ⃗ )4 − ⃗⋅π ⃗, σ ⃗⋅E H [4] = − (⃗ σ⋅π [σ (11.57) 8m3 8m2 while the fifth-order terms contain the anomalous magnetic-moment, ieκ ⃗ ] + eκ {σ ⃗ }} . ⃗⋅π ⃗, σ ⃗⋅E ⃗⋅π ⃗, { σ ⃗⋅π ⃗, σ ⃗⋅B H [5] = − 2 [ σ (11.58) 4m 16m3 The α6 terms, in the general setting, are given as ⃗ )6 (⃗ σ⋅π 5ie 2 ⃗ σ ⃗ ) , {⃗ ⃗⋅π ⃗ }] H [6] = + [(⃗ σ⋅π σ ⋅ E, 16m5 128m4 2 ⃗2 3i e ⃗ ]} + e E . ⃗ )2 , [ σ ⃗⋅π ⃗, σ ⃗⋅E + { (⃗ σ⋅π (11.59) 4 64m 8m3 In the above form, the sixth-order terms in the Hamiltonian are compatible with those used in Ref. [324] for spin-1/2 particles [see Eqs. (36)–(38) of Ref. [324]]. The α6 terms listed in Eq. (11.51) are also equal to those obtained by applying the unitary transformation outlined in Eq. (19) of Ref. [323] to the Hamiltonian given in Eq. (15) of Ref. [323], i.e., to the Hamiltonian obtained by adding the terms given in Eqs. (15) and (20) of Ref. [323]. Also, the result in Eq. (11.51) is equal to the Hamiltonian considered in Eq. (7) of Ref. [325], which in turn has been derived from the NRQED approach outlined in Ref. [326]. The α7 terms attain the following structure, eκ e2 κ ⃗ 2 ⃗ ]} ⃗ } + i e κ { (⃗ ⃗⋅π ⃗, σ ⃗ ⋅ ∂t E ⃗ )2 , [ σ ⃗⋅π ⃗, σ ⃗⋅E E − {σ σ⋅π H [7] = 8m3 16m3 16m4 eκ ⃗ }} ]] ⃗ , [⃗ ⃗, { σ ⃗⋅π ⃗, { σ ⃗⋅π ⃗, σ ⃗⋅B [⃗ σ⋅π σ⋅π − 32m5 2 3e κ ⃗ }} }} + i e κ [ σ ⃗ {⃗ ⃗ . ⃗ , {⃗ ⃗, { σ ⃗⋅π ⃗, { σ ⃗⋅π ⃗, σ ⃗⋅B ⃗ ⋅ E, ⃗, σ ⃗ ⋅ B}] − {⃗ σ⋅π σ⋅π σ⋅π 5 256m 16 m4 (11.60) Finally, the α8 terms are derived from Eq. (11.53) and read as follows, 2 5 i e2 ⃗ σ ⃗ + 7 e [⃗ ⃗ σ⋅π ⃗ ⃗ )8 − ⃗ ⋅ ∂t E] ⃗, σ ⃗ ⋅ E][⃗ ⃗, σ ⃗ ⋅ E] (⃗ σ⋅π [⃗ σ ⋅ E, σ⋅π 7 4 128m 32m 192m5 2 3 e2 ⃗ σ⋅π ⃗ − e [⃗ ⃗ 2 ]] ⃗ ⃗ ⃗ ⃗ ⃗ , [⃗ ⃗ , (⃗ − {⃗ σ ⋅ π , σ ⋅ E}{⃗ , σ ⋅ E} σ⋅π σ⋅π σ ⋅ E) 64m5 24m5 e ⃗ ⃗ )3 , σ ⃗ ⋅ ∂t E} + β{(⃗ σ⋅π 48m5 5ie ⃗ ⃗ , [⃗ ⃗ , [⃗ ⃗ , [⃗ ⃗ , [⃗ ⃗ , σ ⋅ E]]]]] − [⃗ σ⋅π σ⋅π σ⋅π σ⋅π σ⋅π 1024m6 ie ⃗ ⃗ , {⃗ ⃗ , [⃗ ⃗ , [⃗ ⃗ , [⃗ ⃗ , σ ⋅ E]]]}} − {⃗ σ⋅π σ⋅π σ⋅π σ⋅π σ⋅π 32m6 ie ⃗ ⃗ , {⃗ ⃗ , {⃗ ⃗ , {⃗ ⃗ , [⃗ ⃗ , σ ⋅ E]}}}} {⃗ σ⋅π σ⋅π σ⋅π σ⋅π σ⋅π . (11.61) − 48m6 Again, we stress that we restrict the discussion to terms linear in κ. Also, while we indicate general results for the FW transformed Hamiltonians to order α8 , we restrict the discussion to order α6 in the applications discussed below.

H [8] = −

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In the following, we shall discuss four applications of the Foldy–Wouthuysen Hamiltonian given in Eq. (11.54) in the regime of bound states. The first of these will concern a plain Coulomb field, in which case the electric field is static and its time derivative vanishes. We shall derive explicit expressions, with comments on the applicability to bound states in regard to reduced-mass corrections, in each order in the Zα-expansion. A second application will concern magnetic-field coupling, where we shall put special emphasis on the spin-dependent terms. A third application concerns the extraction of the transition current, including relativistic corrections, from the Foldy–Wouthuysen transformed Hamiltonian in the presence of (conceivably quantized) fields. Finally, a fourth application goes beyond the treatment restricted to the F2 form factor, and includes form-factor derivatives as well as the Dirac F1 form factor. 11.3 11.3.1

Applications Coulomb Field Coupling

One of the important applications of the Hamiltonian (11.54) concerns Coulombic bound states. In this case, one has a situation where all fields are purely classical, e A0 = V = −

Zα , r

A⃗ = 0⃗ ,

⃗ = 0⃗ , B

⃗ = −e ∇A ⃗ 0 = −∇V ⃗ , eE

⃗ = p⃗ . π (11.62)

One ends up with the following leading term, H [2] =

p⃗ 2 Zα p⃗ 2 +V = − , 2m 2m r

(11.63)

which is the Schr¨ odinger–Coulomb Hamiltonian given in Eq. (4.1), in the limit of an infinite nuclear mass. We recall that the Dirac formalism is inherently a one-particle formalism and does not allow for the inclusion of reduced-mass effects, which have to be treated on the basis of a manifestly two-body formalism. For the evaluation of the fourth-order corrections given by Eq. (11.57), one needs the identities, Zα Zα r⃗ ⃗ = −∇V ⃗ =∇ ⃗ eE =− 3 , r r

⃗ = −∇ ⃗ 2V − 2 σ ⃗ × p⃗) . (11.64) ⃗ ⋅ E] ⃗ ⋅ (∇V ie [⃗ σ ⋅ p⃗, σ

According to Eq. (11.57), the leading relativistic corrections in the Foldy– Wouthuysen Hamiltonian in this case reads as follows, 1 1 p⃗ 4 ⃗ 2V + ⃗ × p⃗) ⃗ ⋅ (∇V + σ ∇ 3 2 8m 8m 4m2 p⃗ 4 π (Zα) (3) Zα ⃗. ⃗⋅L = − + δ (⃗ r) + σ 3 2 8m 2m 4m2 r3

H [4] = −

(11.65)

This Hamiltonian contains the familiar zitterbewegung term (Darwin term), proportional to a Dirac-δ, and the spin-orbit coupling, both of which receive anomalous

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magnetic-moment corrections. From Eq. (11.58), it follows that the fifth-order (in Zα) terms reads as follows, 1 1 ⃗ 2V + ⃗ × p⃗)] ⃗ ⋅ (∇V ∇ σ 8m2 4m2 π (Zα) (3) Zα ⃗ . ⃗ ⋅ L] = 2κ [ δ (⃗ r) + σ 2 2m 4m2 r3

H [5] = 2κ [

(11.66)

The anomalous magnetic-moment corrections simply multiply the potential terms in Eq. (11.65) by an additional factor 2κ. For the sixth-order corrections, we note the following useful identity which follows from Eq. (11.62), ⃗ = −{⃗ ⃗ } = −i[⃗ ⃗ ⋅ E} ⃗ ⋅ ∇V e{⃗ σ ⋅ p⃗, σ σ ⋅ p⃗, σ p 2, V ] .

(11.67)

Thus, the sixth-order corrections attain the following form, 3 3 p⃗ 6 ⃗ 2V } − ⃗ × p⃗)} { p⃗ 2 , ∇ { p⃗ 2 , σ ⃗ ⋅ (∇V − 5 4 16m 64m 32m4 ⃗ )2 (∇V 5 2 2 [⃗ p , [⃗ p , V ]] + . + 128m4 8m3 This result has to be compared to the one used in Ref. [312], H [6] =

(11.68)

p⃗ 6 3 1 3 ⃗ 2V } − ⃗ × p⃗)} + {⃗ {⃗ ⃗ ⋅ (∇V p 2, ∇ p 2, σ − [⃗ p 2 , [⃗ p 2 , V ]] . 5 4 4 16 m 64 m 32 m 128 m4 (11.69) We use the Schr¨ odinger equation given Eq. (4.1) in the non-recoil limit, where the reduced mass is equal to the electron mass, H

[6]

=

p⃗2 +V , HS ∣φ⟩ = E ∣φ⟩ , (HS − E) ∣φ⟩ = 0 . (11.70) 2m One can show the following identity fulfilled by diagonal matrix elements of the Schr¨ odinger eigenstates, HS =

⃗ )2 ∣φ⟩ . ⟨φ∣[⃗ p2 , [⃗ p2 , V ]]∣φ⟩ = −4m ⟨φ∣⟨(∇V

(11.71)

One thus has the relation 5 1 ⃗ ) 2 ∣φ⟩ ⟨φ∣ [⃗ p 2 , [⃗ p 2 , V ]] ∣φ⟩ + ⟨φ∣(∇V 128m4 8m3 1 1 ⃗ )2 ∣φ⟩ . = ⟨φ∣ [⃗ p 2 , [⃗ p 2 , V ]] ∣φ⟩ = − ⟨φ∣⟨(∇V (11.72) 128 m4 32m3 So, with reference to Eqs. (11.68) and (11.69), one can show that, on the level of diagonal matrix element, the Hamiltonians H [6] and H ⟨φ∣H [6] ∣φ⟩ = ⟨φ∣H 6

[6]

∣φ⟩ .

[6]

are equivalent, (11.73)

On the level of the α corrections, the direct approach for the Foldy–Wouthuysen transformation chosen here, and the modified approach used in Refs. [312, 322], are thus equivalent.

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A few remarks are in order. For hydrogen and positronium, and other twoparticle bound systems, the evaluation of the zitterbewegung terms Eqs. (11.65) and (11.66) (those proportional to Dirac-δ) is immediate. However, if one generalizes the approach to helium, the evaluation of the diagonal matrix element of the zitterbewegung term can be far less trivial, as has been noticed in Ref. [327]. This fact has inspired the development [328] of a scheme where the expectation value of a singular Dirac-δ operator is replaced by less singular momentum operators. It is interesting to observe that the identities derived in Ref. [328] can be traced to the relation ⃗ 2 V = {⃗ 4π(Zα)δ (3) (⃗ r) = [⃗ p, [⃗ p, V ]] = −∇ p2 , V } − 2⃗ p V p⃗ . (11.74) On the left-hand side of this equation, we have a Dirac-δ, on the right-hand side, some standard momentum operators and potentials. Taking the expectation value of the expression on the right-hand side, and using the Hermiticity of the momentum operators, one arrives at less singular expressions. This approach, though, meets some limitations in higher-order. This and similar approaches are commonly referred to as “Drachmanization” in view of the author of Ref. [328]. Some additional remarks on the sixth-order corrections are in order. Let us consider the expectation value of the second term on the right-hand side of Eq. (11.68), ⃗ 2 V } ∣φ⟩ . ⟨φ∣ {⃗ p 2, ∇ (11.75) 2 When evaluated on a Schr¨ odinger eigenstate ψ, one can replace p⃗ → 2m(E − V ), where E is the reference-state energy. For V = −Z α/r, this leads to a matrix element proportional to ⟨φ∣(1/r) δ (3) (⃗ r)∣φ⟩, which is divergent for S states which have a nonvanishing probability density at the origin. The application of relations such as those given in Eq. (11.74) cannot cure the divergence. Indeed, for the exact Coulomb potential V = −Z α/r, several matrix elements of H [6] become singular for reference S states. A more far-reaching approach is needed. One possibility, as shown in Ref. [44], consists in the regularization (at short range) of the Coulomb potential, Zα Zα (1 − e−λ r ) . →− (11.76) V =− r r The original Coulomb potential is obtained in the limit λ → ∞, while for any finite λ, the divergence of the Coulomb potential at r = 0 is regularized and made finite. Using this approach, one can obtain (see Ref. [44]) the (Zα)6 contributions to the energies of the bound states, denoted as E [6] , on the basis of the formula ′ 1 E [6] = ⟨H [4] ( ) H [4] ⟩ + ⟨H [6] ⟩ . (11.77) E−H Here, both the Hamiltonians H [4] as well as H [6] are understood to be formulated with the regularized potential given in Eq. (11.76). The idea is to recover the full (Zα)6 result in the limit λ → ∞, the limit being taken after all integrations are complete. In particular, the corrections to the wave function, due to the correction term proportional to e−λ r , need to be taken into account consistently (see Ref. [44]). The energy at O(Zα)6 can also be regularized dimensionally, as was done for positronium in Ref. [329].

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Magnetic Field Coupling and Electron Moment

The Hamiltonian (11.54) has a rich structure. A second illustrative application concerns an otherwise free electron in a homogeneous, spatially and temporally ⃗ A constant magnetic field is described by the vector constant magnetic field B. 1 ⃗ ⃗ potential A = 2 (B × r⃗). We summarize the following relations, e ⃗ ⃗ × r⃗) , ⃗ = 0⃗ , ⃗ = p⃗ − (B A⃗ = 12 (B eA0 = V = 0 , E π × r⃗) . (11.78) 2 Furthermore, one has the relation [see also Eq. (11.56)] ⃗ = p⃗ 2 − e σ ⃗ − eL ⃗ ⋅B ⃗+ ⃗ )2 = π ⃗2 − eσ ⃗⋅B ⃗⋅B (⃗ σ⋅π

e2 ⃗ (B × r⃗)2 . 4

(11.79)

⃗ 2 describes the leading diamagnetic interaction. The The term proportional to (⃗ r×B) [2] sum of the terms H and H [3] from the Foldy–Wouthuysen Hamiltonian (11.54) becomes, ignoring higher-order corrections in the momenta, 2 p⃗ 2 e ⃗ +L ⃗ ⋅ B} ⃗ + e (⃗ ⃗ 2 . (11.80) {(1 + κ) σ ⃗⋅B r × B) − 2m 2m 8m ⃗ matrices, defined in Eq. (4.343), are proportional to the We note that the Pauli σ ⃗ /2. The 2 × 2-matrices s⃗ describe the electron spin electron spin operator s⃗ = σ angular momentum [see also Eq. (7.102)], and they fulfill the commutator relations [si , sj ] = iijk sk which characterize the Lie algebra of the rotation group. From Eq. (11.80), it is obvious that only the coupling of the spin to the magnetic field (as opposed to the coupling of the orbital angular momentum to the magnetic field) receives a correction due to the anomalous magnetic moment. The electron spin g factor is related to the spin-flip frequency (or spin-flip energy) of a free electron in a homogeneous and constant magnetic field. In order to clarify the definition of the g factor, in Eq. (11.80), one isolates the electron-spin dependent term, and writes the energy shift incurred upon a spin flip, as a function of the magnetic field strength. There is a little subtlety involved in the sign of the electron g factor, in view of its negative charge. This subtlety has given rise to some confusion, and, in particular, to proposals of redefining the sign of the electron’s g factor [330]. Let us go into detail and isolate the spin-dependent terms in Eq. (11.80),

HFW ≈ H [2] + H [3] =

1 e ⃗ ≡ −g e s⃗ ⋅ B ⃗, ⃗. s⃗ ⋅ B s⃗ = σ (11.81) 2m 2m 2 Here, e = −∣e∣ is the electron charge, and the expression following the “≡” sign serves ⃗ /2. Note that a as a definition of the electron g factor. The spin operator is s⃗ = σ matching with Eq. (11.81) leads to the identification HFW = −2(1 + κ)

g = 2(1 + κ)

(11.82)

for the electron g factor. This result should be compared to Eqs. (1.10a), (1.10b) and (1.11). It is intuitively preferable to define the electron g factor as a positive quantity, and this desire is fulfilled in Eq. (11.82) as well as in the literature, in

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general (see, however, Ref. [330]). We must now compare the definition given in Eq. (11.82) to the well-known interaction energy of a magnetic dipole with magnetic ⃗ ⃗ in an external magnetic field; the latter reads as E = −⟨⃗ moment µ µ ⋅ B⟩. This formalism is consistent with Eq. (11.82) if we identify, for the electron, ⃗ , E = −⟨⃗ µ ⋅ B⟩

⃗ = g (−µB ) s⃗ , µ

µB =

∣e∣ e =− . 2m 2m

(11.83)

̵ Here, µB is the Bohr magneton, which, in SI mksA units, is equal to ∣e∣ h/(2mc), where ∣e∣ is the positron charge, and m is the electron mass. Note that, in order for the g factor of the electron to come out positive, one has to define the relationship ⃗, the spin operator, and the Bohr magneton µB , by of the magnetic moment µ ⃗ = −g µB s⃗, where s⃗ introducing a minus sign, as explicitly written in the relation µ ⃗ be equal to g µB s⃗ for the electron, is the spin matrix. If one were to insist that µ then the sign of the electron g factor would be negative. 11.3.3

Magnetic Fields and Electrostatic Potentials

Now, we turn our attention to the problem of an electron coupled to a constant, uniform magnetic field, which is simultaneously bound to an electrostatic field. The canonical momentum is given as e ⃗ ⃗ = p⃗ − (B × r⃗) . (11.84) π 2 Furthermore, because the electric field is static, and thus equal to the gradient of a potential, we have the relation ⃗ = eE ⃗∥ = −∇V ⃗ , eE

V = e A0 .

(11.85)

In the combined fields, the sum of the terms H [2] and H [3] is just H [2] + H [3] =

2 p⃗ 2 e ⃗ ⃗ e ⃗ + e (B ⃗ × r⃗)2 . ⃗⋅B +V − L⋅B− (1 + κ) σ 2m 2m 2m 8m

(11.86)

The expression for H [4] simplifies as follows, 2 ⃗4 π e⃗ π2 ⃗− e B ⃗2 + 1 ∇ ⃗ 2V ⃗⋅B + σ 3 3 8m 4m 8m3 8m2 1 e ⃗ × r⃗)) , ⃗ × p⃗) − ⃗ × (B ⃗ ⋅ (∇V ⃗ ⋅ (∇V + σ σ 4m2 8m2

H [4] = −

(11.87)

where, using Eq. (11.79), one derives the relation 2 4 ⃗ ⋅B ⃗ p⃗ 2 + e {⃗ ⃗ × r⃗)2 } + e2 (L ⃗ ⋅ B) ⃗ 2 + e (B ⃗ × r⃗)4 . ⃗ 4 = p⃗ 4 − 2eL π p 2 , (B 4 16

(11.88)

We recall that, in the current calculation, we assume a strong magnetic field, of the order given in Eq. (11.41). Hence, we keep higher powers of the magnetic ⃗ 4 term in field, commensurate with the desired order of the expansion. E.g., the π Eq. (11.87) contains terms of fourth order in the magnetic fields.

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For the fifth-order Hamiltonian H [5] , we need the following relation, which is valid for a constant, uniform magnetic field, ⃗ }} = 4B ⃗ ⋅ p⃗ σ ⃗ (B ⃗ ⋅ p⃗) . ⃗⋅π ⃗, { σ ⃗⋅π ⃗, σ ⃗⋅B ⃗ ⋅ p⃗ + 2e ((⃗ {σ σ × r⃗) ⋅ B) (11.89) This implies that

κ eκ κ 2 ⃗ × r⃗)) ⃗ V + ⃗ × p⃗) − ⃗ × (B ⃗ ⋅ (∇V ⃗ ⋅ (∇V ∇ σ σ 2 2 4m 2m 8m2 eκ ⃗ eκ ⃗ B ⃗ ⋅ p⃗ . ⃗ ⋅ p⃗ + + [(⃗ σ × r⃗) ⋅ B] (11.90) B ⋅ p⃗ σ 4m3 8m3 Again, we stress that, in view of the assumed scaling relations given in Eq. (11.41), we keep terms quadratic in the magnetic fields. In the combined magnetic and electrostatic fields, the sixth-order corrections summarized in H [6] evaluate as follows, ⃗ )6 (⃗ σ⋅π 5ie 2 ⃗ σ ⃗ ) , {⃗ ⃗⋅π ⃗ }] H [6] = + [(⃗ σ⋅π σ ⋅ E, 16m5 128m4 ⃗ 2 3i e ⃗ ]} + (∇V ) . ⃗ )2 , [ σ ⃗⋅π ⃗, σ ⃗⋅E { (⃗ σ⋅π (11.91) + 4 64m 8m3 From Eq. (11.56), one can derive the following relation, ⃗ 3=π ⃗2 π ⃗π ⃗2 σ ⃗. ⃗ )6 = (⃗ ⃗ 6 + 3e2 B ⃗2 − 3 e σ ⃗⋅B ⃗ 4 − e3 B ⃗⋅B (⃗ σ⋅π π 2 − e⃗ σ ⋅ B) (11.92) H [5] =

⃗ 6 (without Pauli matrices), one finds For the higher-order term π 3

2 ⃗ × r⃗)2 ] . ⃗ ⋅B ⃗ + e (B ⃗ = [⃗ (11.93) π p − eL 4 Two further useful relations can be given which enable us to simplify the sixth-order terms. The first of these is as follows, 2 ⃗ σ ⃗ ⋅B ⃗ + e (B ⃗ × r⃗)2 , [⃗ ⃗ ⋅ B, ⃗ V ]] . (11.94) ⃗ )2 , {⃗ ⃗⋅π ⃗ }] = −ie [⃗ e [(⃗ σ⋅π σ ⋅ E, p 2 − eL p 2 − eL 4 We also have the following relation, 2 ⃗ = i {⃗ ⃗ ⋅B ⃗ + e (B ⃗ × r⃗)2 , ⃗ )2 , [⃗ ⃗, σ ⃗ ⋅ E]} e {(⃗ σ⋅π σ⋅π p 2 − e (⃗ σ + L) 4 e ⃗ ⃗ 2 V + 2⃗ ⃗ × [⃗ ∇ σ ⋅ ∇V p − (B × r⃗)]} . (11.95) 2 Hence, the result for the sixth-order terms of the Foldy–Wouthuysen transformed Hamiltonian, in the combined magnetic and electrostatic fields, reads as follows, ⃗ 2 1 6 2 ⃗2 2 4 3 ⃗2 ⃗π ⃗ + (∇V ) (⃗ ⃗ ⃗ ⃗ ⃗ H [6] = π + 3e B π − 3 e σ ⋅ B − e B σ ⋅ B) 16m5 8m3 2 5 ⃗ ⋅B ⃗ + e (B ⃗ × r⃗)2 , [⃗ ⃗ ⋅ B, ⃗ V ]] + [⃗ p 2 − eL p 2 − eL 128m4 4 6

2

2 3 e ⃗ 2 ⃗ ⋅B ⃗ + e (B ⃗ × r⃗)2 , ∇ ⃗ 2 V + 2⃗ ⃗ × [⃗ {⃗ p − e (⃗ σ + L) σ ⋅ ∇V p − (B × r⃗)]} . 64m4 4 2 (11.96) Expressions for the seventh-order and eighth-order terms in the combined magnetic and electrostatic fields. can be obtained from Eqs. (11.60) and (11.61) under the ⃗ × r⃗), and eE ⃗ → −∇V ⃗ . ⃗ → p⃗ − (e/2)(B substitutions π

−

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Transition Current

We now investigate the question of how to define the transition current operator J µ , which is to be extracted from the Foldy–Wouthuysen transformed Hamiltonian (11.54). The answer to this question can be derived from a consideration of the Lagrangian of the classical electromagnetic field, coupled to a classical fourvector current distribution J µ , as given in Eq. (8.84) of Ref. [81], 1 (11.97) L = − F µν Fµν − J µ Aµ . 4 For the fully quantized theory, we have encountered the Lagrangian in Eq. (10.1). It can be written as the sum of the free Lagrangian L0 of the quantized electromagnetic and Dirac fields, given in Eq. (10.2), and the interaction Lagrangian LI , given in Eq. (10.3). Let us remember that, according to Eq. (10.4), the quantum electrodynamic interaction Hamiltonian has the structure HI = J µ Aµ = e j µ Aµ = e j 0 A0 − e⃗j ⋅ A⃗ ,

(11.98)

where in the latter form, we write out the time-like and spatial components explicitly. We define the current operator j µ = J µ /e, where e is the electron charge. In the context of the Foldy–Wouthuysen transformation, we should remember that the momentum and position operators in Eq. (11.54) act on the first-quantized wave function describing the electron. In the second-quantized formalism, the current becomes an operator, i.e., j µ = ψ γ µ ψ, where ψ and ψ = ψ + γ 0 are the field operators of the Dirac field. The functional form of the total Hamiltonian given in Eq. (11.54) does not change if we assume that the components Aµ of the vector potential are quantized rather than classical fields (see also Chap. 2). If we assume that the electron wave functions are taken in the first-quantized formalism, then we can extract, from the interaction part HI of the Foldy–Wouthuysen Hamiltonian (11.54), the time-like and spatial components j 0 and ⃗j of the current operator by the identification HIγ = e j 0 A0γ − e ⃗j ⋅ A⃗γ ,

(11.99)

where the additional subscripts γ in HIγ and Aµγ refer to the four-vector potentials of a quantized field, describing a photon. The reason for singling out the four-vector potentials of the photon field, distinguished by the subscript γ, will become apparent in the following. We will see that in many cases, the elements of the current, j 0 and ⃗j, are in fact operator-valued in the nonrelativistic limit (they contain momentum operators). Now, one has to be careful: The momentum and gradient operators in j 0 and ⃗j could either act on the quantum mechanical wave functions between which the interaction Hamiltonian HI is “sandwiched”, or, on the four-vector potential Aµ . Indeed, we shall encounter both cases. Let us now discuss how to extract the current from the interaction Hamiltonian. To this end, we consider a typical transition matrix element of the form 0 0 ⃗ ⃗ ⟩, which describes ⃗ ⟨φf , 0∣HIγ ∣φi , 1kλ ⃗ ⟩ = e ⟨φf ∣j ∣φi ⟩ ⟨0∣Aγ ∣1kλ ⃗ ⟩ − e ⟨φf ∣j∣φi ⟩ ⋅ ⟨0∣Aγ ∣1kλ

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a photon annihilation process, namely, the annihilation of one photon in the state with wave vector k⃗ and polarization λ, with a concomitant transition of the electron from the initial state φi to the final state φf . The physical interpretation is that the electron-wave-function part of the given matrix element describes the current which couples to the four-vector potential. These considerations imply that the current operator is the coefficient of the linear terms in Aµ in the interaction Hamiltonian. Some additional remarks on the choice of gauge are in order. For the discussion of bound-state problems, the Coulomb gauge is known to be very practical. This fact is being exploited in the following Chap. 12, when we derive higher-order corrections to the interaction potential between quantum mechanical Dirac particles. It is also well known that the Coulomb gauge is suitable for the discussion of the low-energy part of the self-energy (see Chap. 4). As we saw in Chap. 2, the quantization of the electromagnetic field in the Coulomb gauge proceeds only for the spatial components of the vector potential operator. However, we saw in Sec. 9.3 that the (µ = 0)(ν = 0)-component (henceforth denoted as the 00-component) of the photon propagator is still nonvanishing in the Coulomb gauge. For us, it means that we cannot ignore the time-like part j 0 of the current operator. The Foldy–Wouthuysen transformed Hamiltonian given in Eq. (11.54) contains ⃗ which in turn depends on the spatial compo⃗ = p⃗ − e A, the canonical momentum π ⃗ nents of the vector potential A. We shall use the Hamiltonian given in Eq. (11.54) under the following kinematic conditions, ⃗ = p⃗ − e A⃗ext − e A⃗γ = π ⃗ext − e A⃗γ , π eA = 0

e A0ext

+ e A0γ

=V

+ e A0γ

,

(11.100a)

V =

e A0ext .

(11.100b)

Furthermore, we assume that the action of the gradient operator on the four-vector potential describing the photon, generates a photon wave vector whose magnitude is of the order of ∇i Aµγ ∼ (Zα)2 m Aµγ . That is, we allow for an additional external four-vector potential Aµext , while the quantized electromagnetic field is described by the four-vector potential Aµγ . The action of the gradient operator on the photon four-vector potential results in a prefactor proportional to the photon wave vector, which, for a typical atomic transition, is of order (Zα)2 . Here, Z is the nuclear charge number. Alternatively, we can say that, for an optical photon, one has ⃗ ∼ (Zα)2 . The magnetic field operator therefore of the order of B ⃗ ∼ k⃗ × A⃗ ∼ k = ∣k∣ ⃗ In Eq. (11.100), we implicitly define the quantities π ⃗ext and V . (Zα)2 A. For the nuclear Coulomb field, we have, as an example, A0ext = −

Ze Ze =− , 4π0 r 4πr

V = e A0ext = −

Ze2 Zα =− . 4πr r

(11.101)

⃗=E ⃗tot is a sum of a longitudinal component, due to the The total electric field E scalar potential of both the external field and the photon field, and a transverse

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component, due to the time derivative of the photon field, ∂ ⃗tot = − ∇V ⃗ext + e E ⃗γ , ⃗ − e∇A ⃗ 0γ − e A⃗γ = e E eE ∂t ⃗ext = − ∇V ⃗ , eE

(11.102a) (11.102b)

∂ ⃗ [1] + E ⃗ [2] , ⃗γ = − ∇A ⃗ 0γ − A⃗γ = E (11.102c) E γ γ ∂t ⃗γ[1] = −∇A ⃗γ[2] = − ∂ A⃗γ . Fields with a subscript γ denote photon ⃗ 0γ and E where E ∂t ⃗tot is fields. The total magnetic field B ⃗=B ⃗tot = ∇ ⃗ext + B ⃗γ , ⃗ × A⃗ext + ∇ ⃗ × A⃗γ = B B (11.103) ⃗ext and B ⃗γ . We use the Coulomb gauge. The fields Aµ where we implicitly define B γ

enter the relation (9.73), µν ⟨0∣T Aµγ (t, r⃗)Aνγ (t′ , r⃗′ )∣0⟩ = iDC (x − x′ ) .

(11.104)

Under the kinematic conditions summarized in Eq. (11.100), we can write the Foldy– Wouthuysen Hamiltonian given in Eq. (11.54) as follows (we keep terms up to the fourth-order terms in the momenta), HFW = H [2] + H [3] + H [4] (⃗ πext − e A⃗γ ) 2 πext − e A⃗γ ) 4 e ⃗tot − (⃗ ⃗⋅B + V + e A0γ − (1 + κ) σ 2m 2m 8m3 e ⃗tot + σ ⃗tot × (⃗ ⃗tot )] ⃗ ⋅E ⃗ ⋅ (E − (1 + 2κ) [∇ πext − eA⃗γ ) − (⃗ πext − eA⃗γ ) × E 8m2 (R) (R) (00) (ij) = HFW + e j 0 A0γ − e ⃗j ⋅ A⃗γ = HFW + HIγ + HIγ . (11.105) =

(R)

Here, HFW is the rest term, which contains terms without Aγµ and the seagull term e2 A⃗2γ /(2m2 ) which of second order in the quantized field. Also, we define the (00)

following Hamiltonians HIγ

(00)

HIγ

(ij)

and HIγ ,

= e j 0 A0γ ,

(ij) HIγ = −e ⃗j ⋅ A⃗γ .

(11.106)

Of course, the superscript (ij) does not indicate a tensor structure, but rather, the origin of the Hamiltonian from the spatial components of the current, which are usually denoted by Latin as opposed to Greek indices. We shall, in the following, attempt to extract the components j µ (µ = 0, 1, 2, 3) of the transition current, up to order (Zα)3 . For µ = 0, we obtain the timelike component of the current, which we extract from the Hamiltonian e (00) ⃗ [1] + σ ⃗ [1] × π ⃗ [1] )] ⃗ ⋅E ⃗ ⋅ (E ⃗ext − π ⃗ext × E HIγ = e A0γ − (1 + 2κ) [∇ γ γ γ 8m2 e (1 + 2κ) 2 0 e (1 + 2κ) ⃗ Aγ − ⃗ 0γ ) × π ⃗ ⋅ [(−∇A ⃗ext ] ∇ σ = e A0γ + 8m2 4m2 1 + 2κ 2 0 1 + 2κ ⃗ 0γ × π ⃗ Aγ + ⃗ ⋅ (∇A ⃗ext )] = e [A0γ + ∇ σ 8m2 4m2 1 + 2κ 2 1 + 2κ ⃗ − ⃗ A0γ = e j 0 A0γ , ⃗ ⋅ (⃗ σ πext × ∇)] (11.107) = e [1 + ∇ 8m2 4m2

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where the latter expression reflects the definition given in Eq. (11.106), and we appeal to Eq. (11.102c). There is a small subtlety in the formulation of the current operator j 0 . Namely, it contains gradient operators which act on the time-like component A0γ of the photon field. By contrast, the momentum operator which is ⃗ acts on the quantum mechanical wave function, between which ⃗ext = p⃗−e A, part of π the interaction Hamiltonian is being sandwiched. With this understanding, we can write 1 + 2κ 2 1 + 2κ ⃗ − ⃗ . ⃗ ⋅ (⃗ ∇ σ πext × ∇) (11.108) j0 = 1 + 8m2 8m2 If we transform the vector potential into momentum space, then we can use the ⃗ where k⃗ is the photon momentum. This ⃗ and thus ∇ ⃗ ⇔ ik, correspondence k⃗ ⇔ −i∇, replacement is valid only for a photon annihilation process. In order to see this, let us recall Eq. (2.71) for the transverse vector potential in Coulomb gauge, 3 2 ⃗ (aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ) , ⃗ r⃗) = ∑ ∫ √ d k ˆλ (k) A(t, λ 2ω (2π)3 λ=1

(11.109)

⃗ with the fundamental field commutator (2.70), where ω = ∣k∣, ⃗ a+′ (k⃗′ )] = δ (3) (k⃗ − k⃗′ ) δλλ′ . [aλ (k), λ

(11.110)

For a photon annihilation process, the vector potential therefore is proportional to ⃗r ⃗ e−i k⋅x = ˆλ (k) ⃗ e−i ω t+ik⋅⃗ ⃗ r⃗) ∝ ˆλ (k) A(t, ,

⃗. ω = ∣k∣

(11.111)

In the following, we shall assume that the vector potential is of the form (11.111), ⃗ r⃗) about the origin (⃗ and evaluate correction terms for an expansion of A(t, r = ⃗0). ⃗ ⃗ This naturally gives rise to an expansion of j in terms of k. One observes that the Fourier transform of the current only depends on the spatial components of the ⃗ so that we can write photon momentum k, i(1 + 2κ) 1 + 2κ ⃗2 ⃗ ⋅ (k⃗ × π ⃗ext ) − k . (11.112) σ 4m2 8m2 This result is important because, e.g., for the electron-electron interaction, the timelike (µ = ν = 0) component of the photon propagator (in Coulomb gauge) leads to a nonvanishing and important contribution (see Chap. 12). Let us now turn our attention to the spatial components of the current operator. We first consider the problem in lower order and approximate the Hamiltonian (11.105) as follows, j0 = 1 +

HFW ≈

(⃗ πext − e A⃗γ )2 e ⃗tot . ⃗⋅B − (1 + κ) σ 2m 2m

(11.113)

We here omit the (Zα)3 terms from Eq. (11.119) and (11.126), and keep only the (Zα)2 terms. In Eq. (11.113), the relevant replacement for the extraction of ⃗ext . the current from the canonical momentum square is (⃗ πext − e A⃗γ )2 → −2eA⃗γ ⋅ π ⃗ ⋅ A⃗γ . With In Coulomb gauge, we may ignore terms proportional to [⃗ p, A⃗γ ] ∝ ∇

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reference to Eq. (11.106), one therefore has the following result for the interaction (ij) Hamiltonian HI , as derived from the spatial components of the current, e e (ij) ⃗ × A⃗γ ) ⃗ext − ⃗ ⋅ (∇ HIγ ≈ − A⃗γ ⋅ π (1 + κ) σ m 2m ⃗ext 1 + κ π ⃗ ⋅ A⃗γ = −e ⃗j ⋅ A⃗γ . ⃗ × ∇) = −e ( + σ (11.114) m 2m In coordinate space, this implies that ⃗ext 1 + κ ⃗j = π ⃗. ⃗×∇ + σ (11.115) m 2m The addition of the anomalous-magnetic-moment term (correction of order κ) generates part of the α3 correction. One might argue that if one term of order α3 is included, then one should include all terms in this order. However, we get this term “for free”, namely, as an additional multiplicative (radiative) correction, and thus include it here. Finally, after a Fourier transformation, the current is given as follows, ⃗ext i (1 + κ) ⃗ . ⃗j = π + (⃗ σ × k) (11.116) m 2m Now, let us analyze the corrections to the leading-order term given in Eq. (11.115). To this end, we recall Eq. (11.105), HFW =

(⃗ πext − e A⃗γ ) 2 πext − e A⃗γ ) 4 e ⃗tot − (⃗ ⃗⋅B + V + e A0γ − (1 + κ) σ 2m 2m 8m3 e ⃗tot + σ ⃗tot × (⃗ ⃗tot )] . ⃗ ⋅E ⃗ ⋅ (E (1 + 2κ) [∇ − πext − eA⃗γ ) − (⃗ πext − eA⃗γ ) × E 8m2 (11.117)

The relevant interaction terms for the extraction of the spatial current ⃗j, in the (ij) Hamiltonian HIγ , read as follows, (ij)

2 ⃗ext π ⃗ext ⃗ext } e {A⃗γ , π {A⃗γ , π } e ⃗ × A⃗γ ) ⃗ ⋅ (∇ +e (1 + κ) σ − 3 2m 4m 2m ∂ A⃗γ e e ⃗ × A⃗γ ) ⃗ ⃗ext ) − (1 + 2κ) ⃗ ⋅ (∇V + (1 + 2κ) σ ⋅ ( ×π σ 4 m2 ∂t 4 m2 = − e⃗j ⋅ A⃗γ . (11.118)

HIγ = −

⃗γ[2] = −∂ A⃗γ /∂t The first of the terms in the second line of the above relation involves E as defined in Eq. (11.102c), while the second term in the second line of the above ⃗ext , also according to the definition given in Eq. (11.102), and relation involves E ⃗ the term −e Aγ from the canonical momentum inside the cross product. We now remember that to lowest order in α, we can replace the factors exp(±ik⃗ ⋅ r⃗) in the ⃗ext . Finally, we find field operator (11.109) by unity, so that A⃗γ commutes with π for the spatial components of the current, ⃗ ⃗ext 1 + κ ⃗ π ⃗2 ⃗×π ⃗ext ∂ σ ⃗ × ∇V π σ ⃗j = π ⃗ − ext ext + ⃗×∇ + σ + . (11.119) 3 2 m 2m 2m 4 m ∂t 4 m2

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⃗ext acts on the atomic degrees of freedom, while Here, the momentum operator π the differential operators (gradient operator and time derivative) act on the vector potential. The terms on the right-hand side of Eq. (11.119) are, respectively, of order α and α2 , while the last three terms are of order α3 . After a transformation into Fourier space, one has 2 ⃗ ⃗ext π ⃗ext ⃗ × ∇V ⃗ext i(1 + κ) π iω σ ⃗j = π ⃗ × k⃗ − ⃗×π ⃗ext + + σ − σ , 3 2 m 2m 2m 4m 4 m2 where ω is the photon frequency.

11.3.5

(11.120)

Nonrelativistic Transition Current and Multipoles

One might think that, at this point, we have derived all formulas necessary for a convenient evaluation of transition current matrix elements in the nonrelativistic approximation. This is, however, not the case. Namely, our considerations up to this point ignore the fact that the phase factors in the vector potential operator lead to multipole corrections to the transition current, which cannot be ignored. Let us consider, once more, the Hamiltonians given in Eq. (11.106), (00)

HIγ

= e j 0 A0γ ,

(ij) HIγ = −e ⃗j ⋅ A⃗γ .

(11.121)

Let us also remember that the vector potential operator carries exponential factors according to Eq. (11.109), which we recall for convenience, 3 2 ⃗ (aλ (k) ⃗ e−i k⋅x + a+ (k) ⃗ ei k⋅x ) . ⃗ r⃗) = ∑ ∫ √ d k ˆλ (k) A(t, λ 3 2ω (2π) λ=1

(11.122)

The timelike and spatial current operators are given in Eqs. (11.112) and (11.120), respectively, 1 + 2κ ⃗2 i(1 + 2κ) ⃗ ⋅ (k⃗ × π ⃗ext ) , k + σ 8m2 4m2 2 ⃗ ⃗ext π ⃗ext ⃗ × ∇V ⃗ext i(1 + κ) π σ iω ⃗j = π ⃗ × k⃗ − ⃗ ⃗ + σ − σ × π + . (11.123) ext m 2m 2 m3 4 m2 4 m2 The characteristic exponential factor in the (annihilation part of the) vector potential operator is j0 = 1 −

⃗ A⃗γ ∝ e−i k⋅x = e−i ω t+ik⋅⃗r ,

⃗. ω = ∣k∣

(11.124)

Let us define a nonrelativistic current ⃗jNR by the matching condition ⃗j ⋅ A⃗γ (⃗ ⃗ e−iωt , r) = ⃗jNR ⋅ A⃗γ (⃗ r = 0) ⃗r ik⋅⃗

(11.125)

so that ⃗jNR contains the effect of the exponential factor e . For atomic-physics calculations, the vector potential of the photon, given in Eq. (11.111), can be expanded in powers of k⃗ ⋅ r⃗. This is because r⃗, which measures the electron-nucleus distance, is of the order of the Bohr radius, or, of the order of 1/(Zα), when expressed in ̵ = c = 0 = 1). Also, terms of the electron Compton wavelength (i.e., in units with h ⃗ k is of the order of an optical transition photon, and thus, of the order (Zα)2 in

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natural units. We aim to expand the transition current ⃗jNR up to order (Zα)3 . The ith component of the transition current can thus be written as i jNR =

i ⃗2 1 πi 1 πi πi π πext + { ext , i k⃗ ⋅ r⃗} − { ext , (k⃗ ⋅ r⃗)2 } − ext 3ext m 2 m 4 m 2m 1 i i ⃗ i (1 + i k⃗ ⋅ r⃗) − i ω (⃗ ⃗ )i . (⃗ (⃗ σ × k) σ × πext ) + σ × ∇V + 2m 4 m2 4 m2

(11.126)

The interpretation of the terms is as follows. The first three terms summarize multipole corrections to the transition current pi /m, the fourth term is a relativistic kinetic correction, the fifth term is a magnetic interaction, with retardation effects, while the sixth and seventh terms describe the covariant coupling of the spin-orbit interaction to the quantized electromagnetic field of the photon, the latter being described by the vector field A⃗γ . The analogous expression for the time-like nonrelativistic current is 1 + 2κ ⃗2 i(1 + 2κ) 1 0 ⃗ ⋅ (k⃗ × π ⃗ext ) . k + σ jNR = 1 + ik⃗ ⋅ r⃗ − (k⃗ ⋅ r⃗)2 − 2 8m2 4m2

(11.127)

A final remark is in order. One might ask if the anticommutators in the current operator (11.126) are really necessary. Specifically, one might ask what would happen if one were to write these terms as (1 + i k⃗ ⋅ r⃗ − 12 (k⃗ ⋅ r⃗)2 ) pi /m, i.e., if one were to commute the momentum operator to the right. Starting from Eq. (11.109), the additional term would involve the commutator 2

[pi , Aj (⃗ r)] = k i ∑ ∫ √ λ=1

d3 k 2ω (2π)3

⃗ (aλ (k) ⃗ e−i k⋅x − a+ (k) ⃗ ei k⋅x ) . jλ (k) λ

(11.128)

The factor k i vanishes when contracted with the Coulomb-gauge photon propagator, because the latter is proportional to δ ij − k i k j /k⃗ 2 [see Eq. (9.73)]. For a self-energy calculation in Coulomb gauge, therefore, p⃗ and A⃗ effectively commute, and the ordering of pi /m and (1 + i k⃗ ⋅ r⃗ − 21 (k⃗ ⋅ r⃗)2 ) in Eq. (11.126) does not matter. We note that it was precisely the assumption of Coulomb gauge which let us indicate the results given in Eqs. (11.112) and (11.120); otherwise, one would have had to ⃗ext and A⃗γ . indicate the anticommutator of the canonical momentum π The corrections to the current, listed in Eq. (11.126), are exactly those terms that have to be used in the low-energy part of the self-energy (Lamb shift), to i supplement the leading-order term πext /m [see Eqs. (4.302) and (4.314)]. 11.4 11.4.1

Dirac Form Factor and Bound-State Radiative Energy Shifts Foldy–Wouthuysen Transformation and Form Factors

So far, we have considered the anomalous-magnetic-moment corrections in the FWtransformed Hamiltonians, which can be traced to the Pauli form factor. As we turn our attention to bound-state energy shifts, we need to generalize the treatment and

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consider, in addition, corrections due to the Dirac form factor. We thus take a step back and recall the radiatively corrected Dirac Hamiltonian given in Eq. (11.37), ⃗ r)] + β m + F1 (∇ ⃗ 2 ) A(⃗ ⃗ 2 ) eA0 (⃗ ⃗ ⋅ [⃗ HR (⃗ r) = α p − e F 1 (∇ r) e ⃗ r) − β Σ ⃗ ⋅ B(⃗ ⃗ r)] . ⃗ 2) [i γ⃗ ⋅ E(⃗ (11.129) + F2 (∇ 2m In Eq. (11.38), we made the following approximations, α ⃗ 2 ) ≈ F1 (0) = 1 , ⃗ 2 ) ≈ F2 (0) ≡ κ ≈ F1 ( ∇ F2 (∇ . (11.130) 2π For the discussion of the high-energy part of the bound-electron self-energy, the ⃗ 2 ) ≈ F1 (0) = 1 is not sufficient. We recall at this stage the approximation F1 (∇ result given in Eq. (10.103), ⃗ 2) = 1 + F1 (∇

⃗2 ⃗4 α 1 m 1 ∇ ∇ [ ln ( ) − ] 2 + O ( 4 ) . π 3 λ 8 m m

(11.131)

In dimensional regularization, one has according to Eq. (10.190a), ⃗ 2) = 1 + F1 (∇

⃗2 ⃗4 ∇ ∇ α 1 1 1 m2 [− − + ln ( 2 )] 2 + O ( 4 ) . π 8 6ε 6 µ m m

(11.132)

According to Eqs. (10.100) and (10.190b), the F2 form factor is infrared convergent, F2 (q 2 ) =

1 α ∫ dy 2π 0 1−

1 q2 y(1 − y) m2

=

α 1 q2 1 q4 q6 [1 + + + O ( 6 )] . (11.133) 2 4 2π 6m 30 m m

⃗ 2 /m2 , because The form factors may be expanded in powers of their argument ∇ an expansion in the gradient operator corresponds to an expansion in Zα, in the ⃗ 2 ∼ (Zα)2 . Under the kinematic conditions listed in Eq. (11.62), with sense that ∇ κ ≡ F2 (0), the new Hamiltonian takes the form p⃗ 4 p⃗ 2 ⃗ 2V − + [1 + F1′ (0)] ∇ 2 m 8 m3 1 ⃗ 2V + 2 σ ⃗ × p⃗)] , ⃗ ⋅ (∇V + (1 + 2 κ) [∇ (11.134) 8 m2 where we keep the potential V in general form. Of course, we shall consider the Coulomb potential V = −Zα/r in the following. We here restrict the discussion to terms up to order α5 . We should be well aware that the form-factor treatment has its limits, but it is sufficient to calculate the high-energy part of the bound-electron self-energy. Additional higher-order corrections are described in Refs. [199, 312], and explored further in Chap. 15. HFF =

11.4.2

High-Energy Part in Photon Mass Regularization

Let us isolate, from Eq. (11.134), those operators which are relevant to the highenergy part (HEP) of the electron self-energy in the order α5 , κ ⃗ 2V + ⃗ 2 V + 2⃗ ⃗ × p⃗)] = ∆V1 + ∆V2 , [∇ σ ⋅ (∇V (11.135) HHEP = F1′ (0) ∇ 4 m2

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⃗ 2 V and, with reference to where we define the effective potentials ∆V1 = F1′ (0) ∇ Sec. 4.6.5, κ ⃗ 2 V + 2⃗ ⃗ × p⃗)] [∇ σ ⋅ (∇V ∆V2 = 4 m2 ⃗ ⃗⋅L α 1 π (Zα) δ (3) (⃗ r) α (Zα) σ = + . (11.136) 2 2 3 π 2 m π 4m r The one-loop value for κ is κ = α/(2π), according to Eq. (11.130). Here, in going from the first to the second line, we use properties of the Coulomb potential V = −Zα/r. It is perhaps surprising that the effective potential ∆V2 does not capture the reduced-mass dependence correctly, and in fact, one needs to redefine ∆V2 as follows, ∆V2 → ∆V2 ≡

⃗ ⃗⋅L r) α (Zα) σ α 1 π (Zα) δ (3) (⃗ + . π 2 m2 π 4 m µ r3

(11.137)

We here correct the reduced-mass dependence on account of the fact that the selfenergy actually is derived from the two-body Breit1 Hamiltonian, to be discussed in more detail in Sec. 12.5.3 (a keyword to remember in this context is the proton’s convection current). We remind the reader of our notation for the reduced mass µ of the two-body bound system and for photon mass λ. With the proper reduced-mass dependence, the result in Eq. (11.137) confirms the expression previously indicated in Eq. (4.342). Recalling the formalism outlined in Sec. 4.6.5 for the evaluation of matrix elements of Schr¨ odinger–Pauli wave functions, we confirm a result previously indicated in Eq. (4.347), namely, ∆E2 = ⟨∆V2 ⟩ =

µ 3 δ`0 1 µ 2 α (Zα)4 m [( ) + (− ) ( ) (1 − δ`0 )] . π n3 m 2 2κ (2 ` + 1) m (11.138)

Now, we consider the challenging part of the calculation, which is connected to the effective potential π Zα δ (3) (⃗ r) , (11.139a) m2 (Zα)4 m µ 3 ∆E1 = ⟨∆V1 ⟩ = 4 [m2 F1′ (0)] ( ) δ`0 , (11.139b) n3 m where we appeal to the definition of the “standard potential” in Eq. (4.344) and use the same conventions for the bound-state quantum numbers as in Chap. 4. According to Eqs. (10.103) and (11.131), one has ⃗ 2 V = 4 [m2 F1′ (0)] ∆V1 = F1′ (0) ∇

m 3 α [ln ( ) − ] (11.140) 3π λ 8 in photon mass regularization. When comparing to the low-photon-energy contribution indicated in Eq. (4.325), we are faced with a problem. The form-factor m2 F1′ (0) =

1 Gregory

Breit, transcribed from Russian: Grigory Alfredovich Breit-Shneider (1899–1981).

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slope indicated in Eq. (11.140) is given as a function of the photon mass λ, which in this case acts as an infrared regulator. By contrast, it will turn out that the photon energy cutoff  in Eq. (11.146) acts as an ultraviolet regulator. In principle, one should assume that the dependence on the infrared and ultraviolet regulators should cancel at the end of the calculation, but how should the quantities  and λ be matched against each other? For the record, we recall that ε in Eq. (11.159) indicates the regularized dimension according to d = 4−2ε,  in Eq. (11.146) denotes the (noncovariant) photon energy cutoff, and λ is the photon mass, while µ is the reduced mass of the two-body system for which the Lamb shift is being calculated. In order to compare  and λ, one may recalculate the Dirac form factor slope using photon energy regularization, as outlined in Chap. 7 of Ref. [2]. This leads to a conversion of the high-energy potential ∆V1 from photon mass to photon energy regularization. However, these is an easier way, which we would like to describe in the following. Namely, we can alternatively concentrate on the low-energy part of the bound-electron self-energy and reevaluate it in photon mass regularization. Let us consider the penultimate step in Eq. (4.311), modified for a massive photon. Based on Eq. (12b) and the discussion in the Appendix of Ref. [331], one can show that the appropriate formula reads as follows, ki kj ij RRR RR d3 k δ − k⃗2 +λ2 1 e2 RRRpi j RRR √ √ ⟨φ p RR φ⟩ . (11.141) = 2 ∫ RRR m (2π)3 2 k⃗2 + λ2 RR E − HS − k⃗2 + λ2 RRRR √ k⃗2 + λ2 is the energy of the massive photon, and the Here, the quantity Schr¨ odinger Hamiltonian HS is given in Eq. (4.1). In view of the presence of an infrared regulator in Eq. (11.141), with the condition that (λ) ∆ELEP

m ≫ λ ≫ (Zα)2 m ,

(11.142)

we can actually expand the expression ⎛ ⎞ 1 1 1 HS − E √ = −√ + +O √ , 2 2 ⃗ ⎝ E − HS − k⃗2 + λ2 ( k⃗2 + λ2 )3/2 ⎠ k⃗2 + λ2 k + λ

(11.143)

and use the second term in Eq. (11.141). The first term is absorbed in a mass renormalization inherent to the low-energy part (see Chap. 4). With the result i

j

k k d3 k δ ij − k2 +λ2 1 2 δ ij 2 5 √ = [ln ( ) − ] , lim ∫ λ→0 0 (2 π)3 2 k 2 + λ2 k 2 + λ2 3 4 π2 λ 6 we can conclude that 

(λ)

2 e2 δ ij 2 5 [ln ( ) − ] ⟨pi (HS − E) pj ⟩ 2 2 3 m 4π λ 6 1 α 2 5 ⃗ 2V ⟩ = [ln ( ) − ] ⟨∇ 3 πm2 λ 6 4α (Zα)4 m µ 3 2 5 ( ) [ln ( ) − ] δ`0 . = 3 3π n m λ 6

(11.144)

∆ELEP =

(11.145)

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This result has to be compared to Eq. (4.325), which we recall for convenience, ()

∆ELEP =

2 4 α (Zα)4 m µ 3 ( ) [ln ( ) δ`0 − ln k0 (n`)] . 3π n3 m (Zα)2 µ

(11.146)

We can argue as follows. The result given in Eq. (11.145) depends on both , which takes the role of an ultraviolet regulator, and the photon mass λ, which is an infrared regulator. This is natural, because the divergence for large  is inherent to the formulation of the low-energy part of the self-energy and cannot be cured by an infrared regulator. The low-energy part can be converted to photon mass regularization, though, on account of the following consideration. The high-energy effective potential given in Eq. (11.139), in photon mass regularization, contains the entire interval ω ∈ (0, ∞), which is made possible by the fact that the infrared divergence is cut off at the photon mass scale. By contrast, the low-energy part given in Eq. (11.146) comprises the photon energy interval ω ∈ (0, ). Hence, if one were to add the contributions given in Eq. (11.139) and Eq. (11.146), then one would double-count the integration interval ω ∈ (0, ). This double counting is avoided if one subtracts, from the low-energy part given in Eq. (11.146), the result of an integration over the integration interval ω ∈ (0, ), with the latter one being evaluated in photon mass regularization — the latter quantity is precisely given in Eq. (11.145). Hence, the low-energy part, in photon mass regularization, is given as follows, ()

(λ)

∆ELEP = ∆ELEP − ∆ELEP =

λ 5 4 α (Zα)4 m µ 3 ( ) {[ln ( ) + ] δ`0 − ln k0 (n`)} . 3 2 3π n m (Zα) µ 6

(11.147)

In view of Eqs. (11.139b) and (11.140), one has 4α m 3 (Zα)4 m µ 3 [ln ( ) − ] ( ) δ`0 , (11.148) 3π λ 8 n3 m where λ is the photon mass and µ is the reduced mass of the two-body system. With ∆E2 being given in Eq. (11.138), and on account of the algebraic identity 5/6 − 3/8 + (3/4) × (1/2) = 5/6, the total result for the self-energy shift is ∆E1 =

∆E = ∆E1 + ∆E2 + ∆ELEP =

4 α (Zα)4 m µ 3 m 5 ( ) {[ln ( ) + ] δ`0 3π n3 m (Zα)2 µ 6 + (−

1 m ) (1 − δ`0 ) − ln k0 (n`)} . 2κ (2 ` + 1) µ

(11.149)

For S states, this expression reduces to the result given in Eq. (4.352), while for non-S states, one obtains the result given in Eq. (4.353). Note that the matching indicated in Eq. (11.147) was a very difficult problem in the early days of quantum electrodynamics, as evident from a footnote on page 777 of [332]. The complete correction in this order was obtained first in the years

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1948 and 1949 (see Refs. [332–335]). The full reduced mass dependence (which we indicate in all detail) was clarified much later in Ref. [336] by Sapirstein2 and Yennie.3 The matching of the low-energy part in photon energy and photon mass regularization in higher orders has been discussed in Refs. [312,337]. Indeed, higherorder terms in the Zα-expansion of the self-energy were evaluated over the past decades. The correction term in the order α(Zα)5 m was found in Refs. [338–340], and the higher-order binding corrections in the order α(Zα)6 m were calculated in Refs. [163, 173, 337, 341–352]. 11.4.3

High-Energy Part in Photon Energy Regularization

The result given in Eq. (11.147) has been obtained by converting the low-energy part of the bound-electron self-energy to photon mass regularization. As evident from the discussion in Chap. 4, the most natural way to calculate the low-energy part of the self-energy proceeds via photon energy regularization. One can alternatively convert the high-energy part given in Eq. (11.139b) to photon energy regularization as follows. One first observes that the high-energy part, in photon-mass regularization, comprises all of the interval ω ∈ (0, ∞). However, if one uses photon energy regularization, then the contribution given in Eq. (11.145), which precisely describes the integration interval ω ∈ (0, ), should be subtracted, because the highenergy part, in photon energy regularization, corresponds to the integration interval ω ∈ (, ∞). () Hence, ∆E1 , in photon energy regularization, reads as ()

(λ)

∆E1 = ∆E1 − ∆ELEP =

m 11 4α (Zα)4 m µ 3 ( ) [ln( ) + ] δ`0 , 3 3π n m 2 24

(11.150) (λ)

which precisely reproduces the result given in Eq. (4.341). The quantity ∆ELEP is given in Eq. (11.145), while ∆E1 is given in Eq. (11.148). Adding ∆E2 and the result for the low-energy part given in Eqs. (4.325) and (11.146), one easily reproduces the total result for the self-energy shift given in Eq. (11.149). The latter result is independent of the regularization procedure and demonstrates that the physical energy shift does not depend on whether photon energy or photon mass regularization is used for a calculation. () Note that, as one converts ∆E1 given in Eq. (11.148) to ∆E1 given in Eq. (11.150), one can establish the following equivalence, ln (

m 5 m ) ↔ ln( ) + , λ 2 6

(11.151)

which provides the matching between the infrared photon mass cutoff λ and the noncovariant photon energy cutoff . The precise form of this matching, though, depends on the process under study and the nonlogarithmic term changes as one considers, e.g., higher-order multipole corrections to the Bethe logarithms. 2 Jonathan 3 Donald

Robert Sapirstein (b. 1951). Robert Yennie (1924–1993).

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11.4.4

Self–Energy in Dimensional Regularization

As a last step, beyond photon mass and photon energy regularization, we discuss dimensional regularization, which has already been used in various contexts in this treatise (see Secs. 10.3.4, 10.3.6, and 10.4.2). Note that the slope of the Dirac form factor, in dimensional regularization, is given in Eq. (11.159). We will ignore reduced-mass corrections and set the reduced mass equal to the electron mass. We work in D = 3 − 2ε space dimensions [see Eq. (10.141)] and one time dimension, so that the dimension of space-time is d = 4 − 2ε [see Eq. (10.142)]. Also, we recall that, according to Eq. (10.173), e2 = (4π)1−ε α µ2ε eγE ε .

(11.152)

Using the symmetry of the integrand in D dimensions, we start from the penultimate step in Eq. (4.311), and use some obvious symmetry relations of the integrand, to show that ∆ELEP = e2 ∫

R

= e2 ∫ = e2

D

1 pj ki kj pi dD k ij (δ − ) ⟨ ⟩ (2π)D 2k k2 mE−k−H m

1 pk k i k j δ ij pk dD k (δ ij − 2 ) ⟨ ⟩ D (2π) 2k k D m E−k−H m

k ∞ 1 D − 1 ΩD 1 D−2 p dk k ⟨ pk ⟩ , ∫ D D (2π) 2 0 m E−k−H

(11.153)

where ΩD is given in Eq. (10.138). We now introduce a separation parameter Λ, which cancels at the end of the calculation, to separate the small-momentum from the large-momentum regions. The key to the calculation shall be that dimensional regularization is only required for the integration region k ∈ (Λ, ∞), ∆ELEP = e2 + = e2 +

k ∞ 1 pk D − 1 ΩD 1 D−2 p dk k ⟨ ⟩ ∫ D (2π)D 2 Λ m E−k−H m Λ 2α pk 1 pk ⟩ ∫ dk k ⟨ 3π 0 m E−k−H m k ∞ 1 H − E (H − E)2 pk D − 1 ΩD 1 D−2 p dk k ⟨ (− + − ) ⟩ ∫ D (2π)D 2 Λ m k k2 k3 m Λ 2α pk 1 pk ⟩. ∫ dk k ⟨ 3π 0 m E−k−H m

(11.154)

Using Eq. (10.173), we can show that e2

10 4 µ D − 1 ΩD 1 2 = + ε [ + ln ( )] . D (2π)D 2 3 9 3 2

(11.155)

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We can now use the following results (D = 3 − 2ε), ignoring terms generated by the upper limit at k = ∞ in the spirit of dimensional regularization, ∫ ∫

∞ Λ ∞ Λ

dk k D−3 = − Λ + O(ε) , dk k D−5 =

1 + O(ε) , Λ

∫ ∫

∞ Λ ∞ Λ

dk k D−4 =

1 − ln(Λ) + O(ε) , 2ε

(11.156a)

dk k D−6 =

1 + O(ε) , 2Λ2

(11.156b)

and so on. We see that higher-order terms for large k in the expansion of 1/(E−H−k) give rise to vanishing contributions in the limit of large Λ. We now specialize the discussion to the limit of infinite nuclear mass, where the reduced mass is equal to the electron mass m, in order to be able to concentrate on the essential elements of the discussion. The intermediate expression for the low-energy part is found as follows, ∆ELEP =

α 2 10 4 µ 1 { + ε [ + ln ( )]} [( − ln(Λ)) ⟨pk (H − E) pk ⟩ + Λ ⟨⃗ p 2 ⟩] π 3 9 3 2 2ε −

4α (Zα)4 m 2α 2Λ Λ ⟨⃗ p 2⟩ + (ln ( ) − ln k0 (n, `)) . 3π 3π n3 (Zα)2 m

(11.157)

For the definition of the Bethe logarithm ln k0 (n, `), we refer to the corresponding r)⟩ → 2(Zα)4 /n3 in discussion in Chap. 4. Using ⟨pk (H − E) pk ⟩ = ⟨2πZα δ (D) (⃗ the limit D → 3, we can now see that Λ cancels in the sum of the two above contributions (both the linear as well as the logarithmic term), and that also the ln(2) terms cancel. We finally obtain ∆ELEP =

4α (Zα)4 m 1 5 µ [( + + ln ( )) δ`0 − ln k0 (n, `)] . 3 3π n 2ε 6 (Zα)2 m

(11.158)

This result is strictly of order α(Zα)4 m and does not contain any higher-order terms, in contrast to photon energy regularization, which was discussed in Sec. 4.7.3. This behavior is due to a separation of the kinematic variables (the powers of Zα) from the divergences (i.e., from the divergence for large virtual photon energy). In photon energy regularization, the two mechanism for the generation of divergences are intertwined, and therefore the nonrelativistic dipole term generates divergent contributions in arbitrarily high order in the (Zα)-expansion. Another advantage of the dimensional regularization is that all ln(Zα) terms come from the low-energy part, to which they are physically associated. Let us recall the Dirac form-factor slope in dimensional regularization from Eq. (11.132) [see also Eq. (10.190a)], F1′ (0) =

1 1 1 m2 α (− − + ln ( 2 )) . 2 πm 8 6ε 6 µ

From Eq. (10.190b), we have F2 (0) = α/(2π).

(11.159)

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We write ∆EHEP = ∆E1 + ∆E2 as the sum of two contributions. According to Eqs. (11.139a) and (11.159), the high-energy potential ∆V1 and its associated energy shift are given by (again, in the limit of infinite nuclear mass) ∆V1 =

(3) r)⟩ 4α 1 1 1 m2 π Zα ⟨δ (⃗ [− − + ln ( 2 )] , 2 π 8 6ε 6 µ m

(11.160a)

∆E1 =

4 3 1 1 m2 α (Zα)4 m [− − + ln ( 2 )] δ`0 . 3 8 2ε 2 µ π n3

(11.160b)

After adding the results in Eqs. (11.158) and (11.160b), both the dependence on the ε parameter as well as the dependence on the renormalization scale µ cancel. For the record, let us consider the energy shift ∆E2 listed in Eq. (11.138), in the limit of infinite nuclear mass, ∆E2 =

1 α (Zα)4 m δ`0 [ + (− ) (1 − δ`0 )] . π n3 2 2κ (2 ` + 1)

(11.161)

Adding the results from Eqs. (11.158), (11.160b) and (11.161), one obtains for the self-energy shift ∆ESE , ∆ESE = ∆EHEP + ∆ELEP =

5 1 4 α (Zα)4 m {[ln(Zα)−2 + ] δ`0 + (− ) (1 − δ`0 ) − ln k0 (n`)} , 3 3π n 6 2κ (2 ` + 1) (11.162)

which is precisely equal to the result given in Eq. (11.149) in the limit of infinite nuclear mass. We have avoided the discussion of the nuclear-mass dependence in the context of dimensional regularization in order to keep formulas compact. It is a good exercise to restore this dependence in intermediate steps. 11.4.5

Vacuum Polarization

For S states, there is a further correction which is determined by the vacuum polarization correction to the Coulomb potential, which generates an effective potential [see Eq. (10.235)] 4 δ (3) (⃗ r) ≡ ∆V3 , (11.163) α(Zα) 15 m2 to supplement ∆V1 given in Eq. (11.139a) and ∆V2 given in Eq. (11.137). The vacuum-polarization potential (11.163) leads to an energy shift ∆VVP = −

4 α (Zα)4 m 4 α (Zα)4 m µ 3 ( ) δ ≈ − δl0 , (11.164) l0 15 π n3 m 15 π n3 where the latter form is valid in the limit of infinite nuclear mass. Adding this result to the self-energy shift (4.352) for S states, we obtain for the sum of self-energy and vacuum-polarization corrections, for S states, ∆EVP = ⟨n` ∣∆V3 ∣ n`⟩ = −

∆ESE+VP (nS) =

α (Zα)4 m 4 38 4 ( ln [(Zα)−2 ] + − ln k0 (n, `)) 3 π n 3 45 3

(11.165)

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457

as the total Lamb shift in the order α(Zα)4 m (again, in the non-recoil approximation). For states with ` ≠ 0, the result (4.353) is already complete in the order α(Zα)4 m. The reduced-mass dependence adds an overall multiplicative factor (µ/m)3 , according to Eqs. (4.352) and (11.164). 11.5

Foldy–Wouthuysen Transformation and Gravity

We recall the Dirac–Schwarzschild (DS) Hamiltonian HDS from Eq. (8.298), rs rs 1 α ⋅ p⃗, (1 − )} + βm (1 − ) , (11.166) HDS = {⃗ 2 r 2r where rs = 2 G M . This Hamiltonian describes the coupling of a Dirac particle to the gravitational field surrounding a black hole. Below, we recall a transformation [255] which describes the expansions to first order in χ = rs /r and to higher orders order in ξ = p⃗/m or ξ = 1/(m r). Terms are calculated up to order ξ 4 if there is no gravitational interaction, and up to order ξ 3 χ if there is a gravitational interaction. Terms of order χ2 (second order in the gravitational interaction) are ignored. The Foldy–Wouthuysen transformed Hamiltonian is found as follows [255], HFW = β (m +

p⃗ 4 m rs p⃗ 2 − )−β 2m 8m3 2r

⃗ ⋅L ⃗ 3rs 1 3πrs (3) 3rs Σ {⃗ p 2, } + δ (⃗ r) + ). (11.167) 3 8m r 4m 8m r As pointed out in Ref. [255], the very last term on the right-hand side of Eq. (11.167) describes the gravitational spin-orbit coupling. The classical result for the interaction of a spinning classical particle with the gravitational field is otherwise known as the geodesic precession or Fokker precession [353–355], or de Sitter precession. On the classical level, it has been verified in the Gravity Probe B experiment [356]. When comparing the spin-orbit term to Eq. (26) of Ref. [355], one might notice a prefactor of 3/2 instead of 3/8. However, one should take note that the spin ⃗ and that there is operator used in Ref. [355] carries a factor one half (S⃗ = 21 Σ), an additional factor 2 in the definition of the Schwarzschild radius. The prefactor β in Eq. (11.167) describes the particle-antiparticle symmetry in the gravitational interactions, which cannot be obtained based on classical considerations [353–355]. In particular, gravitationally coupled relativistic Dirac theory makes the unique prediction that the equivalence principle holds for antimatter [251, 291, 357, 358]. Finally, we quote the gravitational corrections to the transition current from Ref. [255], + β (−

i jFW =

−

1 pi pi p⃗ 2 i pi 1 pi ⃗ ⃗ i (1 + i k⃗ ⋅ r⃗) (⃗ + + { , ik ⋅ r⃗} − { , (k⃗ ⋅ r⃗)2 } − σ × k) 3 m 2 m 4 m 2m 2m

i ⃗ i 1 rs 3 pi rs rs (⃗ σ × r⃗)i 1 ⃗ pi rs 3irs (⃗ σ × k) ⃗ ⋅ r⃗), p } . ⃗ { , }+ − {i k ⋅ r , { , }} − + { (i k 4 m r 2r m r2 2 m r 4r m 4 r m (11.168)

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The well-known corrections to the relativistic transition current, given in Eqs. (11.112) and (11.120), are supplemented by the gravitational kinetic correction, and gravitational corrections to the magnetic coupling. 11.6

Further Thoughts

Here are some suggestions for further thought. (1) Alternative Foldy–Wouthuysen Transformations. Consider the free Dirac particle. Verify that the modified Foldy–Wouthuysen transformation U = exp(iS), given as U = eiS ,

S=−

iβ (⃗ α ⋅ p⃗)3 (⃗ α ⋅ p⃗ − ) = S1 + S2 , 2m 3m2

(11.169)

transforms HFD , given in Eq. (11.8), directly into HFW , given in Eq. (11.17), and captures the relativistic corrections up to and including fourth order in the momenta. In addition, show, using the Campbell–Baker–Hausdorff formula, that if iβ iβ ⃗ ⋅ p⃗ , α S2 = (⃗ α ⋅ p⃗)3 , (11.170) U = ei(S1 +S2 ) , S1 = − 2m 6m3 then U = ei(S1 +S2 ) = U1 U2 = eiS1 eiS2 .

(11.171)

Hint: This calculation is still perturbative in the momentum operators; you may discard higher-order terms in intermediate steps. (2) One-Step Transformations. Consider H∞ as given in Eq. (11.15), and HFD as given in Eq. (11.8). Find a one-step unitary transformation S∞ for which H∞ = eiS∞ HFD e−iS∞ .

(11.172)

Hint: Try the ansatz ⃗ ⋅ p⃗ θ(⃗ exp(iS) = exp[β α p)] = cos[∣⃗ p∣ θ(⃗ p)] +

⃗ ⋅ p⃗ βα sin[∣⃗ p∣ θ(⃗ p)] . ∣⃗ p∣

(11.173)

Can this approach be easily generalized easily to more complicated scenarios? (3) Antiparticle Hamiltonians. Show that the effective antiparticle Hamiltonian, corresponding to the physical system described by Eq. (11.54), can be obtained from Eq. (11.54) by the replacement e → −e. (4) Foldy–Wouthuysen Transformation. Compare the result given in Eq. (11.54) to the treatment in Chap. 4 of Ref. [316]. In particular, show the identity ⃗ ×π ⃗ = 2E ⃗ ×π ⃗ ⃗ ×E ⃗−π ⃗×E ⃗ + i∇ E

(11.174)

and compare our result to the one given in Eq. (4.5) of Ref. [316]. Hint: You may have to consider the limit κ → 0.

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(5) Comparison to the Literature. In Ref. [359], a Foldy–Wouthuysen transformation is being used with the operator U = exp(iS) , S = S0 + δS , 1 e i 3 ⃗ , ⃗⋅π ⃗− ⃗) + ⃗ ⋅ E} {β α β (⃗ α⋅π α S0 = − 2 2m 3m 2m i eκ ⃗ − e κ [⃗ ⃗ ⋅B ⃗ ]} . ⃗, β Σ δS = − { i⃗ α⋅E α⋅π 2m 2m 4m2

(11.175a) (11.175b) (11.175c)

In Eq. (2.8) of Ref. [312], it is precisely the Foldy–Wouthuysen transformation in the form given in Eq. (11.175) which is being used in order to derive the α6 and α7 Hamiltonians. In Ref. [322], a Foldy–Wouthuysen transformation is being used which is a little different from the one used in Refs. [359] and [312] and is given as i 1 1 ⃗⋅π ⃗− ⃗ )3 + ⃗ , e A0 − i ∂t ] + δY } , {β α β (⃗ α⋅π [⃗ α⋅π 2m 3 m2 2m (11.176a) β β e i e ⃗ ⃗˙ + [⃗ ⃗ , [⃗ ⃗, α ⃗ ⋅ E]] ⃗ )5 − ⃗⋅E α⋅π α⋅π δY = (⃗ α⋅π α 5 m4 4 m2 24 m3 ie ⃗ +α ⃗ (⃗ {(⃗ ⃗ )2 α ⃗⋅E ⃗⋅E ⃗ )2 } . α⋅π α⋅π (11.176b) − 3 m3 S= −

In terms of the (Zα)-expansion, one has δY ∼ (Zα)5 . The breakdown of individual contributions and iterative transformations is being discussed in Ref. [322]. The corresponding sixth-order Hamiltonian is H [6] =

⃗ )6 (⃗ σ⋅π e ⃗˙ − i e [⃗ ⃗ ⃗, σ ⃗ ⋅ E} ⃗ , [⃗ ⃗ , [⃗ ⃗, σ ⃗ ⋅ E]]] − {⃗ σ⋅π σ⋅π σ⋅π σ⋅π 5 3 16 m 16 m 128 m4 ie ⃗ . {(⃗ ⃗ )2 , [⃗ ⃗, σ ⃗ ⋅ E]} + σ⋅π σ⋅π (11.177) 16 m4

One notes, for a Coulomb field, the identities given in Eqs. (11.64) and (11.67). The sixth-order terms in Eq. (11.177) start with the term propor⃗ )6 . Our result, given in Eq. (11.59), differs in the sense that tional to (⃗ σ⋅π ⃗˙ is absent, and also, the term propor⃗, σ ⃗ ⋅ E} the term proportional to {⃗ σ⋅π

⃗ ⃗ , [⃗ ⃗ , [⃗ ⃗, σ ⃗ ⋅ E]]] tional to [⃗ σ⋅π σ⋅π σ⋅π is absent, while the term proportional to 2 ⃗ {(⃗ ⃗ ) , [⃗ ⃗, σ ⃗ ⋅ E]} carries a different prefactor. σ⋅π σ⋅π Compare the following α6 Hamiltonians: (i) the Hamiltonian given in Eq. (11.177), together with the modified FW transformation from Eq. (11.176), (ii) the Hamiltonian given in Ref. [312], used in its Coulomb-field form in Eq. (11.69), together with the transformation given in Eq. (11.175), (iii) our result (11.59), obtained by the straightforward application of the iterative FW method, where subsequently, one eliminates the odd operators in the transformed Dirac Hamiltonians. Consider both diagonal as well as off-diagonal matrix elements.

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(6) Gauge Invariance and Hamiltonians. Strictly speaking, the Hamiltonian (11.54) is not gauge invariant, as it contains the gauge-dependent scalar and vector potential. However, this observation is equally true of the original ⃗ = p⃗ − e A⃗ (it is Hamiltonian (11.40) which contains the canonical momentum π gauge covariant, but not gauge invariant). Show, however, that energy eigenvalues will be gauge invariant. Hint: Consider the equation H ψ = E ψ and investigate if the gauge-transformed wave function ψ ′ [see Eq. (8.2)] fulfills H ′ ψ ′ = E ψ ′ , where H ′ is the gauge-transformed Hamiltonian. (7) Bound-Electron g Factor. Consider the bound-electron g factor on the basis of Eq. (11.54), for the 1S state of a hydrogenlike bound state, in which ⃗ by zero. Treat case you can replace the orbital angular momentum operator L 6 all terms which contribute to order (Zα) . Repeat the exercise in the length gauge, i.e., with the form of the interaction Hamiltonian obtained after a Power–Zienau transformation. (8) Corrections to the Transition Current. Consider additional corrections to the photon transition current, induced by the addition of an additional vector potential A⃗hfs . Use Eq. (22) of Ref. [360] as an inspiration. Hint: A nonrelativistic motivation can be derived by replacing the Hamiltonian p⃗ ∣e∣A⃗hfs (⃗ p − e A⃗ − e Ahfs )2 → −eA⃗ ⋅ − eA⃗ ⋅ , (11.178) 2m m m ⃗ and matching with the coefficient of the vector potential A. (9) Exponential Tail and Local Potential. Match Eq. (11.163) with the exponential factors in Eq. (10.246).

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Chapter 12

Relativistic Interactions for Many-Particle and Compound Systems

12.1

Overview

In the current chapter, we shall discuss the derivation of time derivative operators (Hamiltonians), and time evolution operators, for systems consisting of many particles. At first, one might conjecture that it is next-to-impossible, from a relativistic point of view, to calculate the effective Hamiltonian for a two-body, or three-body system, and for quantum mechanical systems or more than three particles, because the proper time cannot be uniquely defined for this system. Each of the two (or more) particles has its own proper time. It might be that the only way to define a Hamiltonian in a relativistically invariant way would involve a time derivative with respect to the proper time of each particle in question. How can this question be solved? The key is to observe that we are not calculating an energy level of a particular, isolated particle, but energy levels of a compound system consisting of many particles. We can first formulate a Hamiltonian for the compound system, i.e., a Hamiltonian that contains terms for all involved particles. Then, we go into the center-of-mass frame of the compound system, and define the Hamiltonian as the time derivative operator with respect to the proper time in the center-of-mass system. This approach will prove to be extremely useful. We shall proceed as follows. In Sec. 12.3, we describe the derivation of the many-body Breit Hamiltonian based on the matching of the forward scattering amplitude with the effective Hamiltonian, in the no-retardation (zero-frequency) limit of the photon. The derivation is heavily based on previous work carried out in Chap. 11. Namely, from the Foldy–Wouthuysen transform of the single-particle Dirac Hamiltonian, we shall first derive the transition current operators which we need in order to write down the two-particle, forward scattering amplitude, which is formulated as an expectation value of the many-particle wave function. The calculation culminates in a general formula, given in Eq. (12.64), for the two-particle interaction Hamiltonian of atomic electrons, including the anomalous-magneticmoment corrections. The atomic nucleus (of arbitrary spin) is treated in Sec. 12.4.2. An orbiting particle of arbitrary spin is treated in Sec. 12.4.3. In these treatments, we need to generalize the Pauli spin matrices; this is accomplished in detail for the 461

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spin-1 case. Applications of the derived formalism to two-body systems will allow us to derive the exact reduced-mass dependence of a number of tiny, but important energy shifts for bound systems (see Sec. 12.5). Finally, in Sec. 12.6, we shall discuss the derivation of Hamiltonians applicable to more complicated scenarios, such as the interaction of a compound system with an external magnetic field. 12.2

Interatomic Interactions in Covariant Formalism

12.2.1

General Paradigm of the Matching

Let us discuss the general paradigm underlying the matching of an effective Hamiltonian with the S matrix. First, it needs to be observed that the S matrix describes the time evolution of the system from the infinite past to the infinite future. In turn, of course, the time evolution of a quantum system is described by its Hamiltonian (time derivative operator). Hence, there must be a connection. The S matrix is given as the time-ordered exponential [see Eq. (3.41)] S = T exp [−i ∫

∞ −∞

dt′ HI (t′ )] ≈ 1 − i ∫

∞ −∞

dt′ HI (t′ ) .

(12.1)

Here, HI (t) is the interaction Hamiltonian in the interaction representation [see Eq. (2.108)], HI (t, r⃗) = ei H0 t HI (⃗ r) e−i H0 t ,

HI ≡ HI (t = 0) .

(12.2)

Now, let HI = HI (⃗ r) be an effective, time-independent potential. The S matrix element describes the scattering of an initial state ∣i⟩ to a final state ∣f ⟩. Following the discussion in Chap. 3, we assume that Ef = Ei = E and write Sf i = ⟨f ∣1 − i ∫ = ⟨f ∣i⟩ − i ∫

∞ −∞ ∞

−∞

dt ei H0 t HI e−i H0 t ∣ i⟩

(12.3)

dt ⟨f ∣ei E t HI e−i E t ∣ i⟩ = δf i − i T ∫ d3 r φf (⃗ r) HI (⃗ r) φi (⃗ r) .

The matching relation thus is as follows, Sf i = δf i − i T ∫ d3 r φf (⃗ r) HI (⃗ r) φi (⃗ r) .

(12.4)

Or, if we ignore the Kronecker-δ term, and focus on the first-order contribution to the S matrix (the Kronecker-δ would be the zeroth-order term), we can write, in a somewhat shorthand notation, ⟨S(⃗ r)⟩[1] = −i T ⟨U (⃗ r)⟩ ,

(12.5)

where, in a typical case, some interaction potential HI (⃗ r) = U (⃗ r) is matched against the S matrix element.

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|nA i

|nB i

|mA i

|mB i

|nA i

|nA i

|nB i

|nB i

(a)

|mA i

|mB i

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|nA i

|nB i

(b)

Fig. 12.1 We investigate the two-photon exchange between two active electrons in two atoms, labeled A and B. Diagram (a) is the ladder exchange, while diagram (b) constitutes the crossedladder photon exchange. Atom A makes a virtual transition from the reference states ∣nA ⟩ to the virtual state ∣mA ⟩, and back. Atom B undergoes an analogous transition.

12.2.2

Scattering Amplitudes and Interatomic Interactions

In order to illustrate the considerations outlined in Sec. 12.2.1, let us consider a possible application of the propagator formalism introduced in Chaps. 7 and 9, to the calculation of the interatomic interaction, which we had considered in the Schr¨ odinger picture in Sec. 5.4. The problem with the treatment in Sec. 5.4 is that no time variable is introduced in the description of photon exchange. Hence, it is difficult to correctly assess the placement of the propagator poles in the denominators. It is very interesting to realize that the calculation performed in Sec. 5.4 becomes a lot easier when one resorts to time-ordered perturbation theory and just calculates the forward scattering amplitude for the two-atom system, rather than calculating the Hamiltonian on the basis of time-independent, Schr¨odinger-picture field operators, as in Sec. 5.4. Following Ref. [208], we here illustrate, briefly, how to derive the starting expression for the self-energy and the Casimir–Polder interaction. We are also inspired by the derivation outlined in Chap. 85 of Ref. [3]. The calculation described in Ref. [208] roughly proceeds as follows. One con′ siders two atoms in states ψA (⃗ rA ) and ψB (⃗ rB ) which scatter into states ψA (⃗ rA ) ′ and ψB (⃗ rB ) under the action of a potential U (∣⃗ rA − r⃗B ∣). If we denote the initial state by i and the final state by the subscript f , then the corresponding S-matrix element reads as follows, ′∗ ′∗ Sf i = − i ∫ d3 rA ∫ d3 rB ψA (⃗ rA ) ψB (⃗ rB ) U (∣⃗ rA − r⃗B ∣) ψA (⃗ rA ) ψB (⃗ rB )

× ∫ dt exp[−i (E1 + E2 − E1′ − E2′ ) t] ′∗ ′∗ = − i T ∫ d3 rA ∫ d3 rB ψA (⃗ rA ) ψB (⃗ rB ) U (⃗ rA − r⃗B ) ψA (⃗ rA ) ψB (⃗ rB ) , (12.6)

where we have assumed energy conservation (E1 + E2 = E1′ + E2′ ).

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One uses time-dependent quantum electrodynamic (QED) perturbation theory, where the interaction is formulated in the interaction picture [2, 116]. This means that the second-quantized operators in the interaction Hamiltonian have a time dependence which is generated by the action of the free Hamiltonian [112]. The designation of an interaction Hamiltonian being formulated in the interaction picture is not redundant [112]. The two relevant Feynman diagrams in Fig. 12.1 (as opposed to the twelve diagrams in Fig. 5.1) describe all possible time orderings. We use, specifically, the temporal gauge for the photon propagator, outlined in Eq. (9.133). We recall that the 00 component of the photon propagator vanishes in the temporal (Weyl) gauge, ki kj 1 (δij − 2 ) . (12.7) k 2 + i ω Because the scalar potential vanishes, the interaction Hamiltonian can then be restricted to the dipole interaction with the (spatial components of the) electric field, DF00 (k) = 0 ,

DFij (k) =

⃗ rA , t)⋅d⃗A (t)−E(⃗ ⃗ rB , t)⋅d⃗B (t) ≈ −E( ⃗ R ⃗ A , t)⋅d⃗A (t)−E( ⃗ R ⃗ B , t)⋅d⃗B (t) , (12.8) V (t) = −E(⃗ where d⃗i = e r⃗i is the dipole operator for atom i (for atoms with more than one ⃗ A and electron, one has to sum over all the electrons in the atoms i = A, B). By R ⃗ B , we denote the positions of the atomic nuclei. R We use the relation (9.122) in order to relate the time-ordered product of field operators to the Hamiltonian. This differs, by an overall minus sign, from the conventions used in Eq. (10) of Ref. [208]. In the mixed frequency-position representation, one has Dik (ω, r⃗) = − (δ ik +

∇i ∇k ei∣ω∣r ) , ω2 4πr

(12.9)

√ where ∣ω∣ ≡ ω 2 + i, and the branch √ cut of the square-root function chosen along the positive real axis, so that Im( ω 2 + i) > 0 in the spirit of Eq. (9.34). We carry out the differentiations with the result, ∇i ∇ k

ei∣ω∣r 1 i ei∣ω∣r ri rk 3 3i ei∣ω∣r = ω 2 δ ik (− 2 2 + ) + ω2 2 ( 2 2 − − 1) . 4πr ω r ∣ω∣ r 4πr r ω r ∣ω∣ r 4πr (12.10)

The temporal-gauge photon propagator in the mixed representation becomes Dik (ω, r⃗) = − [αik + β ik ( αik = δ ik −

ri rk , r2

1 ei∣ω∣r i − 2 2 )] , ∣ω∣r ω r 4πr β ik = δ ik − 3

ri rk . r2

(12.11)

Throughout this chapter, we shall depart from the notation (2.58) and use ri rather than xi for the ith component of the position operator r⃗. This is necessitated by

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the fact that, for composite systems, a differentiation of the position vector r⃗ and ⃗ becomes convenient [see Eq. (12.140)]. The photon the center-of-mass coordinate x ⃗ can be translated propagator, which is the propagator for the vector potential A, into the propagator for the electric field by differentiation with respect to the time, ik DE (x1 − x2 ) =

=

∂ ∂ ⟨0 ∣T [Ai (⃗ rA , t1 ) Aj (⃗ rB , t2 )]∣ 0⟩ ∂t1 ∂t2 ∂ ∂2 ∂ (i Dik (x1 − x2 )) = −i 2 Dik (x1 − x2 ) . ∂(t1 − t2 ) ∂(t2 − t1 ) ∂t1 (12.12)

If we work in the mixed representation, we can implement the differentiation with respect to time in the Fourier integral as follows, ik ⟨0 ∣T [E i (⃗ rA , t1 ) E j (⃗ rB , t2 )]∣ 0⟩ = DE (x1 − x2 ) = i ∫

dω 2 ik ω D (ω, r⃗) e−iω(t1 −t2 ) . 2π (12.13)

This result will prove useful in the following. The fourth-order contribution to the S-matrix is S (4) =

(−i)4 ∫ dt1 ∫ dt2 ∫ dt3 ∫ dt4 T[V (t1 )V (t2 )V (t3 )V (t4 )] . 4!

(12.14)

According to the Wick theorem, the time-ordered product is equal to the normal ordered product, plus all contractions. So, 4 (−i)4 × 3 × ∏ ∫ dt` ⟨φ ∣(⟨0 ∣T[V (t1 )V (t2 )∣ 0⟩ ⟨0 ∣T[V (t3 )V (t4 )∣ 0⟩)∣ φ⟩ . 4! `=1 (12.15) We denote by ∣φ⟩ the reference state of the atoms, and by ∣0⟩ the vacuum of the electromagnetic field. Now,

S (4) =

⃗ rA , t1 ) ⋅ d⃗A (t1 )) (E(⃗ ⃗ rB , t2 ) ⋅ d⃗B (t2 ))]∣ 0⟩ ⟨0 ∣T[V (t1 )V (t2 )∣ 0⟩ ∼ ⟨0 ∣T [(E(⃗ ⃗ rB , t1 ) ⋅ d⃗B (t1 )) (E(⃗ ⃗ rA , t2 ) ⋅ d⃗A (t2 ))]∣ 0⟩ , + ⟨0 ∣T [(E(⃗ (12.16) where by ∼ we denote the omission of operators which pertain to the self-energy of the two atoms (these would otherwise contain two A’s or two B’s in the indices), and keep only the terms relevant for the interaction energy. The time ordering of the electric-field operators results in the expression, ⟨0 ∣T[V (t1 )V (t2 )∣ 0⟩ ∼ ⟨0 ∣T [E i (⃗ rA , t1 ) E j (⃗ rB , t2 )]∣ 0⟩ diA (t1 ) djB (t2 ) + ⟨0 ∣T [E i (⃗ rB , t1 ) E j (⃗ rA , t2 )]∣ 0⟩ diB (t1 ) djA (t2 ) ,

(12.17)

and analogously for t1 → t3 , and t2 → t4 , in the last line of Eq. (12.15). After the time ordering of the atomic dipole operators [the T symbol in Eq. (12.16) refers to

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both field as well as dipole operators], for the times t1 , t2 , t3 and t4 , we end up with four contributions, C1 ≡ ⟨nA ∣T diA (t1 ) dkA (t3 )∣nA ⟩ ⟨nB ∣T djB (t2 ) d`B (t4 )∣nB ⟩ ,

(12.18a)

C2 ≡ ⟨nA ∣T diA (t1 ) d`A (t4 )∣nA ⟩ ⟨nB ∣T djB (t2 ) dkB (t3 )∣nB ⟩ ,

(12.18b)

djA (t2 ) dkA (t3 )∣nA ⟩ , djA (t2 ) d`A (t4 )∣nA ⟩ .

(12.18c)

C3 ≡ ⟨nB ∣T

diB (t1 ) d`B (t4 )∣nB ⟩ ⟨nA ∣T

C4 ≡ ⟨nB ∣T

diB (t1 ) dkB (t3 )∣nB ⟩ ⟨nA ∣T

(12.18d)

Contributions C2 and C4 correspond to the crossed-ladder diagram (in the language of Feynman diagrams), whereas C1 and C3 correspond to the two-photon ladder exchange (see also Fig. 12.1). The contributions of atoms A and B to the atomic reference state are denoted as ∣nA ⟩ and ∣nB ⟩, respectively (see Fig. 12.1). With all terms C1 , C2 , C3 , and C4 leading to equivalent contributions, we finally arrive at S (4) =

1 4 i rA , t1 ) E j (⃗ rB , t2 )]∣ 0⟩ ⟨0 ∣T [E k (⃗ rA , t3 ) E ` (⃗ rB , t4 )]∣ 0⟩ ∏ ∫ dt` ⟨0 ∣T [E (⃗ 2 `=1 × ⟨nA ∣T diA (t1 ) dkA (t3 )∣ nA ⟩ ⟨nB ∣T djB (t2 ) d`B (t4 )∣ nB ⟩ .

(12.19)

The time-ordered product of electric-field operators can be evaluated as follows, ik DE (x1 − x2 ) = ⟨0 ∣T [E i (⃗ rA , t1 ) E k (⃗ rB , t2 )]∣ 0⟩ ,

(12.20)

ik where DE (x1 − x2 ) is given in Eq. (12.12). Now, let us proceed to the time-ordered product of dipole operators, which is given as follows (for atom A),

αA,ik (t1 − t2 ) = i ⟨nA ∣T(diA (t1 ) dkA (t2 )∣ nA ⟩ ,

ik αA (t) = ∫

∞ −∞

dω −i ω t ik e αA (ω) , 2π (12.21)

so that ∞

dω −i ω (t1 −t2 ) ik e αA (ω) . (12.22) −∞ 2π The time-ordered product of dipole operators can be evaluated in terms of the polarizability of the atom, with the pole being displaced according to the Feynman prescription, ik ⟨nA ∣T diA (t1 ) dkA (t2 )∣ nA ⟩ = −i αA (t1 − t2 ) = −i ∫

ik αA (ω) = ∫

∞

ik dt eiωt αA (t) = i ∫

−∞ ∞

=i∫

0

∞ −∞

dt eiωt ⟨nA ∣T(diA (t) dkA (0)∣ nA ⟩

dt eiωt ⟨nA ∣diA (t) dkA (0)∣ nA ⟩ + i ∫

⎛ ⟨nA ∣diA ∣ mA ⟩ ⟨mA ∣dkA ∣ nA ⟩ + Emn,A − ω − i mA ⎝

= ∑

0

dt eiωt ⟨nA ∣dkA (0) diA (t)∣ nA ⟩

−∞ ⟨nA ∣diA ∣ mA ⟩

⟨mA ∣dkA ∣ nA ⟩ ⎞

Emn,A + ω − i

⎠

, (12.23)

where Emn,A = EmA − EnA . We recognize the polarizability, with poles displaced according to the Feynman prescription. The attentive reader will have noticed that the sequence of the indicated i and k in the last term has been inverted in going

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from the second to the third line of Eq. (12.23). This requires an explanation. For a nonrelativistic atomic state, one can show that, taking into account the summation over the mA , and using the formalism of reduced matrix elements outlined Chap. 6, ⟨nA ∣dkA ∣ mA ⟩ ⟨mA ∣diA ∣ nA ⟩ Emn,A + ω − i

=

⟨nA ∣diA ∣ mA ⟩ ⟨mA ∣dkA ∣ nA ⟩ Emn,A + ω − i

.

(12.24)

Details are left as an exercise to the reader. The positive-energy poles occur at ω = Emn,A − i, while the negative-energy poles are at ω = −Emn,A + i. If the virtual state is displaced toward lower energy, i.e., Emn,A < 0, then the pole at ω = −Emn,A + i migrates into the first quadrant of the complex plane. Note that the last transformation in Eq. (12.23) holds provided the matrix elements in the numerator are real, in which case we can formulate the Hermitian adjoint. We now reformulate Eq. (12.15), with the help of Eq. (12.22), and approximate ⃗ A , and r⃗B ≈ R ⃗ B , as well as R ⃗=R ⃗A − R ⃗ B , for the interaction in the dipole r⃗A ≈ R approximation, S (4) =

1 4 i ⃗ j ⃗ k ⃗ ` ⃗ ∏ ∫ dt` ⟨0 ∣T [E (R A , t1 ) E (RB , t2 )]∣ 0⟩ ⟨0 ∣T [E (RA , t3 ) E (RB , t4 )]∣ 0⟩ 2 `=1 × ⟨nA ∣T diA (t1 ) dkA (t3 )∣ nA ⟩ ⟨nB ∣T djB (t2 ) d`B (t4 )∣ nB ⟩

=

4 1 4 dωs 2 2 ij ⃗ Dk` (ω2 , R) ⃗ αik (ω3 ) αj` (ω4 ) ω1 ω2 D (ω1 , R) ∏ ∫ dt` ∏ ∫ A B 2 `=1 2π s=1

× e−iω1 (t1 −t2 )−iω2 (t3 −t4 )−iω3 (t1 −t3 )−iω4 (t2 −t4 ) .

(12.25)

One now carries out the dti integration one after the other, with the prescriptions ∫ dt2 → 2πδ(ω1 − ω4 ), then ∫ dt3 → 2πδ(ω2 − ω3 ), and ∫ dt4 → 2πδ(ω2 + ω4 ). As a result, the condition ω1 = ω4 = −ω2 = −ω3 is implemented in the final result, 1 dω1 2 ⃗ Dk` (−ω1 , R) ⃗ αik (−ω1 ) αj` (ω1 ) ω (−ω1 )2 Dij (ω1 , R) ∫ dt1 ∫ 2 2π 1 T dω 4 ij ⃗ Dk` (ω, R) ⃗ αik (ω) αj` (ω) . ω D (ω, R) (12.26) = ∫ A B 2 2π

S (4) =

The matching of the effective potential with the scattering amplitude according to Eq. (12.5) results in the interaction energy ∞ dω i ⃗ Dk` (ω, R) ⃗ αik (ω) αj` (ω) ω 4 Dij (ω, R) ∫ A B 2 −∞ 2π ∞ dω ⃗ Dk` (ω, R) ⃗ αik (ω) αj` (ω) , ω 4 Dij (ω, R) = i∫ A B 2π 0

∆E =

(12.27)

where the latter equality surprisingly holds because of perfect ω ↔ −ω symmetry for the integrand; both the photon propagator as well as the atomic polarizability have that symmetry. We note that the Feynman prescription is crucial in establishing the perfect symmetry of αA,ik (ω) under the transformation ω ↔ −ω.

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The polarizabilities in Eq. (12.27) are used according to the Feynman prescription, j j ⎛ ⟨nA ∣diA ∣ mA ⟩ ⟨mA ∣dA ∣ nA ⟩ ⟨nA ∣diA ∣ mA ⟩ ⟨mA ∣dA ∣ nA ⟩ ⎞ ij αA (ω) = ∑ + , (12.28a) Em,A − En,A − ω − i Em,A − En,A + ω − i ⎠ mA ⎝ j j ⎛ ⟨nB ∣diB ∣ mB ⟩ ⟨mB ∣dB ∣ nB ⟩ ⟨nB ∣diB ∣ mB ⟩ ⟨mB ∣dB ∣ nB ⟩ ⎞ + . Eq,B − En,B − ω − i Eq,B − En,B + ω − i ⎠ mB ⎝

ij αB (ω) = ∑

(12.28b) Here, nA and nB are the atomic reference states, and mA and mB are the virtual states (which can be of lower energy). The tensor structure in Eq. (12.27) amounts to 2i∣ω∣r 2i 5 6i 3 ⃗ Dji (ω, R) ⃗ = 2e Dij (ω, R) (1 + − − + ) . (12.29) 2 2 3 r ∣ω∣ R (ωR) (∣ω∣R) (ωR)4 ij ij For ground-state atoms, with αA (ω) = δ ij αA (ω) and αB (ω) = δ ij αB (ω), a Wick rotation of the expression (12.27) leads to ∞ e−2ωR ω 4 2 5 6 3 ∆E = − ∫ dω (1 + + + + ) αA (iω) αB (iω) , 2 2 3 R ω R (ωR) (ωR) (ωR)4 0 (12.30) confirming the result in Eq. (5.117). For excited-state atoms, the Feynman method dictates the positioning of the poles, which leads to long-range tails [209, 212, 361]. We have demonstrated the power of the Feynman formalism for the calculation of the effective interaction between atoms.

12.3 12.3.1

From the One-Body to the Two-Body Hamiltonian Relevant Hamiltonians

Now, we apply the matching formalism (introduced in Sec. 12.2.1) to the interaction between two charged particles. We start from the effective Hamiltonian obtained by adding the terms H [2] and H [3] given in Eq. (11.55), and the terms H [4] and H [5] obtained in Eqs. (11.57) and (11.58), respectively. Using the relation (11.56) and assuming a spatially slowly varying magnetic field, which justifies the approximation ⃗4 ⃗ )4 eκ e (⃗ σ⋅π ⃗ σ ⃗ ≈− π ⃗ π ({⃗ ⃗⋅π ⃗ }) , ⃗ ⃗ ⃗ ⃗ 2 } + κ {⃗ π ⋅ B, + {⃗ σ ⋅ π , {⃗ σ ⋅ π , σ ⋅ B}} + σ ⋅ B, − 8m3 16m3 8m3 8m3 (12.31) 4 one can write the following general Hamiltonian, which contains all the α terms as well as the α5 corrections due to the anomalous magnetic moment, ⃗2 ⃗4 π e ⃗− π ⃗⋅B HFW = + e A0 − (1 + κ) σ 2m 2m 8m3 e ⃗ +σ ⃗ ×π ⃗ ⃗ ⋅E ⃗ ⋅ (E ⃗−π ⃗ × E)] (1 + 2κ) [∇ − 8m2 e ⃗ π ⃗ π ({⃗ ⃗ 2 } + κ {⃗ ⃗⋅σ ⃗ }) . σ ⋅ B, π ⋅ B, (12.32) + 8m3

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⃗ and B ⃗ fields are related to the scalar and vector potentials as follows, The E ⃗ = − ∇A ⃗ 0 − ∂t A⃗ , E ⃗=∇ ⃗ × A⃗ . B

(12.33)

We take into account the lack of an explicit time dependence in Eq. (12.32), except for the electromagnetic fields. The interaction Hamiltonian is used in the representation given in Eq. (11.98), ⃗ ′ ) Aµ (t′ , r⃗′ )∣ ′ = e j µ (⃗ HI = e j µ Aµ = e j µ (⃗ r, i∂t′ , −i∇ r, i∂x′ ) Aµ (x′ )∣x′ =x . (12.34) r⃗ =⃗ r Our notation in Eq. (12.34) is a little more precise than in Eq. (11.98). Namely, we indicate the fact that the current operators in Eqs. (11.108) and (11.119) contain differential operators acting on the photon vector potential. After the differentiation with respect to the arguments of the vector potential Aµ , the locality of the current dictates that we set the primed coordinates equal to the unprimed ones. The differ⃗ ′ ), and ential operators with respect to the arguments of Aµ are denoted as (i∂t′ , −i∇ summarized in the expression i∂x′ . It is, nevertheless, necessary to distinguish the differentiation with respect to the primed coordinates from the unprimed ones, because the details of the functional dependence determine the behavior in momentum space [see Eq. (12.37)]. 12.3.2

Matching the Two-Particle Interaction Hamiltonian

The matching of the scattering amplitude for the interaction of two electrons, labeled a and b, against the effective Hamiltonian, is a little harder to derive than one might otherwise expect. We start from a second-order matrix element and consider the transition of electron a from the initial state ∣ia ⟩ to the final state ∣fa ⟩, and the transition of electron b from the initial state ∣ib ⟩ to the final state ∣fb ⟩. The procedure is slightly complicated by the fact that the Foldy–Wouthuysen transformed current operators contain momentum operators acting on the four-vector potential of the photon. Following Eq. (12.34), the interaction Hamiltonian is (nota bene the distinction of J µ versus j µ , and the fact that the differential operator i∂x′ denotes the corresponding four-vector with components i∂/∂x′ρ , where ρ = 0, 1, 2, 3) H = Jaµ (x, i∂x′ )Aµ (x′ ) + Jbµ (x, i∂x′ )Aµ (x′ )∣x′ =x .

(12.35)

The transition currents read as follows, Jaµ (x, i∂x′ ) = e φ+fa (x) j µ (⃗ r, i∂x′ ) φia (x) = e e−i(Eia −Efa )t φ+fa (⃗ r) j µ (⃗ r, i∂x′ ) φia (⃗ r) , Jbµ (x, i∂x′ )

=

(12.36a)

e φ+fb (x) j µ (⃗ r, i∂x′ ) φib (x)

= e e−i(Eib −Efb )t φ+fb (⃗ r) j µ (⃗ r, i∂x′ ) φib (⃗ r) .

(12.36b)

Once more, we understand that the primed operators act on the four-vector potential, before setting the primed coordinates equal to the unprimed ones.

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One considers the emergence of two equivalent terms because of the two terms in the interaction Hamiltonian, and writes the second-order S matrix element as follows, Sf i = 2 × {−

1 µ 4 4 µ ∫ d xa ∫ d xb T [Ja (xa , i∂x′a ) Jb (xa , i∂x′b )] 2!

× ⟨0 ∣T[Aµ (x′a ) Aν (x′b )]∣ 0⟩ ∣ = − ∫ d4 xa ∫ d4 xb ∫

x′a =xa ,x′b =xb

}

d4 k T [Jaµ (xa , k) Jbν (xb , −k)] iDµν (k) e−ik(xa −xb ) (2π)4

d4 k Dµν (k) Qµν (k) , (2π)4 where we have used Eq. (9.120) in the form = −i ∫

(12.37)

⟨0 ∣T [Aµ (xa ) Aν (xb )]∣ 0⟩ = iDµν (xa − xb ) = i ∫

d4 k Dµν (k) e−ik⋅(xa −xb ) , (12.38) (2π)4

and we define Qµν (k) = ∫ d4 xa ∫ d4 xb e−ik(xa −xb ) T [Jaµ (xa , k) Jbν (xb , −k)] .

(12.39)

We write the four-vectors as xµa = (ta , r⃗a ) and xµb = (tb , r⃗b ). Then, ⃗

Qµν (k) = Θ(ta − tb )e2 ∫ dta ∫ dtb ∫ d3 ra∫ d3 rb φ+fa (⃗ ra )j µ (⃗ ra , k)eik⋅⃗ra φia (⃗ ra ) ⃗

× φ+fb (⃗ rb )j µ (⃗ rb , −k)e−ik⋅⃗rb φib (⃗ rb )e−i(Eia −Efa )ta e−i(Eib −Efb )tb e−iω(ta −tb ) ⃗

+ Θ(tb − ta ) e2 ∫ dta ∫ dtb ∫ d3 ra∫ d3 rb φ+fb (⃗ rb )j ν (⃗ rb , −k)e−ik⋅⃗rb φib (⃗ rb ) ⃗

ra )e−i(Eia −Efa )ta e−i(Eib −Efb )tb e−iω(ta −tb ) ra )j µ (⃗ ra , k)eik⋅⃗ra φia (⃗ × φ+fa (⃗ = T ∫ d3 ra∫ d3 rb

ra ) j µ (xa , k)φia (⃗ ra )φ+fb (⃗ rb )j ν (⃗ rb , −k)φib (⃗ φ+fa (⃗ rb )

⃗

eik⋅⃗rab i(Eia − Efa + ω − i) φ+f (⃗ rb )j ν (⃗ rb , −k)φib (⃗ rb )φ+fa (⃗ ra )j µ (xa , k)φia (⃗ ra ) ik⋅⃗ ⃗ + T ∫ d3 ra∫ d3 rb b e rab , −i(Eia − Efa + ω + i) (12.40)

where ω = k 0 . Furthermore, we define r⃗ab = r⃗a −⃗ rb . The Jacobian of the substitutions (ta , tb ) → (τ, ta − tb ) ,

(ta , tb ) → (τ, tb − ta ) ,

(12.41)

is just equal to unity. The long time interval is ∫ dτ = T (as opposed to the time ordering operator T). We have used the fact that, in order to generate the factor T which we need for the matching with the effective Hamiltonian, we need the relations Efa − Eia = Eib − Efb ,

Efa + Efb = Eia + Eib .

(12.42)

Recalling Eq. (12.37), one matches with the effective Hamiltonian Hab proceeds using the formula, Sf i = −i T ⟨φf ∣Hab ∣φi ⟩ ,

(12.43)

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and obtains i d4 k (−ie2 ∫ Dµν (k) Qµν (k)) = −ie2 ∫ d3 ra ∫ d3 rb T (2π)4 + ra ) j µ (⃗ ra , k) φia (⃗ ra ) φ+fb (⃗ rb ) j ν (⃗ rb , −k) φib (⃗ rb ) d4 k ⃗ rab φfa (⃗ ik⋅⃗ D (k) e [ ×∫ µν (2π)4 ω − (Efa − Eia ) − i φ+f (⃗ rb ) j ν (⃗ rb , −k) φib (⃗ rb ) φ+fa (⃗ ra ) j µ (⃗ ra , k) φia (⃗ ra ) − b ]. (12.44) ω − (Efa − Eia ) + i

⟨φf ∣Hab ∣φi ⟩ =

One now closes the integration contour over the photon energy in the upper half of the complex plane, and ignores the pole due to the photon propagator which otherwise leads to a self-energy type effect contributing to the recoil correction Furthermore, one neglects the resonance frequency Efa − Eia of the transition. This can be justified as follows. Picking up the poles given in Eq. (12.44) results in photon propagator expressions of the form 1/[(Efa − Eia )2 − k⃗2 ]. The Coulomb ⃗ ∼ (Zα) m, while interaction is mediated by virtual photons with a momentum ∣k∣ atomic-state energy difference are of the order of Efa − Eia ∼ (Zα)2 m (in natural ̵ = c = 0 = 1). Hence, we can ignore (Ef − Ei )2 ≪ k⃗2 to the order units with h a a of approximation we are interested in. This corresponds to the approximation of a static interaction, and leads to the Breit Hamiltonian [see also task (1) in Sec. 12.7]. The result, in view of the residue theorem, is ⟨φf ∣Hab ∣φi ⟩ = ∫ d3 ra ∫ d3 rb φ+fa (⃗ ra ) φ+fb (⃗ rb ) × {e2 ∫

d3 k ⃗ Dµν (k)eik⋅⃗rab j µ (⃗ ra , k) j ν (⃗ rb , −k)}∣ φia (⃗ ra )φib (⃗ rb ). (2π)3 k0 =0 (12.45)

We can read off the effective two-body interaction Hamiltonian Hab = e2 ∫

d3 k ⃗ ra −⃗ rb ) ν ⃗ j µ (k) ⃗ ei k⋅(⃗ ⃗ Dµν (k) jb (−k)∣ , a (2π)3 k0 =0

(12.46)

where we use the short-hand notations defined in Eqs. (11.112) and (11.120). This expression will be the basis for our evaluation of the interelectronic and electronnucleus interaction. 12.3.3

Photon Exchange and Breit Hamiltonian

In order to proceed with the evaluation of Hab given in Eq. (12.46), we use the photon propagator in Coulomb gauge, and recall Eq. (9.73) for convenience, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⃗2 ⎪ ⎪ ⎪k 1 k k Dµν (k) = ⎨ (δij − ⃗i 2 j ) ⎪ ⎪ 2 2 ⃗ ⎪ k ω − k + i ⎪ ⎪ ⎪ ⎪ 0 ⎩

µ=ν=0 µ = i, ν = j µ≠ν.

(12.47)

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We employ the definition r⃗ab = r⃗a − r⃗b . The Fourier transform Dµν (⃗ rab ) = ∫

d3 k ⃗ rab ⃗ ei k⋅⃗ Dµν (k) (2π)3

(12.48)

is found as follows, ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 4πrab ⎪ ⎪ ⎪ i j Dµν (⃗ rab ) = ⎨ − 1 (δ ij − ra rb ) ⎪ 2 ⎪ ⎪ rab 8πrab ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0

µ=ν=0 µ = i, ν = j .

(12.49)

µ≠ν

We recall, from Eq. (11.112), the time-like component of the current, 1 + 2κ ⃗2 ⃗ = 1 + i (1 + 2κ) σ ⃗ ⋅ (k⃗ × π ⃗) − k . j 0 (k) 2 4m 8m2

(12.50)

⃗ext → π ⃗ , identifying the quantity π ⃗ In comparison to Eq. (11.112), we have replaced π with a canonical momentum that contains a vector potential from external sources. Finally, one obtains the matrix element of the effective Hamiltonian, (00)

Hab

d3 k ⃗ ra −⃗ rb ) 0 ⃗ j 0 (k) ⃗ ei k⋅(⃗ ⃗ D00 (k) jb (−k) a (2π)3 ⃗ ⃗ d3 k ⎡⎢ ei k⋅(⃗ra −⃗rb ) i (1 + 2κ) ei k⋅(⃗ra −⃗rb ) ⃗×π ⎢ ⃗ ⃗ = e2 ∫ σ ⋅ ( k ) + a a (2π)3 ⎢⎣ 4m2 k⃗ 2 k⃗ 2 ⃗ ⎤ 1 + 2κ i k⋅(⃗ ei k⋅(⃗ra −⃗rb ) i (1 + 2κ) ⃗ ra −⃗ rb ) ⎥ ⃗×π ⎥ ⃗ ⃗ σ ⋅ (− k ) − e + b b ⎥ 4m2 4m2 k⃗ 2 ⎦ 1 i (1 + 2κ) 1 ⃗a × π ⃗a ⋅ (−i∇ ⃗a ) = 4πα [ + σ 4π∣⃗ ra − r⃗b ∣ 4m2 4π∣⃗ ra − r⃗b ∣

= e2 ∫

+

i (1 + 2κ) 1 1 + 2κ (3) ⃗a × π ⃗b ⋅ (i∇ ⃗b ) σ − δ (⃗ ra − r⃗b )] . 2 4m 4π∣⃗ ra − r⃗b ∣ 4m2

(12.51)

⃗ and the convention that We have repeatedly used the correspondence k⃗ ⇔ −i ∇, ⃗ the operators ∇a = ∂/∂ r⃗a act on the explicit r⃗a -dependence of the potential factors, ⃗a and π ⃗b act on the electron (wave function) coordinates. One while the operators π may consult Eqs. (12.34) and (11.108) for the distinction between the “primed” ⃗ a , which act on the potentials) and the “unprimed” gradient operators (here, the ∇ ⃗a , which act on the wave functions). Using, again, the definition ones (here, the π r⃗ab = r⃗a − r⃗b , we have πα α(1 + 2κ) α ⃗b ) − σ ⃗a ⋅ (⃗ ⃗a )] . rab ) + [⃗ σb ⋅ (⃗ rab × π rab × π − 2 (1 + 2κ) δ (3) (⃗ 3 rab m 4m2 rab (12.52) We recall the spatial components of the current from Eq. (11.120), (00)

Hab

=

⃗ i (1 + κ) ⃗ = π ⃗ . ⃗j(k) + (⃗ σ × k) m 2m

(12.53)

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These components generate an effective Hamiltonian Hab , (ij)

d3 k ⃗ ra −⃗ rb ) j ⃗ j i (k) ⃗ ei k⋅(⃗ ⃗ Dij (k) jb (−k) a 3 (2π) ⃗ ra −⃗ ⎡ i k⋅(⃗ rb ) d3 k πai i (1 + κ) k i k j ⎤⎥ ij ⃗ i ) ⎢⎢− e = e2 ∫ ( + (⃗ σ × k) (δ − )⎥ ⎢ (2π)3 m 2m k⃗ 2 k⃗ 2 ⎥⎦ ⎣

Hab = e2 ∫

×

⎛ πbj i (1 + κ) ⃗ j ⎞ = T1 + T2 + T3 + T4 . + (⃗ σ × (−k)) 2m ⎠ ⎝m

(12.54)

The expression for T1 is obtained immediately, while the terms T2 , T3 and T4 require a more elaborate treatment of the Fourier (back-)transformation from momentum to coordinate space, ⎡ j ⎤ j π i ⎢ 1 ⎛ ij rai rb ⎞⎥⎥ πb δ − 2 ⎥ , T1 = e2 a ⎢⎢− m ⎢ 2rab ⎝ rab ⎠⎥ m ⎣ ⎦ ⃗ ra −⃗ ⎡ i k⋅(⃗ ⎤ j rb ) d3 k i (1 + κ) 2 i ⎢ e ⊥,ij ⎥ πb ⃗ ⎢ ⎥ T2 = e ∫ − ( (⃗ σ × k) ) δ , a ⎢ ⎥ m (2π)3 2m k⃗ 2 ⎣ ⎦ ⃗ d3 k πai ⎡⎢ ei k⋅(⃗ra −⃗rb ) ⊥,ij ⎤⎥ i (1 + κ) 2 ⃗ j) , ⎢− T3 = e ∫ δ ⎥( (⃗ σb × (−k)) ⎥ 2 ⃗ (2π)3 m ⎢⎣ 2m k ⎦ ⃗ ra −⃗ ⎤ ⎡ i k⋅(⃗ rb ) 3 i (1 + κ) d k i ⎢ e ⊥,ij ⎥ 2 ⃗ ⎥ ⎢ T4 = e ∫ ( (⃗ σ × k) ) − δ a ⎥ ⎢ (2π)3 2m k⃗ 2 ⎦ ⎣ i (1 + κ) ⃗ j) . ×( (⃗ σb × (−k)) (12.55) 2m Here, δ ⊥,ij = δ ij − k i k j /k⃗ 2 . A useful integral for the calculation of T2 is ⃗ ra −⃗ ⎡ i k⋅(⃗ rb ) ⃗b d3 k k i k j ⎤⎥ j (⃗ σa × r⃗ab ) ⋅ π i⎢ e ij ⃗ ⎢ − (⃗ σ × k) . (12.56) I=∫ (δ − )⎥ πb = −i a 3 ⎢ ⎥ 2 2 ⃗ ⃗ (2π)3 4πr k k ab ⎣ ⎦ The corresponding evaluation, adapted to the term T3 , reads as follows, ⃗ ⃗a d3 k i ⎡⎢ ei k⋅(⃗ra −⃗rb ) ij k i k j ⎤⎥ σb × r⃗ab ) ⋅ π ⃗ j = i (⃗ ⎢ J =∫ π − (δ − )⎥ (⃗ σb × (−k)) . (12.57) 3 (2π)3 a ⎢⎣ 4πrab k⃗ 2 k⃗ 2 ⎥⎦ For T4 , we find it useful to calculate ⃗ ra −⃗ ⎡ i k⋅(⃗ rb ) d3 k k i k j ⎤⎥ ij ⃗ i ⎢⎢− e ⃗ j K=∫ (⃗ σ × k) (δ − )⎥ (⃗ σ × (−k)) a ⎢ ⎥ b 2 2 ⃗ ⃗ (2π)3 k k ⎣ ⎦ ⃗ ra −⃗ ⎡ ei k⋅(⃗ rb ) ⎤ d3 k ⎥ ⃗2 − σ ⃗σ ⃗ ⎢⎢− (⃗ ⎥ ⃗ ⃗ ⃗ σ ⋅ σ k ⋅ k ⋅ k) = −∫ a b a b ⎢ ⎥ 2 ⃗ (2π)3 k ⎣ ⎦ 1 (3) ⃗a σ ⃗ a) ⃗a ⋅ σ ⃗b δ (⃗ ⃗b ⋅ ∇ =σ rab ) + (⃗ σa ⋅ ∇ 4πrab =

j i ⎞ σ i σ j ⎛ δ ij rab rab 2 ⃗a ⋅ σ ⃗b δ (3) (⃗ σ rab ) − a b − 3 . 3 5 3 4π ⎝ rab rab ⎠

(12.58)

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Armed with the results for I and J, and K, we can calculate the “spatial” contribution to the Breit Hamiltonian as follows, ⎡ j ⎤ j i (1 + κ) −(1 + κ)2 πi ⎢ 1 ⎛ ij rai rb ⎞⎥⎥ πb i (1 + κ) (ij) δ + 2 ⎥ + I+ J+ K. Hab = 4πα a ⎢⎢− 2 2 m ⎢ 8πrab ⎝ rab ⎠⎥ m 2m 2m 4m2 ⎣ ⎦ (12.59) (ij)

So, Hab

is eventually found as follows,

(ij)

Hab = − −

j i ⎞ j α(1 + κ) rab rab α i ⎛ ij ⃗b − σ ⃗b × r⃗ab ⋅ π ⃗a ] π δ + π + [⃗ σa × r⃗ab ⋅ π a 2 3 2m2 rab rab ⎠ b 2m2 rab ⎝ j j i ⎞ rab rab 2π(1 + κ)2 α σai σb ⎛ ij (3) ⃗ ⃗ σ ⋅ σ δ (⃗ r ) + δ − 3 , a b ab 3 2 2 2 3m 4m rab ⎝ rab ⎠

(12.60)

(00)

which is purely an α4 contribution. Together with Hab from Eq. (12.52), which we recall for convenience, πα α(1 + 2κ) α (00) ⃗b ) − σ ⃗a ⋅ (⃗ ⃗a )] , − (1 + 2κ) δ (3) (⃗ rab ) + [⃗ σb ⋅ (⃗ rab × π rab × π Hab = 3 rab m2 4m2 rab (12.61) (00) (ij) we obtain Hab as the sum of Hab and Hab . Finally, the Breit Hamiltonian for a many-electron system is obtained as the sum of the one-particle, and the manyparticle Hamiltonians, HBP = ∑ Ha + ∑ Hab . a

(12.62)

a>b

The single-particle Hamiltonian is obtained from Eq. (12.32). However, we perform two adjustments in order to allow for a coupling to an external field, by replacing e A0 → Va + e A0a = −Zα/ra + e A0a , where Va = −Zα/ra is the Coulomb potential, A0a ⃗ → −∇ ⃗a . Using the ⃗ aV + e E is an external vector potential, evaluated at r⃗a , and e E electron coordinate r⃗a of the ath electron, one obtains ⃗ 2 Zα ⃗a4 π e π(Zα) ⃗a − π ⃗a ⋅ B Ha = a − + e A0a − (1 + κ) σ + (1 + 2κ) δ (3) (⃗ ra ) 2m ra 2m 8m3 2m2 Zα(1 + 2κ) r⃗a e(1 + 2κ) ⃗a + σa ⋅ (E ⃗a × π ⃗a )] ⃗ ⋅E [∇ ⃗a ⋅ ( 3 × π ⃗a ) − ⃗a − π ⃗a × E + σ 2 4m ra 8m2 e ⃗a , π ⃗a , π ({⃗ ⃗a ⋅ σ ⃗a }) . ⃗a2 } + κ {⃗ + σa ⋅ B πa ⋅ B (12.63) 8m3 The two-particle interaction Hamiltonian is the sum of Eqs. (12.60) and (12.61), Hab =

j i ⎞ j πα rab α i ⎛ δ ij rab α π rab ) − + πb − 2 (1 + 2κ) δ (3) (⃗ a 3 rab 2m2 r r m ⎝ ab ab ⎠ j i j i ⎞ rab rab 2πα α 2 σa σb ⎛ ij 2 (3) ⃗ ⃗ (1 + κ) σ ⋅ σ δ (⃗ r ) + (1 + κ) δ − 3 a b ab 3 3 2 2 3m 4m rab ⎝ rab ⎠ α ⃗a ) ⃗b − σ ⃗b ⋅ r⃗ab × π + [2(1 + κ) (⃗ σa ⋅ r⃗ab × π 3 4m2 rab

−

⃗b − σ ⃗a ⋅ r⃗ab × π ⃗a )] . + (1 + 2κ) (⃗ σb ⋅ r⃗ab × π

(12.64)

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These terms are identified and interpreted as follows. First, we have a plain Coulomb repulsion and a magnetic photon exchange, then, a zitterbewegung term corrected by the anomalous magnetic moment, a magnetic photon exchange, a spin-spin contact interaction, a higher-multipole correction to the spin-spin interaction, and finally, the spin-other-orbit and spin-orbit term. Note that we have kept the canonical ⃗a in the most general form possible. One might argue that the vector momenta π ⃗ ra ) in Eq. (12.32) was absorbed in the photon propagator ⃗a = p⃗a − e A(⃗ potential in π in Eq. (12.37). The idea is that A⃗ could be written as the sum of two terms, one of ⃗a might which describes the quantized field, while the second vector potential in π correspond to additional external fields A⃗a (e.g., constant magnetic fields) which ⃗a . contribute to π 12.3.4

Breit Hamiltonian for Unequal Particles

With the same conventions as used in Sec. 12.3, we start from Eq. (12.46) and use the photon propagator in Coulomb gauge [see Eq. (12.47)]. However, unlike Eq. (12.50), we distinguish, in the timelike component of the current, the anomalous magnetic moment κa and the mass ma , 1 + 2κa ⃗2 ⃗ = 1 + i (1 + 2κa ) σ ⃗ ⋅ (k⃗ × π ⃗) − k , (12.65) ja0 (k) 4ma 8m2a and analogously for a → b. The Hamiltonian will be denoted by Hab instead of Hab . After going through analogous steps as the ones discussed in Sec. 12.3.3, the effective Hamiltonian is calculated as follows, d3 k ⃗ ra −⃗ (00) rb ) 0 ⃗ j 0 (k) ⃗ ei k⋅(⃗ ⃗ Hab = ea eb ∫ D00 (k) jb (−k) a 3 (2π) ⃗ ⃗ d3 k ⎡⎢ ei k⋅(⃗ra −⃗rb ) i (1 + 2κa ) ei k⋅(⃗ra −⃗rb ) ⃗ ⎢ ⃗ ⃗ + = ea eb ∫ σ ⋅ ( k × π ) a a (2π)3 ⎢⎣ 4m2a k⃗ 2 k⃗ 2 ⃗ ⎤ ei k⋅(⃗ra −⃗rb ) i (1 + 2κb ) 1 + 2κa 1 + 2κb i k⋅(⃗ ⃗ ⎥ ⃗b ⋅ (−k⃗ × π ⃗b ) − ( + σ + ) e ra −⃗rb ) ⎥ . 2 2 2 ⎥ 4mb 8ma 8mb k⃗ 2 ⎦ (12.66) We use the convention ea eb αab = , (12.67) 4π and note that, e.g., for a negatively charged orbiting particle of charge e, and a nucleus of charge −Z e, one has ea = e , eb = −Z e , αab = −Zα . (12.68) After a Fourier transformation, one obtains αab αab (1 + 2κa ) (00) ⃗a ⋅ (⃗ ⃗a ) Hab = − σ rab × π 3 rab 4m2a rab +

αab (1 + 2κb ) παab 1 + 2κa 1 + 2κb ⃗b ⋅ (⃗ ⃗b ) − σ rab × π ( + ) δ (3) (⃗ rab ) . 2 3 4mb rab 2 m2a m2b (12.69)

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Hence, (00)

Hab

=

⃗b ⋅ (⃗ ⃗b ) ⃗a ⋅ (⃗ ⃗a ) αab αab σ rab × π σ rab × π + 3 [(1 + 2κb ) − (1 + 2κa ) ] 2 2 rab 4rab mb ma −

παab 1 + 2κa 1 + 2κb ( ) δ (3) (⃗ rab ) . + 2 m2a m2b

(12.70)

The spatial components of the current are given by Eq. (12.53), but with the distinction for the ath particle mass ma and its anomalous magnetic moment κa , ⃗ i (1 + κa ) ⃗ = π ⃗ , ⃗ja (k) + (⃗ σ × k) ma 2ma

(12.71)

and analogously for a → b. The effective Hamiltonian from the spatial elements of the current reads as (ij)

Hab = ea eb ∫

d3 k ⃗ ra −⃗ rb ) j ⃗ j i (k) ⃗ ei k⋅(⃗ ⃗ Dij (k) jb (−k) a 3 (2π)

πai i (1 + κa ) d3 k ⃗ i) ( + (⃗ σ × k) 3 (2π) ma 2ma ⃗ ra −⃗ j ⎡ ei k⋅(⃗ rb ) k i k j ⎤⎥ ⎛ πb i (1 + κb ) ⎢ ⃗ j⎞ . (δ ij − )⎥ × ⎢− + (⃗ σ × (−k)) ⎥ ⎢ 2 2 ⃗ ⃗ 2mb ⎠ k k ⎦ ⎝ mb ⎣

= ea eb ∫

(12.72)

(ij)

Using Eqs. (12.56), (12.57) and (12.58), one can obtain Hab as (ij)

Hab = −

⎛ ri rj ⎞ αab 2π(1 + κa )(1 + κb )αab ⃗a ⋅ σ ⃗b δ (3) (⃗ πai δ ij + a2 b πbj − σ rab ) 2ma mb rab ⎝ rab ⎠ 3ma mb

σi σj ⎛ ri rj ⎞ αab (1 + κa )(1 + κb ) a3 b δ ij − 3 ab2 ab 4ma mb rab ⎝ rab ⎠ αab ⃗a ⋅ r⃗ab × π ⃗b − (1 + κb ) σ ⃗b ⋅ r⃗ab × π ⃗a ] . + 2[(1 + κa ) σ 3 4ma mb rab

+

(00)

We recall the result for Hab (00)

Hab

=

(12.73)

from Eq. (12.70),

⃗b ⋅ (⃗ ⃗b ) ⃗a ⋅ (⃗ ⃗a ) αab αab σ rab × π σ rab × π + 3 [(1 + 2κb ) − (1 + 2κa ) ] 2 2 rab 4rab mb ma −

παab 1 + 2κa 1 + 2κb ( + ) δ (3) (⃗ rab ) . 2 m2a m2b

(12.74) (00)

(ij)

The full interaction Hamiltonian Hab is the sum of Hab and Hab . Because we treat all interactions on the same footing, we do not single out the electron-nucleus interaction as in Eq. (12.63), but rather, the single-particle Hamiltonian is a “free” Hamiltonian for the particle, plus coupling terms, to the ⃗ and B ⃗ fields, and to the external vector potential. It is obtained from external E

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Eq. (12.63) by setting Zα → 0, and using masses and anomalous magnetic moments specific to particle a, Ha =

⃗a2 ⃗4 π π e ⃗a ⃗a ⋅ B − a3 + e A0a − (1 + κa ) σ 2ma 8ma 2ma e ⃗a + σa ⋅ (E ⃗a × π ⃗a )] ⃗ ⋅E ⃗a − π ⃗a × E (1 + 2κa ) [∇ − 8m2a e ⃗a , π ⃗a , π ({⃗ ⃗a2 } + κa {⃗ ⃗a ⋅ σ ⃗a }) . + σa ⋅ B πa ⋅ B 8m3a (00)

The two-particle interaction Hamiltonian is the sum of Hab Hab =

(12.75)

(ij)

and Hab ,

j i ⎞ j παab 1 + 2κa 1 + 2κb (3) rab αab πai ⎛ ij rab αab − δ + π − ( ) δ (⃗ rab ) + 2 rab 2ma mb rab ⎝ rab ⎠ b 2 m2a m2b

−

2παab ⃗a ⋅ σ ⃗b δ (3) (⃗ (1 + κa )(1 + κb ) σ rab ) 3ma mb

+

σi σj ⎛ ri rj ⎞ αab (1 + κa ) (1 + κb ) a3 b δ ij − 3 ab2 ab 4ma mb rab ⎝ rab ⎠

+

αab 2 ⃗a ⋅ r⃗ab × π ⃗b − (1 + κb ) σ ⃗b ⋅ r⃗ab × π ⃗a ) [ ((1 + κa ) σ 3 4 rab ma mb

+ (1 + 2κb )

⃗a ⋅ r⃗ab × π ⃗a ⃗b ⋅ r⃗ab × π ⃗b σ σ − (1 + 2κa ) ]. m2b m2a

(12.76)

This interaction Hamiltonian is valid for arbitrary spin-1/2 particles, with masses ma and mb , and anomalous magnetic moments describes by κa and κb . 12.4 12.4.1

Applications and Generalizations for Many-Particle Systems Application to Many-Electron Systems

In Sec. 12.3.3, we have discussed an approach where we would describe the energy levels of an atom by the sum of “one-particle” Hamiltonians Ha given in Eq. (12.63), which describe the electron–nucleus interaction (in the non-recoil limit), and the electron-electron interaction, which is given by the Hamiltonian (12.64). In Sec. 12.3.4, for a general N -particle system with unequal particles a, b = 1, . . . , N , we add the free one-particle Hamiltonians given in Eq. (12.75) to the interactions given in Eq. (12.76). The latter approach includes the mass dependence of the many-body system exactly. However, one might ask as well where the connection is, i.e., in which limit the two approaches become identical. To this end, we investigate an N -particle system with one nucleus of charge −Z e and N − 1 electrons of charge e (typically, N − 1 = Z for a neutral atom). This system can be described, according to the approach described in Sec. 12.3.4, by “oneparticle” Hamiltonians (for the electrons) which are the sum of the single-particle Hamiltonians Ha given in Eq. (12.75), the electron-nucleus Hamiltonians Han (with

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b = n), and electron-electron interaction Hamiltonians Hab , which are obtained from Eq. (12.76) for ma = mb = m, αab = α, and κa = κb = κ, where κ ≈ α/(2π) describes the anomalous contribution to the electron g factor [see Eqs. (1.11) and (11.38)]. We should mention that the latter approach from Sec. 12.3.4 is, strictly speaking, only valid if the nucleus is a spin-1/2 particle, because we have derived our general Hamiltonian for the interaction of two particles with masses ma and mb and g factors 2(1 + κa ) and 2(1 + κb ) under the assumptions of a spin-1/2 character. It is interesting to verify the consistency of the approach outlined in Sec. 12.3.4 with the approach from Sec. 12.3.3, and with the single-particle formalism outlined in Sec. 11.3.1. First, one verifies, easily, that the interaction Hamiltonian (12.76), for the case ma = mb = m, αab = α, and κa = κb = κ, reproduces the interaction Hamiltonian in Eq. (12.64). The (somewhat trivial) details are left as an exercise to the reader (see Sec. 12.7). The next step is to calculate the limit of infinite nuclear mass (mn → ∞) of the Hamiltonian given in Eq. (12.76), under the conditions ea → e ,

eb → −Ze ,

κa → κ ,

r⃗ab = r⃗an → r⃗a ,

αab → −Zα , mb = mn → ∞ .

(12.77)

One obtains lim Han = −

mn →∞

r⃗a Zα Zα πZα ⃗a )] . + (1 + κ) δ (3) (⃗ ra ) + (1 + 2κ) [⃗ σa ⋅ ( 3 × π ra 2m2 4m2 ra (12.78)

The second term in Eq. (12.78) corresponds to the term −

παab 1 + κa 1 + κb ( + ) δ (3) (⃗ rab ) 2 m2a m2b

(12.79)

in Eq. (12.76). While the third term in Eq. (12.78) corresponds to the spin-orbit coupling term 4αrab3 [((1 + 2κb ) σ⃗b ⋅⃗rmab2×⃗πb − (1 + 2κa ) σ⃗a ⋅⃗rmab2×⃗πa )] in Eq. (12.76), the lata ab b ter taken under the replacements given in Eq. (12.77). Furthermore, the Hamiltonian (12.78) is equal to the sum of the terms H [2] +H [4] +H [5] given in Eqs. (11.63), (11.65) and (11.66), under the kinematic conditions given in Eq. (11.62). The latter fact ramifies the general wisdom that Dirac theory, augmented by the Foldy– Wouthuysen method, constitutes a single-particle theory and cannot, in principle, account for reduced-mass corrections. The new single-particle Hamiltonian Ha is the sum of the single-particle Hamiltonian Ha given in Eq. (12.75), which contains the interactions with the external fields and the electron-nucleus Hamiltonian Han taken from Eq. (12.78), Ha =

⃗a2 ⃗4 π e π ⃗a − e (1 + 2κ) [∇ ⃗a ⃗ ⋅E ⃗a ⋅ B − a3 + e A0a − (1 + κ) σ 2ma 8ma 2m 8m2 ⃗a × π ⃗a )] + e ({⃗ ⃗a , π ⃗a , π ⃗a2 } + κ {⃗ ⃗a − π ⃗a × E ⃗a ⋅ σ ⃗a }) σa ⋅ B + σa ⋅ (E πa ⋅ B 8m3 Zα πZα Zα r⃗a ⃗a )] . − + (1 + κ) δ (3) (⃗ ra ) + (1 + 2κ) [⃗ σa ⋅ ( 3 × π (12.80) ra 2m2 4m2 ra

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This has to be compared to the result obtained for the single-particle Hamiltonian (12.63). A closer inspection reveals that the Hamiltonians (12.63) and (12.80) are identical. Reassuringly, the Hamiltonian given in Eq. (12.63), if one sets the ⃗a → 0, ⃗a → 0, ⃗ B ⃗ π ⃗a → p⃗a ) becomes identiexternal fields equal to zero (A0a → 0, E cal to the Foldy–Wouthuysen Hamiltonian for the Dirac–Coulomb problem given in Eqs. (11.63), (11.65) and (11.66). Hence, we have clarified that the approaches outlined in Secs. 12.3.3 and 12.3.4 are in mutual agreement, in the non-recoil limit of infinite nuclear mass mN → ∞. The approach outlined in Sec. 12.3.4 treats the atomic nucleus on a more equal footing as compared to the approach described in Sec. 12.3. Let us illustrate the advantage of the latter approach by deriving the so-called mass polarization contribution to the spectrum of helium. We have three particles, the electrons labeled as 1 and 2, and the helium nucleus N . We take the nonrelativistic kinetic-energy contributions from the two electrons and the nucleus, in the spirit of the approach taken in Sec. 12.3.4, p⃗ 2 p⃗ 2 p⃗12 + 2 + n , (12.81) 2m 2m 2mn which really is the sum over a = 1, 2 and a = N of the free+field Hamiltonian given ⃗a2 → p⃗2a , assuming a in Eq. (12.75), taking the first term under the replacement π vanishing external vector potential. In the centre-of-mass frame, one has p⃗N = −⃗ p1 − p⃗2 , and thus HNR =

HNR =

p⃗12 p⃗ 2 (−⃗ p1 − p⃗2 )2 p⃗12 p⃗22 p⃗1 ⋅ p⃗2 + 2 + = + + . 2m 2m 2mN 2µ 2µ 2mN

(12.82)

Here, µ = m mN /(m + mN ) is the reduced mass of the electron-nucleus system. The last term in Eq. (12.82) is just the mass-polarization term. 12.4.2

Single-Particle Hamiltonian for Arbitrary Spin

It is very interesting to consider the question of how to generalize the Hamiltonian (12.75) to the case of arbitrary nuclear spin. An example would be an atomic nucleus such as the deuteron, which constitutes a spin-1 particle. Let si be the spin matrices of the appropriate structure. For s = 12 , we can choose si = 21 σ i , where the ⃗ of spin matrices for s = 1 σ i are the Pauli matrices [see Eq. (4.343)]. The vector S reads as follows (see Chap. 5 of Ref. [81]), jk

(Si )

= −i ijk ,

(12.83)

where Si is the spin matrix for the ith Cartesian direction, and its components are labeled as jk. An explicit representation has been given in Eq. (8.340), and we recall it here for convenience,

S

1

⎛0 0 0 ⎞ = ⎜ 0 0 −i ⎟ ⎝0 i 0 ⎠

S

2

⎛0 0 i⎞ = ⎜ 0 0 0⎟ , ⎝ −i 0 0 ⎠

S

3

⎛ 0 −i =⎜i 0 ⎝0 0

0⎞ 0⎟ . 0⎠

(12.84)

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The commutator of the spin-1 matrices has been given in Eq. (8.341) as [ Si , Sj ] = i ijk Sk , where the Einstein summation convention is used for the sum over k = 1, 2, 3 on the right-hand side. The eigenvalues and eigenvectors λ of S3 are manifest in the spherical basis vectors e⃗λ , which fulfill S3 e⃗λ = λ e⃗λ and are given by √ ⎛ −1/√ 2 ⎞ ⎛0⎞ 1 ⎟, ⎜0⎟ , ⃗ e⃗+1 = − √ (ˆ ex + i eˆy ) = ⎜ e = e ˆ = (12.85a) −i/ 2 0 z ⎜ ⎟ 2 ⎝1⎠ ⎝ 0 ⎠ √ ⎛ 1/ √2 ⎞ 1 ⎟ e⃗−1 = √ (ˆ ex − i eˆy ) = ⎜ ⎜ −i/ 2 ⎟ . 2 ⎝ 0 ⎠

(12.85b)

A number of useful formulas can be found in Ref. [16] (see also Chap. 6). We go even one step further and denote a general spin matrix as ⎧ 1 ⎪ ⃗a sa = 1/2 ⎪ ⎪2σ ⃗sa = ⎨ . ⎪ ⃗a ⎪ S s = 1 ⎪ a ⎩ We define the g factor for particle a, denoted as ga , as follows, ga ≡ 2(1 + κa ) .

(12.86)

(12.87)

This definition deserves some comments. One substitutes the expression for the ath particle, in Eq. (12.75) as follows, e ⃗a → −ga ea ⃗sa ⋅ B ⃗a . ⃗a ⋅ B (1 + κa ) σ (12.88) − 2ma 2ma Here, we recall that ea is the physical charge of the ath particle, and ma is its mass. We had included some discussion of this convention around Eqs. (11.81) and (11.82), and has remarked that, with this definition, the g factor of the electron (and, by the way, also that of the positron), comes out as positive (cf. Ref. [330]). We had also remarked that the positive sign of the electron g factor hinges upon the negative sign in the relation that connects the electron’s magnetic moment with the Bohr magneton, as in Eq. (11.83). According to the substitution (12.88), the leading-order interaction of the ath particle’s spin with the magnetic field is given by the Hamiltonian ea ⃗a . ⃗sa ⋅ B (12.89) Ha = −ga 2ma For electrons (and muons), this Hamiltonian is in agreement with the discussion surrounding Eqs. (11.81) and (11.82), and with the usual conventions from the literature. Let us now investigate if the g factor defined according to Eq. (12.87) is in agreement with the usual conventions used for nuclei. To this end, let us first recall the definition of the Bohr magneton µB and the nuclear magneton µN , µB =

∣e∣ , 2m

µN =

∣e∣ , 2mp

(12.90)

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where m is the electron mass, mp is the proton mass, and ∣e∣ is the positron (proton) charge. Let us now assume that a = p is a proton. Then, ea = ∣e∣, and because the proton carries a spin-1/2, the spin-magnetic interaction Hamiltonian becomes Hp = −gp

∣e∣ ⃗a , ⃗sp ⋅ B 2mp

⃗sp =

1 2

⃗p . σ

(12.91)

The usual convention of the nuclear g factor, for a nucleus N , connects the nuclear ⃗N with the nuclear spin operator ⃗sN via the relation magnetic moment µ ⃗N = gN µN ⃗sN , µ

⃗sN = I⃗N ,

(12.92)

where ⃗sN = I⃗N is the nuclear spin operator for the nucleus N . This leads to the spin-magnetic Hamiltonian ⃗ = −gN µN ⃗sN ⋅ B ⃗. HN = −⃗ µN ⋅ B

(12.93)

Here, gN is the nuclear g factor in the usual convention. With the help of Eqs. (12.90) and (12.92), one can easily show that the Hamiltonians HN and Hp given in Eqs. (12.93) and (12.91) are in agreement with the general Hamiltonian Ha given in Eq. (12.89), for a = p being a proton. The situation changes for nuclei other than the proton. Here, the relation (12.92) is still valid in the commonly accepted conventions for the nuclear g factor gN , but the nucleon’s mass is not necessarily equal (or close) to the proton mass. In order to use the general formulas given below for nuclei other than the proton, one needs to equate gN

∣e∣ eN = gN µN = gN , 2mN 2mp

(12.94)

where a = N is a nucleus, gN is the nuclear g factor in the usual conventions, while gN is the g factor according to the definition (12.87), and eN is the nucleon charge. For the neutron, N = n, the nuclear g factor, according to the definition (12.92), reads as gn = −3.826 085 45(90) [60, 61]. The neutron, though, carries a charge en = 0. Hence, in order to apply any of the formulas given below to the neutron, one has to assume an infinitesimal neutron charge en = . The gn factor of the neutron assumes the large value gn = gn (∣e∣/) (mn /mp ), and one keeps the product gn  = gn ∣e∣ (mn /mp ) constant throughout the calculation. In the end, one takes the limit  → 0, keeping the value of the product gn  constant. For the deuteron N = d, the nuclear g factor, according to the definition (12.92), reads as gd = 0.847 438 2338(22) [60,61]. while the g factor defined according to Eq. (12.87) reads as gd = gd (md /mp ) = 1.714 . . . . It is disputable which definition of the g factor is the most appropriate one for the neutron and deuteron, but the above discussion clarifies the potential ambiguities.

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After this detour on the conventions of the g factor, we substitute in Eq. (12.75), for a particle of general spin, as follows, e ⃗a → −ga ea ⃗sa ⋅ B ⃗a , ⃗a ⋅ B − (1 + κa ) σ (12.95a) 2ma 2ma e ⃗a ⃗ ⋅E (1 + 2κa ) ∇ − 8m2a e ⃗a − e ⟨r2 ⟩ ∇ ⃗a , ⃗ ⋅E ⃗ ⋅E → − 2 (ga − 1) δsa ,1/2 ∇ (12.95b) 8ma 6 E a e ⃗a × π ⃗a ) ⃗a − π ⃗a × E − (1 + 2κa )σa ⋅ (E 8m2a e ⃗a × π ⃗a ) . ⃗a − π ⃗a × E (12.95c) → − 2 (ga − 1) ⃗sa ⋅ (E 4mN Here, we use the fact that according to Ref. [362], the so-called zitterbewegung term, ⃗a , vanishes for a spin-1 nucleus. It has been a topic of ⃗ ⋅E proportional to δsa ,1/2 ∇ discussion of whether to include, or exclude, the Darwin–Foldy term in the nuclear radius. Specifically, in Refs. [205, 363], the exclusion of this term from the nuclear radius is advocated, while in Ref. [364], the authors advocate to include this term. If 2 ⟩ = 3/(4m2N ), this term were included, then for a point particle, we would have ⟨rE reproducing the zitterbewegung term which is otherwise only present for a spin1/2 nucleus. However, since this term is absent for a spin-1 particle [362], we should better exclude the Darwin term from the definition of the nuclear radius, in 2 ⟩a is accordance with Refs. [205, 363]. Note that the mean-square charge radius ⟨rE defined as the slope of the Sachs GE form factor [see Eq. (11) of Ref. [205]]. One can also replace, in Eq. (12.75), e ⃗a , π ⃗a , π ({⃗ ⃗a ⋅ σ ⃗a }) ⃗a2 } + κ {⃗ πa ⋅ B σa ⋅ B 8m3a e ⃗a , π ⃗a , π [2 {⃗sa ⋅ B ⃗a ⋅ ⃗sa }] . ⃗a2 } + (ga − 2) {⃗ πa ⋅ B (12.95d) → 8m3a For a particle with an internal structure, there can be an additional Hamiltonian term (12.96) due to a quadrupole moment, where e HQ = − (sia sja )(2) QEa (Ea )i,j , 6

(Ea )i,j ≡

∂(Ea )i , ∣ ∂rj r⃗=⃗ra

(12.96)

and QEa is the electric quadrupole constant of the ath particle. The traceless quadrupole spin tensor reads as follows,

sia sja + sja sia

δ ij ⃗sa ⋅ ⃗sa . (12.97) 2 3 For completeness, we also add an interaction quadratic in the magnetic fields, (sia sja )(2) =

HM = −

−

e2a i j M ij B B [αa δ + γaM (sia sja )(2) ] , 2

(12.98)

where αaM is the magnetic susceptibility of the ath particle, while γaM is the magnetic quadrupole constant.

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The superscript (2) denotes the selection of the (` = 2)-component of the interaction. Our final effective Hamiltonian for an arbitrary-spin particle (interacting with external fields) is obtained as a simplified version of the Hamiltonian given in Eqs. (1)–(4) of Ref. [324]; the Hamiltonian in Ref. [324] also comprises sixth-order terms in the momenta. Our Hamiltonian ⃗2 ⃗4 π π ea ⃗a Ha′ = a + ea A0a − a3 − ga ⃗sa ⋅ B 2ma 8ma 2ma ea ⃗a − ea ⟨r2 ⟩ ∇ ⃗a ⃗ ⋅E ⃗ ⋅E − (ga − 1) δsa ,1/2 ∇ E a 2 8ma 6 ea ⃗a × π ⃗a ) − ea (si sj )(2) QEa (Ea )i ⃗a − π ⃗a × E − (ga − 1) ⃗sa ⋅ (E a a ,j 4m2a 6 ea ⃗a , π ⃗a , π [2 {⃗sa ⋅ B ⃗a ⋅ ⃗sa }] ⃗a2 } + (ga − 2) {⃗ πa ⋅ B + 8m3N e2a i j M ij B B [α δ + γaM (sia sja )(2) ] , 2 a a a still has quite a rich structure, which we will explore in the following. −

12.4.3

(12.99)

Interaction Hamiltonian for General Spin

In addition to the single-particle Hamiltonian (12.99), the generalization of the interaction Hamiltonian given in Eq. (12.76) to particles of general spin is equally interesting. It is obvious that we should replace, in Eq. (12.76), σai → 2 sia , as well as 2(1 + κa ) → ga , to describe the spin vector and g factor of particle a. Using substitutions akin to those applied in Eq. (12.95), we derive the interaction Hamiltonian (12.64) for particles of a general spin quantum number as follows, ′ Hab =

j i ⎛ δ ij rab ⎞ j 2παab rab αab αab (⟨⃗ π − rE2 ⟩a + ⟨⃗ rE2 ⟩b ) δ (3) (⃗ rab ) − πai − 3 rab 2ma mb rab ⎠ b 3 ⎝ rab παab παab − (ga − 1) δsa ,1/2 δ (3) (⃗ rab ) − (gb − 1) δsb ,1/2 δ (3) (⃗ rab ) 2m2a 2m2b

⎛ ⎝

−

2παab ga gb αab ga gb ⃗sa ⋅ ⃗sb δ (3) (⃗ rab ) + 3ma mb 4ma mb r3

+

ga αab gb ⃗sa ⋅ (⃗ ⃗b ) − ⃗sb ⋅ (⃗ ⃗a ) [ rab × π rab × π 3 2 rab ma mb ma mb

+

gb − 1 ga − 1 ⃗sb ⋅ (⃗ ⃗b ) − ⃗sa ⋅ (⃗ ⃗a )] rab × π rab × π m2b m2a

sia δ ij − 3

j i ⎞ j rab rab s 2 rab ⎠ b

j i ij 2 ⎫ ⃗ab ⎪ αab ⎪ i j (2) i j (2) 3 rab rab − δ r [QEa (sa sa ) QEb (sb sb ) ] + ⎬. (12.100) 5 ⎪ 6 rab ⎪ ⎭ This Hamiltonian is not quite equivalent to the ones given in Eq. (3) of Ref. [365] and in Eq. (17) of Ref. [366]. Namely, a closer inspection reveals that the Hamiltonian given in Eq. (3) of Ref. [365] and in Eq. (17) of Ref. [366] uses the radii ⟨⃗ rE2 ⟩a and

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⟨⃗ rE2 ⟩b in the conventions of Ref. [364], i.e., with the Darwin term included in the definition of the nuclear radius. However, the Darwin term actually is excluded from the nuclear radius in atomic-physics conventions, and also, canonically, in the analysis of scattering experiments, as explained in detail in Ref. [205]. The Hamiltonian given in Eq. (12.100) is compatible with a definition of the radii in terms of the derivative of the Sachs GE form factor [see Eq. (11) of Ref. [205]]. This is the accepted definition, and it is implemented in Eq. (12.100). In Eq. (12.100), the electric quadrupole terms are relevant only for spin-1 particles (QE = 0 for spin-1/2). Once more, we reemphasize that the radii of the spin-1/2 particles are defined in terms of the slope of the Sachs GE form factor [see Eq. (11) of Ref. [205]]. 12.5 12.5.1

Application to Two-Body Bound Systems General Aspects

Considerable simplifications of the outlined formalism are possible for two-body systems such as hydrogen. We investigate a system of two spin-1/2 particles, of masses m1 and m2 , without any external fields. Indeed, in some sense, atomic hydrogen constitutes the most imminent application of the formalism outlined in Sec. 12.3.4, in general. The total Hamiltonian is the sum of the single-particle Hamiltonians given in Eq. (12.75), but without coupling to external fields, i.e., A0a → ⃗a → 0, ⃗a → 0, ⃗ B ⃗ π ⃗a → p⃗a , for a = 1, 2, and of the interaction Hamiltonian (12.76), 0, E evaluated for a = 1, b = 2. Also, we assume that e1 = e ,

e2 = −Ze ,

α12 = −Zα ,

r⃗12 = r⃗ ,

(12.101)

where e is the electron charge. Typically, we will choose m1 to be the mass of the lighter (“orbiting”) particle, while m2 is the mass of the heavier particle. Then, H = H1 + H2 + H12 = T1 + T2 ,

(12.102)

and it is useful to group the terms as follows. The term T1 contains the spinindependent relativistic effects, p⃗ 2 p⃗ 4 p⃗ 4 Zα πZα 1 + κ1 1 + κ2 p⃗ 2 } − 13 − 23 + ( + ) δ (3) (⃗ r) T1 = { 1 + 2 − 2m1 2m2 r 8m1 8m2 2 m21 m22 Zα ri rj pi1 (δ ij + 2 ) pj2 , 2m1 m2 r r while T2 contains the spin-dependent terms, ⃗1 ⋅ r⃗ × p⃗1 Zα ⃗2 ⋅ r⃗ × p⃗2 Zα σ σ T2 = + 3 (1 + 2κ1 ) − 3 (1 + 2κ2 ) 2 4r m1 4r m22 Zα [(1 + κ2 )⃗ σ2 ⋅ r⃗ × p⃗1 − (1 + κ1 )⃗ σ1 ⋅ r⃗ × p⃗2 ] + 2m1 m2 r3 +

+

(12.103)

(12.104)

σi σj Zα 8π ri rj ⃗1 ⋅ σ ⃗2 δ (3) (⃗ (1 + κ1 )(1 + κ2 ) [ σ r) − 1 3 2 (δ ij − 3 2 )] . 4m1 m2 3 r r

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Now, with reference to suggestion 1 in Sec. 4.8, as well as Secs. 13.2.2 and 16.2.1, we go into the center-of-mass frame, with p⃗1 = −⃗ p2 = p⃗ and P⃗ = p⃗1 + p⃗2 = ⃗0, and ⃗ introduce the orbital angular momentum L = r⃗ × p⃗, H= {

p⃗ 2 Zα p⃗ 4 p⃗ 4 πZα 1 + κ1 1 + κ2 − }− − + ( + ) δ (3) (⃗ r) 3 3 2µ r 8m1 8m2 2 m21 m22

Zα ri rj Zα 1 + 2κ1 ⃗ + 1 + 2κ2 σ ⃗ ⃗1 ⋅ L ⃗2 ⋅ L) σ pi (δ ij + 2 ) pj + 3 ( 2m1 m2 r r 4r m21 m22 Zα ⃗ + (1 + κ2 )⃗ ⃗ [(1 + κ1 )⃗ + σ1 ⋅ L σ2 ⋅ L] 2m1 m2 r3 −

+

σi σj Zα 8π ri rj ⃗1 ⋅ σ ⃗2 δ (3) (⃗ (1 + κ1 )(1 + κ2 ) [ σ r) − 1 3 2 (δ ij − 3 2 )] . 4m1 m2 3 r r (12.105)

This Hamiltonian can be written as the sum H = HS + Hrel + Hfs + Hhfs ,

(12.106a)

where HS is the Schr¨ odinger Hamiltonian, and the other terms read as follows. The nonrelativistic Schr¨ odinger–Coulomb term HS and the spin-independent relativistic corrections are p⃗ 2 Zα − , (12.106b) 2µ r p⃗ 4 πZα 1 + κ1 1 + κ2 (3) Zα ri rj p⃗ 4 + ( + ) δ (⃗ r) − pi (δ ij + 2 ) pj . Hrel = − 3 − 3 2 2 8m1 8m2 2 m1 m2 2m1 m2 r r (12.106c) HS =

Fine-structure and hyperfine-structure effects are described by the Hamiltonians Hfs and Hhfs , Zα Zα ⃗+ ⃗ (1 + 2κ1 )⃗ σ1 ⋅ L (1 + κ1 )⃗ σ1 ⋅ L 2 3 4 m1 r 2m1 m2 r3 ⃗ ⃗1 ⋅ L m1 Zα σ = [1 + 2κ1 + 2 (1 + κ1 )] , 2 3 m2 4 m1 r ⃗ ⃗2 ⋅ L Zα σ 1 m1 (1 + 2κ2 )] Hhfs = [1 + κ2 + 2 m2 2 m1 m2 r3 Hfs =

+

(12.106d)

σi σj ri rj Zα 8π ⃗1 ⋅ σ ⃗2 δ (3) (⃗ (1 + κ1 )(1 + κ2 ) [ σ r) − 1 3 2 (δ ij − 3 2 )] . 4m1 m2 3 r r (12.106e)

Here, HS is the Schr¨ odinger–Coulomb Hamiltonian (which involves the reduced mass µ = m1 m2 /(m1 + m2 ). In particular, the calculation clarifies one more time that the reduced mass should be used in the evaluation of the spectrum of the Schr¨ odinger–Coulomb Hamiltonian. Moreover, Hrel describes the spin-independent relativistic correction terms (these do not contribute to the fine-structure). The

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⃗1 of the orbiting Hamiltonian Hfs contains all the terms which contain the spin σ particle; these terms contribute to the fine-structure (e.g., of hydrogenic P states, i.e., to the splitting of P1/2 and P3/2 states). Furthermore, Hfs enables us to calculate the reduced-mass dependence of the terms exactly, i.e., to all orders in the mass ratio (yet, we remember that the Breit Hamiltonian is valid only up to order α4 , or, with anomalous-magnetic-moment corrections, one might point out that a part of the order-α5 corrections are also contained in it. Finally, Hhfs contains the ⃗2 ), nuclear-spin-dependent (or “other-particle-spin-dependent”) terms (operator σ and it also includes the complete reduced-mass dependence. 12.5.2

Definition of the Lamb Shift

In order to calculate the Lamb shift L of atomic energy levels in a two-body bound systems, it is necessary to have a precise definition of the effect. This question alone is non-trivial, because in higher orders, even the spin of the nucleus plays a decisive role in calculating the spectrum [362]. Let us write the energy levels of a two-body system as E = µ [f (n, j) − 1] −

µ2 [f (n, j) − 1]2 + L + Ehfs . 2(m1 + m2 )

(12.107)

Here, µ = m1 m2 /(m1 + m2 ) is the reduced mass of the system. This definition was given by Sapirstein and Yennie in Ref. [336] and has been adopted in many later works, e.g., in Eq. (67) of Ref. [173]. It is very instructive (and nontrivial) to investigate the origin of the idea behind Eq. (12.107). The dimensionless Dirac energy f (n, j) determines the energy levels of a twobody bound system, in the non-recoil limit, according to Eq. (8.49e) as −

E = m f (n, j) ,

⎛ ⎞ (Zα)2 √ f (n, j) = 1 + 2 2 2 ⎝ [nr + κ − (Zα) ] ⎠

1 2

nr = n − ∣κ∣ = n − j − 12 .

,

(12.108) (12.109)

Here, nr is the “reduced” principal quantum number. The analytic form of the result (12.107) is motivated as follows: By expansion of f (n, j) in Zα, one obtains the Schr¨ odinger spectrum as follows, − 12

(Zα)2 f (n, j) − 1 ≈ (1 + ) (n − ∣κ∣ + ∣κ∣)2

− 12

(Zα)2 − 1 = (1 + ) n2

−1≈−

(Zα)2 . (12.110) 2 n2

The inclusion of the reduced mass in the Schr¨odinger spectrum has been discussed in Sec. 4.2.2. However, in the context of the Dirac equation, it is actually impossible to include reduced-mass effects from first principles, because the Dirac–Coulomb Hamiltonian (8.9), in contrast to the Breit Hamiltonian, constitutes a one-particle operator. Still, one can use the Breit Hamiltonian given in Eq. (12.106) in order to rewrite the reduced-mass dependence of the energy levels of the two-particle system into a unified formula, as far as possible.

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Let us explore to which extent this is possible. First, we observe that hyperfine effects are described by the radiatively corrected hyperfine Breit Hamiltonian given in Eq. (12.106e) and therefore, are absorbed in Ehfs in Eq. (12.107). Let us write Eqs. (12.106c) and (12.106e) without radiative corrections, i.e., in the limit κ1 → 0, ̃ and κ2 → 0. This leads to the total Hamiltonian H, ̃ = HS + H ̃rel + H ̃fs , H

(12.111a)

where HS is the Schr¨ odinger Hamiltonian [see Eq. (4.1)], and 4 4 Zα ri rj ̃rel = − p⃗ − p⃗ + πZα ( 1 + 1 ) δ (3) (⃗ r) − pi (δ ij + 2 ) pj , H 3 3 2 2 8m1 8m2 2 m1 m2 2m1 m2 r r (12.111b)

⃗ ⃗1 ⋅ L ̃fs = (1 + 2 m1 ) Zα σ H . 2 3 m2 4 m1 r

(12.111c)

One can show that, up to order (Zα)4 , the energy levels of the Hamiltonian (12.111) are given by the formula E = µ [f (n, j) − 1] − +

µ2 [f (n, j) − 1]2 2(m1 + m2 )

(Zα)4 1 1 ( − ) (1 − δ`0 ) , 2n3 m22 j + 1/2 ` + 1/2

(12.112)

including the complete reduced-mass dependence. Again, the formula (12.112) includes all the (Zα)4 terms, as well as a part of the higher-order corrections. The notion behind the decomposition in Eq. (12.112) is that all terms generated by the Breit Hamiltonian (12.111) which respect the (n, j) degeneracy, are still grouped together and written in terms of the dimensionless Dirac energy f (n, j), while the terms in Eq. (12.112) which break the (n, j) degeneracy are known as the Barker– Glover corrections (see Ref. [367]) and read as follows, HBG ∼

1 1 (Zα)4 ( − ) (1 − δ`0 ) , 2n3 m22 j + 1/2 ` + 1/2

(12.113)

and thus are part of the Lamb shift L, according to the definition (12.107). These considerations explain the motivation behind the definition (12.107). 12.5.3

Application to Hydrogenlike Ions

One application which is particularly important in the discussion of the hydrogen spectrum (but actually more general) concerns the contribution of the anomalous magnetic moment of the electron to the Lamb shift, and the reduced-mass dependence of the terms.

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For hydrogenlike ions (and related systems, the discussion actually is more general), we set α ⃗1 → σ ⃗e , ⃗2 → σ ⃗N , m1 = m , m2 = mN , σ σ κ1 = κ ≈ , (12.114) 2π and assume that m ≪ mN . However, we keep the nuclear charge number Z as a variable. Let us isolate, from Hfs defined in Eq. (12.106d), Hfs ∼ [1 + 2κ + 2

⃗ ⃗e ⋅ L m Z ασ (1 + κ)] , 2 3 mN 4m r

(12.115)

those terms which contribute in the order α(Zα)4 . One obtains [5]

⃗ ⃗ ⃗e ⋅ L ⃗e ⋅ L m Zα σ m Zα σ κ) = (1 + )κ 2 3 2 3 mN 4m r mN 2m r ⃗ ⃗ ⃗e ⋅ L ⃗e ⋅ L Zα σ Zα σ m + mN )κ =κ . =( mN m 2 m r3 2 m µ r3

HA ∼ (2κ + 2

(12.116)

We use the relation ⃗ ⃗e ⋅ L Zα σ ⃗ ⟨ Zα ⟩ ⟩ = ⟨⃗ σe ⋅ L⟩ ⟨ 3 r r3 =

[j(j + 1) − `(` + 1) − s(s + 1)] 2(Zα)4 µ3 (1 − δ`0 ) ` (2` + 1) (` + 1) n3

=

(Zα)4 µ3 2 (− ) (1 − δ`0 ) , 3 n κ(2` + 1)

(12.117)

for a hydrogenic state of symmetry n2S+1 Lj . Here, κ is not the anomalous contribution to the electron’s magnetic moment, which, otherwise, we denote as κ ≠ κ [see Eq. (1.9)], but the Dirac quantum number κ = (−1)j+`+1/2 (j + 1/2), defined in Eq. (4.348). From Eqs. (12.116) and (12.117), we can derive the anomalous magnetic moment correction to the Lamb shift, [5]

Eκ = ⟨HA ⟩ = =

2 κ (Zα)4 µ3 (− ) (1 − δ`0 ) 2mµ n3 κ(2` + 1)

1 α µ 2 (Zα)4 m ( ) (− ) (1 − δ`0 ) , π m n3 2κ(2` + 1)

(12.118)

where we approximate κ ≈ α/(2π) [see Eqs. (1.11) and (11.38)]. Here, an important observation is made: The dependence on the reduced mass of the anomalous-magnetic-moment contribution to the self-energy for states with ` ≠ 0 is given by the factor (µ/m)2 , not (µ/m)3 , as it would otherwise be expected for typical contributions to the Lamb shift (see Sec. 11.4). The correct reducedmass dependence of the anomalous-magnetic moment correction to the Lamb shift can only be found within a two-body formalism; this consideration finally explains the replacement made in Eq. (11.137). In Refs. [173, 336], the peculiar reduced-mass dependence is ascribed to “a consequence of the proton’s convection current”. In Ref. [336], we refer to a remark

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on page 569, in between Eqs. (2.4) and (2.5a). How should this remark be interpreted? Let us consider Eqs. (12.104) and (12.116). One realizes that a term ⃗1 ⋅ r⃗× p⃗2 is the origin of the anomalous magnetic-moment correction proportional to σ to the Lamb shift; it leads to the peculiar reduced-mass dependence described by the factor (µ/m)2 . This term, in turn, contains the proton’s momentum operator p⃗2 (as opposed to p⃗1 ), which corresponds to the convection current as opposed to the spin current. In the center-of-mass frame with p⃗1 = −⃗ p2 , of course, the distinction between the electron’s and the proton’s convection current becomes unnecessary. Now, we isolate, from Eq. (12.106c), specifically, from the term πZα 1 + 2κ (3) Hrel ∼ ( ) δ (⃗ r) , (12.119) 2 m2 an anomalous-magnetic moment corrections which contributes to S states, κ [5] r) . (12.120) HB = πZα 2 δ (3) (⃗ m ′ The energy correction induced by Hα5 is κ α 1 µ 3 (Zα)4 m (3) ⟨δ (⃗ r )⟩ = ( ) δ`0 . (12.121) m2 π 2 m n3 For κ = −1 (which is the Dirac angular quantum number for S states), and µ = m (infinite nuclear mass limit), the formulas (12.118) reduce to (12.121); however, for a finite nuclear mass, one needs to distinguish between S and non-S states. Two remarks are in order: (i) In regard to the discussion of the effective Dirac equation (with form factors) from Eq. (11.37), we should mention that there is a slight danger of double counting the anomalous magnetic-moment contributions to the Lamb shift. One needs to realize that the radiatively corrected Breit Hamiltonian describes the same effect as the F2 (0) term in Eq. (11.37). Double counting should be avoided; once the anomalous magnetic moment has been taken into account on the level of the two-particle formalism, one cannot invoke it in addition, on the basis of a single-particle Foldy–Wouthuysen transformation. (ii) One might wonder about the term corresponding to Eq. (12.119), but with the proton mass as opposed to the electron mass in the denominator. This term is present in Eq. (12.105) but it forms part of the nuclear self-energy and is thus omitted here. For hydrogenlike systems, the hyperfine splitting operator from Eq. (12.106e) reads as ⃗ ⃗N ⋅ L Zα σ 1 m (1 + 2κN )] Hhfs = [1 + κN + 2 mN 2 m mN r 3 [5]

Eκ′ = ⟨HB ⟩ = πZα

+

σi σj Zα 8π ri rj ⃗e ⋅ σ ⃗N δ (3) (⃗ (1 + κ)(1 + κN ) [ σ r) − e 3 N (δ ij − 3 2 )] . 4 m mN 3 r r (12.122)

This operator can be brought into a much more familiar form by using the identity ⃗N ⃗N α 1 e2 σ ∣e∣ Z∣e∣ ∣e∣ µ ⃗N = Z (1 + κN ) σ 2Z(1 + κN ) = (1 + κN ) s⃗N = , 2mN 4π 2mN 2 4π 2mN 4π (12.123)

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⃗N /2 is the proton spin operator, and gN is the proton g factor. We shall where s⃗N = σ now explain the reasoning which leads to the identity (12.123) in greater detail. To this end, we first observe that, by definition, the nuclear magnetic moment is related to the nuclear spin operator as follows [see the discussion surrounding Eq. (12.90)], ⃗N = Z(1 + κN ) µ

⃗N ⃗N σ ∣e∣ ∣e∣ σ s⃗N ≡ gN = gN µN . 2mN 2mp 2 2

(12.124)

We had initially calculated the hyperfine Hamiltonian, from the assumption that the nucleus is a point-like spin-1/2 particle, of charge ∣q∣ = Z∣e∣. Thus, a priori, its ⃗ = (1 + κN ) ∣q∣/(2mN )⃗ magnetic moment would be µ sN , according to our definition of the magnetic component of the current in Eq. (12.71). This would be valid for an internally structureless nucleus. Indeed, if the proton or other nucleus were a point-like, spin-1/2 particle of charge Z ∣e∣, then we would have κN = Zκ, and this term, indeed, figures in a properly defined nuclear self-energy for S states, according to Eq. (28) of Ref. [205]. However, nuclear magnetic moments are, by definition, measured in terms of the nuclear magneton µN , which involves the proton mass, and this is the definition adopted in Eq. (12.124). The formula ⃗N σ ⃗N = gN µN , (12.125) µ 2 which measures nuclear magnetic moments in terms of the nuclear spin operator ⃗N /2, also defines the nuclear g factor, which we denote as gN . With these s⃗N = σ definitions, the general hyperfine Hamiltonian (12.106e) for an orbiting electron and ⃗N takes the form a nucleus with magnetic moment vector µ Hhfs =

⃗ ⃗N gN − 1 m L m∣e∣ µ ⋅ [(1 + ) 2 3 4π gN mN m r + (1 + κ)

σe ⋅ r⃗) 4π δ (3) (⃗ r) 1 + κ r⃗ (⃗ ⃗e ⃗e )] . σ + ( −σ 3 m2 2m2 r3 r2

(12.126)

Let us assume that the nucleus is a proton (N = p), and that the quantization axis is the z axis. The matrix element of the z component of the proton’s magnetic moment is gp µN ⟨±∣⃗ µp ⋅ ˆez ∣±⟩ = ± , (12.127) 2 where ∣±⟩ are states with the up or down projections of spin, onto the z axis. Irrespective of the spin orientation, the magnitude of the proton’s magnetic moment amounts to gp µN J ∣⟨±∣⃗ µp ∣±⟩∣ = ≈ 1.4106 × 10−26 , (12.128) 2 T while the nuclear magneton is µN =

∣e∣ J ≈ 5.0507 × 10−27 , 2mp T

(12.129)

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leading to a proton g factor of gp = 2

∣⟨±∣⃗ µp ∣±⟩∣ ≈ 5.58569 . µN

(12.130)

One notes the factor 2, which is somewhat non-obvious. For S states, the Hamiltonian given in Eq. (12.126) can be reduced to only one term (a contact interaction), ⟨Hhfs ⟩ =

4π m∣e∣ δ (3) (⃗ r) ⃗e (1 + κ) ⟨⃗ µN ⋅ σ ⟩ 2 4π 3 m

∣e∣ ⃗e δ (3) (⃗ gN µN ⟨⃗ σN ⋅ σ r)⟩ . (12.131) 6m Now, for S states, the total angular momentum is F⃗ = s⃗e + s⃗N , i.e., the sum of the spin angular momenta, = (1 + κ)

2 ⃗e ⟩ = 4 ⟨⃗ ⟨⃗ σN ⋅ σ sN ⋅ s⃗e ⟩ = 2 (F⃗ 2 − s⃗N − s⃗e2 ) .

(12.132)

For F = 1 states, one has 2 X1 = F⃗ 2 − s⃗N − s⃗e2 = F (F + 1) − sN (sN + 1) − se (se + 1) = 2 −

3 4

−

3 4

=

1 2

. (12.133)

For F = 0 states, one has 2 X2 = F⃗ 2 − s⃗N − s⃗e2 = F (F + 1) − sN (sN + 1) − se (se + 1) = 0 −

3 4

−

3 4

= − 32 . (12.134)

The difference between F = 1 and F = 0 states thus is X1 − X2 = 2. Denoting the hyperfine-structure difference of the sublevels of the ground state of hydrogenlike ions by the symbol ⟨⟨⋅⟩⟩, we have ⃗e ⟩⟩ = 2 (X1 − X2 ) = 4 . ⟨⟨⃗ σN ⋅ σ

(12.135)

Then, ∣e∣ ⃗e δ (3) (⃗ gN µN ⟨⟨⃗ σN ⋅ σ r)⟩⟩ 6m 2∣e∣ gN µN ∣φ(⃗0)∣2 . (12.136) = (1 + κ) 3m This is the familiar result [see Eq. (12) of Ref. [37]]. It includes the leading correction due to the anomalous magnetic moment of the electron. The reduced-mass dependence is captured in the formula ⟨⟨Hhfs ⟩⟩ = (1 + κ)

⟨⟨Hhfs ⟩⟩ = (1 + κ)

e2 (Zαµ)3 gN 3 m mp πn3

= (1 + κ) (

µ 3 4α (Zα)3 m2 ) gN . m 3n3 mp

(12.137)

This formula describes the so-called 21 cm line that one observes in astrophysical spectra; it corresponds to the F = 0 to F = 1 transition among the hyperfine sublevels of the hydrogen ground state. The derivation has been given in great detail, in order to elucidate the origin of all terms, including the anomalous magnetic moment, and the reduced-mass dependence.

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Compound System in a Homogeneous Magnetic Field Power–Zienau Transformation for a Magnetic Field

In Secs. 12.3.3 and 12.3.4, we have set up the formalism for the treatment of a system of particles which interact both with external electromagnetic fields, as well as with each other. The ultimate aim of the endeavor, of course, is to be able to accurately describe the properties of compound systems, consisting of electrons and nuclei. We are now in the position to apply this formalism to the calculation of atomic interactions with external fields. In this, we are inspired by Refs. [365,366,368–370]. The entire setting is a nontrivial problem because one has to separate the interaction of the compound system with the external fields, from the internal interactions. Here, we will concentrate on the case of an external magnetic field, which we assume to be constant and uniform over the scale of the quantum system under study, and analyze its interaction with a compound system, as an example of the interaction of a compound system with an external field. We assume arbitrary spin symmetries of the constituent particles and start from the Hamiltonian ′ H = ∑ Ha′ + ∑ Hab . a

(12.138)

a,b>a

Here, Ha′ is given by Eq. (12.99), in the sense of Eq. (12.86)]. We assume that there ⃗a → 0), ⃗ while we have a uniform magnetic field are no external electric fields (E ′ ⃗a = B). ⃗ Furthermore, H is given by Eq. (12.100). (B ab Our aim is to find a suitable unitary transformation φ, so that the transformed Hamiltonian H = e−iφ H eiφ

(12.139)

is brought into a form suitable for further calculations of the magnetic-field interaction of the compound system, with the external field. Let us go into the center-of-mass frame, with ⃗, ⃗ = ∑ ma r⃗a , ⃗a = r⃗a − R (12.140a) x R a M ma ⃗ q⃗a = p⃗a − P, P⃗ = ∑ p⃗a , (12.140b) M a where M = ∑a ma . The letter q here is reserved for the relative momenta. The covariantly coupled total momentum of the compound system is ⃗ = ∑π ⃗ R)] ⃗ = P⃗ − Q A( ⃗ R) ⃗ , ⃗a = ∑ [⃗ Π pa − ea A( a

a

Q = ∑ ea ,

(12.141)

a

where Q is the total charge. In the following, we assume an exclusively spatially dependent vector potential, restricting the discussion to a uniform external magnetic field. Effects connected with external electric fields like the R¨ontgen term (see Ref. [371]) are excluded from the discussion.

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In the following, it is very important to realize that we shall denote the compo⃗a are labeled xka (we had used different nents of r⃗a as rak , while the components of x conventions at other places in this book). The commutators are easily calculated as follows (see also Sec. 12.7), [xia , qbj ] = [xia , pjb ] = i δ ij (δab −

mb ), M

[Ri , P j ] = i δ ij , [xia , P j ]

= [R

i

, qaj ]

(12.142a) (12.142b)

= 0.

(12.142c)

We now employ a gauge transformation inspired by the Power–Zienau method [176], φ = ∑ ea ∫ a

1 0

⃗ R ⃗ + ux ⃗a ⋅ A( ⃗a ) , du x

(12.143)

⃗ + 1 xi xj Ai (R) ⃗ + 1 xi xj xk Ai (R) ⃗ + ⋯) . = ∑ ea (xia Ai (R) a a ,j 2! 3! a a a ,jk a

(12.144)

Here, the partial derivatives are ⃗ = ∂ Ai ∣ Ai,j (R) . ⃗ ∂rj r⃗=R

(12.145)

⃗ field, one has for general argument r⃗, For a constant B ⃗ r ) = 1 (B ⃗ × r⃗) , A(⃗ 2

1 Ai = i`m B ` rm , 2

1 Ai,j = i`j B ` . 2

(12.146)

For a homogeneous magnetic field, the gauge phase φ can be formulated as follows, ⃗ × r⃗) into the formula for φ, by directly inserting the expression A⃗ = 21 (B 1

1 1 ⃗ R ⃗ + ux ⃗ × (R ⃗ + ux ⃗a ⋅ A( ⃗a ) = ∑ ea ∫ du x ⃗a ⋅ B ⃗a ) du x 2 0 0 a a 1 1 1 ⃗ ×R ⃗ + ∑ ea 1 ∫ du u x ⃗ ×x ⃗a ⋅ B ⃗a ⋅ B ⃗a = ∑ ea ∫ du x 2 0 2 0 a a 1 ⃗ ×R ⃗ + 0 = 1 ∑ ea x ⃗ ×R ⃗. ⃗a ⋅ B ⃗a ⋅ B = ∑ ea x (12.147) 2 2 a a

φ = ∑ ea ∫

Other important results are ∂xic ∂ mb i mb ik = k (rci − ∑ rb ) = δ ik δac − ∑ δ δab k ∂ra ∂ra b M b M ma ik ma = δ ik δac − δ = δ ik (δac − ), M M ∂Ri ∂ mb i mb ik ma = k∑ rb = ∑ δ δab = δ ik . k ∂ra ∂ra b M M b M

(12.148a) (12.148b)

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It is helpful to calculate the commutator [ ∂r∂k , φ], a

∂xic i`m ` m ∑ ec k  B R ∂ra c

m 1 1 ∂ i i`m ` ∂R φ = + e x  B ∑ c c ∂rak 2 2 c ∂rak 1 ma i`m ` m 1 ∂Rm = ∑ ec δ ik (δac − )  B R + ∑ ec xic i`m B ` 2 c M 2 c ∂rak ma ec ik i`m ` m ec i i`m ` mk ma 1 δ  B R + xc  B δ ) = ∑ (ec δ ik δac i`m B ` Rm − 2 c M 2 2 M ea ⃗ ⃗ k Q ma ⃗ ⃗ k 1 ma ⃗ ⃗ k = (B × R) − (B × R) + (D × B) . (12.149) 2 2 M 2 M Here,

⃗ = ∑ ea x ⃗a D

(12.150)

a

is the dipole moment of the charge distribution, measured in reference to its centerof-mass. The Power–Zienau transformation serves to eliminate the vector potentials in favor of the fields (we need to differentiate the summation indices b and c from ⃗a under investigation). One finds for the particle index a for the momentum π −i φ k iφ ′k ⃗ ⃗ ⃗a = −i∇a − ea A(⃗ e πa e = πa where π ra ), ∂ ⃗ + [ ∂ , φ] − ea Ak (⃗ ra )) exp (iφ) = −i ∇ka − ea Ak (⃗ xa + R) k ∂ra ∂rak 1 ⃗ Q ma ⃗ ⃗ k 1 ma ⃗ ⃗ k ⃗ a )k − = pka − ea (B ×x (B × R) + (D × B) 2 2 M 2 M k Q ⃗ ⃗ k 1 ma ⃗ ma k ma ⃗ ⃗a + P }+ [P k − (B × R) ] + [(ea x D) × B] = {pka − M M 2 2 M ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

πa′k = exp (−iφ) (−i

k =qa

k 1 ma ⃗ ⃗ + ma [P k − Q (B ⃗ × R) ⃗ k ] = πk + ma Πk . ⃗a + D) × B] [(ea x a 2 M M 2 M ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ k k ≡Π ≡πa (12.151)

= qak +

We have just established the result ma ⃗ Π, M ma ⃗ ⃗ a = ea x ⃗a + D, D M

⃗a + ⃗a eiφ = π e−i φ π

⃗a = q⃗a + π

1 ⃗ ⃗, Da × B 2

(12.152a) (12.152b)

⃗ has been defined in Eq. (12.150), and q⃗a in Eq. (12.140b). With the help where D of Eq. (12.152), the kinetic energy transforms as follows (see also Sec. 12.7), e−i φ (∑ a

⃗ ⃗2 ⃗2 π ⃗a2 π Π Π ⃗ × B) ⃗ . ) eiφ = ∑ a + + ⋅ (D 2ma 2M M a 2ma

(12.153)

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The following identity has been used, ⃗ = ∑ (⃗ ⃗a ⋅ Π qa + ∑π a

a

1 ma ⃗ ⃗ ⋅Π ⃗ ⃗a + (ea x D) × B) 2 M

⃗ + [ 1 ( ∑ ea x ⃗ ⋅Π ⃗ ⃗a ) × B] = (∑ q⃗a ) ⋅ Π 2 a a ma ⃗ ⃗ ⃗ 1 ⃗ × B) ⃗ ⋅Π ⃗. (D × B) ⋅ Π = (D + ∑ 2 a M

(12.154)

One finds that the Hamiltonian H, defined in Eq. (12.138), of the compound system, interacting with the external magnetic field, can be split into three contributions: (i) the nonrelativistic term H0 , (ii) the relativistic interaction, still in terms of the internal coordinates of the system HR , and (iii) the overall kinetic term HΠ , which describes the interaction of the compound system with the external magnetic field. The terms read as follows [see Eq. (12.139)], H = e−i φ H e−iφ = H0 + HR + HΠ , H0 = ∑ a

⃗a2 π 2ma

HR = ∑ {− a

−

⃗2 ⃗ ea αab ⃗ + Π + Π ⋅ (D ⃗ × B) ⃗ , −∑ ga ⃗sa ⋅ B r 2m 2M M ab a a a>b,b

+ ∑

(12.155a) (12.155b)

⃗a4 π ea ⃗ + (ga − 2) {π ⃗ π ⃗a2 ⃗sa ⋅ B ⃗a ⋅ B, ⃗a ⋅ ⃗sa }] [4 π + 2m3a 8m3a

e2a i j M ij ̃′ . B B [αa δ + γaM (sia sja )(2) ]} + ∑ H ab 2 a>b

(12.155c)

̃′ is obtained from H′ We recall the definition of αab from Eq. (12.67). Here, H ab ab ⃗a . For a spin-1/2 particle, it ⃗a → π given in Eq. (12.100) under the replacement π reduces to the Hamiltonian given in Eq. (12.76). So, we have, specifically, j i ij ̃′ = αab − αab πi ⎛ δ − rab rab ⎞ πj − 2παab (⟨⃗ H rE2 ⟩a + ⟨⃗ rE2 ⟩b ) δ (3) (⃗ rab ) ab 3 rab 2ma mb a ⎝ rab rab 3 ⎠ b παab παab − (ga − 1) δsa ,1/2 δ (3) (⃗ rab ) − (gb − 1) δsb ,1/2 δ (3) (⃗ rab ) 2 2ma 2m2b j i ⎞ j rab rab 2παab ga gb αab ga gb i ⎛ ij ⃗sa ⋅ ⃗sb δ (3) (⃗ rab ) + s δ − 3 s a 2 3 3ma mb 4ma mb r rab ⎠ b ⎝ ⎡ αab ⎢⎢ ga gb ⃗b ) − ⃗a ) ⃗sa ⋅ (⃗ ⃗sb ⋅ (⃗ + 3 ⎢ rab × π rab × π 2 rab ⎢ ma mb ma mb ⎣ ⎤ ⎥ gb − 1 ga − 1 ⃗ ⃗ ⃗sb ⋅ (⃗ ⃗sa ⋅ (⃗ + rab × πb ) − rab × πa )⎥⎥ m2b m2a ⎥ ⎦ j i ij 2 ⃗ab 3 r r αab ab ab − δ r [QEa (sia sja )(2) QEb (sib sjb )(2) ] . (12.156) + 5 6 rab

−

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One obtains the following initial expression for HΠ as defined in Eq. (12.155a) [see Eq. (32) of Ref. [366]] ⃗2 ⎛ ⃗a2 π 1 ⃗4 1 ⃗2 ⃗ ⃗ ⃗ Π αab ⎞ HΠ = − Π − Π Π ⋅ ( D × B) − + ∑ ∑ 3 3 2 8M 2M 2M ⎝ a 2ma a>b,b rab ⎠ j i ⎞ αab rab rab 1 Πi Πj ⎛ πia πja ⃗ ⃗ ⋅ Π) ⃗ 2 (π + −∑ π ∑ ∑ 3 2M 2 ⎝ a 2ma a>b,b rab ⎠ a 2M m2a a a ea ⃗ 2 ⃗sa ⋅ B ⃗ + (ga − 2) (Π ⃗ ⋅ B) ⃗ (Π ⃗ ⋅ ⃗sa )] + ∑ ea [2Π +∑ 2m 2 4M a a 4M ma a ⃗ sa ⋅ B) ⃗ + (ga − 2)(Π ⃗ ⋅ B)( ⃗ π ⃗ ⋅ ⃗sa )(π ⃗ ⃗a ⋅ Π)(⃗ ⃗a ⋅ ⃗sa ) + (ga − 2) (Π ⃗a ⋅ B)] × [4(π ⎤ ⎡ j i ij ⎥ ⎢ 1 ⃗ − 1 Πi ⎛ δ + rab rab ⎞ πj ⎥ . (⃗ ⃗ s ⋅ r × Π) + ∑ αab ⎢⎢ a ab b⎥ 3 3 2M mb rab ⎠ ⎥ ⎝ rab ⎢ 2M ma rab a>b,b ⎦ ⎣ (12.157) However, after several unitary transformations, similar to those employed in Refs. [365, 366], one may simplify HΠ to read ⃗2 ⎛ ⃗4 ⃗2a π Π αab ⎞ Π − + ∑ HΠ = − ∑ 8M 3 2M 2 ⎝ a 2ma a>b,b rab ⎠

−

gJ − 2 ⃗ ⃗ ⃗ ⃗ Q ⃗2 ⃗ ⃗ (J ⋅ Π) (B ⋅ Π)] . (12.158) [Π (J ⋅ B) + 3 2M 2 The angular momenta are defined as follows, Q ea ⃗ (`a + ga ⃗sa )] , ⃗a × q⃗a . (12.159) gJ J⃗ = ∑ [ `⃗a = x J⃗ = ∑(`⃗a + ⃗sa ) , M m a a a Here, J⃗ describes the sum of the orbital and spin angular momentum of the compound system. A closer inspection implies that our result in Eq. (12.158) and the result given in Eq. (33) of Ref. [366], are in agreement. Based on Eqs. (12.152) and (12.155b), and a few transformations (see Sec. 12.7), we can write the nonrelativistic Hamiltonian, inclusive of the magnetic interaction, in a frame of reference ⃗ = ⃗0, as follows, ⃗ and R with P⃗ = 0, 2 p⃗ αab ea ⃗ ⃗ + ∑ 1 (D ⃗ 2 . (12.160) ⃗ a × B) H0 = ∑ a + ∑ −∑ (`a + ga ⃗sa ) ⋅ B 2m r 2m 8 m a a a a a a a>b,b ab On the occasion, we recall Eqs. (12.67) and (12.152b). The total nonrelativistic magnetic interaction terms of the compound system are thus given by ea ⃗ ⃗ + ∑ 1 (D ⃗ 2. ⃗ a × B) (`a + ga ⃗sa ) ⋅ B (12.161) HM = − ∑ 2m 8 m a a a a The term quadratic in the magnetic field describes the magnetic susceptibility. e2a ⃗ 2 + 3 (D ⃗ × B) ⃗ 2 = ∑ 1 (D ⃗ 2. ⃗ a × B) (⃗ xa × B) (12.162) ∑ 8M a 8 ma a 8 ma The term ea ⃗ ⃗ = − Q gJ J⃗ ⋅ B ⃗ Hm = − ∑ (`a + ga ⃗sa ) ⋅ B (12.163) 2m 2M a a gives the magnetic moment including the finite-nuclear-mass corrections [we appeal to Eq. (12.159)]. +

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12.6.2

Reduced-Mass Corrections to the Zeeman Effect

We now attempt to make contact with the literature. While the result given for HM in Eq. (12.161) is entirely nonrelativistic, it incorporates the reduced-mass correction to the magnetic interaction, in the compound (many-particle) system. It is instructive to compare our result to Ref. [368], where an analogous calculation is carried out, but formulated in terms of relative coordinates of the electrons (with respect to the nucleus) and the nucleus is assumed to be spinless. We thus investigate the magnetic moment term (12.163). We assume an N -particle system, with the particles numbered as a = 1, . . . , N , and the case a = N represents the (spinless) nucleus. One writes the relative coordinates ∂ , (12.164a) r⃗ab = r⃗a − r⃗b , p⃗ab = −i ∂ r⃗ab ⃗ ′ = ∑′ r⃗aN × p⃗aN , L a

′ S⃗′ = ∑ ⃗sa , a

′

N −1

∑ ≡ ∑, a

(12.164b)

a=1

where r⃗N is the coordinate of the atomic nucleus, and the summation index a runs over the electrons (a = 1, . . . , N − 1). One would like to include reduced-mass corrections of first order in m/M . We start from the orbital interaction terms in Eq. (12.163), which, for the spinless case under discussion, we denote as ea ea ⃗ ⃗ (1) ⃗, (⃗ xa × q⃗a ) ⋅ B (12.165a) `a ⋅ B = − ∑ Hm = −∑ a 2ma a 2ma (1) ′(1) (1),N Hm = Hm + Hm ,

(12.165b) ⃗ ⃗ ⃗a × q⃗a has been defined in Eq. (12.159), and x ⃗a = r⃗a − R according to where `a = x Eq. (12.140). Furthermore, the orbital coupling term (12.165a) will be split into an (1),N ′(1) electron term Hm and a nucleus term Hm , to be specified below. The center-of-mass coordinate of the system is written as ⃗ = ∑′ m r⃗a + mN r⃗N , R (12.166) M a M where the primed sum is over the electron coordinates, m is the electron mass, and M is the total mass of the system, which is assumed to be neutral (the total charge is Q = 0). One then goes into a frame with [see Eqs. (12.140a) and (12.140b)] ⃗ = 0⃗ , ⃗, ⃗a = r⃗a = r⃗aN + r⃗N , (12.167) R P⃗ = ∑ p⃗a = 0 q⃗a = p⃗a − P⃗ = p⃗a , x a

where a = N denotes the nucleus, and r⃗aN = r⃗a − r⃗N . ′(1) Let us concentrate on the term Hm that is exclusively due to the electrons. One can write it as follows, mN ′ e ′ m ′(1) ⃗ Hm = −∑ [⃗ ra − ∑ r⃗b − r⃗N ] × p⃗a ⋅ B M a 2m b M ′

e mN ′ m ′ m ⃗, [⃗ raN − ∑ r⃗bN − ∑ r⃗N + (1 − ) r⃗N ] × p⃗a ⋅ B 2m M M M b b

′

e ′ m ⃗, [⃗ raN − ∑ r⃗bN ] × p⃗a ⋅ B 2m M b

= −∑ a

= −∑ a

(12.168)

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where we have used the relations r⃗b = r⃗bN + r⃗N and M = mN + ∑ b m = mN + Z m. ⃗ = 0⃗ [see Eq. (12.167)], the nucleus term is In the frame with P⃗ = ⃗0 and R eN (1),N ⃗ × p⃗N ⋅ B ⃗ = − eN (⃗ ⃗ (⃗ Hm = − rN − R) rN × p⃗N ) ⋅ B 2mN 2mN 2 eN ′ m ⃗ = eN ( m ) ∑′ (⃗ ⃗ . (12.169) (− ∑ r⃗b × p⃗N ) ⋅ B = − rb × p⃗N ) ⋅ B 2M 2m M b M b This term is of relative order (m/M )2 in comparison to the leading term and can thus be neglected. Finally, one has, with reference to Eq. (12.165b), (1) ′(1) (1),N Hm = Hm + Hm = −∑ a

′

e ′ m ⃗. (⃗ raN − ∑ r⃗bN ) × p⃗a ⋅ B 2m b M

(12.170)

Under the variable change r⃗a → r⃗aN = r⃗a − r⃗N ,

r⃗N → r⃗N ,

(12.171)

the nuclear coordinate only acts as a constant shift, and one has j j ∂rN ∂ ∂ ∂ ∂ ∂ ′ ∂rbN ′ = + = ∑ δ ij δab j = i . ∑ j j i i i ∂ra ∂ra ∂rbN ∂ra ∂rN ∂rbN ∂raN b b

(12.172)

Hence, we can write e ′ m ⃗ [(⃗ raN − ∑ r⃗bN ) × p⃗aN ] ⋅ B a 2m b M m e ′ ⃗ + e ∑′ (⃗ ⃗. ) (⃗ raN × p⃗aN ) ⋅ B rbN × p⃗aN ) ⋅ B = − ∑ (1 − 2m a M 2M a≠b,b ′

(1) Hm = −∑

Under the addition of the spin term for the electrons and the nucleus, ea (2) ⃗, ⃗sa ⋅ B Hm = −ga ∑ 2m a a

(12.173)

(12.174)

and appealing to the definitions in Eq. (12.164b), one obtains for the interaction given in Eq. (12.163), ea ⃗ (1) (2) ⃗ Hm = Hm + Hm = −∑ (`a + ga ⃗sa ) ⋅ B (12.175) 2m a a e m ⃗′ ⃗ − gN eN sN ⋅ B ⃗. ⃗ + e ∑ (⃗ [(1 − ) L + gS S⃗′ ] ⋅ B raN × p⃗bN ) ⋅ B = − 2m M 2M a≠b,b 2mN The gS = 2(1 + κ) factor is the spin g factor for the electrons in the system. We have ⃗ ′ = ∑′ r⃗aN × p⃗aN and S⃗′ = ∑′ ⃗sa outlined in Eq. (12.164). used the conventions L a a Our result, given in Eq. (12.175), generalizes a result given in the last (unnumbered) equation on page 1803 of Ref. [368], to an arbitrary nuclear spin, and clarifies that the summations over i ≠ j in the last (unnumbered) equation on page 1803 of Ref. [368], in the conventions used in Ref. [368], should be understood as summations over i and j, where i runs over all values i > j. The result for the reduced-mass correction (the reduction factor 1 − m/M ) reflects the fact that the orbital momentum of the nucleus partially compensates that of the electrons.

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It is interesting to remark that the mixed term proportional to (⃗ rbN × p⃗aN ) in HM leads to a second-order energy shift ∆E = (

′ e 2 ⎛ ′ ⃗ ⎞⟩ , (12.176) ⃗ ⎞ ( 1 ) ⎛ ∑′ (⃗ raN × p⃗bN ) ⋅ B ) ⟨ ∑ (⃗ raN × p⃗bN ) ⋅ B 2M ⎝a≠b,b ⎠ E − H ⎝a≠b,b ⎠

and contributes to the magnetic susceptibility. However, it is of second order in the electron-to-nucleus mass ratio. From the sum of H0 given in Eq. (12.155b) and HR given in Eq. (12.155c), one ⃗ = 0) the following result for the interaction Hamiltomay extract (still setting Π nian HM of the compound system with the magnetic field, including relativistic corrections. One notes that in a weak magnetic field, it is possible to ignore terms ⃗ One finds a generalization of a result reported in of Ref. [369], which quadratic in B. was originally derived under the assumption of a spin-1/2 nucleus (see Ref. [366]), ea ⃗ ⃗ + ∑ 1 (D ⃗ 2 ⃗ a × B) (`a + ga ⃗sa ) ⋅ B a 8 ma a 2ma 1 ⃗ + 2ea q⃗2 ⃗sa ⋅ B ⃗ + ea (ga − 2) (⃗ ⃗ ⃗ a ⋅ (⃗ [⃗ +∑ qa2 D qa × B) qa ⋅ ⃗sa ) (⃗ qa ⋅ B)] a 3 a 4ma ⎡ j i ⎢ ⎞ ⃗ rab αab i ⎛ δ ij rab ⃗ j (Db × B) qa + + ∑ ⎢⎢− 3 4ma mb ⎝ rab rab ⎠ a≠b,b ⎢ ⎣ ga αab ⃗ . ⃗ − αab ga − 1 (⃗sa × r⃗ab ) ⋅ (D ⃗ a × B)] ⃗ b × B) (⃗sa × r⃗ab ) ⋅ (D + 3 3 4rab ma mb 4rab m2a (12.177)

HM = − ∑

⃗ a of particle a had been The reduced-mass-corrected dipole moment operator D defined in Eq. (12.152b).

12.6.3

Nuclear Magnetic Shielding

It is instructive to consider, in Eq. (12.177), the term a = N which describes the atomic nucleus. Indeed, nuclear magnetic resonance (NMR) spectroscopy investigates the spin-flip frequency of the nucleus, as modified by its environment, i.e., by atomic or molecular electrons. It is also known that NMR spectroscopy can be used as a diagnostic tool for the environmental conditions of an atom, e.g., within solvants, or in a chemical compound [372–374]. Hence, it is imperative to formulate an accurate theoretical description of the effect of the surrounding electrons on the nuclear magnetic moment. The Hamiltonian (12.177) provides for a convenient basis to study this effect. One sets a = N in Eq. (12.177) and isolates the nucleus term, including the nonrelativistic spin coupling. After an isolation of the terms which depend on the nuclear

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spin ⃗sN , one may write the interaction Hamiltonian as follows, eN (1) ⃗ + eN [2 q⃗2 ⃗sN ⋅ B ⃗ + (gN − 2) (⃗ ⃗ HN = − gN ⃗sN ⋅ B qN ⋅ ⃗sN ) (⃗ qN ⋅ B)] N 2mN 4m3N +∑

′

b

gN eN eb ⃗sN × r⃗N b ⃗ b × B) ⃗ − gN − 1 (D ⃗ N × B)] ⃗ . (D ⋅[ 3 4π 4rN m m m2N N b b

(12.178)

There is also a second-order energy shift which generates, surprisingly, a term linear (not quadratic) in the magnetic field. The first of these is the orbital coupling term ea ⃗ ⃗ (2) HN = − ∑ `a ⋅ B (12.179) a 2ma from Eq. (12.155c). We recall that according to Eq. (12.152), one has ⃗a = q⃗a + π

1 ⃗ ⃗. Da × B 2

(12.180)

⃗ in the ⃗a × B ⃗a → 21 D The last term in Eq. (12.178) is obtained by the replacement π corresponding term in Eq. (12.177). However, there is another term generated if we ⃗a → q⃗a in Eq. (12.177), and it reads replace π (3)

HN = ∑ b

′

eN eb ⃗sN × r⃗N b q⃗N q⃗b ⋅ [gN − (gN − 1) ]. 3 4π 2mN rN m m b N b

(12.181) (2)

(3)

The second-order energy shift generated by the combined action of HN and HN ⃗ is proportional to ⃗sN ⋅ B. Combining all terms, one can derive the so-called shielding constant σ for the interaction of the nuclear magnetic moment with the external, homogenous magnetic field, as follows, ⃗ (1 − σ) . ∆E = −gN µN ⟨⃗sN ⋅ B⟩

(12.182)

Here 1 − σ is an “attenuation factor” for the interaction of the nucleus with the external magnetic field, due to the orbiting electrons, and is commonly referred to as the shielding constant in the chemical literature [375], because it describes the way in which the nuclear spin is “shielded” by the atomic electrons. There is a certain caveat, though, in our derivation up to this point. Namely, our derivation of the magnetic interaction terms, starting from Eq. (12.139), has ⃗ field generates the only been valid only under the assumption that the external B nonvanishing vector potential in the physical system. A closer inspection reveals that in doing so, we risk overlooking the dominant contribution to the nuclear magnetic shielding. The nuclear magnetic moment itself [see Eq. (12.125)] ⃗N = gN µN ⃗sN µ

(12.183)

generates, at the point r⃗a of the ath electron, a vector potential ⃗N × r⃗aN µ , A⃗I (⃗ ra ) = 3 4πraN

(12.184)

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which exists in addition to the vector potential [see Eq. (12.146)] 1 ⃗ A⃗ext (⃗ ra ) = (B × r⃗aN ) , 2

(12.185)

⃗ field. Here, one might wonder why one which describes the constant external B uses the argument r⃗aN for the vector potential describing the constant external magnetic field, and not r⃗a . The answer to this question is subtle. One can show that the only consistent way to define the vector potential corresponding to the constant external magnetic field is to adjust the origin of the coordinate system so ⃗ × (⃗ ra − r⃗0 )⟩ where ⟨⃗ ra − r⃗0 ⟩ = ⃗0 in an electronic state. The latter that A⃗ext (⃗ ra ) = 12 (B condition is fulfilled for r⃗0 = r⃗N . From the seagull term A⃗2 /(2ma ) in the original Hamiltonian (12.75) [see also Eqs. (12.99)], one has the mixed term ′

HS = ∑ e2a a

(A⃗I (⃗ ra ) + A⃗E (⃗ ra ))2 A⃗I (⃗ ra ) ⋅ A⃗E (⃗ ra ) ′ → ∑ e2a , 2ma m a a

(12.186)

one obtains, for S states, ′

e2a gN µN 1 ⃗ × r⃗aN )⟩ ⟨ 3 (⃗sN × r⃗aN ) ⋅ (B 4π 2ma raN

′

e2a gN µN 1 ⃗ r⃗ 2 − (⃗sN ⋅ r⃗aN )(B ⃗ ⋅ r⃗aN )]⟩ ⟨ 3 [(⃗sN ⋅ B) aN 4π 2ma raN

⟨HS ⟩ = ∑ a

=∑ a

′

= (∑ a

e2a 1 1 ⃗ , ⟨ ⟩) gN µN ⟨⃗sN ⋅ B⟩ 4π 3ma raN

(12.187)

implying that (in leading order) σ= ∑ a

′

e2a 1 1 ⟨ ⟩. 4π 3ma raN

(12.188)

Let us now consider S states in an atomic system with ea = e (electron charge) for all orbiting particles (hence, e2a = 4πα for a = 1, . . . , N − 1), and a nucleus a = N with mass mN . The general result for the shielding constant σ can be expressed, in relatively compact form, as (see Ref. [366]) σ=

2 ⟨⃗ ⟩ 2 + 3 gN 1 pN α gN − 1 1 ′ 1 ⟨∑ ⟩+ ( − ) 3me a raN mN 6gN mN 3 gN Zme

+

′ α 1 r⃗c p⃗c gN − 1 p⃗N ⟨ ∑ (⃗ ra × p⃗b ) ( ) ∑ [ 3 ×( − )]⟩ . 3mN a,b≠N,a≠b E − H c≠N rc me gN mN

(12.189) ′

⃗ The first term on This formula is valid in a system with p⃗N = − ∑ a p⃗a , and P⃗ = 0. the right-hand side of Eq. (12.189) is precisely Eq. (12.188). In deriving Eq. (12.189), one also notes the importance of second-order shifts generated by the seagull Hamiltonian HS and the Breit Hamiltonian (12.75) for the electrons.

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Further Thoughts

Here are some suggestions for further thought. (1) Coulomb Potential and Momentum Scales. Show that the main integration region for expectation values of the Coulomb interaction in Schr¨odinger– Coulomb eigenstates is mediated by virtual photons with exchange momentum ⃗ ∼ (Zα)m. ∣k∣ Hint: Consider the expectation value of the Coulomb potential, Zα ) φ(⃗ r ) ∼ α2 m , (12.190) ⟨V ⟩ ∼ ∫ d3 r φ∗ (⃗ r) (− r where r ∼ 1/(Zαm) for distances of the order of the Bohr radius. Furthermore, consider the normalization integral 3 r)∣2 = 1 , ∫ d r ∣φ(⃗

(12.191)

which, since ∫ d3 r ∼ 1/(Zαm)3 , and ∫ d3 k ∼ (Zαm)3 , implies that ∣φ(⃗ r)∣2 ∼ 3 2 −3 ⃗ (Zα) , ∣φ(k)∣ ∼ (Zα) . In momentum space, the expectation value of the Coulomb potential can be formulated as ⃗ (− 4πZα ) φ(k⃗′ ) ∼ (Zα)2 m . ⟨V ⟩ ∼ ∫ d3 k ∫ d3 k ′ φ∗ (k) (12.192) (k⃗ − k⃗′ )2

(2)

(3)

(4)

(5)

Counting, now, the powers of Zα, one has 6 powers from the d3 k and d3 k ′ integrations, and three inverse powers from the wave functions, and one explicit Zα, result in a factor (Zα)4 . These powers, divided by (k⃗ − k⃗′ )2 , should reproduce the two powers of α on the right-hand side of Eq. (12.192). Show that this consideration justifies the order-of-magnitude estimate ∣⃗ q ∣ = ∣k⃗ − k⃗′ ∣ ∼ Zα. Difficult Integral. Fill in the missing intermediate steps in the derivation (12.57). Hint: Why can you ignore the terms k i k j /k⃗2 in the Coulombgauge photon propagator? Think about the properties of the  tensor. Another Difficult Integral. Fill in the missing intermediate steps in the derivation (12.56). Hint: Extract the terms proportional to a Dirac-δ function using Gauss’s theorem. Interaction Hamiltonians. Verify that the interaction Hamiltonian from Eq. (12.76), for the case ma = mb = m, αab = α, and κa = κb = κ, reproduces the interaction Hamiltonian in Eq. (12.64). Nuclear Size Correction in Two-Body Systems. Explain why the replacement 1 2Zα 3 3 πZα 1 H∼ ( 2 + 2 ) δ (3) (⃗ r) ∼ ( 2+ + ⟨r2 ⟩N ) πδ (3) (⃗ r) 2 m1 m2 3 4m1 4m22 (12.193) in Eq. (12.111) is a good description for the effect of an extended nucleus. Clarify the role of the so-called Darwin term in the definition of the nuclearsize effect. Consult and compare Refs. [205, 364, 376].

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(6) Breit Hamiltonian and Muonic Hydrogen. Study the application of the radiatively corrected Breit Hamiltonian from Eq. (12.106) to muonic hydrogen, where m1 = mµ and m2 = mp . Verify the correctness of the treatment of the anomalous-magnetic-moment correction to the Lamb shift in the literature, as discussed in Refs. [377] and [205]. Explore the inclusion of proton radius effects into the radiatively corrected Breit Hamiltonian according to Eq. (7) of Ref. [205]. Study the atomic-physics conventions for the definition of the proton radius, and the role of the so-called Darwin term and its relation to the terms H∼

(7)

(8)

(9)

(10)

(11)

πZα 1 + κ1 1 + κ2 ( + ) δ (3) (⃗ r) 2 m21 m22

(12.194)

in Eq. (12.106). The proton radius excludes all effects which would be ascribed to the proton if it were a point-like spin-1/2 particle. Explain why the anomalous-magnetic-moment term with κ = α/(2π) should be included in the definition of the nuclear self-energy for S states, but not for P states, in muonic hydrogen. Consult Refs. [205, 376]. Numerical Data for Muonic Hydrogen. Study once more the application of the radiatively corrected Breit Hamiltonian from Eq. (12.106) to muonic hydrogen, i.e., set m1 = mµ and m2 = mp . Derive the frequencies for the finestructure, and hyperfine structure frequencies, of muonic hydrogen. In partic̵ in order to convert all formulas to ular, supply the missing prefactors c and h the SI mksA unit system. Compare your results with Refs. [203,204,378–380]. Breit Hamiltonian and Positronium. Apply the two-body Breit Hamiltonian (12.106) to positronium; so m1 = m2 = m and κ1 = κ2 = κ. Derive the positronium hyperfine splitting to lowest order, with a radiative correction due to the anomalous magnetic moment. Study the literature (e.g., Chap. 85 of Ref. [3]) in order to find a proper way to include the annihilation diagram. Hydrogen Hyperfine Transition. Convince yourself that the formula given in Eq. (12.137) describes the transition wavelength among the hyperfine sublevels of the hydrogen ground state, including the reduced-mass dependence. Furthermore, calculate the theoretical prediction with and without the radiative correction of order α, and compare with the terms given in Eq. (7) of Ref. [39], and identify the term a00 in the conventions used in Ref. [39]. Breit Hamiltonian and Massive Photon. Generalize the Breit Hamiltonian to an interaction with a massive photon. Thus, derive the relativistic corrections to the vacuum-polarization energy correction in muonic hydrogen. Consult Refs. [377, 381]. Hyperfine Structure and Nuclear Spin. Generalize the hyperfine splitting Hamiltonian given in Eq. (12.136) to nuclei with spin I = 1, and I = 3/2. Derive, in leading order, the hyperfine frequencies of low-lying states of deuterium, and muonic deuterium.

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(12) Operators for External-Field Interactions. Show the commutator relations in Eq. (12.142). Hint: Consider the following transformations, mb j mb i j P ] = [xia , pjb ] − [x , P ] M M a mc i j mc ij = [xia , pjb ] = [rai − ∑ rc , pb ] = iδ ij δab − i ∑ δ δbc M c c M mb = iδ ij (δab − ). (12.195) M

[xia , qbj ] = [xia , pjb −

Verify that this consideration establishes that [xia , pjb ] = iδ ij (δab −

mb ). M

(12.196)

Then, [xia , P j ] = ∑[xia , pjb ] = iδ ij ∑ (δab − b

b

mb ) = iδ ij (1 − 1) = 0 . M

(12.197)

Show the rest of the commutator relations in Eq. (12.142) in a similar fashion. (13) Transformation of the Kinetic Term. Show Eq. (12.153). Hint: Maybe, you want to consider −i φ

e

2 a ⃗ ⃗a + m (π Π) ⃗a2 π M iφ (∑ )e = ∑ 2ma a 2ma a

=∑

⃗2a π ma ⃗ 2 1 ⃗ +Π ⃗ ⋅π ⃗a ) ⃗ ⋅Π +∑ Π + ∑ (π 2ma a 2M 2 2M a a

=∑

⃗2 ⃗ ⃗2a π Π Π ⃗ × B) ⃗ . + + ⋅ (D 2ma 2M M

a

a

(12.198)

(14) Nonrelativistic Hamiltonian with Magnetic Coupling. Consider the steps which lead from Eq. (12.155b) to Eq. (12.160). Hint: You might proceed as follows. Profiting from the fact that in our frame of reference, we have ∑a p⃗a = ⃗0, start from the relation H0 = ∑ a

e2 p⃗a2 ⃗ 2 + ∑ ma (D ⃗ × B) ⃗ 2 + ∑ a2 (⃗ xa × B) 2 2ma a 8ma a 8M ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ =T1

ea ⃗ ⋅ p⃗a + ∑ 1 {⃗ ⃗ × B)} ⃗ +∑ (⃗ xa × B) pa , (D a 2ma a 4M ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ =0

ea ⃗ ⋅ (D ⃗ × B) ⃗ + ∑ αab − ∑ ea ga ⃗sa ⋅ B ⃗, (⃗ +∑ xa × B) a 2ma a 4M a>b,b rab ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ =T2

(12.199)

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where we implicitly define the terms T1 and T2 . A very easy transformation should lead to the following expression, H0 = ∑ a

e2 p⃗a2 ⃗ 2 + 3 (D ⃗ × B) ⃗ 2 + ∑ a2 (⃗ xa × B) 2ma a 8ma 8M ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ =T1 +T2

ea ea ⃗ ⃗ ea eb ⃗. −∑ ga ⃗sa ⋅ B −∑ `a ⋅ B + ∑ 2m 4π r 2m a ab a a a a>b

(12.200)

Finally, one should obtain p⃗a2 e2 ⃗ 2 + 3 (D ⃗ × B) ⃗ 2 + ∑ a2 (⃗ xa × B) 2m 8m 8M a a a a ea ⃗ ⃗ ea eb ea ⃗. `a ⋅ B + ∑ −∑ ga ⃗sa ⋅ B −∑ a 2ma a 2ma a>b 4π rab

H0 = ∑

(12.201)

On the basis of the identity ∑ a

e2a ⃗ × B) ⃗ 2 = ∑ 1 (D ⃗ 2, ⃗ 2 + 3 (D ⃗ a × B) (⃗ xa × B) 8 ma 8M a 8 ma

(12.202)

this expression should easily be shown to be equivalent to the Hamiltonian given in Eq. (12.160). (15) Magnetic Shielding. Study in some further detail the theory of magnetic shielding of the nucleus on the basis of Refs. [369,382–385]. Attempt to derive at least the leading logarithmic QED correction, inspired by Refs. [375, 386, 387]. (16) Electromagnetic Properties of Diatomic Molecules. Attempt to extend the analysis of Sec. 12.6 to diatomic molecules. Be inspired by Refs. [384,385].

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Chapter 13

Fully Correlated Basis Sets and Helium

13.1

Overview

Helium is the simplest atom with more than one electron. It is a quantum threebody system for which no exact solutions are known, not even on the level of nonrelativistic quantum mechanics. The lack of exact solutions, however, does not prevent an accurate discussion of the spectrum of the helium atom, including the relativistic corrections which are so decisive for an accurate understanding of the spectrum of the system. We proceed as follows. In Sec. 13.2, we review basic ingredients in the theoretical description of many-electron systems, starting from the spin wave functions in the nonrelativistic approximation. Relativistic and radiative corrections are discussed in Sec. 13.3. The derivation of the helium Lamb shift in Sec. 13.3.2 is rather tricky; it involves a combination of the self-energy of the bound electrons individually, as well as a radiative correction to the interelectron interaction (recoil correction). We conclude with a brief description of the actual, numerical calculation of the helium spectrum in Sec. 13.4. Within the Hartree–Fock approximation, the electron wave functions are modeled on the basis of a Slater1 determinant. The restrictions set forth by the Slater determinant are relaxed in the case of a fully correlated basis set, which takes into account the interelectron interaction to all orders right from the start of the calculation and does not rely on the specification of individual electronic orbitals. It is the latter approach which is relevant to helium. 13.2 13.2.1

Essential Ingredients for Many-Electron Systems Spin Wave Functions

In helium, the spin-orbit coupling, which is of order of (Zα)4 m, is much weaker than the interelectron interaction, which is of the order of (Zα)2 m (in natural units, ̵ = c = 0 = 1, and m denotes the electron mass). The spin of the electron enters the h spectrum only at the order of (Zα)4 m, and the spectrum as well as the properties of 1 John

Clarke Slater (1900–1976). 507

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neutral helium are therefore spin-independent to a very good approximation. The decisive degree of freedom in helium therefore is the orbital angular momentum; the electron spins couple independently and interact only weakly with the environment. This is the regime of LS-coupling, where the spins couple independently of the ⃗ the orbital angular angular momenta. First, the spins s⃗i couple to the total spin S, ⃗ and then total spin couples momenta `⃗i couple to the total angular momentum L, to the total angular momentum. There is strong electron correlation, and weak binding to the nucleus. The situation is reversed in jj coupling. First, the spin s⃗i of the ith individual electron couples to its orbital angular momentum `⃗i , to form the individual total angular momentum ⃗ji , and then the total angular momenta ⃗ji from the different electrons couple to each other. One canonically denotes the angular momenta of individual electrons by lowercase letters, whereas the combined momenta are denoted by uppercase letters. This explains the notation jj versus LS coupling. For weak electron correlation, the electrons are strongly bound to the nucleus. An example for jj coupling is the electronic configuration (2p21/2 )J=0 , where both electrons are in states with a total angular momentum j = 1/2, and couple to a state with J = 0. Let us now return to LS coupling. If the total spin wave function is symmetric, then, according to the Pauli exclusion principle, the orbital wave function has to be antisymmetric under particle exchange, and vice versa. This implies that the ground state of helium is a spin singlet. It has an anti-symmetric spin wave function. Within the approximation of individual atomic orbitals, we would say that the two electrons both occupy the 1s ground state, to yield the state 1s2 1 S0 . This is referred to as para-helium. The metastable ground state of ortho-helium (with an S = 1 symmetric spin wave function) has an antisymmetric orbital wave function. The two electrons cannot occupy the same quantum state. The 1s2s 3 S1 ground state of ortho-helium is energetically higher than the ground state of para-helium. The spin wave function of para-helium is given by 1 (13.1) ∣S = 0, M = 0⟩ = √ (∣ ↑ ↓⟩ − ∣ ↓ ↑⟩) , 2 whereas for triplet helium, the spin states are ∣S = 1, M = 1⟩ = ∣ ↑ ↑⟩ , ∣S = 1, M = −1⟩ = ∣ ↓ ↓⟩ , (13.2a) 1 ∣S = 1, M = 0⟩ = √ (∣ ↑ ↓⟩ + ∣ ↓ ↑⟩) . (13.2b) 2 If we assume that the two electrons occupy independent orbitals, labeled 1 and 2, then the two-particle wave functions are given by the following general formulas, 1 1 ΨS=0,M =0 (⃗ r1 , r⃗2 ) = √ (Ψ1 (⃗ r1 ) Ψ2 (⃗ r2 ) + Ψ1 (⃗ r2 ) Ψ2 (⃗ r1 )) √ (∣ ↑ ↓⟩ − ∣ ↓ ↑⟩) . (13.3) 2 2 Here, we have split the wave function into an orbital and a spin part. For triplet states, we have 1 1 ΨS=1,M =0 (⃗ r1 , r⃗2 ) = √ (Ψ1 (⃗ r1 ) Ψ2 (⃗ r2 ) − Ψ1 (⃗ r2 ) Ψ2 (⃗ r1 )) √ (∣ ↑ ↓⟩ + ∣ ↓ ↑⟩) . (13.4) 2 2

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Both of these wave functions are odd under particle exchange. They factorize into a spin and an orbital part. Because the electron spins interact only weakly and spinflip induced transitions are slow, the ground-state of spin-1 helium (ortho-helium) actually is metastable. There are fundamental differences in the excitation spectra of para-helium and ortho-helium, which explains the names for the two different “kinds” of helium. The limitations in the suitability of the two wave functions (13.3) and (13.4) for the description of electron correlation become clear when we look at the identity ⃗ f (∣⃗ L r1 − r⃗2 ∣) = (`⃗1 + `⃗2 ) f (∣⃗ r1 − r⃗2 ∣) = 0 ,

(13.5)

which states that any function of the interelectron distance r12 = ∣⃗ r1 − r⃗2 ∣ belongs to a state of total angular momentum L = 0 (an S state). This suggests the presence of terms that involve r12 = ∣⃗ r1 − r⃗2 ∣ in the two-particle wave function; Eqs. (13.4) and (13.5) lack such terms. As we know from classical electrodynamics [81], an expression of the form 1/r12 gives rise to arbitrarily high multipoles proportional to ∗ Y`m (⃗ r1 ) Y`m (⃗ r2 ) when 1/r12 is expanded in (inverse) powers of r< = min(r1 , r2 ) and r> = max(r1 , r2 ). In order to correctly describe the electron correlation, one would thus have to describe a many-electron state in terms of a linear superposition of orbitals, each with a different orbital momentum, replacing, schematically, Ψ1 (⃗ r1 ) Ψ2 (⃗ r2 ) + Ψ1 (⃗ r2 ) Ψ2 (⃗ r1 ) → a (Ψ1 (⃗ r1 ) Ψ2 (⃗ r2 ) + Ψ1 (⃗ r2 ) Ψ2 (⃗ r1 )) + b (Ψ3 (⃗ r1 ) Ψ4 (⃗ r2 ) + Ψ4 (⃗ r2 ) Ψ3 (⃗ r1 )) ,

(13.6)

with suitable expansion coefficients a and b and different angular symmetries for the Ψj with j = 1, 2, 3, 4. A simple product ansatz consisting of individual atomic orbitals, as given by Eqs. (13.3) and (13.4), will not be sufficient. However, a wave function expanded over a suitable basis set which contains r12 is able to describe electron correlation correctly and to all orders. This is called a fully correlated basis set. We recall that in LS coupling, the spins of the electrons first couple to the total spin S, the orbital angular momenta couple to the total orbital angular momentum L, and then L and S couple to J. A configuration is denoted as 2S+1 LJ . There is strong electron correlation and a comparatively weak binding to the nucleus. This situation is relevant to all outer shells of neutral atoms, because the effective nuclear charge numbers for the outer electron shells are small. An example is the 1s2 1 S0 ground state of neutral helium. By contrast, in jj coupling, the spins si of the individual electrons first couple to the individual angular momenta `i , and then the total angular momenta ji couple to each other. One can then couple the total electron angular momenta together, as in the ground-state configuration (1s21/2 )0 of heliumlike uranium. Here, the subscript denotes the total angular momentum of all electrons in the configuration (J = 0). In highly charged ions, there is weak electron correlation, and strong binding to the atomic nucleus. The transition region between LS and jj coupling can be estimated as follows. Let us consider the

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electron correlation term α/r12 in the Hamiltonian and compare it to the spin-orbit coupling term, Zα α ⃗ ⋅ (⃗ ∼ Z α2 m , σ r × p⃗) ∼ (Zα)4 m . (13.7) r 4 m2 r 3 The first term measures the strength of the electron-electron correlation, the second term measures the strength of the spin-orbit coupling, and we have used the fact that the dimension of the atom reads as r ∼ (Zαm)−1 . The transition occurs at a value of Z given by the condition Z α2 = (Zα)4 ,

Z ≈ 26.57 ,

(13.8)

i.e., in between iron (Fe, Z = 26) and cobalt (Co, Z = 27) in the periodic table of elements. 13.2.2

Two-Body Problem: Center-of-Mass Coordinates

Before we dwell on the helium atom and its non-separability, and the mass polarization term, let us briefly recall how the two-body Schr¨odinger–Coulomb Hamiltonian can be separated into a free Hamiltonian describing the motion of the combined twobody system, and a Schr¨ odinger–Coulomb Hamiltonian in the relative coordinate, and with the reduced mass. We denote the coordinates and momenta of the two particles by subscripts 1 and 2 and define the relative coordinate as r⃗ = r⃗1 − r⃗2 . The ⃗ are defined as follows, center-of-mass coordinates r⃗ and R m m 2 1 ⃗= r⃗1 + r⃗2 , r⃗ = r⃗1 − r⃗2 , M = m1 + m2 , (13.9) R M M where the subscripts 1 and 2 denote the “orbiting particle” and the “nucleus”, respectively. (The discussion is general, and is also valid for, say, positronium with m1 = m2 .) The coordinates of the individual particles can also be obtained from the center-of-mass coordinates, ⃗ + m2 r⃗ = R ⃗ + ξ2 r⃗ , r⃗2 = R ⃗ − ξ1 r⃗ , ξ1 = m1 , ξ2 = m2 . (13.10) ⃗ − m1 r⃗ = R r⃗1 = R M M M M The formalism with the ξi factors (i = 1, 2) becomes important in the discussion of the Bethe–Salpeter equation in Sec. 16.2.1. The total momentum of the system is denoted as P⃗ , while p⃗ is the relative momentum, µ µ m1 m2 p⃗1 − p⃗2 = ξ2 p⃗1 − ξ1 p⃗2 , µ= . (13.11) P⃗ = p⃗1 + p⃗2 , p⃗ = m1 m2 m1 + m2 The individual momenta p⃗1 and p⃗2 can be obtained as follows, m1 ⃗ m2 ⃗ p⃗1 = P + p⃗ = ξ1 P⃗ + p⃗ , p⃗2 = P − p⃗ = ξ2 P⃗ − p⃗ . (13.12) M M The commutation relations read as follows, [P i , Rj ] = [pi , rj ] = −i δ ij ,

[P i , rj ] = [pi , Rj ] = 0 .

(13.13)

The Hamiltonian transforms as follows, H=

p⃗12 p⃗ 2 Zα P⃗ 2 p⃗ 2 Zα + 2 − = + − , 2m1 2m2 ∣⃗ r1 − r⃗2 ∣ 2M 2µ r

(13.14)

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where of course r = ∣⃗ r∣ = ∣⃗ r1 − r⃗2 ∣. The two-body problem is exactly separable in center-of-mass coordinates and there is no mass polarization term. The Hamiltonian is the sum of the kinetic energy term for the composite system, described by the kinetic term P⃗ 2 /(2M ), and the unperturbed Schr¨odinger–Coulomb Hamiltonian ⃗2 HS = p2µ − Zα . This consideration provides for an a posteriori justification of the r starting point for the Schr¨ odinger–Coulomb Hamiltonian given in Eq. (2.1). 13.2.3

Three-Body Problem: Mass Polarization

For a general three-body system, we would have to introduce three masses m1 , m2 and m3 . However, it is customary to restrict the discussion to a three-body system with two particles of mass m (typically, the electron mass), and a third particle of mass mN (typically, the nucleus). The Hamiltonian of the three-body system is given as follows, H=

p⃗ 2 p⃗12 p⃗ 2 Zα Zα α + 2 + N − − + . 2m 2m 2mN ∣⃗ r1 − r⃗N ∣ ∣⃗ r2 − r⃗N ∣ ∣⃗ r1 − r⃗2 ∣

(13.15)

If we are only interested in the “internal degrees of freedom” of the system, then we may actually obtain the three-body Hamiltonian using a simple trick. Namely, we work directly in the center-of-mass system, where the sum of the particle momenta ⃗ and the nucleus is at the origin of the vanishes and we have p⃗1 + p⃗2 + p⃗N = 0, ⃗ ⃗ coordinate system, rN = 0. In view of the relation p⃗N = −⃗ p1 − p⃗2 , the kinetic term becomes H = p⃗12 (

p⃗1 ⋅ p⃗2 p⃗12 p⃗22 p⃗1 ⋅ p⃗2 1 1 1 1 + ) + p⃗22 ( + )+ = + + . 2m 2mN 2m 2mN mN 2µ 2µ mN

(13.16)

The reduced mass µ corresponds to the two-body system of electron and nucleus, m mN µ= . (13.17) mN + m The Hamiltonian of the three-body system, in the center-of-mass system, can therefore be written as H=

p⃗12 p⃗22 Zα Zα α p⃗1 ⋅ p⃗2 + − − + + , 2µ 2µ r1 r2 r12 mN

(13.18)

where r1 = ∣⃗ r1 ∣, r2 = ∣⃗ r2 ∣, and r12 = ∣⃗ r1 − r⃗2 ∣. The last term in Eq. (13.18) is the so-called mass polarization term. It is a consequence of the non-separability of the three-body Hamiltonian. Let us repeat the derivation once more, but this time without going into the center-of-mass system. The total momentum of the system is P⃗ = p⃗1 + p⃗2 + p⃗N . The momenta P⃗1 and P⃗2 in this case have to be defined as mN + m m m P⃗1 = p⃗1 − p⃗2 − p⃗N , mN + 2m mN + 2m mN + 2m mN + m m m P⃗2 = p⃗2 − p⃗1 − p⃗N . (13.19) mN + 2m mN + 2m mN + 2m

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The center-of-mass coordinates can be found as follows, mN r⃗N + m (⃗ r1 + r⃗2 ) . (13.20) mN + 2m The commutation relations of the canonically conjugate variables follow almost immediately, ⃗ 1 = r⃗1 − r⃗N , R

⃗ 2 = r⃗2 − r⃗N , R

ρ⃗ =

[P1i , R1j ] = [P2i , R2j ] = [P i , ρj ] = −i δ ij , [P1i , R2j ] = [P1i , ρj ] = [P2i , R1j ] = [P2i , ρj ] = [P i , R1j ] = [P i , R2j ] = 0 .

(13.21)

Here, ρ⃗ is the canonically conjugate variable of the total momentum P⃗ . The momenta of the individual particles can be obtained in terms of canonically conjugate new variables as follows, m m p⃗1 = P⃗ + P⃗1 , p⃗2 = P⃗ + P⃗2 , (13.22a) mN + 2m mN + 2m mN P⃗ − P⃗1 − P⃗2 . (13.22b) p⃗N = mN + 2m The kinetic term for the entire system can then be separated according to p⃗ 2 p⃗ 2 p⃗12 P⃗ 2 P⃗ 2 P⃗ 2 P⃗1 ⋅ P⃗2 + 2 + N = + 1 + 2 + . (13.23) 2m 2m 2mN 2(mN + 2m) 2µ 2µ mN In the center-of-mass system, then, the helium Hamiltonian reads as α P⃗ 2 P⃗ 2 Zα P⃗22 Zα P⃗1 ⋅ P⃗2 H= + 1 − + − + + . ⃗1 − R ⃗2∣ 2(mN + 2m) 2µ R1 2µ R2 ∣R mN

(13.24)

⃗1 − R ⃗ 2 ∣ → ∣⃗ Upon the identification P⃗1 → p⃗1 , P⃗2 → p⃗2 , R1 → r1 , R2 → r2 , ∣R r1 − r⃗2 ∣ ≡ r, this Hamiltonian is equivalent to the one given in Eq. (13.18). The mass polarization term P⃗1 ⋅ P⃗2 /mN is an essential ingredient to the description of the three-body system. Finally, let us indicate the generalization of Eq. (13.18) to a bound system consisting of n electrons, and one nucleus, n

H = ∑( i=1

n p⃗i2 Zα α p⃗i ⋅ p⃗j − )+ ∑( + ), 2µ ri mN i 2, the quantity ln(k0 /Z 2 ) approximates a constant very well. The definition given in Eq. (13.55) is a natural generalization of the Bethe logarithm defined in Eq. (4.317). Indeed, the formula (4.317) can be obtained from Eq. (13.55) under the replacement (Zα)4 µ3 . (13.56) n3 The indication of a result valid for general Z is useful for heliumlike ions. Note that the definition (13.55) implies that the Bethe logarithms only depend on the orbital symmetry (total orbital quantum number L) and on the total spin angular quantum number S. Here, π(Zα) ⟨δ (3) (⃗ r1 ) + δ (3) (⃗ r2 )⟩ → π(Zα) ⟨δ (3) (⃗ r)⟩ =

⃗ = `⃗1 + `⃗2 = r⃗1 × p⃗1 + r⃗2 × p⃗2 , L

⃗ 2 = L(L + 1) . L

(13.57)

The total spin angular momentum corresponds to the two-particle spin states given in Eqs. (13.1) and (13.2). The two-particle states are labeled as S states for L = 0, and as P states if L = 1. Furthermore, singlet states have S = 0, whereas for triplet states, one has S = 1.

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Quantum Electrodynamics in Action: Atoms, Lasers and Gravity Table 13.1 The Bethe logarithms for neutral helium (nuclear charge number Z = 2) are given for singlet-S, triplet-S, singlet-P and triplet-P states (data are taken from Ref. [390]). The 23 S state is the ground state of ortho-helium. The definition (13.55) implies that the Bethe logarithms only depend on the orbital symmetry (total orbital quantum number L) and on the total spin quantum number S. Bethe Logarithms for Low-Lying States of Helium n1 S

n3 S

n1 P

n3 P

n=1

2.983 865 857(3)

–

–

–

n=2

2.980 118 36(7)

2.977 742 46(1)

2.983 803 46(3)

2.983 690 84(2)

n=3

2.982 870 510(8)

2.982 372 56(2)

2.984 001 39(9)

2.983 939 5(3)

n=4

2.983 596 30(1)

2.983 429 08(5)

2.984 068 8(2)

2.984 039 80(5)

n=5

2.983 857 3(1)

2.983 783 95(8)

2.984 096 18(4)

2.984 080 2(2)

The configurations can be addressed more clearly, in the sense that singlet-S states correspond to the configurations 1s ns 1 S0 , where the two-electron orbital configuration approximates the symmetrized combination of one electron in the 1S state, the other electron being in the ns state. The total angular momentum of orbital and spin angular momenta of both electrons is J = 0, and this is indicated in the spectroscopic notation n1 SJ=0 . Correspondingly, n1 P states qualify themselves as 1s np 1 P1 , while n1 D states would be 1s nd 1 D2 , and n1 F states would be 1s nf 1 F3 . For triplet state, where the orbital wave function is antisymmetrized, one identifies the n3 S state configurations more precisely as 1s ns 3 S1 , while the n3 P configurations split up into 1s np 3 P0,1,2 under the fine-structure splitting (with J = 0, 1, 2), and the n3 D configurations qualify themselves as 1s nd 3 D1,2,3 . TripletF configurations n3 F are qualified as 1s nf 3 F2,3,4 . In general, when we consider triplet states, then, for L ≥ 1, the permissible values of J are J = L − 1, L, L + 1. 13.4 13.4.1

Numerical Calculation of the Helium Spectrum Fully Correlated Basis Sets

The Hartree–Fock approximation still treats the electron wave function as being composed of individual orbitals and therefore is not suited for atomic systems with either a strong interelectronic correlation, or for systems where an exceptionally high degree of accuracy in the theoretical predictions is required. A paradigmatic example for a system of the latter kind is atomic helium, where experiments have meanwhile reached absolutely impressive accuracy [391].

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521

In a fully correlated basis set, the functional dependence on the relative coordinate r = r12 = ∣⃗ r1 − r⃗2 ∣ is taken into account to all orders. The basic paradigm then consists in writing down a suitable set of basis (“trial”) functions, and to evaluate the matrix elements of the helium Hamiltonian on the basis functions. In the limit of an infinite number of basis functions, the basis hopefully becomes complete and the calculated bound-state energies become accurate approximations of the true energy eigenvalues. This difficulty cannot be avoided because no exact, analytic eigenstate solutions of the three-body unperturbed helium Hamiltonian p⃗12 p⃗22 Zα Zα α + − − + (13.58) 2µ 2µ r1 r2 r in the space of spin-symmetrized nonrelativistic wave functions are known. We recall that (13.58) is an approximation to (13.18), neglecting the mass polarization term. In Sec. 6.4.3, we had introduced the spherical biharmonic, H=

`2 Y (ˆ r1 ) Y`2 m2 (ˆ r2 ) . YL`1M (⃗ r1 , r⃗2 ) = ∑ C`LM 1 m 1 `2 m 2 `1 m 1

(13.59)

m1 m2

Here, the two individual angular momenta `1 and `2 are added to give the total angular momentum L. Singlet-state wave functions of total angular momentum L and magnetic projection M have the following structure, `2 ΨS=0 (L, M, r⃗1 , r⃗2 ) = [RnL (r1 , r2 , r12 ) YL`1M (ˆ r1 , rˆ2 ) `2 + RnL (r2 , r1 , r12 ) YL`1M (ˆ r2 , rˆ1 )] ∣00⟩ .

(13.60)

Here, n is a multi-index for the approximate principal quantum numbers, which give the main admixtures to the state. The singlet ground state is (1s2 ) 1 S0 , and higher excited singlet states have the structure 1 ΨS=0 (0, 0, r⃗1 , r⃗2 ) = [Rn0 (r1 , r2 , r12 ) + Rn0 (r2 , r1 , r12 )] ∣00⟩ . (13.61) 4π Triplet state wave functions of total angular momentum J and total magnetic projection M have the following structure, ΨS=1 (J, M, r⃗1 , r⃗2 ) =

` ` JM r1 , rˆ2 ) ∑ CL′ M ′ 1 M ′′ [RnL′ (r1 , r2 , r12 ) YL1′ M2 ′ (ˆ

M ′ M ′′

`2 − RnL′ (r2 , r1 , r12 ) YL`1′ M r2 , rˆ1 )] ∣1 M ′′ ⟩ . ′ (ˆ

(13.62)

3

The triplet ground state is 1s 2s S1 , and triplet S configurations have the general structure ΨS=1 (1, M, r⃗1 , r⃗2 ) = [Rn0 (r1 , r2 , r12 ) − Rn0 (r2 , r1 , r12 )] ∣1 M ⟩ .

(13.63)

Within the nonrelativistic approximation, the triplet (S = 1) state energies only depend on the total orbital angular momentum L′ , and are independent of the value of J = L′ − 1, L′ , L′ + 1 and of course independent of M . We reemphasize that the total angular momentum operator ⃗ = `⃗1 + `⃗2 , L

⃗ RnL (r1 , r2 , r12 ) = 0 , L

(13.64)

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has a zero eigenvalue when acting on any function of the radial coordinates RnL (r1 , r2 , r12 ). In constructing the basis set, the following criteria are relevant: The basis set of trial functions must have sufficient mathematical structure in order to accommodate the particularities of the helium atom, but it must also be simple enough so that the integrals of the terms in the helium Hamiltonian, and in the relativistic corrections, can be easily calculated, preferentially in closed analytic form. Two basis sets have been used widely in calculations of properties of the helium atom, namely, the Hylleraas2 [392–395] and the Korobov3 [396–398] basis sets. Indeed, the different approaches for the treatment of the helium atom are distinguished based on the functional form of the radial component. The Hylleraas ansatz entails the functional form m

m−i m−i−j

i=0

j=0

i j k −α r −β r ∑ aijk (r1 ) (r2 ) (r12 ) e 1 2 ,

RnL (r1 , r2 , r12 ) = ∑ ∑

(13.65)

k=0

where of course the aijk depend on the principal quantum numbers. They are expansion parameters, to be determined in the diagonalization of the Hamiltonian matrix, and the α and β are global parameters which describe the exponential falloff of the basis functions. The terms (r1 )i , (r2 )j and (r12 )k are powers of the radial coordinates, which we denote in a very explicit notation, in order to differentiate k them from the Cartesian components r1i , r2j and r12 = r1k − r2k as the kth component of r⃗12 = r⃗1 − r⃗2 . The parameters α and β are treated as variational parameters in a typical calculation, which are optimized according to a variational principle. E.g., the best approximation to the ground-state energy, given a finite number of basis functions, is naturally determined as the trial function which, among the given set of basis functions, yields the minimum energy. In the Hylleraas basis, trial states are ordered into so-called Pekeris shells of order m with i + j + k = m. A possible modification of the Hylleraas idea consists in a “double basis set” for the exponentials. The “double bases” are distinguished according to the distance from the nucleus which they model. Indeed, the double-basis Hylleraas ansatz for the helium wave functions of excited 1s ns states is given as M −i M −i−j

M

RnL (r1 , r2 , r12 ) = ∑ ∑ i=0

j=0

i j k −α r −β r ∑ aijk r1 r2 r12 e I 1 I 2

k=0

M

M −i M −i−j

i=0

j=0

+∑ ∑

i j k −α r −β r ∑ bijk r1 r2 r12 e II 1 II 2 .

(13.66)

k=0

In a typical calculation with the double basis set (see Refs. [399, 400]), the parameters αI and βI model the short-distance physics. Sometimes, the Hylleraas ansatz is written in terms of the so-called Hylleraas variables s = r1 + r2 , 2 Egil

Andersen Hylleraas (1898–1965). 3 Vladimir Ivanovich Korobov (b. 1956).

t = r1 − r2 ,

u = r12 .

(13.67)

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Counting all orders in the expansion in s, t, and u, the same terms are generated as by the powers of r1 , r2 and r12 . According to Refs. [401, 402], even negative powers of s have to be admitted. Matrix elements of operators in the Hylleraas basis are discussed in Ref. [399]. Calculations using Hylleraas basis sets are usually numerically quite stable, and the separation into a double basis can help to separate short-distance from long-distance physics [403]. For matrix elements in the Hylleraas basis, we refer to [404] and to the early compilation of recurrence relations in Ref. [405]. For the higher-lying Rydberg states, the double basis set has proven to be remarkably stable from a numerical standpoint [404]. In the Korobov basis set (exponential basis set), the radial functions are of the form RnL (r1 , r2 , r12 ) = ∑ ai K(αi , βi , γi , r1 , r2 , r12 ) , i

K(αi , βi , γi , r1 , r2 , r12 ) = exp (−αi r1 − βi r2 − γi r12 ) .

(13.68)

The basis states are conveniently labeled by the parameters α, β and γ which define three regions in which these are optimized. The Korobov basis set is well suited for practical calculations because the integrals of the functions in the basis set assume a compact functional form. One basic integral [406] is 16π 2 e−α r1 −β r2 −γ r12 3 3 = . (13.69) ∫ d r1 ∫ d r2 r1 r2 r12 (α + β)(β + γ)(γ + α) The advantage of the exponential basis set is that the product of two wave functions in the basis set reproduces a mathematical expression of the same structure as a wave function within the basis set, K(α1 , β1 , γ1 , r1 , r2 , r12 ) K(α2 , β2 , γ2 , r1 , r2 , r12 ) = K(α1 + α2 , β1 + β2 , γ1 + γ2 , r1 , r2 , r12 ) .

(13.70)

Integrals of the form e−α r1 −β r2 −γ r12 r1 r2 r12 ∂ k ∂ m ∂ n 16π 2 (13.71) =( ) ( ) ( ) ∂α ∂β ∂γ (α + β)(β + γ)(γ + α) can then be evaluated by repeated differentiation over parameters. k, m, n ≥ −1; for −1 one has to replace differentiation by integration. Calculations made using the Korobov basis typically entail significant losses of numerical precision in intermediate steps, and extended-precision arithmetic [407–409] is often used. 3 3 k m n ∫ d r1 ∫ d r2 r1 r2 r12

13.4.2

Matrix Elements and Nonorthogonal Basis

For actual numerical calculations, it is customary to first transform to dimensionless variables, in a manner consistent with the length dimensions of the system. We start from the helium Hamiltonian given in Eq. (13.58), 1 2 Zα 1 2 Zα α ⃗ − ⃗ − H =− ∇ − ∇ + , (13.72) 2µ 1 r1 2µ 2 r2 r12

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and scale the coordinate as follows, ρ1 , r1 → αµ

r2 →

ρ2 . αµ

(13.73)

The Hamiltonian then becomes (α µ)2 ∂ 2 (α µ)2 ∂ 2 Z α2 µ Z α2 µ α2 µ H= − − − − + 2µ ∂ ρ⃗12 2µ ∂ ρ⃗22 ρ1 ρ2 ρ12 = α2 µ (−

Z 1 ∂2 Z 1 ∂2 1 − − − + ), 2 2 ∂ ρ⃗1 ρ1 2 ∂ ρ⃗22 ρ2 ρ12

(13.74)

which enables us to scale the quantity α2 µ out of the Hamiltonian term. The remaining term, after the redefinition ρ⃗i → r⃗i , can be written as follows, 1 1 2 Z 1 2 Z ⃗ − ⃗ − − ∇ + . (13.75) H0 = − ∇ 2 1 r1 2 2 r2 r12 Here, all quantities are dimensionless, and they assume numerical values of order unity for the helium atom. For definiteness, we shall now specialize to the evaluation of matrix elements of the unperturbed helium Hamiltonian H0 , for singlet S and triplet S states, in the Korobov basis, we have the matrix elements, H(αi , βi , γi , αj , βj , γj ) = ⟨K(αi , βi , γi , r1 , r2 , r12 )∣H0 ∣K(αj , βj , γj , r1 , r2 , r12 )⟩. (13.76) The matrix element of the Hamiltonian is obtained as H(αi , βi , γi , αj , βj , γj ) = H(αi , βi , γi , αj , βj , γj ) ± (βi ↔ βj ) ± (αi ↔ αj ) + (αi ↔ αj , βi ↔ βj )

(13.77)

where the positive sign is to be used for spin singlet states and the negative sign is for spin triplet states. We work in a non-orthogonal basis and therefore define the overlap matrix (with multi-indices i and j which specify the states) S(αi , βi , γi , αj , βj , γj ) = ⟨K(αi , βi , γi , r1 , r2 , r12 )∣K(αj , βj , γj , r1 , r2 , r12 )⟩ , (13.78) with S(αi , βi , γi , αj , βj , γj ) = S(αi , βi , γi , αj , βj , γj ) ± (βi ↔ βj ) ± (αi ↔ αj ) + (αi ↔ αj , βi ↔ βj ) .

(13.79)

Here, S is properly (anti-)symmetrized. We can view the overlap matrix as a matrix which consists of the overlap integrals of the states ∣ψi ⟩ and ∣ψj ⟩, Sij = ⟨ψi ∣ψj ⟩ ,

1 = ∑ ∣ψi ⟩ (S −1 )ij ⟨ψj ∣ , ij

S = LT L ,

(13.80)

where the latter expression identifies the Cholesky decomposition. The above expression for the unit matrix is easily proven by writing out the explicit expression for the square 12 , and using the fact that Sij = ⟨ψi ∣ψj ⟩. The lower-diagonal matrix L has the properties T

−1

(L−1 ) = (LT )

,

but

L−1 ≠ LT .

(13.81)

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Using the fundamental decomposition of an eigenvector ∣ψ⟩ = ∑k ck ∣ψk ⟩, the eigenvalue problem can then be formulated as H∣ψ⟩ = E ∣ψ⟩. After inserting a unit matrix, this reads as

1 H ∑ ck ∣ψk ⟩ = E ∑ ci ∣ψi ⟩ .

(13.82)

i

k

Using the relation (13.80), we then have ∑ ∣ψi ⟩ i

⎛ ⎞ −1 ∑ (S )ij ⟨ψj ∣H∣ψk ⟩ ck = ∑ ∣ψi ⟩ E ci . ⎝ jk ⎠ i

(13.83)

This has to be fulfilled for all i separately since the basis spanned by the ∣ψi ⟩ is non-singular. Thus, −1 ∑ (S )ij ⟨ψj ∣H∣ψk ⟩ ck = E ci ,

S−1 H c = E c .

(13.84)

jk

The latter equation is valid in terms of the column vector c spanned by the ci coefficients and the matrix H spanned by the ⟨ψi ∣H∣ψj ⟩. This can be reformulated as H c = E S c. Inserting the unit operator as 1 = L−1 L on the left-hand side, and using the relation S = LT L on the right-hand side, we obtain

H L−1 L c = E LT L c .

(13.85)

This can trivially be reformulated as −1

((LT )

H L−1 ) d = E d ,

d = Lc.

(13.86)

The Hamiltonian matrix which gives the eigenvalues thus is given by −1

Heff = (LT ) H L−1 ,

Heff d = E d ,

c = L−1 d .

(13.87)

It is still symmetric, but it does not have a band structure. 13.4.3

Numerical Results and Technical Issues

Numerical calculations for helium have a long history. The starting point is the scaled Hamiltonian given in Eq. (13.75), 1 2 Z 1 2 Z 1 ⃗ − ⃗ − H0 = − ∇ − ∇ + . (13.88) 2 1 r1 2 2 r2 r12 Throughout the last two decades, it has been a challenge to determine the ground state of the helium atom to utmost accuracy. We give three recent results, obtained using fully correlated basis sets. The first result is taken from Ref. [410], E0 = −2.903 724 377 034 119 598 30(2) .

(13.89)

The second result is taken from a more recent article by Korobov [398], E0 = −2.903 724 377 034 119 598 311 093 1 .

(13.90)

The record in accuracy is still being held by Refs. [401, 402], and reads as E0 = −2.903 724 377 034 119 598 311 159 245 194 404 446 696 925 309 .

(13.91)

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Here, E0 refers to the energy of the 1 S0 state. The next higher excited singlet and triplet S state reference energies are given as follows [411], E(1s 2s 1 S0 ) = − 2.145 974 046 054 426(18) ,

(13.92a)

E(1s 2s S1 ) = − 2.175 229 378 236 791 23(10) .

(13.92b)

3

The numerical accuracy of these results exceeds that of the relativistic and radiative corrections, and is thus more than needed for a comparison of theory and experiment. Selected values for singlet and triplet states of the helium atom are given in Tables 13.2 and 13.3. For a recent compilation of level energies with relativistic and radiative corrections, one consults Refs. [412, 413]. For 1s nd states, one consults Ref. [414]. Progress toward the evaluation of all α7 m corrections in helium, with tremendous advances being reported for triplet n = 2 states, has recently been reported in Refs. [415–418]. One of the most important boundary conditions, to be integrated in the formalism, concerns the Kato cusp condition [419, 420]. We briefly generalize the discussion, return to a Hamiltonian formalism with dimensionful momenta and coordinates, and consider those terms in the Hamiltonian in a multi-electron atom which describe the kinetic energy of electrons i and j (or, say, charged particles i and j), and the electrostatic interaction between them, p⃗j2 qi qj p⃗i2 + +⋯+ + ⋯ (no sum over i or j) . (13.93) 2mi 2mj 4π rij ̵ = c = 0 = 1. For the electron-electron interaction in the We use natural units with h helium atom, we have i = 1 and j = 2, and m1 = m2 = µ as well as q1 q2 = e2 = 4πα. H=

Table 13.2 The nonrelativistic energies of the singlet states 1 L0 of the helium atom are given for excitations with principal quantum numbers n = 1, . . . , 6, and total orbital angular momenta L = 0, 1, 2, 3. Data are taken from Refs. [403,404]. The singlet states follow the same notation as was used in Table 13.1. Nonrelativistic Binding Energies for Singlet States of Helium n1 S

n1 P

n1 D

n1 F

n=1

-2.903 724 377 034

—

—

—

n=2

-2.145 974 046 054

-2.123 843 086 498

—

—

n=3

-2.061 271 989 741

-2.055 146 362 092

-2.055 620 732 852

—

n=4

-2.033 586 717 031

-2.031 069 650 450

-2.031 279 846 179

-2.031 255 144 382

n=5

-2.021 176 851 574

-2.019 905 989 901

-2.020 015 836 160

-2.020 002 937 158

n=6

-2.014 563 098 447

-2.013 833 979 672

-2.013 898 227 424

-2.013 890 683 816

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Fully Correlated Basis Sets and Helium

Table 13.3 The analog of Table 13.2 for nonrelativistic energies of triplet states 3 LJ of the helium atom involves excitations with principal quantum numbers n = 1, . . . , 6, and total orbital angular momenta L = 0, 1, 2, 3. Data are taken from Refs. [403,404]. The same notation was used as in Table 13.1. Nonrelativistic Binding Energies for Triplet States of Helium n3 S

n3 P

n3 D

n3 F

n=2

-2.175 229 378 237

-2.133 164 190 779

—

—

n=3

-2.068 689 067 472

-2.058 081 084 274

-2.055 636 309 453

—

n=4

-2.036 512 083 098

-2.032 324 354 297

-2.031 288 847 502

-2.031 255 168 403

n=5

-2.022 618 872 302

-2.020 551 187 256

-2.020 021 027 447

-2.020 002 957 377

n=6

-2.015 377 452 993

-2.014 207 958 774

-2.013 901 415 454

-2.013 890 698 348

The eigenfunctions of H contain singularities at points where two charged particles collide, i.e., have zero relative distance, which results in an infinite Coulomb potential. The divergence in the potential energy must be compensated by a corresponding divergence in the kinetic energy. It is generally stated that an analysis of the two-particle collisions in a nonrelativistic Coulomb system leads to the Kato cusp condition, valid for fully correlated basis sets, qi qj ∂ ⟨ψ⟩ = µij ψ(rij = 0) , ∂rij 4π

µij =

mi mj . mi + mj

(13.94)

Here, ⟨ψ⟩ is the wave function averaged over an infinitesimal sphere centered about rij = 0. Let us try to present a streamlined derivation of the Kato cusp condition. To this end, we write near the cusp, ψ ∼ a + b rij ,

(13.95)

where a and b are constants with respect to rij (they can still depend on ri and rj , explicitly). In view of the relation 2 ⃗ i2 (rij ) = p⃗j2 (rij ) = −∇ ⃗ j2 (rij ) = − . (13.96) p⃗i2 (rij ) = −∇ rij The eigenvalues equation near the cusp reads as H ψ ∼ H(a + b rij ) = Eψ = (−

b b a q 1 q2 − + + O(rij )0 ) . mi rij mj rij 4π rij

(13.97)

The term proportional to 1/rij has to vanish if the eigenvalue condition is to hold near the cusp, and this leads to the relation −

a q i qj b + = 0, µij rij 4π rij

ψ(rij ) = a ,

∂ ⟨ψ⟩ = b, ∂rij

(13.98)

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where we use the fact that 1/mi +1/mj = 1/µij . Equation (13.98) immediately leads to the Kato cusp condition (13.94). With trial wave functions that fail to obey the cusp condition, there will typically be some fluctuations around the exact eigenenergy as the size of the basis is increased. The wave functions fulfilling the cusp condition occupy a small portion of configuration space. Variational calculations of the helium spectrum will generally converge much more slowly unless one implements the Kato cusp condition; sometimes, the Kato cusp condition is implemented by hand in all trial wave functions in the basis. A few final words are in order regarding the evaluation of matrix elements of singular operators (e.g., Dirac-δ’s and relativistic corrections involving higher powers of the momentum operators), such as the ones appearing in the Breit Hamiltonian [see Eqs. (13.29), (13.30), (13.31) and (13.32)]. One of the most successful strategies consists in rewriting the singular operators in terms of other operators that probe the wave function over longer distances, a process which is sometimes called “Drachmanization” [see Ref. [421]]. Let us reproduce the derivation here. We start from a Hamiltonian ⃗2 qi qj ∇ qi qZ r1 , . . . , r⃗N ) , V (⃗ r1 , . . . , r⃗N ) = ∑ +∑ , (13.99) H = − ∑ i + V (⃗ 2m 4πr 4πr i i ij i i 1. Surprisingly, the

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spatial extent of the nucleus prevents this breakdown; the Coulomb potential is altered nonperturbatively. The conclusions of Ref. [504] indicated that one might eventually encounter new physics effects connected with the overlap of the most tightly bound electronic level with the Dirac sea when Z = 173, not Z = 137. In Ref. [505], despite the enormously large number of electrons, attempts were made to determine the electronic ground-state configurations, based on average level (AL) Dirac–Fock calculations, for a number of elements in the range Z ≈ 140 and Z ≈ 170. The ground state of the elements with Z = 171 was predicted to be the [Rn]5f 14 6d10 7s2 7p6 8s2 8p6 7d10 5g 18 6f 14 9s2 9p configuration, with J = 1/2. For the element with Z = 172, a closely related configuration was found to have lowest energy, with 9s2 9p replaced by 9s2 9p2 , and J = 2. For the 9s2 9p3 ground-state configuration of the element with Z = 173, the authors of Ref. [505] find J = 3/2. The following best fit for the nuclear radii [506] was used in Ref. [505], R = (r0 +

r2 r1 + ) A1/3 , A2/3 A4/3

(14.70)

with r0 = 0.891(2) fm, r1 = 1.52(3) fm, and r2 = −2.8(1) fm. An independent ansatz based on the MCDHF procedure is given by the Flexible Atomic Code (FAC) described in Ref. [507]. Multireference (MR) Møller– Plesset (MP) perturbation theory is implemented in the MR–MP code described in Ref. [508]. The Jena Atomic Calculator (JAC) is described in Refs. [509, 510]. It is based on the JULIA language (see Ref. [511]), which combines the advantages of the high speed of a compiled computer language with the flexibility of modern computer algebra systems. Finally, the QEDMOD package described in Ref. [428] represents the QED selfenergy operator as an “almost” local potential, modeled as a Yukawa potential with a range equal to the Compton wavelength of the electron, and a prefactor adjusted to reproduce the known diagonal elements of the self-energy operator. It also contains expressions for Uehling and Wichmann–Kroll potentials and thus offers a reliable implementation of QED effects for the Dirac–Hartree–Fock orbitals. 14.4

Systems with a Large Number of Electrons

For systems with a large number of electrons, it is clear that further simplifications of the Hartree–Fock equations are desirable. Indeed, ideas to simplify the exchange term in the Hartree–Fock equations have been around for many decades; for example, in Eq. (6) of Ref. [512], it was proposed to approximate the exchange term with an average exchange energy, averaged over the charge density of the atom. In Eq. (14) of Ref. [512], a further simplification is proposed, based on the exchange term for a free electron gas, somewhat foreshadowing the development of density-functional theory [513, 514]. A few words on density-functional theory are thus indicated. It is sometimes being said that in density-functional theory, one writes the energy of a system

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exclusively in terms of the probability density n(⃗ r) = ∑i ni ∣ψi (⃗ r)∣2 of the electrons, where i runs over all charged particles in the system, and ni is the occupation number of the ith orbital. However, an inspection reveals that the kinetic term Ekin in the energy functional, Ekin = ∑ ni ∫ d3 r ψi∗ (⃗ r) [− i

⃗2 ∇ ] ψi (⃗ r) , 2m

(14.71)

cannot be written as a function of the density alone. In order to see this, one ob⃗ 2 ∣ψi (⃗ serves that any conceivable ansatz proportional to the volume integral of ∇ r)∣2 2 2 ⃗ vanishes, in view of the fact that ∇ ∣ψi (⃗ r)∣ is the divergence of the gradient ⃗ i (⃗ ∇∣ψ r)∣2 . The divergence theorem then indicates that the volume integral vanishes. In fact, this problem may have motivated the investigations reported in Ref. [515]. It could be solved via a generalization of the functional derivative, or, via the introduction of a two-coordinate density n(⃗ r, r⃗′ ) = ∑i ni ψi∗ (⃗ r) ψi (⃗ r′ ) [see the unnumbered equation between Eqs. (5) and (6) of Ref. [515]]. According to Ref. [516], however, one can show that at least, a so-called density functional exists which could be used in place of the exchange term in the Hartree– Fock equations and still reproduce the spectrum of the fully interacting system. The project to formulate the entire system based on the probability density alone is relaxed in Ref. [516] via the introduction of one-particle orbitals ψi referred to as the Kohn12 –Sham13 orbitals [Ref. [517]]. One arrives at equations of the type [Eq. (2.8) of Ref. [517]] called the Kohn–Sham equations {−

′ ⃗2 e2 ∇ 3 ′ n(r ) + V (⃗ r) + + Vxc (n(⃗ r))} ψi (⃗ r) = i ψi (⃗ r) . ∫ d r 2m 4π ∣⃗ r − r⃗′ ∣

(14.72)

The exchange term functional is Vxc (n(⃗ r)). Here, i has the physical interpretation of a one-particle (“orbital”) energy, and V (⃗ r) is the external potential. The term ′ e2 3 ′ n(r ) ∫ d r 4π ∣⃗ r − r⃗′ ∣

(14.73)

is the direct term in the electron-electron interaction (“Hartree term”). The essential difference between Eq. (14.72) and the Hartree–Fock equations is that the exchange term is formulated in terms of a functional of the electron density n(⃗ r), and does not require any knowledge of the wave functions. The computational effort is now only linearly related with the number of electrons in the system. One might still hope to do away with the Kohn–Sham orbitals, a field of research now known as orbital-free density functional theory. The original idea came from the so-called Thomas–Fermi model, formulated in Refs. [518–520]. The assumption is that all energy levels in the interacting system up to the Fermi momentum are ¨ cker14 in Ref. [521]. Von filled. This model has been augmented by von Weizsa Weizs¨ acker argued that the Thomas–Fermi model needs to be augmented because it 12 Walter

Kohn (1923–2016). Jeu Sham (b. 1938). 14 Carl Friedrich von Weizs¨ acker (1912–2007). 13 Lu

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effectively assumes a sudden drop-off of the particle density at the Fermi momentum. Let us consider once more the kinetic-energy term, ̵ 2∇ ̵2 ⃗2 h h 3 ∗ 3 ⃗ r) (− ) ψi (⃗ r) = ∑ r)∣2 . (14.74) ∑ ni ∫ d r ψi (⃗ i (⃗ ∫ d r ∣∇ψ 2m 2m i i The gradient of the wave function at the boundary of the nucleus, for which von Weizs¨ acker had developed his theory, enters the expression for the kinetic energy. In fact, Eq. (14.74) is the starting point [in an unnumbered equation] for the considerations of Ref. [521]. By assuming a more elaborate model for the particle wave functions as compared to the original Thomas–Fermi approximation, von Weizs¨acker obtained the energy functional E=

⃗ r)]2 h2 3h2 3 2/3 3 5/3 3 [∇n(⃗ ( ) d r [n(⃗ r )] + d r ∫ ∫ 40m π 32π 2 m n(⃗ r) 2 ′ e r) n(⃗ r) 3 3 ′ n(⃗ + ∫ d3 r n(⃗ r) VN (⃗ r) + . ∫ d r∫ d r 8π ∣⃗ r − r⃗′ ∣

(14.75)

The first two terms in this functional serve as approximations to the kinetic energy of the ensemble of particles. Although “orbital-free density functional theory” could in principle lead to a much simplified formulation, it still entails quite considerable approximations. A more elaborate ansatz involves Kohn–Sham quantum orbitals. Implementations of density-functional theory include the GAUSSIAN package [522, 523], originally developed under the guidance of Pople,15 which has been developed over a number of decades. The GAUSSIAN package uses atomic basis functions whose radial part is proportional to a functional form exp(−α r2 ), where α is a constant and r is the radial variable. These functions are different from Slater functions which are proportional to the functional form exp(−ζ r). The use of the Gaussian ansatz, augmented by powers of the radial variable in order to correctly describe the wave functions near the atomic nuclei, has some computational advantages. GAUSSIAN can be regarded as one of the most powerful densityfunctional-theory inspired molecular structure packages available today [513, 514]. The DIRAC program [524,525] also constitutes a very powerful tool for the analysis of molecular electronic structure. One feature of the DIRAC package is that relativistic basis sets for the notoriously problematic lanthanide series of elements are available [526] and have been used for the determination of ground-state spectroscopic constants for YbF [526] with good success. Also, the DIRAC package has been used to analyze the role of spin-orbit coupling in the notoriously problematic actinide cations, and evidence has been found to support the hypothesis that the x-ray photoelectron spectra (XPS) of ionic actinide materials may provide direct information about the angular momentum coupling within the 5f shell [527]. Finally, the General Atomic and Molecular Electronic Structure System (GAMESS) is a program for ab initio molecular quantum chemistry [528]. It has also evolved over a number of decades [529,530]. A variety of atomic and molecular 15 Sir

John Anthony Pople (1925–2004).

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properties, including dipole moments and frequency-dependent hyperpolarizabilities, may be computed. Finally, the AMBER package [531] is a relatively recent, highly powerful addition to the family of molecular structure packages currently available. One important focus of AMBER lies on biomolecules [532, 533]. 14.5

Further Thoughts

Here are some suggestions for further thought. (1) Relativistic Breit Interaction. Review the derivation of the relativistic Breit interaction (14.45) as a generalization of the approach used in Chap. 12, but with the propagator sandwiched between relativistic Dirac bispinors. Try to derive its nonrelativistic limit, and leading corrections, and compare with the treatment in Chap. 12. (2) Exchange Term for Filled Subshells. Assume that subshells are filled and that an average is taken over all magnetic projections. Derive the simplification considered in Eq. (14.17), r2 ) U (⃗ r2 , r⃗1 ) φa (⃗ r2 ) φn (⃗ r1 ) − ∑ ∫ d3 r2 φ+n (⃗ n≠a

2

r`+1 a`a (14.76) and thus, confirm expressions obtained in Refs. [438, 439]. (3) Interactions and Correlation Orbitals. Apply the decompositions (14.52) and (14.59) to the calculation of matrix elements (reduction to radial integrals) between correlation orbitals of the form T ψna κa µa (⃗ r1 ) = [fna κa (r1 )χ−κa µa (ˆ r1 ), i gna κa (r1 )χκa µa (ˆ r1 )] and ψnb κb µb (⃗ r2 ) = T [fnb κb (r2 )χ−κb µb (ˆ r2 ), i gnb κb (r2 )χκb µb (ˆ r2 )] . (4) Hartree–Fock–Roothaan Equations. Start from Eq. (14.15) and assume a basis set expansion of the orbital wave functions in a basis, e.g., in the form ̃ν (⃗ φn (⃗ r ) = ∑ Cν n φ r) , (14.77) →−

ν

̃ν (⃗ where the φ r) are the basis functions (ν = 1, . . . , K where K is the size of the basis). Try to define a suitable one-particle operator h so that, with ̃+ (⃗ ̃ν (⃗ hµν = ∫ d3 r φ r) h(⃗ r) φ r) , µ

̃+ (⃗ ̃ r) , Sµν = ∫ d3 r φ µ r ) φν (⃗ you can write the Hartree–Fock eigenvalue problem as hC = SCE . ∑ hµν Cνi = ∑ Sµν Cνi Ei , ν

(14.78) (14.79)

ν

Attempt to discuss similarities and differences to the approach outlined in Sec. 13.4.2, notably, Eq. (13.84), and attempt to clarify why the Hartree–Fock method is said to entail a so-called self-consistent field.

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(5) Tensors and Symmetries. Use Eqs. (8.4.10) and (8.4.17) of Ref. [16] to ab establish the relation of sba JM and sJM . (6) QED Effects and Multi-Electron Atoms. Write a program implementing some of the ideas outlined in the current chapter, including QED effects. Be inspired by Refs. [22, 428, 449, 453–458].

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Chapter 15

Beyond Breit Hamiltonian and On-Shell Form Factors

15.1

Overview

The energy corrections to atomic levels described so far could be understood based on the Foldy–Wouthuysen transformation and effective Dirac equations containing on-shell form factors (see Chaps. 10 and 11), and the Breit Hamiltonian (Chap. 12). Beyond this point, one has to resort to different and more sophisticated techniques for the description of bound-state energy levels. We will describe these in the following. In the current chapter, we shall thus dwell into the matching of scattering amplitudes with energy shifts in atoms. We have already learned that Feynman diagrams give us S matrix elements (see Secs. 9.5, 9.6 and 10.2.1), which can be matched against energy shifts [see Eq. (12.43)]. In the current chapter, this concept will be ramified and applied to the calculation of two paradigmatic and important energy shifts which affect bound-state energies in two-particle bound systems in higher order. Because the corrections are mediated by the binding Coulomb field, they are also referred to as “binding corrections”. We shall study these processes based on two examples. The first of these is the α(Zα)5 two-Coulomb-vertex correction to the Lamb shift, originally evaluated in Ref. [340]. The second example, which entails a scattering amplitude, is a recoil correction to energy levels of order (Zα)5 , which is a genuine two-particle effect and cannot, in principle, be derived from the Breit Hamiltonian or the Foldy– Wouthuysen transformation. The recoil correction entails a loop integration, and necessitates the evaluation of retardation effects which cannot be described by a static Hamiltonian. Alternatively, the recoil correction is an effect which we would like to refer to as the Salpeter correction [534]. Results for hydrogenic states with principal quantum numbers n = 1 and n = 2 were derived in Ref. [534], before the results were generalized for an arbitrary reference state in Refs. [346, 347, 535, 536]. The correction entails the evaluation of a matrix element of the form 1/r3 , which is divergent when evaluated on a reference S state and thus, in need of a regularization. This regularization is accomplished by a distribution proposed in Refs. [537, 538]. The result for the recoil correction will also be relevant for our derivation of the helium Lamb shift in Chap. 13. 557

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15.2

Furry Picture, Scattering Amplitude and Self–Energy

15.2.1

Furry Picture

In Chap. 10, we have studied quantum electrodynamics based on free states, with interactions happening “in the middle” of the diagram, while incoming and outgoing states are eigenstates of the free Dirac Hamiltonian. However, for bound-state calculations, it is often advantageous and in fact, appropriate, to start from a different formulation of QED, based on the Furry1 picture [539]. The Lagrangian of Furry-picture QED is 1 L = ψ¯ [γ µ (i∂µ − e Aµ ) − γ 0 V − m] ψ − F µν Fµν . 4

(15.1)

In contrast to Eq. (10.1), we have incorporated the nuclear potential (Coulomb potential) V = −Zα/r as a classical potential in the Lagrangian. The four-vector potential Aµ is that of the quantized photon field. The free Lagrangian (10.2) is modified to read as follows, 1 L0 = ψ¯ (i γ µ ∂µ − γ 0 V − m) ψ − F µν Fµν . 4

(15.2)

The eigenstates of the fermionic part of the corresponding Hamiltonian are the bound and continuum eigenstates of the Dirac–Coulomb Hamiltonian, discussed in Chap. 8. The appropriate quantum numbers to label the states are the principal quantum number n, the Dirac angular quantum number κ [see Eq. (4.348)], and the magnetic projection µ. One can of course equivalently exchange the quantum number n for the bound-state energy E. (Recall that κ summarizes the values of the total angular quantum number j and of the orbital quantum number ` into a single quantum number.) Given κ and µ, specification of n is tantamount to specifying E. In the Furry picture, the interaction Lagrangian is the same as in Eq. (10.3), LI = −e ψ¯ γ µ Aµ ψ .

(15.3)

In ordinary QED, we had encountered the fermionic field operators in Eqs. (7.140) and (7.141). In the Furry picture, one writes the field operator and its Dirac adjoint as a sum of contributions from positive-energy and negative-energy states, ψ(x) = ψ(x) =

∑

bEκµ ψEκµ (⃗ r) e−iEt +

∑

b+Eκµ ψ Eκµ (⃗ r) eiEt +

(E>0)κµ

(E>0)κµ

∑

d+E ′ κ′ µ′ ψ−E ′ κ′ µ′ (⃗ r) eiE t ,

(15.4a)

dE ′ κ′ µ′ ψ−E ′ κ′ µ′ (⃗ r) e−iE t .

(15.4b)

′

(E ′ >0)κ′ µ′

∑

(E ′ >0)κ′ µ′

′

The canonical conjugate field momentum is obtained from Eq. (15.1) as π(x) = i ψ(x) γ 0 . 1 Wendell

Hinkle Furry (1907–1984).

(15.5)

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For two space-time points with equal time coordinates, xµ1 = (t, r⃗1 ) and xµ2 = (t, r⃗2 ), the canonical anticommutation relation (7.161), {ψ(x1 ), π(x2 )}∣t1 =t2 = i δ (3) (⃗ r1 − r⃗2 ) ,

(15.6)

is fulfilled provided the eigenstates (see Sec. 8.2.2) are normalized as follows [see Eq. (8.88)] 3 + r) ψE2 κ2 µ2 (⃗ r) = δE1 E2 δκ1 κ2 δµ1 µ2 . ∫ d r ψE1 κ1 µ1 (⃗ The interaction Lagrangian density in the Furry picture is LI = −e ψ¯ γ µ Aµ ψ = −J µ Aµ , J µ = e ψ¯ γ µ ψ .

(15.7) (15.8)

The Feynman rules in the Furry picture are given in Table 15.1. The double lines denote the electron bound inside the Coulomb field. The Feynman rule for the fermion propagator involves the Dirac–Coulomb Feynman propagator, which we had already encountered in Sec. 8.2.5, 1 , (15.9) SFDC = 0 p
− m + γ V + i in the slightly different form GDC . The form GDC discussed in Sec. 8.2.5 does not specify the infinitesimal part in the denominator. The following conversion formulas are relevant in order to convert the Dirac–Coulomb Green function between the different conventions, in analogy to Eq. (7.175) for the free-electron case, DC 0 GDC F = iSF γ ,

0 SFDC = −iGDC F γ ,

GFDC

GDC F

=

iGDC F ,

SFDC = − GFDC γ 0 ,

=

−iGFDC ,

GFDC = −SFDC γ 0 .

(15.10a) (15.10b) (15.10c)

All conventions will be used in the following. 15.2.2

Matching for the Bound-State Self–Energy

In order to derive the starting expression for the bound-state self-energy shift, to all orders in the Coulomb field, we will carry out the matching in the Furry picture outlined above (see also Sec. 12.2 for details of the matching procedure). Let us consider the diagram in Fig. 15.1. From the Feynman rules in the Furry picture, one obtains the following forward-scattering S-matrix element, S = ∫ d4 x ∫ d4 x′ i DF µν (x − x′ ) ψ¯n (x) (−ieγ µ ) i S DC (x, x′ ) (−ieγ ν ) ψn (x′ ) F

= e2 ∫ d4 x ∫ d4 x′ DF µν (x − x′ ) ψ¯n (x) γ µ SFDC (x, x′ ) γ ν ψn (x′ ) .

(15.11)

Recalling that ψn (x) = ψn (⃗ r) exp(−i En t) and separating the covariant fourdimensional integrals into their components, we can write d4 k S = e2 ∫ dt ∫ d3 r ∫ dt′ ∫ d3 r′ ∫ (2π)4 ′ ′ dE DC × ψ¯n (⃗ r) eiEn t γ µ ∫ SF (E, r⃗, r⃗′ ) e−iE(t−t ) γ ν ψn (⃗ r′ ) e−iEn t 2π ′ ⃗ −iω⋅(t−t′ ) ik⋅(⃗ ⃗ × DF µν (ω, k) e e r−⃗r ) , (15.12)

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Fig. 15.1 In the Feynman diagram for the bound-electron self-energy in the Furry picture, the double line denotes a Dirac–Coulomb propagator given in Eq. (15.9).

where ω = k 0 . We change variables according to (t, t′ ) → (t, τ = t − t′ ), with unit Jacobian, to obtain d4 k dE ⃗ S = e2 ∫ d3 r ∫ d3 r′ ∫ dt ∫ dτ ∫ DF µν (ω, k) ∫ 4 (2π) 2π ² =T ′ ⃗ DC × ψ¯n (⃗ r) γ SF (E, r⃗, r⃗′ ) γ ν ψn (⃗ r′ ) eik⋅(⃗r−⃗r ) e−i(E−En +ω)τ dE d4 k ⃗ e2 T ∫ d3 r ∫ d3 r′ ∫ DF µν (ω, k) ∫ (2π)4 2π µ

=

⃗

× ψ¯n (⃗ r) γ µ SFDC (E, r⃗, r⃗′ ) γ ν ψn (⃗ r′ ) eik⋅(⃗r−⃗r ) (2π) δ(E − En + ω) . ′

(15.13)

Integrating out the Dirac-δ function, one obtains d4 k ⃗ r −⃗ r′ ) ⃗ ik⋅(⃗ DF µν (ω, k)e 4 (2π)

S = e2 T ∫ d3 r ∫ d3 r′ ∫

× ψ¯n (⃗ r) γ µ SFDC (En − ω, r⃗, r⃗′ ) γ ν ψn (⃗ r′ ) = −i T ⟨ψ + ∣∆HSE ∣ψ⟩ ,

(15.14)

where the latter identification with the self-energy Hamiltonian HSE is due to the matching given in Eq. (12.43). After all, the matching has resulted in the following equation for the self-energy Hamiltonian ∆HSE , ∆ESE = ⟨ψ + ∣∆HSE ∣ψ⟩ = ie2 ∫ d3 r ∫ d3 r′ ∫

d4 k ⃗ DF µν (ω, k) (2π)4

⃗ ⃗ ′ × ψ¯n (⃗ r) γ µ eik⋅⃗r SFDC (En − ω, r⃗, r⃗′ ) γ ν e−ik⋅⃗r ψn (⃗ r′ ) .

(15.15)

Using Eq. (15.10), in the form GFDC (En − ω, r⃗, r⃗′ ) = −SFDC (En − ω, r⃗, r⃗′ ) γ 0 , and defining the matrices αµ = γ 0 γ µ , we can alternatively write ∆ESE = ⟨ψ + ∣∆HSE ∣ψ⟩ = −ie2 ∫ ⃗

d4 k ⃗ ∫ d3 r ∫ d3 r′ DF µν (ω, k) (2π)4 ⃗

× ψn+ (⃗ r) αµ eik⋅⃗r GFDC (En − ω, r⃗, r⃗′ ) αν e−ik⋅⃗r ψn (⃗ r) . ′

(15.16)

This latter representation is useful in numerical evaluations of the one-loop boundstate self-energy [257, 258, 267]. For the role of the mass counter term, we refer to Sec. 15.6.

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561

Table 15.1 The Feynman rules of Furry-picture quantum electrodynamics can be expressed in terms of a correspondence of a mathematical expression and a diagram. The building blocks are given below; they are slightly different from those given in Table 10.1. The Coulomb potential is incorporated in the unperturbed Dirac propagator and breaks translational symmetry of wave functions and propagators. Energy is still conserved. Therefore, the Feynman propagator in the Furry picture, in momentum space, has the arguments (p0 , p⃗1 , p⃗2 ), or, in the mixed energy-coordinate representation, the arguments (p0 , r⃗1 , r⃗2 ). The mathematical energy variable, required for momentum conservation at the vertex, is being denoted as p0 , and it corresponds to the energy conservation at the vertex (denoted by a dot), when the energy flow is applied in the indicated direction. The energy variable E is understood to be positive. By contrast, the physical energy of a positron flows in the opposite direction. Symmetry factors for an outgoing positron, the interchange of a field operator, and a closed fermion loop, are the same as for ordinary quantum electrodynamics (see Table 10.1).

p0 = E Incoming electron

ψEκµ

p0 = E Outgoing electron

ψ Eκµ

Incoming positron from the past

ψ −Eκµ

Outgoing positron into the future

ψ−Eκµ

Incoming photon

µ kλ

Outgoing photon

µ kλ

Photon line

µν iDF

Fermion line

DC iSF

Fermion-photon vertex

−i e γµ

p0 = −E p0 = −E k

k

p

(p0 , p~1 , ~p2 )

15.3 15.3.1

Binding Correction to the Lamb Shift Two-Coulomb-Vertex Scattering Amplitude

We start from Eq. (15.15). After Fourier transformation, using the photon propagator in Feynman gauge according to Dµν (k) = −gµν /(k 2 + i) [see Eq. (9.121)], and

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(a)

(b)

(c)

(d)

Fig. 15.2 The integrals for the two-Coulomb vertex, one-loop forward scattering amplitudes, to be discussed below, are actually two-loop integrals, because the three-momentum flowing into the diagrams has to flow out again. Coulomb interactions are denoted by dashed lines, with a space-like momentum exchange (0, q⃗). The photon in the loop carries all polarizations, and has a four-momentum kµ .

taking into account Eq. (15.9), one finds ∆ESE = e2 ∫

d4 k 1 1 γµ ∣ ψ⟩ . ⟨ψ ∣γ µ 0 (2π)4 i k 2 + i p
− k
− m + γ V + i

(15.17)

Here, we recall that V is the Coulomb potential. The α(Zα)5 correction is described by the exchange of high-energy Coulomb photons, which is why we can expand the propagator in powers of the potential V , which amounts to an expansion in an increasing number of Coulomb vertices. Specifically, in Fourier space, one has V = −4πZα/⃗ q 2 with q⃗2 ∼ m2 . Diagram (a) in Fig. 15.2 gives the energy shift ∆Ea = e2 ∫

CF

d4 k 1 1 1 1 ⟨ψ ∣γ µ γ0 V γ0 V γµ ∣ ψ⟩ . 4 2 (2π) i k p p p
− k
− m
− k
+
q − m
− k
− m (15.18)

We here explicitly specify the Feynman contour CF , which allows us to suppress the infinitesimal shifts +i in the propagators, making the notation more compact. Both the momenta of the Coulomb photons exchanged as well as the four-momenta of the virtual photon in the loop are hard, i.e., of the order of the electron mass. The

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momentum from the wave function entering the diagram is of the order of the boundstate momentum Zαm ≪ m and can therefore be ignored to good approximation. This implies that we can approximate pµ = (E, p⃗) ≈ (m, ⃗0) ≡ tµ

(15.19)

in all propagator denominators in Eq. (15.18). We can also approximate the relativistic wave function ψ by the value of the nonrelativistic wave function φ at the ⃗ = φ(⃗ ⃗ multiplied by a fundamental bispinor us , and write origin, φ(0) r = 0), ψ(⃗ r) ≈ φ(⃗0) us ,

⃗ , φ(⃗ p) ≈ (2π)3 δ (3) (⃗ p) φ(0)

(15.20)

where φ denotes the nonrelativistic (Schr¨odinger–Coulomb) wave function (see Chap. 4). Here and in the following, we will implement the convention that ⃗ denotes the value of the nonrelativistic wave function at the origin φ(⃗0) = φ(⃗ r = 0) of coordinate space, not momentum space. The fundamental particle bispinors us with s =↑, ↓ (spin up and spin down) are given as u↑ = (1, 0, 0, 0)T ,

u↓ = (0, 1, 0, 0)T ,

(15.21)

and they fulfill the following relation, relevant to the averaging over the electron spins, γ 0 + 1 1 12×2 02×2 1 = ( ). ∑ us ⊗ us = 2 s=↑,↓ 4 2 02×2 02×2

(15.22)

For an S state, in the nonrelativistic approximation, the final result for the resulting energy shift must be independent of the spin state. Hence, we can write ∆Ea =

d4 k d3 q1 d3 q2 d3 p 1 1 2 ∑ e ∫ ∫ ∫ ∫ 2 s=↑,↓ (2π)3 (2π)3 (2π)3 k 2 CF (2π)4 i 4πZα 1 γ 0 (− 2 ) q⃗1 t − k
+
q 1 +
q 2 − m 1 4πZα 1 × γ 0 (− 2 ) γµ φ(⃗ p) us . q⃗2 t − k
+
q 2 − m t − k
− m

× φ+ (⃗ p + q⃗1 + q⃗2 ) us γ µ

(15.23)

The Feynman slashes of the Coulomb exchange momenta are given as
q 1 = −⃗ γ ⋅ q⃗1 , and
q 2 = −⃗ γ ⋅ q⃗2 . There is no energy exchange in a Coulomb photon, and hence (q1 )0 = (q2 )0 = 0. One uses Eq. (15.20) and obtains ⃗ ∫ ∆Ea = e2 ∣φ(0)∣ 2

CF

d3 q1 1 d4 k 3 3 ∫ ∫ d q2 ∫ d p 2 (2π)4 i (2π)3 k

× δ (3) (⃗ p + q⃗1 + q⃗2 ) Tr [γ µ ×

4πZα 1 γ 0 (− 2 ) q⃗1 t − k
+
q 1 +
q 2 − m

4πZα 1 1 γ 0 + 1 (3) γ 0 (− 2 ) γµ ] δ (⃗ p) , q⃗2 4 t − k
+
q 2 − m t − k
− m

(15.24)

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and carries out the ∫ d3 p and ∫ d3 q2 integrations, 3 d4 k 1 1 ⃗ 2∫ d q ∫ ∆Ea = 64 π 3 α (Zα)2 ∣φ(0)∣ (2π)3 CF (2π)4 i q⃗ 4 k 2 1 1 1 γ0 + 1 γ0 γ0 γµ ] 4 t − k
− m t − k
+
q − m t − k
− m α (Zα)5 m µ 3 ( ) δ`0 (4πKa ) , (15.25) = π n3 m where we implicitly define the integral Ka . We have also used the relation e2 = 4πα. Based on the well-known value of the S-state wave function at the origin, (Zαµ)3 2 ∣φ(⃗0)∣ = δ`0 , (15.26) πn3 we can write the dimensionless integral Ka as d3 q d4 k 1 m2 Ka = ∫ ∫ 3 ⃗ 4 k2 (2π) CF π 2 i q × Tr [γ µ

1 1 1 γ0 + 1 γ0 γ0 γµ ], (15.27) 4 t − k
− m t − k
+
q − m t − k
− m where we have used the trivial algebraic identity 64π 2 /(2π)4 = 4/π 2 . The reducedmass dependence of the energy shift has been obtained consistently. In order to show that the integral Ka is dimensionless, we observe that the propagator denominators t − k
− m and t − k
+
q − m carry powers of energy, respectively, which are canceled against the ∫ d3 q integral. The factor m2 /k 2 is dimensionless, as is the expression ∫ d4 k/⃗ q 4. × Tr [γ µ

15.3.2

Forward Scattering Amplitudes

In Eq. (15.25), we have derived the energy shift ∆Ea , unrenormalized, for the diagram in Fig. 15.2(a). The total energy shift, due to the subdiagrams (a)–(d), is given as α (Zα)5 m µ 3 ( ) δ`0 (4πK) , K = Ka + Kb + Kc + Kd , (15.28) ∆E = π n3 m where the integral Ka is given in Eq. (15.27). We compile the unrenormalized expressions for the dimensionless terms Ka and Kb , corresponding to the subdiagrams (a) and (b) in Fig. 15.2, respectively, employing both an infrared regulator mass λ as well as an ultraviolet regulator Λ, in the photon propagator, d3 q 1 d4 k 1 1 − ) Ta (q, k) , Ka = ∫ ( ∫ 3 4 (2π) q⃗ CF π 2 i k 2 − λ2 k 2 − Λ2 Ta (q, k) = m2 Tr [γ µ Kb = ∫

1 1 1 γ0 + 1 γ0 γ0 γµ ], 4 t − k
− m t − k
+
q − m t − k
− m

(15.29a)

1 1 1 γ0 + 1 γµ γµ γ0 ]. t +
q − m t − k
+
q − m t +
q − m 4

(15.29b)

d4 k 1 d3 q 1 1 ( 2 − 2 ) Tb (q, k) , ∫ 4 3 2 2 (2π) q⃗ CF π i k − λ k − Λ2

Tb (q, k) = m2 Tr [γ 0

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The expressions for subdiagrams (c) and (d) of Fig. 15.2 are obtained by rearranging the order of the Dirac γ matrices appropriately, Kc = ∫

1 1 d4 k d3 q 1 ( − ) Tc (q, k) , ∫ 3 4 (2π) q⃗ CF π 2 i k 2 − λ2 k 2 − Λ2

Tc (q, k) = m2 Tr [γ 0 Kd = ∫

1 1 1 γ0 + 1 γµ γ0 γµ ], t +
q − m t − k
+
q − m t − k
− m 4

(15.29c)

1 1 γ0 + 1 1 γ0 γµ γ0 ]. 4 t − k
− m t − k
+
q − m t +
q − m

(15.29d)

d3 q 1 d4 k 1 1 ( 2 − 2 ) Td (q, k) , ∫ 3 4 2 2 (2π) q⃗ CF π i k − λ k − Λ2

Td (q, k) = m2 Tr [γ µ

Here, we recall that tµ = (m, 0, 0, 0) is a constant time-like vector [see Eq. (15.19)], ⃗ while q µ = (0, q⃗) is the Coulomb k is the photon momentum four-vector k µ = (ω, k), exchange momentum. One derives the following identity, (t − k + q)2 − m2 = −(k⃗ − q⃗ )2 − X 2 , X 2 = 2mω − ω 2 . (15.30) Furthermore, one can separate the Dirac traces into numerator and denominator structures, Na Nb Nc Nd Ta = , Tb = , Tc = , Td = . (15.31) Da Db Dc Dd The mass dimension of all given terms Ta through Td is −1. They can be written as the ratio of a numerator and a denominator, of mass dimensions five and six, respectively. For the diagram in Fig. 15.2(a), we have Na = − 2k⃗2 m3 + 8m5 + 6k⃗2 m2 ω − 12m4 ω + 6m3 ω 2 + 2m2 ω 3 − 4m3 k⃗ ⋅ q⃗ − 4m2 ω k⃗ ⋅ q⃗ , Da = (k⃗2 + X 2 )2 [−(k⃗ − q⃗)2 − X 2 ] .

(15.32) (15.33)

The expressions for the diagram in Fig. 15.2(b) reads as follows, Nb = 8m5 + 2m3 q⃗2 + 8m4 ω + 2m2 q⃗2 ω − 8m3 k⃗ ⋅ q⃗ , Db = q⃗ 4 [−(k⃗ − q⃗)2 − X 2 ] .

(15.34) (15.35)

For the diagram in Fig. 15.2(c), one has Nc = − 4k⃗2 m3 + 8m5 + 2m3 q⃗2 − 8m4 ω − 2m2 q⃗2 ω − 4m3 ω 2 + 4m2 ω k⃗ ⋅ q⃗ , Dc = − q⃗ (−k − X ) [−(k⃗ − q⃗) − X ] . 2

⃗2

2

2

2

(15.36) (15.37)

The expressions for the diagram in Fig. 15.2(d) are the same as for the diagram in Fig. 15.2(c), and hence, Nd = Nc ,

Dd = D c .

(15.38)

What remains is to do the renormalization, and the integration over the photon loop momentum.

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15.3.3

Dispersion Relation and Subtractions

The structure of the K term defined in Eq. (15.28) is as follows, K= ∫ =∫

d4 k 1 1 d3 q 1 ( 2 − 2 ) T (q, k) ∫ 3 4 2 2 (2π) q⃗ CF π i k − λ k − Λ2 d3 q 1 R(⃗ q2 ) . (2π)3 q⃗4

(15.39)

Here, T = Ta +Tb +Tc +Td is the sum of the terms Ta through Td , given in Eq. (15.31), including renormalizations. Furthermore, we have defined d4 k 1 1 ( 2 − 2 ) T (q, k) (15.40) 2 2 k −λ k − Λ2 CF i π as the expression obtained after the photon four-momentum integral. We recall that we define q µ = (0, q⃗) as the four-momentum pertaining to the Coulomb exchange. Therefore, the result for R(⃗ q 2 ) only depends on the spatial components. Furthermore, for symmetry reasons, R(⃗ q 2 ) only depends on the modulus of the momentum transfer. The idea now is that we consider what would happen in case q⃗2 = −Q2 becomes negative. This would correspond to a time-like (as opposed to space-like) fictitious Coulomb photon entering the diagram. In this case, the virtual photon and/or the virtual electron (if Q2 is sufficiently large) in the diagrams in Fig. 15.2 could become on-shell, and the energy shift would acquire an imaginary part. This consideration allows us to write a dispersion relation for R(⃗ q 2 ). We use a counter-clockwise contour C, and encircle the branch cut for negative q⃗2 in the positive sense. The contour extends from q⃗2 = −∞ to the origin, above the cut. The second part of the contour extends from q⃗2 = 0 to −∞, below the cut. We then complete the contour around the pole and obtain R(⃗ q2 ) = ∫

2 ∫ d(Q ) C

R(Q2 ) = 2πi R(⃗ q2 ) , Q2 − q⃗2

(15.41)

where the position of q⃗2 in the complex plane in this formula is anywhere except on the negative real axis. But then, the contribution from the circle at infinite radius does not contribute, because the integrand falls off sufficiently fast. Then, with an infinitesimal imaginary part i, one has (see Fig. 15.3) 2 ∫ d(Q ) C

2 2 0 0 R(Q2 ) 2 R(Q + i) 2 R(Q − i) = d(Q ) − d(Q ) . ∫ ∫ Q2 − q⃗2 Q2 − q⃗2 Q2 − q⃗2 −∞ −∞

(15.42)

We make the change of variable Q2 → −Q2 and normalize the cut of the integrand as follows, 1 [R(−Q2 + i ) − R(−Q2 − i )] , RC (Q2 ) = Q2 > 0 . (15.43) 2πi Taking all information from Eqs. (15.41), (15.42) and (15.43) into account, one can write the relation ∞ RC (Q2 ) R(⃗ q2 ) = − ∫ d(Q2 ) 2 . (15.44) q⃗ + Q2 0

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■♠✭◗✷ ✮ ✻ ❨ ❈

✙❈

◗✷ ❂ ⑦q ✷ ✂

✲ ✛

❥

❈

✲ ❘❡✭◗✷ ✮

❈ ✯

Fig. 15.3 The integration contour C used in Eqs. (15.41) and (15.42) encircles the pole at Q2 = q 2 in the mathematically positive sense.

It comes as a surprise that, in order to calculate the energy shift according to Eq. (15.28), one needs to carry out additional subtractions. Namely, the formula (15.39) implies that the scaled dimensionless integral K, being proportional to 3 q 4 ) R(⃗ q ), contains lower-order terms corresponding to double Coulomb ex∫ d q (1/⃗ change. We recall that the Coulomb potential, in momentum space, is proportional to 1/⃗ q 2 . We have ignored the momentum of the incoming bound-state wave function, which otherwise acts as an infrared regulator in the calculation. Hence, the infrared divergences are naturally cut off at the bound-electron momentum scale, which is Zα m. The integration of the infrared divergent terms with infrared cutoff ∞ q ∣ ∣⃗ q ∣−2 ∝ 1/(Zα m), would otherwise lead to contributions of proportional to ∫Zαm d∣⃗ which is of lower order in the (Zα)-expansion. Hence, we need to replace K by the subtracted integral d3 q 1 [R(⃗ q 2 ) − R(0)] , (15.45) K(1) = ∫ (2π)3 q⃗4 R(⃗ q 2 ) − R(0) = ∫

∞ 0

d(Q2 )

q⃗2 RC (Q2 ) . Q2 (⃗ q 2 + Q2 )

(15.46)

The subtracted integral K(1) evaluates to ∞ d3 q 1 RC (Q2 ) K(1) = ∫ d(Q2 ) 2 2 ∫ 2 3 (2π) q⃗ (⃗ q +Q ) Q2 0

∞ ∞ 4π 1 RC (Q2 ) d(Q2 ) ∫ d∣⃗ q∣ 2 ∫ 3 2 8π q⃗ + Q Q2 0 0 ∞ ∞ 4π π RC (Q2 ) 1 RC (Q2 ) = . = 3 ∫ dQ (2Q) dQ ∫ 8π 2Q Q2 2π 0 Q2 0

=

(15.47)

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There is another subtraction to be done in the Q-integral, which is also generated by a spurious infrared-divergent lower-order contribution. Infrared divergences pro∞ portional to ∫ dQ Q2 /Q4 → ∫Zα m dQ Q−2 ∝ 1/(Zα m) are cut off at the scale of the bound-state momentum. The infrared divergences describe lower-order terms in the Zα-expansion. The end result is the double-subtracted integral ∞ dQ 1 [RC (Q2 ) − RC (0)] . ∫ 2π 0 Q2

K(2) =

(15.48)

The energy shift (15.28) becomes α (Zα)5 m µ 3 ( ) δ`0 (4πK(2) ) . π n3 m Our task now is to calculate K(2) , and that is not completely trivial. ∆E =

15.3.4

(15.49)

Example Integral

Let us specialize Eq. (15.40) to diagram (c) of Fig. 15.2, Rc (⃗ q2 ) = ∫

CF

d4 k 1 1 ( − ) Tc (q, k) , i π 2 k 2 − λ2 k 2 − Λ2

(15.50)

where Tc is given in Eqs. (15.31), (15.36) and (15.37). The diagram in Fig. 15.2(a) is ultraviolet finite. For the corresponding expressions for diagrams Fig. 15.2(b), Fig. 15.2(c), and Fig. 15.2(d), one has to subtract suitable integral representations of the mass counter term δm, and of the vertex renormalization Z1 . After the renormalizations, one obtains ultraviolet convergent as well as infrared safe expressions, so that we can replace 1 1 1 1 − → 2 = . (15.51) 2 k 2 − λ2 k 2 − Λ2 k ω − k⃗2 From diagram (c), one obtains a particularly instructive example integral whose cut, for negative q⃗2 = −Q2 , we would like to calculate for illustration. Let us define 1 1 1 1 J = 2 ∫ d4 k 2 2 2 2 2 ⃗ ⃗ iπ CF ω − k X + k X + (k⃗ − q⃗)2 1 1 1 1 = 2 ∫ d4 k 2 , (15.52) 2 2 iπ CF k (t − k) − m (t − k + q)2 − m2 where we recall the definition of t from Eq. (15.19). We now use Feynman parameterization (see Sec. 10.3.5) in the form given in Eq. (10.159), 1 x 1 x 1 1 1 = 2 ∫ dx ∫ dy = 2 dx dy , ∫ ∫ 3 AB C [A(1 − x) + B(x − y) + Cy] [D(x, y)]3 0 0 0 0 (15.53)

where we implicitly define the denominator D(x, t). In our case, one has A = k2 ,

B = (t − k)2 − m2 ,

(15.54)

C = (t + q − k) − m = (p − k) − m , 2

2

2

2

p ≡ t+q,

(15.55)

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where we define p for our calculation. The denominator structure in Eq. (15.53) becomes, alternatively, D = k 2 (1 − x) + [(k − t)2 − m2 ](x − y) + [(k − p)2 − m2 ]y = k 2 (1 − x + x − y + y) + (−2t ⋅ k)(x − y) + (−2k ⋅ p + p2 − m2 ]y = k 2 − 2k ⋅ (t(x − y) + py) + (p2 − m2 ).

(15.56)

We now define ̃ k = k − (t(x − y) + py) ,

(15.57)

to rewrite the denominator as D=̃ k 2 − [t(x − y) + py]2 + (p2 − m2 )y =̃ k 2 − m2 x2 + (p2 − m2 )y − (p2 − m2 )y 2 =̃ k 2 − m2 x2 + (m2 − q⃗2 − m2 )y − (m2 − q⃗2 − m2 )y 2 =̃ k 2 − [m2 x2 + q⃗2 y (1 − y)] .

(15.58)

We have used the relations t2 = m2 ,

t ⋅ p = m2 ,

p2 = m2 − q⃗2 .

(15.59)

With the help of Eq. (10.138), one obtains the following integral for the total solid angle in Euclidean four-dimensional space, ∞ k3 1 2π 2 = 2π 2 , dkE 2 E 3 = . (15.60) Ω4 = ∫ Γ(2) (kE + Y ) 4Y 0 After a Wick rotation, ∫ d4̃ k → i ∫ d4̃ kE using the formula ̃ k 2 = −̃ k 2 , one obtains a E

convenient integral representation for J, 1 x 2 1 J = 2 ∫ dx ∫ dy ∫ d4̃ k 3 iπ 0 0 {̃ k 2 − [m2 x2 + q⃗2 y (1 − y)]} = −

1 x 2 1 4 kE ∫ dx ∫ dy ∫ d ̃ 3 2 2 + [m2 x2 + q π 0 0 {̃ ⃗2 y (1 − y)]} kE

x 2(2π 2 ) 1 1 ∫ dx ∫ dy 2 2 ⃗2 2 4π m x + q y (1 − y) 0 0 1 1 1 = − ∫ dx ∫ ds 2 . (15.61) m x + q⃗2 s (1 − xs) 0 0 In the last step, we have substituted y = xs, so that the integration interval becomes symmetric in x and s. It is clear that, for q⃗2 = q 2 = −Q2 < 0, the above integral develops an imaginary part. In the same spirit as given by Eqs. (15.41)–(15.43), we can write a dispersion relation 1 [J(−Q2 + i ) − J(−Q2 − i )] , JC (Q2 ) = Q2 > 0 , (15.62a) 2πi ∞ JC (Q2 ) . (15.62b) J(q 2 ) = − ∫ d(Q2 ) 2 q + Q2 0

= −

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Let us evaluate JC (Q2 ) with care, with the help of the Sochocki–Plemelj prescription (4.356) (see Chap. 4), JC (Q2 ) = − =− =∫ =∫

1 1 1 1 − (i → −i)) ∫ dx ∫ ds ( 2 2 2πi 0 m x + (−Q + i) s (1 − xs) 0 1 1 1 1 − (i → −i)) ∫ dx ∫ ds ( 2 2πi 0 m x − Q2 s (1 − xs) + i 0 1 0 1 0

dx ∫ ds ∫

1 0 1

0

ds δ(m2 x − Q2 s (1 − xs)) dx δ(x(m2 + Q2 s2 ) − sQ2 )) .

(15.63)

We have used the fact that s (1 − xs) ≥ 0 inside the integration interval. Now, let us see. In the integral over x, the Dirac-δ function peaks at sQ2 1 , δ(x(m2 + Q2 s2 ) − sQ2 )) = 2 δ(x − x0 ) . (15.64) 2 2 2 m +Q s m + Q2 s2 The question now is whether or not the point x0 still lies inside the integration interval x ∈ (0, 1). We thus need to consider the function x = x0 =

JC (Q2 ) = ∫

0

1

ds

sQ2 1 Θ (1 − ). m2 + Q2 s2 m2 + Q2 s2

(15.65)

Here, Θ is the step function. Let us consider the expression f (s, Q) =

m2

sQ2 , + Q2 s2

f (0, Q) = 0 ,

f (1, Q) =

Q2 ≤ 1. + Q2

m2

(15.66)

This means that, for s = 0 as well as s = 1, the Θ function in Eq. (15.65) is equal to unity. However, for sufficiently large Q, there is a region s1 < s < s2 where the expression f (s, Q) is larger than unity. One finds that √ √ 1 1 4m2 1 1 4m2 s = s1 = − 1− 2 , s = s2 = + 1− 2 . (15.67) 2 2 Q 2 2 Q Interestingly, the solutions exist only above the fermionic pair production threshold Q > 2m. This means that we must exclude, from the integration interval s ∈ (0, 1), the interval s ∈ (s1 , s2 ), but only if Q > 2m. So, s2 1 1 ds 2 − Θ(Q − 2m) ∫ . 2 2 +Q s m + Q2 s2 s1 0 The result can be expressed as follows, 1 m 1 2m JC (Q2 ) = arccot ( ) − Θ(Q − 2m) arccos ( ). mQ Q mQ Q The formula √ i ⎛ y − i 1 − y2 ⎞ √ arccos(y) = ln 2 ⎝ y + i 1 − y2 ⎠

JC (Q2 ) = ∫

1

ds

m2

is useful in the final simplifications.

(15.68)

(15.69)

(15.70)

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All integrals pertaining to the renormalized integrand R(⃗ q 2 ) can be evaluated using generalizations of the above result. Let us take, for example, the integral 4 ω k⃗ ⋅ q⃗ 1 1 ̃ q2 ) = ∫ d k , J(⃗ 2 2 2 π i ω − k [k⃗ 2 + X 2 ]2 (k⃗ − q⃗)2 + X 2

(15.71)

which is obtained if one substitutes, for T = Ta + Tb + Tc + Td in Eq. (15.40), terms from Ta = Na /Da , given in Eqs. (15.32) and (15.33). The cut of J̃ is obtained as 2 2 ̃ ̃ J̃C (Q2 ) = (2πi)−1 [J(−Q + i ) − J(−Q − i )], where √ Q2 Q2 m 4m2 2 2 ̃ JC (Q ) = − JC (Q ) + + Θ(Q − 2m) 2 1 − 2 . (15.72a) ⃗2) 4m Q Q 4m(m2 + Q For other integrals, a partial fraction decompositions like (ω 2

ω 1 1 1 = ( + ) 2 2 2 2 2 2 ⃗ ⃗ ⃗ ⃗ 2m − k ) (k + X ) ω −k k + X2

(15.72b)

help in simplifying some of the integrals. 15.3.5

Final Integration and A50 Coefficient

One finally obtains the following results for the cut functions, defined in Eq. (15.44), for the diagrams in Fig. 15.2. The diagram is indicated as a superscript in the results (a)

RC (Q2 ) = − m + (b)

RC (Q2 ) = −

8m6 + 18m4 Q2 + 12m2 Q4 + 3Q6 Q2 (m2 + Q2 )2

(15.73)

4m(−4m4 − m2 Q2 + Q4 ) Θ(Q − 2m) + m(8m2 − Q2 ) JC (Q2 ) , √ 4m2 4 Q 1 − Q2 m3 (8m4 + 10m2 Q2 + Q4 ) , Q2 (m2 + Q2 )2

(15.74) 2

(c) RC (Q2 )

4m m3 (4m2 + 5Q2 ) 4m (1 − Q2 ) Θ(Q − 2m) 2m3 (Q2 − 4m2 ) − = − JC (Q2 ) , √ 2 (m2 + Q2 )2 Q2 (Q2 /m2 ) 1 − 4m Q2

(15.75) (d)

(c)

RC (Q2 ) = RC (Q2 ) .

(15.76)

The JC function is given in Eq. (15.69). The function (a)

(b)

(c)

(d)

RC = RC + RC + RC + RC

(15.77)

has mass dimension one and is given as RC (Q2 ) = − m +

4(Q2 − 3m2 )(Q2 + 4m2 )Θ(Q − 2m) 16m4 + 20m2 Q2 + 3Q4 √ + m Q2 (m2 + Q2 ) Q4 1 − (4m2 )/Q2

m(16m4 + 4m2 Q2 − Q4 ) JC (Q2 ) . Q2

(15.78)

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It has the property RC (Q2 ) = −

16 m + O(Q2 ) , 3

(15.79)

which enters the formula (15.48). We recall from Eq. (15.48) that K(2) =

∞ dQ 1 [RC (Q2 ) − RC (0)] . ∫ 2π 0 Q2

(15.80)

The energy shift (15.49) can be written as ∆E =

α (Zα)5 m µ 3 ( ) A50 , π n3 m

(15.81)

where the dimensionless coefficient A50 is A50 = 4πK(2) δ`0 = 2δ`0 ∫

∞ 0

dQ [RC (Q2 ) − RC (0)] . Q2

(15.82)

The notation adopted for the coefficient A50 is explained in the following Sec. 15.4. The coefficient A50 can be expressed as a sum of a contribution X1 from the integration interval 0 < Q < 2m, and a contribution X2 from the integration interval 2m < Q < ∞, 5 9 + arctan(2) + 2i χ2 (2i) , 3 2 5 139π 9 − arctan(2) − 2π ln(2) − 2i χ2 (2i) . X2 = − + 3 32 2 X1 =

(15.83) (15.84)

Here, the Legendre χ function has been used, χν (z) =

1 [Liν (z) − Liν (−z)] . 2

(15.85)

In the end result, all the terms proportional to the Legendre χ cancel, and we have A50 = (X1 + X2 ) δ`0 = 4π δ`0 (

139 1 − ln(2)) = δ`0 9.29112 . . . , 128 2

(15.86)

a result first derived in Ref. [340]. 15.4

Higher-Order Terms

In the preceding discussion, we have obtained the self-energy binding correction to the Lamb shift of order α(Zα)5 in Eq. (15.86), based on the analysis of the Feynman diagrams in Fig. 15.1, using dispersion relations. Technical difficulties mount as the power of Zα is being increased (this concerns the number of Coulomb vertices in the high-energy part, and the order of the multipole expansion in the low-energy part of the bound-state self-energy). The higher-order binding corrections in the order α(Zα)6 m, for a number of excited states, were calculated in Refs. [163, 173, 337, 341–352].

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We here concentrate on the n = 1 and n = 2 states, just to give a brief review of the results obtained over the last couple of decades. The (real part of the) energy shift ∆ESE can be written as follows [540], α (Zα)4 m F (nLj , Zα) , (15.87) π n3 where we assume the non-recoil limit. The semi-analytic expansion of the scaled selfenergy function F (nLj , Zα) about Zα = 0 for a general atomic state with quantum numbers nLj (in spectroscopic notation) is ∆ESE =

F (nLj , Zα) = A41 (nLj ) ln[(Zα)−2 ] + A40 (nLj ) + (Zα) A50 (nLj ) + (Zα)2 [A62 (nLj ) ln2 [(Zα)−2 ] + A61 (nLj ) ln[(Zα)−2 ] + GSE (nLj , Zα)] .

(15.88)

Here, GSE (nLj , Zα) is the so-called self-energy remainder function. In the limit of a vanishing nuclear charge, Z → 0, one has GSE (nLj , Zα) → A60 (nLj ), which is another state-dependent coefficient. The A coefficients have two indices, the first of which denotes the power of Zα [including those powers implicitly contained in Eq. (15.87)], while the second index denotes the power of the logarithm ln[(Zα)−2 ]. For P states, the coefficients A41 , A50 and A62 vanish, and we have F (nPj , Zα) = A40 (nPj ) + (Zα)2 [A61 (nPj ) ln[(Zα)−2 ] + GSE (nPj , Zα)] . (15.89) The higher-order binding corrections A62 and A61 were evaluated in [163, 341–349]. Together with results compiled in Chap. 4 and in Sec. 15.3, we have for the 1S ground state of hydrogen, A41 (1S1/2 ) =

4 , 3

A50 (1S1/2 ) = 4π [

A40 (1S1/2 ) = 139 1 − ln 2] , 128 2

10 4 − ln k0 (1S) , 9 3 A62 (1S1/2 ) = −1 ,

(15.90) A61 (1S1/2 ) =

28 21 ln 2 − . 3 20

The numerical value of the Bethe logarithm ln k0 (1S) is given in Eq. (4.1) (to nine decimals) and in Eq. (4.337) to a very large numerical accuracy. For the metastable 2S (excited) state, we have A41 (2S1/2 ) =

4 , 3

A50 (2S1/2 ) = 4π [

A40 (2S1/2 ) = 139 1 − ln 2] , 128 2

10 4 − ln k0 (2S) , 9 3 A62 (2S1/2 ) = −1

(15.91) A61 (2S1/2 ) =

16 67 ln 2 + . 3 30

for the numerical value of ln k0 (2S), we refer to Eq. (4.337) and to Table 4.1. For the 2P1/2 and 2P3/2 states, we have the coefficients [173] 1 4 − ln k0 (2P ) , 6 3 1 4 A40 (2P3/2 ) = − ln k0 (2P ) , 12 3 A40 (2P1/2 ) = −

103 , 108 29 A61 (2P3/2 ) = . 90

A61 (2P1/2 ) =

(15.92) (15.93)

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Quantum Electrodynamics in Action: Atoms, Lasers and Gravity Table 15.2 We present a table of A60 coefficients for higher excited atomic states with positive Dirac angular quantum number κ (i.e., j = ` − 1/2). Data are taken from Ref. [352]. All decimal figures shown are significant. A60 Coefficients for States with κ > 0 n

A60 (nP1/2 )

A60 (nD3/2 )

A60 (nF5/2 )

A60 (nG7/2 )

2

−0.998 904 402

–

–

–

3

−1.148 189 956

0.005 551 573

–

–

4

−1.195 688 142

0.005 585 985

0.002 326 988

–

5

−1.216 224 512

0.006 152 175

0.002 403 151

0.000 814 415

6

−1.226 702 391

0.006 749 745

0.002 531 636

0.000 827 468

7

−1.232 715 957

0.007 277 403

0.002 661 311

0.000 857 346

Table 15.3 In the analog of Table 15.1 for states with negative κ, we combine data from Ref. [352] with calculations reported in Ref. [312]. A60 Coefficients for States with κ < 0 n

A60 (nS1/2 )

A60 (nP3/2 )

A60 (nD5/2 )

A60 (nF7/2 )

A60 (nG9/2 )

1

−30.924 149 46

–

–

–

–

2

−31.840 465 09

−0.503 373 465

–

–

–

3

−31.702 501

−0.597 569 388

0.027 609 989

–

–

4

−31.561 922

−0.630 945 795

0.031 411 862

0.007 074 961

–

5

−31.455 393

−0.647 013 508

0.033 077 570

0.008 087 015

0.002 412 929

6

−31.375 130

−0.656 154 893

0.033 908 493

0.008 610 109

0.002 748 250

7

−31.313 224

−0.662 027 568

0.034 355 926

0.008 906 989

0.002 941 334

The evaluation of the A60 coefficient presented nearly insurmountable technical challenges in its evaluation. Ground-breaking work by Pachucki2 (Ref. [163]) led the pathway to a resolution. The calculation of the A60 coefficient necessitates, among other things, the evaluation of so-called relativistic Bethe logarithms, which are generated by the relativistic corrections to the wave function. In order to derive the corresponding expressions, one consults Chap. 11 for the relativistic corrections to the Schr¨ odinger–Coulomb Hamiltonian and applies the corrections to the wave function, and the Hamiltonian and reference-state energy in the propagator of the 2 Krzysztof

Pachucki (b. 1963).

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nonrelativistic bound-state self-energy, as discussed in Sec. 4.6.2. From Refs. [267, 312], we have the numerically precise results A60 (1S) = −30.924 149 46(1) ,

A60 (2S) = −31.840 465 09(1) .

(15.94)

From Ref. [351], we have the following results for P states, A60 (2P1/2 ) = −0.998 904 402(1) ,

A60 (2P3/2 ) = −0.503 373 465(1) .

(15.95)

At nuclear charge Z = 1, we have the following results for n = 1 and n = 2 states, from the high-precision numerical study reported in Ref. [540]: GSE (1S1/2 , Z = 1) = − 30.290 24(2) ,

(15.96a)

GSE (2S1/2 , Z = 1) = − 31.185 15(9) ,

(15.96b)

GSE (2P1/2 , Z = 1) = − 0.973 5(2) ,

(15.96c)

GSE (2P3/2 , Z = 1) = − 0.486 5(2) .

(15.96d)

For higher excited S states, results have been indicated in Ref. [541], while for higher excited P states, one consults Ref. [542]. The calculation of the nonperturbative remainder function, in the expansion parameter Zα, necessitates the use of advanced numerical techniques [543] and a suitable formulation of the numerical problem, in terms of virtual photon integration contours [257, 258]. 15.5 15.5.1

Relativistic Recoil Correction Retardation and Salpeter Correction

The physical essence of the relativistic recoil correction to atomic energy levels, which was calculated by Salpeter [534], is understood as follows. We had already stressed in our derivation of the Breit Hamiltonian in Sec. 12.3.3 [see the discussion following Eqs. (12.47)] that the poles of the photon propagator are being ignored in the derivation of the effective Hamiltonian. This leads, in the end, to a static Hamiltonian, beyond which we encounter self-energy and recoil effects. Expressed differently, the Fourier transform of a constant in frequency space has a Dirac-δ dependence in time, i.e., it is proportional to δ(t − t′ ), i.e., to a non-retarded, instantaneous interaction. Retardation effects are induced by an expansion about the case of a constant in frequency space and correspond to the nontrivial ω-dependence of the propagator denominators. In our discussions of the van der Waals and Casimir–Polder interactions and of the short-range and long-range forms of the atom-surface interactions (see Chap. 5), we took into account the fact that the atom needs to undergo a virtual transition to an excited state in order to be subject to any interaction at all; all results were proportional either to the static, or to the dynamic, polarizability. The necessity to consider retardation arose from the failure of the expansions of the “retardation exponentials” exp(−ωR/c) and exp(−ωz/c) in terms of their arguments, in the long-distance limits (see Sec. 5.4.1). In the case of photon exchange among two

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constituent particles of an atom, the situation is different. The “retardation exponential” is of the form exp(ik⃗ ⋅ r⃗), where k⃗ is the modulus of the wave vector of the exchanged photon, and r⃗ is an atomic coordinate. For a Coulomb exchange, we have ⃗ ∼ Zαµ and ∣⃗ ∣k∣ r∣ ∼ 1/(Zαµ), and thus, the “retardation expansion” is unavailable for the higher-order terms beyond the Breit interaction. Note that the low-energy part of the self-energy correction discussed in Sec. 4.6 is mediated by even softer ⃗ ∼ (Zα)2 µ, whereas the Breit interaction and the dominant part of photons, with ∣k∣ ⃗ ∼ Zαµ. the Salpeter correction correspond to photons with ∣k∣ The question then is, to which extent we can understand the Salpeter correction as the retardation correction to the Breit interaction. The answer is that the retardation expansion here comes from the fact that the exchange photon energy is large as compared to a typical virtual transition frequency of the atom. In the propagator denominators, we can thus expand 1 H −E 1 → − + ⋯, (15.97) H −E+ω ω ω2 ⃗ ≡ k ∼ Zαµ is the virtual photon frequency. We had already pointed out where ω = ∣k∣ that within the retardation expansion, which is relevant to the middle-energy part of ⃗ r) in powthe recoil correction, the multipole expansion, i.e., the expansion of exp(ik⋅⃗ ers of its argument, remains unavailable. However, instead, the expansion (15.97) leads to a tremendous simplification of the calculation. We aim to calculate the energy shift of order (Zα)5 µ3 /(m1 m2 ), beyond the Breit interaction, as given by the diagrams in Fig. 15.4. This recoil correction is exclusively given by one- and two-photon exchange between the constituent particles of the system. We have to use a manifestly two-body QED Hamiltonian and analyze the exchange diagrams between the two particles. We denote by e = −∣e∣ the electron charge and consider one particle of charge q1 = e, and assume there is another spin1/2 particle (the nucleus) of charge q2 = −Ze. For the low-energy part of the effect, we can use the total Hamiltonian of the two-body system as follows, ⃗ r1 ))2 (⃗ ⃗ r2 ))2 Zα (⃗ p1 − eA(⃗ p2 + ZeA(⃗ + − H= 2m1 2m2 r e ⃗ r1 ) + Ze σ ⃗ r2 ) + ∑ k a+⃗ a⃗ . ⃗1 ⋅ B(⃗ ⃗2 ⋅ B(⃗ (15.98) − σ kλ kλ 2m1 2m2 ⃗ kλ Here, the masses of the two particles are m1 and m2 , and we define r = ∣⃗ r∣ where ⃗1 and σ ⃗2 are the Pauli spin matrices for the two particles. The r⃗ = r⃗1 − r⃗2 . Also, σ ⃗ The unperturbed Hamiltonian is photon energy is k = ∣k∣. p⃗ 2 p⃗ 2 Zα H0 = 1 + 2 − + ∑ k a+kλ (15.99) ⃗ . ⃗ akλ 2m1 2m2 r ⃗ kλ

We now follow the discussion in Chap. 4 [see Eq. (4.1)] and in Secs. 13.2.2 and 13.2.3 and express H0 in center-of-mass coordinates (⃗ p1 = −⃗ p2 ) and the sum of the Schr¨ odinger and field Hamiltonians, p⃗2 Zα H0 = − + ∑ k a+kλ (15.100) ⃗ = HS + HF . ⃗ akλ 2µ r ⃗ kλ

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(a)

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(b)

Fig. 15.4 The recoil correction to the Lamb shift involves a simple exchange of a virtual photons, but the two exchanges happen at manifestly different points in time; the two-body system is in a virtual excited state in between the exchanges, as displayed by the non-vertical photon lines. If the lines are made vertical and the contributions from the two diagrams are added, then the transverse part of the Breit interaction is recovered.

The reduced mass is µ = m1 m2 /(m1 +m2 ), and p⃗ = −i ∂∂r⃗ is the canonical momentum for the relative coordinate. Once a matrix element is formulated in terms of r⃗ only (no explicit appearance of r⃗1 and r⃗2 ), we can immediately replace p⃗1 → p⃗ and p⃗2 → −⃗ p. The normalized eigenfunctions φn (⃗ r) ≡ ψn`m (⃗ r) [see Eq. (4.60)], have the bound-state energies (n is a multi-index in our notation) En = −

(Zα)2 µ . 2 n2

(15.101)

The interaction Hamiltonian for a system of two spin-1/2 particles is given as follows, 2 2 e ⃗ r1 ) + Ze p⃗2 ⋅ A(⃗ ⃗ r2 ) + e A(⃗ ⃗ r1 )2 + (Ze) A(⃗ ⃗ r2 )2 p⃗1 ⋅ A(⃗ m1 m2 2m1 2m2 e ⃗ r1 ) + Ze σ ⃗ r2 ) , ⃗1 ⋅ B(⃗ ⃗2 ⋅ B(⃗ − σ (15.102) 2m1 2m2

HI = −

where we courageously ignore the necessity to hermitize the interaction by anticom⃗ r1 ) → 1 {⃗ ⃗ r1 )}, which would otherwise lead to a more clumsy mutators, p⃗1 ⋅ A(⃗ p1 , A(⃗ 2 notation. However, the sequence of the operators actually does not matter in the derivations to follow (up to the order of approximation relevant to our studies here). The details are left as an exercise to the reader. 15.5.2

Low-Energy Part and Dipole Approximation

The unperturbed state consists of the atom in the state φn (⃗ r) and zero photons in the radiation field. We know that the second-order perturbation with two couplings ⃗ r1 ) and p⃗1 ⋅A(⃗ ⃗ r1 ) affecting the same charged particle, yield self-energies already p⃗1 ⋅A(⃗ discussed in Sec. 4.6.2. The low-energy part is mediated by long-wavelength photons where we can still use the dipole approximation. In the extreme low-energy region, we can ignore the magnetic couplings from Eq. (15.102). The exchange contribution

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from the diagram in Fig. 15.4(a) is e 1 Ze ⃗ r1 )) ⃗ r2 ))⟩ . ∆Ea = ⟨(− p⃗1 ⋅ A(⃗ ( p⃗2 ⋅ A(⃗ (15.103) m1 E0 − H0 m2 We recall the vector field operator given in Eq. (4.309), within the dipole approximation, ⃗ (aλ (k) ⃗ + a+ (k)) ⃗ . ⃗ r) ≈ A(⃗ ⃗ r = 0) ⃗ =∑√ 1 A(⃗ ˆλ (k) (15.104) λ 2ω V ⃗ ⃗ k kλ In full analogy to Eqs. (4.289) and (4.311), one then obtains the low-energy part of the energy shift ⃗ j (k) ⃗ ⟨φn ∣pi1 ∣ φm ⟩ ⟨φm ∣pj2 ∣ φn ⟩ iλ (k) Ze2 λ ∆Ea = − ∑∑ m1 m2 m kλ 2kV En − Em − k ⃗ j i Ze2 d3 k 1 k i k j ⟨φn ∣p1 ∣ φm ⟩ ⟨φm ∣p2 ∣ φn ⟩ ij (δ − ) ∑∫ m1 m2 m (2π)3 2 k En − Em − k k⃗2 2 Zα 1 i = pi ∣ φn ⟩ . (15.105) ∫ dk k ⟨φn ∣p 3 πm1 m2 En − HS − k The second diagram leads to an equal energy shift ∆Eb = ∆Ea . We take the integration limits for the photon energy as zero and Zαµη, where η is a dimensionless scale-separation parameter. So, the whole low-energy part is ∆EL = ∆Ea + ∆Eb with Zαµη 1 4 Zα dk k ⟨φn ∣pi pi ∣ φn ⟩ ∆EL = − ∫ 3 πm1 m2 0 HS − En + k Zαµη 4 Zα k + HS − En − (HS − En ) i = − p ∣ φn ⟩ dk ⟨φn ∣pi ∫ 3 πm1 m2 0 HS − En + k

= −

=

ZαµK 4 Zα (HS − En ) i dk ⟨φn ∣pi p ∣ φn ⟩ + Ren ∫ 3 πm1 m2 0 HS − E n + k

=

Zαµη 4 Zα ⟨φn ∣pi (HS − En ) ln ( ) pi ∣ φn ⟩ 3 πm1 m2 ∣HS − En ∣

=

8 (Zα)5 µ3 Zαµη n3 ⟨φn ∣[pi , [HS − En , pi ]]∣ φn ⟩ [ln ( ) 3 πn3 m1 m2 (Zα)2 µ/2 4µ3 (Zα)4

∣HS − En ∣ ) pi ∣ φn ⟩] (15.106) (Zα)2 µ/2 where Ren are physically irrelevant renormalization terms as introduced in Sec. 4.6.2, and a principal-value prescription is understood for the k-integration. Now, we use the fact that a Dirac-δ potential is nonvanishing only for S states, n3 ⟨φn ∣[pi , [HS − En , pi ]]∣ φn ⟩ = δ`0 . (15.107) 3 4µ (Zα)4 We recall the definition of the Bethe logarithm from Eq. (4.317) and write the low-energy part as 8 (Zα)5 µ3 2η [ln ( ) δ`0 − ln k0 (n, `)] . (15.108) ∆EL = 3 πn3 m1 m2 Zα −

n3

2µ3 (Zα)4

⟨φn ∣pi (HS − ES ) ln (

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For systems with a heavy nucleus like hydrogen, the above correction is small, because m2 (the proton mass) exceeds m1 (the electron mass) by orders of magnitude. However, for m1 = m2 (positronium), the low-energy part of the one-photon exchange is numerically as important as the self-energy. 15.5.3

Middle-Energy Part and Araki–Sucher Distribution

In the one-photon exchange correction, the photon is emitted and absorbed at two ⃗ ⃗ different places, and the factors eik⋅⃗r1 (for absorption) and e−ik⋅⃗r2 (for emission) therefore do not compensate each other as for the self-energy. The resulting ex⃗ ⃗ ponential eik⋅(⃗r1 −⃗r2 ) = eik⋅⃗r serves as an ultraviolet cutoff for the photon energy integration and implies that we can eliminate the scale-separation parameter η, already by considering the middle-energy part ∆Em , which is given by photons with k ≥ Zαµ η. The magnetic term could be important but we will see it is not. Convergence for large photon energies is generated by a different mechanism as compared to the Lamb shift. In the interaction Hamiltonian (15.102), we have included both the minimal p⃗⋅ A⃗ ⃗ coupling of the spin to the magnetic field of the exchange ⃗ ⋅B coupling as well as the σ photon. Employing the same formalism as in the derivation of Eq. (15.105), but dropping the dipole approximation and adding the magnetic coupling, we obtain the expression d3 k 1 ki kj Ze2 ij (δ − ) ∫ m1 m2 (2π)3 2 k k⃗2 1 i i ⃗ r1 ⃗ r2 ⃗ i ) eik⋅⃗ ⃗ j ) e−ik⋅⃗ σ1 × k) (pj2 − (⃗ σ2 × k) ∣ φn ⟩ . × ⟨φn ∣(pi1 + (⃗ 2 En − HS − k 2 (15.109)

∆Ea′ = −

We note the sign for the magnetic coupling of the photon emission operator, which is obtained from the expression 1 i ⃗ r2 ⃗ r2 ⃗ j ) e−ik⋅⃗ ⃗ 2 )j ) e−ik⋅⃗ (pj2 + (⃗ σ2 × ∇ = (pj2 − (⃗ σ2 × k) . (15.110) 2 2 It is left as an exercise to the reader to calculate the corresponding expression for virtual photon absorption, which also enters Eq. (15.110). The inclusion of the magnetic coupling is necessary because we are working in a regime where the exchange photon momentum is of order p ∼ k ∼ Zαµ, so that the expansion (15.97) is applicable, 1 1 HS − En 1 =− + +O( 3) . (15.111) 2 En − HS − k k k k The term −1/k leads to the expression ̃a = δE

Ze2 d3 k 1 ki kj ij (δ − ) ∫ m1 m2 (2π)3 2 k 2 k⃗2 i i ⃗ r1 ⃗ r2 ⃗ i ) eik⋅⃗ ⃗ j ) e−ik⋅⃗ × ⟨φn ∣(pi1 + (⃗ σ1 × k) (pj2 − (⃗ σ2 × k) ∣ φn ⟩ . 2 2

(15.112)

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This evaluates to ij i j ⃗1 ⋅ σ ⃗2 (⃗ σ1 ⋅ r⃗) (⃗ σ2 ⋅ r⃗) ̃a = Zα ⟨φn ∣pi ( δ + r r ) pj ∣ φn ⟩ − Zα ( σ −3 ) δE 1 2 4m1 m2 r r3 8m1 m2 r3 r5 πZα ⃗1 ⋅ σ ⃗2 δ (3) (⃗ + σ r) . (15.113) 3m1 m2 The negative sign in front of the first term is obtained in the center-of-mass frame where the momentum operator assumes the form p⃗ ≡ p⃗1 = −⃗ p2 . The two remaining terms are easily identified as the spin-spin interaction, and the spin-spin contact interaction, which were already given for the helium system in Eqs. (13.33d) and (13.33e). As compared to the electron-electron interaction in helium, we here replace α → −Zα. Furthermore, a factor 2 is supplied by the diagram in Fig. 15.4(b). We have thus explicitly verified that the first term in Eq. (15.111) exactly corresponds to a lower-order term from the Breit interaction, which has already been taken into account. In the middle-energy part, the next-to-leading term in Eq. (15.111) is of relative order Zα and contributes to the recoil correction. We can thus express the middle-energy part generated by diagram (a) in Fig. 15.4 as ∞ d3 k 1 ki kj Ze2 ij (δ − ∆Ea = − ) ∫ m1 m2 Zαµη (2π)3 2k 3 k⃗2 i i ⃗ r1 ⃗ r2 ⃗ i ) ei k⋅⃗ ⃗ j ) e−i k⋅⃗ (HS − En ) (pj2 − (⃗ ∣ φn ⟩ . σ1 × k) σ2 × k) × ⟨φn ∣(pi1 + (⃗ 2 2 (15.114) We can now use the equation (HS − En )∣φn ⟩ = 0 and the commutator relation A B A = 12 A2 B + 21 B A2 + 12 [A, [B, A]], and the fact that the reference state fulfills (HS − En )∣φn ⟩ = 0, to write the matrix element in Eq. (15.114) in the form i i ⃗ r1 ⃗ r2 ⃗ i ) ei k⋅⃗ ⃗ j ) e−i k⋅⃗ σ1 × k) (HS − En ) (pj2 − (⃗ σ2 × k) ⟩ M = ⟨(pi1 + (⃗ 2 2 1 i i ⃗ r1 ⃗ r2 ⃗ i ) ei k⋅⃗ ⃗ j ) e−i k⋅⃗ = ⟨[(pi1 + (⃗ σ1 × k) , [HS − En , (pj2 − (⃗ σ2 × k) ] ]⟩ . 2 2 2 (15.115) In the latter step, we suppress the reference state φn in the notation. Here, the ⃗i × k⃗ is of the order of Zαµ, so it not photon wave vector k⃗ in the expression σ a priori permissible to neglect the magnetic terms. However, after the evaluation ⃗ r2 ⃗ j e−i k⋅⃗ of the commutator [(HS − En ) , i (⃗ σ2 × k) ], we are left with an expression that is only a function of r⃗2 , and therefore, the commutator of the spin-dependent interaction terms with the “left” expression in Eq. (15.115) vanishes. Using the formula [A, BC] = [A, B] C + B [A, C] repeatedly, one finally obtains 1 ⃗ ⃗ M = ⟨[pi1 ei k⋅⃗r1 , [HS − En , pj2 e−i k⋅⃗r2 ] ]⟩ 2 1 Zα j p⃗ 2 ⃗ ⃗ ⃗ ⃗ , p2 ] ] + [pi1 ei k⋅⃗r1 , pj2 [ 2 , e−i k⋅⃗r2 ]⟩ = ⟨[pi1 ei k⋅⃗r1 , e−i k⋅⃗r2 [− 2 r 2m2 =

1 i k⋅⃗ Zα j 1 i k⋅⃗ Zα ⃗ ⃗ ⃗ ⟨e r1 [pi1 , [− , p2 ] ] e−i k⋅⃗r2 ⟩ = ⟨e r [pi , [pj , − ]]⟩ . 2 r 2 r

(15.116)

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The resulting expression for ∆Ea′ reads as ∆Ea′ =

∞ 1 d3 k 1 ki kj Zα Ze2 ⃗ ) { ⟨ei k⋅⃗r [pi , [pj , (δ ij − ] ]⟩} . ∫ 3 3 m1 m2 Zαµη (2π) 2k 2 r k⃗2 (15.117)

It is left as an exercise to the reader to verify that the diagram in Fig. 15.4(b) leads to an equal contribution, so that the middle-energy part reads as ∆EM =

∞ d3 k 1 ki kj Ze2 Zα ⃗ ij (δ − ] ] ∣ φn ⟩ . ) ⟨φn ∣ei k⋅⃗r [pi , [pj , ∫ 3 3 2 ⃗ m1 m2 Zαµη (2π) 2k r k (15.118)

This expression has to be handled with extreme care, because [pi , [pj ,

Zα Zα Zα 3 xi xj 4π ij ] = ∇i (∇j (− )) = 3 (δ ij − )+ δ Zα δ 3 (r) , (15.119) 2 r r r r 3

and the matrix element of 1/r3 is divergent for a reference S state at the origin. We thus face a rather peculiar situation and choose a different sequence for the d3 k and d3 r-integrations in Eq. (15.118), depending on the angular symmetry of the reference state. For S states, it is more convenient to do the evaluation of ⃗ because the wave function is radially the matrix element first, as a function of k, symmetric, and it is therefore possible to do the angular momentum algebra easily. For P states and higher angular momenta, by contrast, one does the k-integration first, which leads to a simple expression in coordinate space, yet, to a matrix element which is convergent only for states with ` ≠ 0. Let us consider the 1S state first. We apply the ansatz ⃗

N ij = ⟨1S ∣ei k⋅⃗r [pi , [pj , ⃗

= ⟨1S ∣ei k⋅⃗r [

Zα ] ] ∣ 1S⟩ r

4π ij Zα ij 3 xi xj (δ − )+ δ Zα δ (3) (⃗ r)]∣ 1S⟩ r3 r2 3

= A(k) k⃗2 δ ij + B(k) k i k j ,

(15.120)

where A(k) and B(k) can be determined based on suitable projection operations. We multiply both sides of Eq. (15.120) by δ ij and obtain the relation S = N ii = ⟨1S ∣4π Zα δ (3) (⃗ r)∣ 1S⟩ = 4 (Zα)4 µ3 = (3 A + B) k⃗2 .

(15.121)

Second, we multiply both sides of Eq. (15.120) by k i k j and obtain a second relation for A and B, ⃗

T = k i k j N ij = ⟨1S ∣ei k⋅⃗r (

Zα ⃗2 3 (k⃗ ⋅ r⃗)2 4π ⃗2 (k − )+ k Zα δ 3 (r))∣ 1S⟩ = (A+B) k⃗4 . 3 2 r r 3 (15.122)

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Even if the radial symmetry of the S state wave function leads to a mutual cancelation of the divergence at the origin in between the terms k⃗2 − 3 (k⃗ ⋅ r⃗)2 /r2 , the matrix element T is still challenging to evaluate. We write it as Zα 3 (k⃗ ⋅ r⃗)2 4π ⃗2 ⃗ T = ⟨1S ∣ei k⋅⃗r ( 3 (k⃗2 − )+ k Zα δ (3) (⃗ r))∣ 1S⟩ 2 r r 3 ∞ dr 1 2π 4(Zα)4 µ3 ⃗2 2 dφ ∫ du ∫ k + Zα k⃗2 ∫ [φ1S (⃗ r)] (1 − 3 u2 ) ei kru 3 r 0 −1 0 2Zαµ 2Zαµ = 4(Zα)4 µ3 (k⃗2 + 4(Zα)2 µ2 ) [1 − arccot ( )] . (15.123) k k We can now solve Eqs. (15.121) and (15.122) for A and B as a function of S and T , with the result

=

4(Zα)5 µ4 4(Zαµ)2 2Zαµ Zαµ [(1 + ) arccot ( )−2 ], 3 2 ⃗ k k k k 4(Zα)4 µ3 Zαµ 4(Zαµ)2 2Zαµ (Zαµ)2 B(k) = [−3 (1 + ) arccot ( )+6 + 1] . 2 k k k k⃗2 k⃗2 (15.124) A(k) =

After integration over k, the result for ∆EM for the ground state is ∞ d3 k 1 ki kj Zα Ze2 ⃗ (δ ij − ) ⟨ei k⋅⃗r [pi , [pj , ]]⟩ ∫ 3 3 m1 m2 Zαµη (2π) 2k r k⃗2 ∞ Ze2 d3 k 1 ki kj ij ) (A(k) k⃗2 δ ij + B(k) k i k j ) = (δ − ∫ m1 m2 Zαµη (2π)3 2k 3 k⃗2

∆EM (1S) =

8 (Zα)5 µ3 η 4 [− ln ( ) + ] . (15.125) 3 π m1 m2 2 3 In the sum of the low-energy and middle-energy parts, the dependence on η cancels for the ground state, =

∆EL+M (1S) = ∆EL (1S) + ∆EM (1S) =

8 (Zα)5 µ3 4 4 [ln ( ) + − ln k0 (1S)] . 3 π m1 m2 Zα 3 (15.126)

For the weighted difference of S states, and for states with nonvanishing angular momenta, the d3 k integral is carried out first in Eq. (15.118). One has to calculate J ij = ∫ ⃗

∣k∣≥Zαµη

ki kj d3 k 1 ij (δ − ) exp(i k⃗ ⋅ r⃗) = a(r) δ ij r2 +b(r) ri rj . (15.127) 2 ⃗ (2π)3 2k 3 k

Multiplication by δ ij yields J ij δ ij = ∫ ⃗

∣k∣≥Zαµη

d3 k 1 (3 − 1) exp(ik⃗ ⋅ r⃗) = (3a + b) r2 , (2π)3 2k 3

whereas a multiplication by ri rj gives the result d3 k 1 (k⃗ ⋅ r⃗)2 (r2 − ) exp(ik⃗ ⋅ r⃗) = (a + b) r4 . J ij ri rj = 3 3 (2π) 2k k2

(15.128)

(15.129)

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The two equations for two unknowns, a(r) and b(r), can now be solved as follows, a(r) =

γE ln(Zαµ η r) 5 − − , 36π 2 r2 6π 2 r2 6π 2 r2

b(r) =

1 . 12π 2 r2

(15.130)

Both results for a(r) and b(r) are given for small Zαµη, which is sufficient for our purposes. The infrared divergence of a(r) is cut off at k = Zαµη, which leads to the logarithmic divergence in coordinate space for r → 0. We now multiply the entire expression by the term [pi , [pj , −

Zα 3ri rj − δ ij r2 4π ij ]] = −Zα + δ Zα δ (3) (⃗ r) , r r5 3

(15.131)

where the delta term is irrelevant for non-S states. We finally have J ij [pi , [pj , −

Zα 2Zα δ (3) (⃗ r) Zα ]] = − 2 3 + [1 − γE − ln(Zαµ η r)] . r 6π r 3π

(15.132a)

For non-S states (they vanish at the origin), we can replace J ij [pi , [pj , −

Zα Zα ]] → − 2 3 . r 6π r

(15.132b)

For the 1/r3 term, for general hydrogenic states, one may write a regularizing distribution due to Araki3 [537] and Sucher4 [538]. It is canonically denoted as P , and is defined as follows, 1 1 r) [ 3 Θ(r − a) + 4πδ (3) (⃗ ) ∣ψ⟩ = lim ∫ d3 r φ∗ (⃗ r) (γE + ln(a))] ψ(⃗ r) . 3 a→0 r r (15.133) Sandwiched between hydrogen eigenstates, this distribution has a defined expectation value for both S and non-S states. For S states, it has a finite expectation value, and its weighted difference for S states [see Eq. (15.135) below] describes the functional dependence of the 1/r3 matrix element. For non-S states, the expectation value of the Araki–Sucher distribution is just equal to the expectation value of the operator 1/(Zαµr)3 . The Araki–Sucher distribution has compensating terms to cut off the logarithmic divergence of the integral for small radial coordinate and is independent of a in the limit a → 0. This can be verified if one assume that the rest of the integrand, namely, the product φ∗ (⃗ r) ψ(⃗ r), approaches a constant in the limit r⃗ → ⃗0. As a function of the bound-state quantum numbers, we have ⟨φ∣ P (

⟨P (

δ`0 1 1 1 )⟩ ≡ 3 (⟨⟨nS ∣ ∣ nS⟩⟩ − 4 ln 2) + (1 − δ`0 ) ⟨ ⟩ 3 3 (Zαµr) n (Zαµr) (Zαµr)3 4 n 1 1 2 1 − δ`0 = δ`0 3 [ln ( ) − Ψ(n) − γ + − ]+ 3 . n 2 2 2n n ` (` + 1) (2` + 1) (15.134)

3 Huzihiro 4 Joseph

Araki (b. 1932). Sucher (1930–2019).

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(a)

(b)

Fig. 15.5 The seagull contribution to the low-energy part of the recoil correction involves the exchange of four photons, two of which emerging from either fermion line. Diagrams (a) and (b) illustrate the different possible time orderings.

Here, the expectation value of the weighted difference of S states reads as ⟨⟨nS ∣

1 1 1 ∣ nS⟩⟩ = n3 ⟨nS ∣ ∣ nS⟩ − ⟨1S ∣ ∣ 1S⟩ . 3 3 (Zαµr) (Zαµr) (Zαµr)3

(15.135)

The logarithmic term −4 ln(2) in the S state contribution to the P distribution is chosen so that the non-S state expectation value has a particularly simple form. The complete result for the middle-energy part is ∆EM =

4 1 1 8 (Zα)5 µ3 [(− ln η + ) δ`0 − ⟨P ( )⟩] . 3 π m1 m 2 3 4 (Zαµr)3

(15.136)

Adding the low-energy part from Eq. (15.108) and the middle-energy part from Eq. (15.136), we have for ∆EL+M = ∆EL + ∆EM , ∆EL+M =

4 1 1 8 (Zα)5 µ3 2 [(ln ( ) + ) δ`0 − ⟨P ( )⟩ − ln k0 (n, `)] . 3 3 π n m1 m2 Zα 3 4 (Zαµr)3 (15.137)

The scale-separation parameter η has canceled. 15.5.4

Seagull Part

For the one-electron Lamb shift, the so-called seagull terms proportional to A⃗2 in the interaction Hamiltonian (15.102) do not contribute to the energy shift because their contribution can be absorbed in a mass renormalization; the atom is not in an excited, virtual state in between the seagull photon emissions and absorptions. However, for a two-particle system, the seagull contribution (see Fig. 15.5) is nontrivial, because the emission and absorption processes involve two different particles. The seagull term involves two diagrams, each of which yield equal contributions, and so ∆ES = 2 ×

′ e4 1 ⃗ r1 )2 ( ⃗ r2 )2 ∣ φn ⟩ . ⟨φn ∣A(⃗ ) A(⃗ 4m1 m2 E 0 − H0

(15.138)

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The atom undergoes an “instantaneous” emission and absorption process and therefore is not in an excited state. We can thus replace the expression E0 − H0 in Eq. (15.138) by −q1 − q2 , where q1 = ∣⃗ q1 ∣ and q2 = ∣⃗ q2 ∣ are the momenta of the exchanged seagull photons. (H0 contains photon field terms.) Let us motivate the expression for this energy shift. From √ the form of the vector field operator, we know that each photon carries a factor 1/(2q). As already stated, the propagator in the middle is just 1/(E0 − H0 )′ → −1/(q1 + q2 ), from the field part of H0 , because the electron state has not changed, but the photon field undergoes a transition from a zero-photon state to a two-photon, virtual state. We now add a factor 2 because of the presence of two diagrams, and another factor 2 because we can, colloquially speaking, glue the photon lines emerging from the two vertices together in two different ways. The details are left as an exercise to the reader. The two transverse photon propagators and two factor 1/(2m1 ) and 1/(2m2 ) for the two seagull vertices lead to a momentum-space expression M , which reads qi qj qi qj 1 1 Ze2 Ze2 −1 (δ ij − 1 2 1 ) (δ ij − 2 2 2 ) . (15.139) M =2×2× 2q1 2q2 q1 + q2 q1 q2 2m1 2m2 This means that Z 2 e4 d3 q1 d3 q2 d3 p φ∗ (⃗ p − q⃗1 ) φ(⃗ p + q⃗2 ) ES = − ∫ ∫ ∫ 3 3 4m1 m2 (2π) (2π) (2π)3 q1 q2 (q1 + q2 ) × (δ ij −

q1i q1j q2i q2j ij ) (δ − ) q12 q22

q1 − q⃗2 ) d3 q1 d3 q2 ρ(⃗ (⃗ q1 ⋅ q⃗2 )2 Z 2 e4 (1 + ). (15.140) ∫ ∫ 3 3 4m1 m2 (2π) (2π) q1 q2 (q1 + q2 ) q12 q22 Using the paradigm that convolution in momentum space is equivalent to multiplication in coordinate space, we recognize the Fourier transform of the charge density as d3 p ∗ ρ(⃗ q1 + q⃗2 ) = ∫ φ (⃗ p − q⃗1 ) φ(⃗ p + q⃗2 ) . (15.141) (2π)3 The UV divergence is created by the direction where q⃗1 is parallel to q⃗2 , and so we do not need to regularize the second term. The first corresponds to the exchange of two photons, and so it has to be regularized according to 1 1 1 1 Λ2 → − = , Λ = Zαµλ . (15.142) q2 q 2 q 2 + Λ2 q 2 q 2 + Λ2 The latter term would actually vanish were it not for the fact that it leads to a singular contribution at the origin in coordinate space which leads to a nonvanishing contribution for S states. Just as for the middle-energy part, we need to consider the ground state, first. In that case, ρ is just the Fourier transform of the charge density of the ground state of the hydrogen atom and reads as follows, 16(Zαµ)4 ρ1S (⃗ q1 + q⃗2 ) = . (15.143) 2 [(⃗ q1 + q⃗2 )2 + 4(Zαµ)2 ] = −

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We now split the resulting expression into two parts, Z 2 e4 Λ2 Λ2 d3 q1 d3 q1 ρ(⃗ q1 + q⃗2 ) ES (1S) = − [2 ∫ ∫ 4m1 m2 (2π)3 (2π)3 q1 q2 (q1 + q2 ) q12 + Λ2 q22 + Λ2 q12 q22 − (⃗ q1 ⋅ q⃗2 )2 ]=A+B, (15.144) q12 q22 where A corresponds to the first term in brackets. An integration over the angles between q⃗1 and q⃗2 leads to the expression ∞ ∞ 1 8(Zα)2 (Zαµ)4 dq2 A(1S) = − dq 1 ∫ ∫ 2 m1 m2 π q1 + q2 0 0 Λ2 Λ2 1 1 ×( − ). 2 2 2 2 2 2 2 (q1 − q2 ) + 4(Zαµ) q1 + Λ q2 + Λ (q1 + q2 ) + 4(Zαµ)2 (15.145) Only the first term requires a regularization. We now transform the integration variables as follows, q q q1 = + k , q2 = − k , (15.146a) 2 2 q = q1 + q2 , k = 12 (q1 − q2 ) . (15.146b) +

1

1 )∣ = 1 . (15.146c) −1 2 The Jacobian of this transformation is just unity. The integration limits are k ∈ (−q/2, q/2) and q ∈ (0, ∞). In the terms involving the Λ parameters, when replacing q1 and q2 by q and k, we can replace q1 → q/2 and q2 → q/2 (and ignore k), because in a large subregion of the integration region, the difference q1 −q2 = 2k has a negligible effect. In order to convince oneself, one expands in the terms proportional to k, and verifies that the subleading terms in that expansion fall off sufficiently fast with Λ to be ignored in the limit of large Λ. So, J = ∣det ( 21

2

Λ2 ⎛ ⎞ ∞ q/2 8(Zα)2 (Zαµ)4 1 ⎜ 1 ( q2 /4+Λ2 ) 1 ⎟ δ`0 dq dk − A(nS) = − ∫ ∫ ⎜ m1 m2 π 2 n3 q 4 k 2 + (Zαµ)2 q 2 + 4(Zαµ)2 ⎟ 0 −q/2 ⎝ ⎠

1 (Zα)5 µ3 {2 ln (Λ) − 3} δ`0 O ( ) . (15.147) 3 m1 m2 πn Λ In writing Eq. (15.147), we anticipate that the contribution for the 1S state scales as 1/n3 for the excited states, except for a further term, which corresponds to the weighted difference of S states, and which can be evaluated separately. When adding this term to the above result and to the one below given in Eq. (15.148), we obtain the complete result for any state. For the remaining term B, we have the following expression, = −

(⃗ q1 ⋅⃗ q2 ) −1 Z 2 e4 d3 q1 d3 q2 16(Zαµ)4 q12 q22 B(1S) = − . ∫ ∫ 4m1 m2 (2π)3 (2π)3 [(⃗ q1 + q⃗2 )2 + 4(Zαµ)2 ]2 q1 q2 (q1 + q2 ) (15.148) 2

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We use a parameterization of the (⃗ q1 , q⃗2 ) space which can be useful in other contexts, s = q1 + q2 , q1 =

s+t , 2

t = q1 − q2 , q2 =

s−t , 2

u = ∣⃗ q1 − q⃗2 ∣ , d3 q1 d3 q2 = 2π 2 (s2 − t2 ) u ds dt du. (2π)3 (2π)3

(15.149)

The integration limits are 0 ≤ t ≤ u ≤ s ≤ ∞. One first integrates over t in the interval (0, u), then over s in the interval (u, ∞), and finally one integrates u in the whole interval (0, ∞). The latter variable takes the role of a homogeneity parameter. So, u ∞ ∞ 1 (Zα)5 µ3 2 1 2 ds ∫ dt 2π 2 (s2 − t2 ) u 2 du ∫ ∫ 3 4m1 m2 n 0 (u + 4)2 s + t s − t s 0 u ⎡ 2⎤ ⎢ 2 2 2 2 2 s+t 2 s − t 2 ⎥⎥ ) ( ) (u − ( ) −( ) ) ⎥ δ`0 × ⎢⎢( s−t 2 2 ⎢ s+t ⎥ ⎦ ⎣ 5 3 (Zα) µ 8 {ln(2) − 1} δ`0 . (15.150) = − m1 m2 πn3 3

B(nS) = −

For the weighted difference of S states and for any state with a nonvanishing angular momentum, we have ES → C where C is proportional to the expectation value of the operator 1/r3 , 1 (Zα)5 µ3 2(Zα)5 µ3 1 1 {ln(n) − Ψ(n) − γE − + } δ`0 − ⟨ ⟩ (1 − δ`0 ). 3 m1 m2 πn 2n 2 2m1 m2 π (Zαµr)3 (15.151) The complete result for ES = A + B + C is C=−

17 4 2(Zα)5 µ3 1 {ln(n) − Ψ(n) − γE − + ln λ − + ln(2)} δ`0 m1 m2 πn3 2n 3 3 1 (Zα)5 µ3 ⟨ ⟩ (1 − δ`0 ) . (15.152) − 2m1 m2 π (Zαµr)3

∆ES = −

15.5.5

High-Energy Part

The contribution from the scattering amplitude involves the exchange of two photons (see Fig. 15.6). The starting expression for the energy shift can be derived based on the matching of the S-matrix element with the effective Hamiltonian (see Sec. 12.2) and on the averaging over the spin orientations of the S wave as outlined in Sec. 15.3.1. The result is ⃗ 2∫ EH = − e4 ∣φ(0)∣ × {Tr [γµ

0 1 d4 k 1 µ νγ +1 Tr [γ γ ] (2π)4 i k 4 4 t 1 − k
− m1

1 γ0 + 1 1 γ0 + 1 γν ] + Tr [γν γµ ]} . 4 4 t 2 + k
− m2 t 2 − k
− m2

(15.153)

Here, t 1 = γ 0 m1 and t 2 = γ 0 m2 are time-like unit vectors parameterizing the two particles at rest. The loop momentum is on the order of the mass of the bound ⃗ which corresponds particles. One approximates the wave function as φ(⃗ r) ≈ φ(0),

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(a)

(b)

Fig. 15.6 The Feynman diagrams for the high-energy part of the recoil correction involve twophoton exchange. Diagrams (a) and (b) correspond to the two terms in the second line of Eq. (15.153).

to a Dirac-δ in momentum space. Essentially, one thereby ignores the incoming momentum of the wave function, which otherwise acts as an infrared regulator for the diagram. The approximation φ(⃗ r) ≈ φ(⃗0) thus entails the necessity to subtract from the scattering amplitude, certain spurious infrared divergent terms. Because only S state wave functions are nonvanishing at the origin, the entire high-energy part only contributes to S states. In order to do the photon energy integration, one performs a Wick rotation in the integral d4 k k 4 + 2ω 4 128π(Zα)5 m1 m2 µ3 δ `0 ∫ n3 (2π)4 i k4 1 1 1 1 × 2 , k + 2m1 ω k 2 − 2m1 ω k 2 − 2m2 ω k 2 + 2m2 ω

EH = −

(15.154)

⃗ with the result where k µ = (ω, k), 4π(Zα)5 m1 m2 µ3 d3 k 1 δ `0 ∫ 3 3 6 n (2π) k m21 m22 ⎡ (k 4 + 6 k 2 m21 + 8m41 ) m22 (k 4 + 6 k 2 m22 + 8m42 ) m21 ⎤⎥ ⎢ ⎥, × ⎢k 3 + √ + √ ⎢ k 2 + m21 (m21 − m22 ) k 2 + m22 (m22 − m21 ) ⎥⎦ ⎣

EH = −

(15.155)

where now k = ∣k⃗ ∣. Just like in Sec. 15.5.3, various lower-order terms have to be subtracted. The leading subtraction term comes from the two-Coulomb-photon exchange in the on-shell approximation. We recall, from Chap. 5, that, for the exchange of hard photons, we can approximate the Schr¨odinger–Coulomb Green function as follows, ′ 1 1 2µ , (15.156) ) ≈− ≈− E − HS HS k⃗ 2 where the approximation is valid for hard momenta k⃗ on the order of the electron mass. We recall that the Coulomb potential is diagonal in coordinate space, but a

(

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function of the momentum transfer in momentum space, Zα , ∣ r⃗ ∣ 4πZα V (⃗ q − q⃗′ ) = − . (⃗ q − q⃗′ )2

⟨⃗ r∣V ∣⃗ r′ ⟩ = V (⃗ r) δ (3) (⃗ r − r⃗′ ) , ⟨⃗ q ∣V ∣⃗ q ′ ⟩ = V (⃗ q − q⃗′ ) ,

V (⃗ r) = −

(15.157a) (15.157b)

The second-order energy perturbation for the exchange of two hard Coulomb photons can thus be written as ⟨V (

′ 1 d3 k d3 k ′ d3 k ′′ ) V⟩= ∫ ∫ ∫ E−H (2π)3 (2π)3 (2π)3

⃗ (− 4πZα ) (− 2µ ) (− 4πZα ) φ(k⃗′′ ) . (15.158) × φ+ (k) (k⃗ − k⃗′ )2 k⃗′ 2 (k⃗′ − k⃗′′ )2 Due to the hard nature of the Coulomb photons, we replace for the calculation of the subtraction term −

4πZα 4πZα →− , ′ 2 ⃗ ⃗ (k − k ) k⃗′ 2

−

4πZα 4πZα →− . ′ ′′ 2 ⃗ ⃗ (k − k ) k⃗′ 2

(15.159)

The integrations over d3 k and d3 k ′′ thus decouple. After Fourier transformation of the wave function into coordinate space (which results in its value at the origin), we obtain ⟨V (

′ 3 ′ 1 ⃗ 2 ∫ d k (− 4πZα ) (− 2µ ) (− 4πZα ) ) V ⟩ ≈ ∣φ(⃗ r = 0)∣ E−H (2π)3 k⃗′ 2 k⃗′ 2 k⃗′ 2 3 5 4 d k 32π(Zα) µ 1 =∫ (− ) δ`0 . (15.160) 3 (2π) n3 k⃗ 6

We have used the fact that 3 3 ⃗ 2 = (Zα) µ δ`0 . ∣φ(⃗ r = 0)∣ π n3

(15.161)

Comparison of Eq. (15.160) with Eq. (15.155) reveals that the leading 1/k⃗6 asymptotics of the integrand in Eq. (15.155) indeed correspond to two-Coulomb-photon exchange, and need to be subtracted. A further subtraction term in Eq. (15.155) corresponds to the Breit Hamiltonian. The logarithmically divergent term has a factor Λ4 /(k⃗2 + Λ2 )2 with Λ = Zαµλ. Finally, one has ∆EH = −

7 2 m1 (Zα)5 µ3 m2 δ`0 { − 2 ln(Zαλ) + 2 [m21 ln ( ) − m22 ln ( )]} 2 3 m1 m2 n 3 m1 − m2 µ µ (15.162)

as the result for the high-energy part.

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15.5.6

Final Result

The complete recoil correction of order (Zα)5 , in first order in the mass ratio, is obtained as the sum of the low- and the middle-energy part, and of the seagull and high-energy terms. We have from Eqs. (15.108), (15.136), (15.152) and (15.162), ∆E = ∆EL + ∆EM + ∆ES + ∆EH =

1 8 7 1 (Zα)5 µ3 2 { δ`0 ln ( )⟩ ) − ln k0 (n, `) − n3 ⟨P ( πm1 m2 n3 3 Zα 3 6 (Zαµr)3

62 2δ`0 m2 m1 δ`0 − 2 [m21 ln ( ) − m22 ln ( )]} . (15.163) 2 9 m1 − m2 µ µ This result is independent of λ and η, as it should be, and the expectation value of the Araki–Sucher distribution P can be found in Eq. (15.134) for a general state. +

15.6

Further Thoughts

Here are some suggestions for further thought. (1) Bound-Electron Self–Energy and Alternative Representations. With the help of Eq. (9.121), DFµν (k) = −g µν /(k 2 + i), and a suitable identification of the binding potential V , show that Eq. (15.15) can alternatively be written as follows, 1 1 d4 k ⟨ψ¯n ∣γ µ γµ ∣ ψn ⟩ . (15.164) ∆ESE = e2 ∫ (2π)4 i k 2 + i p − k − m − γ0V

This is just the unrenormalized expression for the bound-state self-energy. (2) Spectral Representation and Classical Self–Energy. In a spectral representation, we have ψj (⃗ r) ψj+ (⃗ r′ ) . (15.165) GFDC (En − ω, r⃗, r⃗′ ) = ∑ j Ej (1 − iδ) − En + ω Selecting the virtual state j = n (equal to the reference state), and closing the contour so that the pole at zero photon energy is fully encircled, one obtains ψn (⃗ r) ψn+ (⃗ r) GFDC (En − ω, r⃗, r⃗′ ) → → 2πiψn (⃗ r) ψn+ (⃗ r) . (15.166) −iEn δ + ω Of course, we assume that En > 0. The contribution of the reference-state pole to the self-energy becomes [see Eq. (15.15)] d3 k dω DF µν (ω, r⃗ − r⃗′ ) ⟨ψ + ∣H∣ψ⟩ = − ie2 ∫ ∫ 2π (2π)3 ⃗

⃗

× ⟨ψn+ ∣αµ eik⋅⃗r GF (En − ω, r⃗, r⃗′ ) αν e−ik⋅⃗r ∣ ψn ⟩ → − ie2 ∫ d3 r ∫ d3 r′ ∫ ⃗

× ψn+ (⃗ r) α0 eik⋅⃗r {

′

d3 k ⃗ DF 00 (0, k) (2π)3

1 ⃗ ′ [2πi ψn (⃗ r) ψn+ (⃗ r′ )]} α0 e−ik⋅⃗r ψn (⃗ r′ ) . (15.167) 2π

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591

Beyond Breit Hamiltonian and On-Shell Form Factors

Remembering that the photon propagator is given by [see Eq. (9.121)] gµν gµν ⃗ = 1 DF µν (k) = − 2 =− , DF 00 (0, k) (15.168) 2 2 ⃗ k + i k⃗2 ω − k + i one obtains ⃗

d3 k eik⋅(⃗r−⃗r ) + ⟨ψ ∣H∣ψ⟩ → e ∫ d r ∫ d r ∫ ψn (⃗ r) ψn (⃗ r) ψn+ (⃗ r′ ) ψn (⃗ r′ ) (2π)3 k⃗2 e2 2 2 ∣ψn (⃗ r′ )∣ = ∫ d3 r ∫ d3 r′ ∣ψn (⃗ r)∣ 4π∣⃗ r − r⃗′ ∣ α 2 2 = ∫ d3 r ∫ d3 r′ ∣ψn (⃗ r)∣ ∣ψn (⃗ r′ )∣ . (15.169) ∣⃗ r − r⃗′ ∣ +

2

3

3 ′

′

The multiplying factor is α, not Zα, because it is a self-interaction of the electron, not of the nucleus. (3) Renormalized Self–Energy. How does the Ward identity (10.134) affect the renormalization of the Furry-picture relativistic bound-electron self-energy? It is possible that the renormalized version of the bound-state self-energy reads as follows, ∆ESE = ie2 ∫ d3 r ∫ d3 r′ ∫

d4 k ⃗ DF µν (ω, k) (2π)4

⃗

⃗

× ψ¯n (⃗ r) γ µ eik⋅⃗r SFDC (En − ω, r⃗, r⃗′ ) γ ν e−ik⋅⃗r ψn (⃗ r) − δm ∫ d3 r ψ¯n (⃗ r) ψn (⃗ r) .

′

(15.170)

Here, the mass counter term is the one defined in Eq. (10.261). Show that the renormalization of the bound-state self-energy derived in Sec. 15.2.2 does not require any additional renormalizations proportional to the Z1 or Z2 renormalization constants. (4) Nonrelativistic Self–Energy. Show the equivalence of the relativistic self-energy (as discussed in Sec. 15.2.2) with the nonrelativistic version in Chap. 4, for bound states, in the nonrelativistic limit. Consider, in particular, Eqs. (6.100) and (6.99). (5) Two-Loop Binding Corrections. Consider the generalization of the binding corrections of order α(Zα)5 m, discussed in Sec. 15.3, to two-loop order, namely α2 (Zα)5 m. The binding two-loop corrections have been evaluated by two independent groups, in Refs. [544–547] and in Refs. [548,549]. It is instructive to study the role of the Fried–Yennie gauge in the treatment of infrared divergences. Gauge-invariant sets of diagrams can be treated separately, and the complete analytic form for the corrections of α(Zα)5 m is still unknown, as opposed to very accurate numerical results for the coefficients, which have been evaluated in Refs. [544–549]. An analytic evaluation of the entire correction of order α2 (Zα)5 m would be of interest. (6) Higher-Order Corrections. In Sec. 15.4, we had discussed the evaluation of higher-order binding corrections of sixth order in Zα and beyond.

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An informative review, with some explanatory remarks on particular higherorder terms of interest, is provided by Eides,5 Grotch6 and Shelyuto7 in Refs. [27, 550]. The leading two-loop triple logarithmic binding correction of order α2 (Zα)6 ln3 [(Zα)−2 ] m (for S states) has been derived by Karshenboim8 in Ref. [551]. For two-loop logarithmic corrections of order α2 (Zα)6 lnn [(Zα)−2 ] m, with n = 0, 1, 2, results can be found in Refs. [551–554]. A particular nonlogarithmic correction of order α2 (Zα)6 m, namely, the twoloop Bethe logarithm, constitutes the two-loop generalization of the effect discussed in Sec. 4.6.3. It has been calculated in Refs. [555–557]. Results for logarithmic energy corrections of orders α2 (Zα)7 ln2 [(Zα)−2 ] m, α2 (Zα)8 ln4 [(Zα)−2 ] m and α2 (Zα)9 ln3 [(Zα)−2 ] m, have recently been derived in Refs. [554, 558, 559]. They compare favorably with numerical evaluations of the two-loop self-energy effect [560]. For a recent review on the general status of the higher-order corrections, we also mention Ref. [561]. Note that the two-loop slope of the Dirac form factor of the electron, as elucidated by Barbieri,9 Mignaco10 and Remiddi,11 is infrared finite [562, 563], in contrast to the one-loop slope discussed in Sec. 10.3. The three-loop slope of the Dirac form factor has been evaluated in Ref. [564], and the four-loop slope has recently been evaluated by Laporta12 in Ref. [565]. The twoloop, three-loop, and four-loop slopes give rise to nonlogarithmic corrections of orders αn (Zα)4 m, with n = 2, 3, 4, respectively. An independent confirmation of some of the recently obtained results, in particular, of the logarithmic corrections of orders α2 (Zα)7 ln2 [(Zα)−2 ] m, α2 (Zα)8 ln4 [(Zα)−2 ] m, and α2 (Zα)9 ln3 [(Zα)−2 ] m, with n = 0, 1, 2 (see Refs. [554, 558, 559]), could be of interest. (7) Hamiltonian and Hermiticity. Reconsider the derivation of the relativistic recoil correction, and in particular, the Hamiltonian given in Eq. (15.102). In this expression, as already emphasized, we ignore the necessity to hermitize ⃗ r1 )}, which would ⃗ r1 ) → 1 {⃗ p1 , A(⃗ the interaction by anticommutators, p⃗1 ⋅ A(⃗ 2 otherwise lead to a more clumsy notation. Show that, in the derivation of the relativistic recoil correction, the hermitized version of the Hamiltonian leads to an equivalent result. When investigating the derivation, pay close attention to the transition from Eq. (15.145) to (15.147). (8) Vacuum Polarization and Dispersion Relation. Be inspired by Sec. 15.3 and attempt to formulate a dispersion relation approach to the calculation of vacuum polarization, to supplement the treatment in Sec. 10.4. 5 Mikhail

Iosifovich Eides (b. 1947). Grotch (b. 1940). 7 Valery Aleksandrovich Shelyuto (b. 1955). 8 Savely Grigorievich Karshenboim (b. 1960). 9 Riccardo Barbieri (b. 1944). 10 Juan Alberto Mignaco (1940–2001). 11 Ettore Remiddi (b. 1941). 12 Stefano Laporta (b. 1960). 6 Howard

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Chapter 16

Bethe–Salpeter Equation

16.1

Overview

In earlier sections of this book, we have explored a variety of corrections to the basic description of a two-particle Coulombic bound state contained in the Dirac equation. We worked out higher-order corrections to one-electron energy levels in Chap. 11 through the use of the Foldy–Wouthuysen expansion of the Dirac–Coulomb interaction. Radiative corrections to the coupling of a single photon with one of the fermionic bound-state constituents were introduced through the use of form factors. The two-particle Breit Hamiltonian was described in Chap. 12. Additional twobody interactions were incorporated in Chap. 15, including the effect of radiative corrections to the coupling of two photons to one of the fermionic constituents. The latter aspect might be considered a two-photon form factor. We also discussed, in Chap. 15, recoil corrections coming from the exchange of a transverse photon (with retardation). The Bethe–Salpeter approach allows us to analyze the energy levels of two-body bound systems to any desired order but presents us with an involved technical formalism which we would like to discuss here. The equation reminds us of the fact that the calculation of bound-state energy levels and scattering amplitudes is a rigorous and systematic procedure based directly on quantum field theory. In principle, the Bethe–Salpeter equation (BSEQ) can be used to obtain energies to any desired order of α, including all form factor, retardation, recoil, radiative-recoil, and every other type of correction. In practice, the procedures available to obtain these energies from the BSEQ are somewhat cumbersome, as we will see. In some sense, the BSEQ contains too many variables, being dependent on individual time variables for each constituent instead of simply an overall time. Also important to note is the wealth of scales: infrared (E → 0), low-energy (E ∼ mα2 ), middleenergy (E ∼ mα), high-energy (E ∼ m), and ultraviolet (E → ∞). All of these scales were present in earlier calculations and all are important for the evaluation of contributions using the BSEQ and must be dealt with. The challenge of separating contributions coming from different scales is a prime difficulty encountered in using the BSEQ for high-order calculations. Some early references to the BSEQ and 593

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related matters include [566–568], and reviews and an extensive list of references can be found in [569–571]. 16.2 16.2.1

General Ideas Two-Body Bound States and Green Functions

All methods for finding bound-state energies rely on identifying some quantity that depends on the energy and a procedure for extracting that energy value. So far in this book, we have obtained energy values as eigenvalues of some Hamiltonian operator which enters a Schr¨ odinger equation or a generalization such as the Dirac equation, the Breit equation, or one or these supplemented with effective interactions obtained from scattering amplitudes. In the Bethe–Salpeter approach, the bound state energy can be extracted from information contained in the electronproton to electron-proton (two-particle) Green function (16.1) G′ (x1 , x2 ; y1 , y2 ) = ⟨0∣Tψ1 (x1 )ψ2 (x2 )ψ 1 (y1 )ψ 2 (y2 )∣0⟩. ′ (These are Heisenberg picture operators, so the Green function G includes all radiative corrections but is not renormalized.) We are thinking of our bound state as hydrogen, but the procedure described here would apply as well to other bound states containing two spin-1/2 constituents, such as muonium, and with some modifications, also to positronium. We also assume that the fermions are elementary, so in this chapter we are not considering, say, the internal structure of the proton. The diagrammatic expansion of the 2-to-2 Green function is given in Fig. 16.1(a). Intuitively, if appropriately expressed in terms of the center of mass (CM) momentum, G′ will have a pole when the CM energy equals that of a bound state, corresponding to coherent propagation of a two-body subsystem for an extended period of time. These bound-state poles are completely analogous to the poles found in the Coulomb Green function [see Eq. (4.124)], in the relativistic free-fermion propagator in Eq. (7.179), and in the Dirac–Coulomb propagator in Eq. (8.107). In order to fix ideas, let us note that the energy variable in the Schr¨odinger–Coulomb Green function of Eq. (4.124) is hidden in the denominator factor (k + ` + 1 − ν). This Green function contains an energy pole because of the definition ν 2 = −Z 2 α2 µ/(2E) found in Eq. (4.83). Before displaying the bound-state poles in G′ explicitly, it is useful to describe the transformation we use to go from coordinates for the individual particles to center-of-mass (CM) and relative space-time coordinates, as a generalization of the treatment in Sec. 13.2.2. These are X = ξ1 x1 + ξ2 x2 (CM) , x = x1 − x2 (relative) , (16.2a) x1 = X + ξ2 x , x2 = X − ξ1 x . (16.2b) The transformation from individual particle coordinates x1 , x2 to CM and relative variables X, x has unit Jacobian as long as ξ1 + ξ2 = 1. We will use m2 m1 , ξ2 = , M = m1 + m2 . (16.3) ξ1 = M M

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Bethe–Salpeter Equation

Fig. 16.1 Diagrammatic representation of the quantities that appear in the Bethe–Salpeter equation. Part (a) represents the full 2-to-2 Green function G′ . The electron line is shown at the top and the proton at the bottom. Wiggly lines represent photons. Part (b) shows the product of a fully corrected electron propagator with a fully corrected proton propagator. Corrections on these lines are known as self-energy corrections. Part (c) shows the kernel of the BSEQ, symbolized by a circle with a vertical line through it. The kernel is two-particle irreducible, meaning that it cannot be cut into two disjoint pieces by a single cut on the electron line and a single cut on the proton line. It is also one-particle irreducible. The small vertical lines on the external legs indicate amputation — the kernel is defined with all external lines removed. (The amputated lines are present in the diagram purely for clarity.)

The corresponding CM and relative momenta are P = p1 + p2

(CM) ,

p1 = ξ1 P + p ,

p = ξ2 p 1 − ξ1 p 2

(relative) ,

p 2 = ξ2 P − p .

(16.4a) (16.4b)

The momentum transformation p1 , p2 → P, p also has unit Jacobian, and we note that x1 ⋅p1 + x2 ⋅p2 = X ⋅P + x⋅p, a relation that will prove useful when moving to momentum space. The momentum-space version of the 2-to-2 Green function has the form ′

G (p1 , p2 ; q1 , q2 ) = ∫ d4 x1 d4 x2 d4 y1 d4 y2 ei(p1 ⋅x1 +p2 ⋅x2 −q1 ⋅y1 −q2 ⋅y2 ) G′ (x1 , x2 ; y1 , y2 ) ˜ ′ (X, x; Y, y) , = ∫ d4 X d4 x d4 Y d4 y ei(P ⋅X+p⋅x−Q⋅Y −q⋅y) G

(16.5)

˜ ′ (X, x; Y, y) = G′ (X + ξ2 x, X − ξ1 x; Y + ξ2 y, Y − ξ1 y). Translation invariance where G ˜ ′ (X, x; Y, y) is unaltered when X and Y are in coordinate space tells us that G displaced by the same arbitrary amount. The immediate consequence is that the momentum space 2-to-2 Green function contains the energy-momentum conserving Dirac-δ function δ (4) (P − Q) as a factor. In detail, we find that ′

G (p1 , p2 ; q1 , q2 ) = (2π)4 δ (4) (P − Q) G′ (P ; p, q)

(16.6)

G′ (P ; p, q) = ∫ d4 X d4 x d4 y ei(P ⋅X+p⋅x−q⋅y) G′ (X, x; 0, y).

(16.7)

where

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We are now prepared to display the bound-state poles in the momentum-space representation of G′ . For times such that x01 , x02 > y10 , y20 , we can insert a complete set of states into G′ , among which are the two-particle bound states ∣P⃗ , n, k⟩. These states of mass Mn and energy ωn (P⃗ ) = (Mn2 + P⃗ 2 )1/2

(16.8)

are specified by n, their three-momentum P⃗ , and some additional labels k describing possible degeneracy. (The mass Mn is the physical mass of the bound system including all corrections; n stands for the principal quantum number and all other quantum numbers that are necessary in order to determine the states that have that mass; and k summarizes the additional quantum numbers required to completely specify the state.) The contribution of the two-particle bound states to G′ is d3 P (2π)3 2ωn (P⃗ ) nk × ⟨0∣ Tψ1 (x1 )ψ2 (x2 ) ∣P⃗ , n, k⟩ ⟨P⃗ , n, k∣ Tψ 1 (y1 )ψ 2 (y2 ) ∣0⟩ + ⋯ , (16.9)

G′ (x1 , x2 ; y1 , y2 ) = Θ(X 0 − Y 0 ) ∑ ∫

where the dots represent contributions from other types of states and contributions from other orderings of the time variables that don’t give rise to particle-particle bound state poles. We note that the condition x01 , x02 > y10 , y20 actually implies a more stringent theta function, Θ(min(x01 , x02 ) − max(y10 , y20 )), rather than simply Θ(X 0 − Y 0 ). However, the use of the more exact theta function changes neither the positions nor the residues of the bound-state poles in G′ (see also Sec. 16.5). The matrix elements that have appeared represent a new kind of form factor commonly known as the Bethe–Salpeter wave function: Ψnk (P⃗ ; x1 , x2 ) = ⟨0∣ Tψ1 (x1 )ψ2 (x2 ) ∣P⃗ , n, k⟩ ,

(16.10)

Ψnk (P⃗ ; y1 , y2 ) = ⟨P⃗ , n, k∣ Tψ 1 (y1 )ψ 2 (y2 ) ∣0⟩ .

(16.11)

These are curiously complicated wave functions as they each depend on two time variables. This complication can be reduced somewhat by use of CM and relative coordinates along with the explicit extraction of the dependence of the wave function on the CM coordinate. We make use of the field-theoretical four-momentum operator Pˆ to write Ψnk (P⃗ ; x1 , x2 ) = ⟨0∣Tψ1 (X + ξ2 x) ψ2 (X − ξ1 x)∣P⃗ , n, k⟩ ˆ ˆ ˆ ˆ = ⟨0∣T eiP ⋅X ψ1 (ξ2 x)e−iP ⋅X eiP ⋅X ψ2 (−ξ1 x) e−iP ⋅X ∣P⃗ , n, k⟩ = e−iPn ⋅X Ψnk (P⃗ , ξ2 x, −ξ1 x) ,

(16.12)

where Pn = (ωn (P⃗ ), P⃗ ), and analogously Ψnk (P⃗ ; y1 , y2 ) = eiPn ⋅Y Ψnk (P⃗ ; ξ2 y, −ξ1 y).

(16.13)

So, the most important new feature compared with the usual nonrelativistic twoparticle quantum mechanics is the dependence of the wave function on a relative

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Bethe–Salpeter Equation

time. We make use of the representation of Eq. (7.151) for the theta function (with E = 0 and f (E) = 1) to write (16.9) as dP 0 e−iP (X −Y ) d3 P ∑∫ 0 2π P + i (2π)3 2ωn (P⃗ ) nk −iP ⋅X+iP n n ⋅Y × Ψnk (P⃗ , ξ2 x, −ξ1 x) e Ψnk (P⃗ , ξ2 y, −ξ1 y) + ⋯ 0

0

0

G′ (x1 , x2 ; y1 , y2 ) = i ∫

d4 P e−iP ⋅(X−Y ) (2π)4 2ωn (P⃗ ) P 0 − ωn (P⃗ ) + i nk × Ψnk (P⃗ , ξ2 x, −ξ1 x) Ψnk (P⃗ , ξ2 y, −ξ1 y) + ⋯ ,

= i∑∫

(16.14)

where Pn = (ωn (P⃗ ), P⃗ ) = ((Mn2 + P⃗ 2 )1/2 , P⃗ ) is the physical (mass-shell) momentum and the replacement P 0 → P 0 − ωn (P⃗ ) was made in the P 0 integration variable in order to achieve the final form. The energy pole is most clearly revealed in momentum space. We combine Eqs. (16.7) and (16.14) to find Ψnk (P⃗ ; p)Ψnk (P⃗ ; q) + ⋯, P 0 − ωn (P⃗ ) + i

(16.15)

Ψnk (P⃗ , ξ2 x, −ξ1 x) , √ 2ωn (P⃗ )

(16.16)

Ψnk (P⃗ , ξ2 y, −ξ1 y) Ψnk (P⃗ , q) = ∫ d4 y e−iq⋅y . √ 2ωn (P⃗ )

(16.17)

G′ (P ; p, q) = i ∑ nk

where Ψnk (P⃗ , p) = ∫ d4 x eip⋅x

The bound state pole is at position P 0 → ωn (P⃗ ) = (Mn2 + P⃗ 2 )1/2 , or, in the CM frame with P⃗ = 0, it is at P 0 → Mn . The extra terms represented by the dots do not contain a two-particle bound state pole. We have included the factor (2ωn (P⃗ ))1/2 in the momentum space wave function for convenience later on. We will use a graphical approach to derive the actual Bethe–Salpeter equation, as was done in the original Ref. [566]. The 2-to-2 Green function G′ (P ; p, q) is shown in Fig. 16.2(a) with labels for the momenta of the external legs. The graphical expansion of G′ was shown in Fig. 16.1(a) above. It contains all diagrams having one electron and one proton in the initial state, and one each in the final state, with photons (and vacuum polarization loops) attached in all possible ways. In our notation, the product of an arbitrary pair of 2-to-2 objects, say A and B, is obtained by integration over the internal relative momentum [AB] (P ; p, q) = ∫

d4 k A(P ; p, k)B(P ; k, q) . (2π)4

(16.18)

This is illustrated in Fig. 16.2(b). The inhomogeneous BSEQ is shown in the symbolic equation depicted in Fig. 16.2(c). The idea is that the individual diagrams in G′ implicitly contained in the diagram on the left-hand side of the equation depicted in Fig. 16.2(c) are in one-to-one correspondence with the diagrams found in the sum

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P

P

ξ1 +p

ξ1 +q

ξ1 P+k

ξ1 P+p

A P

ξ2 -p

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ξ2 P-q

(a)

=

ξ2 P-p

ξ1 P+q

B ξ2 P-k (b)

ξ2 P-q

+

(c)

Fig. 16.2 Illustration of the diagrammatic language useful for discussion of the BSEQ. Panel (a) shows the 2-to-2 Green function with momentum labels on the external legs. A CM fourmomentum P flows through the diagram, shared between the lines according to ξ1 P and ξ2 P . The outgoing and incoming relative momenta are p and q. Part (b) represents the product of A and B where k is an integration variable representing the relative momentum of the intermediate state. Part (c) is the inhomogeneous BSEQ itself. The vertical line through the kernel is a reminder that it is two- (and one-)particle irreducible.

of the non-interacting 2-to-2 Green function S ′ of Fig. 16.1(b) and the product of S ′ times the two-particle irreducible kernel K ′ of Fig. 16.1(c) times G′ itself (see also Sec. 16.5). In symbols, the BSEQ can be written as G′ (P ; p, q) = S ′ (P ; p, q) + ∫

d4 ` d4 k ′ S (P ; p, `) K ′ (P ; `, k) G′ (P ; k, q) . (16.19) (2π)4 (2π)4

Because there is no possibility for momentum flow between the electron and proton lines in S ′ , the relative momentum coming into the free propagator must equal that coming out, and we can write [see also Eq. (16.6)] S ′ (P ; p, q) = (2π)4 δ (4) (p − q) S ′ (P ; p),

(16.20)

so that the BSEQ takes the slightly simpler form G′ (P ; p, q) = S ′ (P ; p, q) + S ′ (P ; p) ∫

d4 k K ′ (P ; p, k) G′ (P ; k, q) . (2π)4

(16.21)

As a shorthand, we can represent the BSEQ as G′ = S ′ + S ′ K ′ G′ ,

(16.22)

with the integration over relative momentum implicit. We note that the graphical demonstration of the BSEQ works just as well in the reversed order, so as an alternative form we also have G′ = S ′ + G′ K ′ S ′ .

(16.23)

Up until this point we have been dealing with unrenormalized quantities, but as discussed in Chap. 10, we can renormalize by (i) writing the equations in terms

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Bethe–Salpeter Equation

of the physical charge and mass, and (ii) multiplying by the appropriate electron and proton wave function renormalization constants Z2e and Z2p as defined in Eq. (10.111): G′ = Z2e Z2p G,

S ′ = Z2e Z2p S,

K ′ = (Z2e Z2p )−1 K ,

(16.24)

where the primed quantities are the original bare objects and their unprimed partners are the finite renormalized quantities. It is evident that Z2e and Z2p cancel out and the BSEQ G = S + SKG = S + GKS

(16.25)

has the same form after renormalization that it had before, and we will continue to write it in the same way in terms of G, S, and K as we did in terms of G′ , S ′ , and K ′ . No additional ultraviolet divergences are generated by the integrations implicit in the products contained in the BSEQ. (This is not the case when the BSEQ is used to describe a particle-antiparticle state such as positronium because the kernel contains a term representing the annihilation of the e−− e+ pair to a single photon [572, 573].) The graphical expressions for the quantities entering the BSEQ are best given in terms of the renormalized self-energy, vertex, and vacuum polarization functions. The graphical expansions are greatly compressed in terms of this more efficient language (see Fig. 16.3). As an explicit example, the noninteracting propagator S contains only one term and can be written as S(P ; p) = [

i i ] [ ] . ξ1 P + p
− m1 − Σ1 (ξ1 P + p) + i 1 ξ2 P − p
− m2 − Σ2 (ξ2 P − p) + i 2 (16.26)

The Σi (q) functions are the renormalized self-energies. The subscript labels on the square brackets are reminders that there are two fermion lines, each with its own Dirac indices, which must be distinguished. From now on we will deal exclusively with the renormalized theory. A homogeneous equation for the Bethe–Salpeter wave functions is obtained as the residue of (16.25) at a bound state pole. From Eq. (16.15), we can write the pole term of G as follows, G(P 0 ) → i ∑ k

Ψnk Ψnk , P 0 − Pn0

(16.27)

as P 0 approaches the bound-state energy Pn0 = ωn (P⃗ ) = (Mn2 + P⃗ 2 )1/2 , where Mn is the rest energy of the nth bound state. We now find that the residue of (16.22) [and of (16.25)] for P 0 near the bound-state pole position Pn0 is i ∑ Ψnk Ψnk = S(Pn0 )K(Pn0 ) i ∑ Ψnk Ψnk , k

(16.28)

k

where the dependence of S and K on the CM energy P 0 is made explicit. The degenerate wave functions Ψnk for the various values of the degeneracy label k are assumed to be linearly independent, so we obtain the homogeneous BSEQ Ψnk = S(Pn0 ) K(Pn0 ) Ψnk .

(16.29)

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G=

…

= a)

S=

K=

…

=

b)

c)

Fig. 16.3 Depiction of quantities entering the renormalized BSEQ. The 2-to-2 Green function is shown in (a), the non-interacting two particle Green function in (b), and the amputated twoparticle-irreducible kernel in (c). The fermion and photon lines in this diagrams are adorned with filled dots to indicate that they are fully-corrected and renormalized propagators, while a dot on a fermion-photon vertex stands for a fully-corrected vertex function.

If we make the dependence on relative momentum explicit, the homogeneous BSEQ takes the form d4 ` Ψnk (P⃗ ; p) = S(Pn ; p) ∫ K(Pn ; p, `) Ψnk (P⃗ ; `) . (16.30) (2π)4 In order to derive a normalization condition for Bethe–Salpeter wave functions, and later on, a form of the BSEQ more convenient for calculations and a perturbative expansion for obtaining estimates of the bound state energies, it is necessary to develop some confidence with the manipulation of the BSEQ into a variety of equivalent forms. We can write formal solutions of the inhomogeneous BSEQ as given in Eqs. (16.22) and (16.23) as G = (1 − SK)−1 S = S(1 − KS)−1 .

(16.31)

In our notation, the inverse of an operator A satisfies AA−1 = 1 if the function A−1 (P ; p, q) exists and satisfies ∫

d4 ` A(P ; p, `) A−1 (P ; `, q) = (2π)4 δ (4) (p − q) . (2π)4

(16.32)

Using the general operator formula (AB)−1 = B −1 A−1 , we can express the 2-to-2 Green function as G = (S −1 − K)−1 ,

(16.33)

starting from either form of (16.31). Or, we can expand either form of (16.31) in a geometric series to find G = S + S K S + S K S K S + ⋯.

(16.34)

Consideration of the diagrams shown in Fig. 16.1 will convince the reader that this expansion is correct. Finally, it is sometimes useful to define the truncated

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Green function GT by taking away the disconnected component and amputating the external legs: G = S + S GT S .

(16.35)

Operator manipulations of the sort used above can be used to show that the truncated Green function satisfies BSEQs of its own: GT = K + K S GT = K + GT S K ,

(16.36)

GT = (1 − KS)−1 K = K(1 − SK)−1 .

(16.37)

with solutions

A geometric expansion, or a diagrammatic analysis, leads to the series expression GT = K + K S K + K S K S K + ⋯ .

(16.38)

We are now in a position to derive the orthonormality condition for Bethe– Salpeter wave functions. Ultimately, a normalization of the wave functions is implied by the fact that G satisfies an inhomogeneous equation given in Eq. (16.25). So, the normalization of G and also of its residues is fixed. We start by noting the expansion of G−1 , G−1 (P 0 ) = G−1 (Pn0 ) + (P 0 − Pn0 )

d G−1 (P 0 )∣ + O(P 0 − Pn0 )2 , dP 0 Pn0

(16.39)

where we have made the dependence of G−1 on the CM energy P 0 explicit. We consider the identity G = GG−1 G and write out the residue at the bound-state pole Pn0 of each side, finding i ∑ Ψnk Ψnk = i ∑ Ψnr Ψnr k

r

d G−1 (P 0 ) ∣ i ∑ Ψns Ψns . dP 0 Pn0 s

(16.40)

The homogeneous BSEQ was used to remove a potential second-order pole on the right-hand side. The linear independence of the states Ψnk for various values of k, and similarly of the states Ψnk , allows us to write the orthonormality relation in the final form d δrs = iΨnr (16.41) G−1 (P 0 ) ∣ Ψns , dP 0 Pn0 where G−1 = S −1 − K. To close this introductory section, we will take a look at the leading contribution to the BSEQ when the binding is due to a Coulomb potential. We will make a number of approximations and find in the end that the BSEQ is just an equation for a nonrelativistic Coulomb bound state embellished with relativistic, radiative, recoil, and retardation corrections. The main approximations are based on the facts that the binding energy in a nonrelativistic Coulombic bound state is small (of order µα2 where µ is the reduced mass), the typical momentum is of order µα, and the radiative corrections are small of order α. We have explored these aspects in the preceding chapters. We commence our string of approximations by

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neglecting all self-energy corrections from the non-interacting electron-proton Green function S(P ; p). In this limit, S(P ; p) simply becomes the product of a free Dirac propagator for the electron and one for the proton, S(P ; p) → S0 (P ; p) = [

i i ] [ ] , ξ1 P + p ξ − m + i P − p − 1 2 1

m2 + i 2

(16.42)

where the two particles, electron and proton, are denoted particles 1 and 2. It is important to maintain the distinction between the particles, as the first square bracket carries Dirac indices for particle 1 and the second carries Dirac indices for particle 2 (as denoted by subscripts on the square brackets). We also restrict the kernel to just one term describing the exchange of a single photon, K(P ; p, q) → [−iq1 γµ ]1

−4πiZα γ1µ gµν γ2µ −igµν [−iq γ ] = , 2 ν 2 (p − q)2 + i (p − q)2 + i

(16.43)

where q1 q2 is the product of the charges of the two particles: q1 q2 = −4πZα. At this stage, we have made what is called the ladder approximation. The 2-to-2 Green function G has been approximated by a sum of terms containing 0, 1, 2, etc., photons connecting the electron and proton without crossing (like the rungs of a ladder). We make two additional approximations at this point. To begin with, we anticipate that the Bethe–Salpeter wave function has “large” and “small” components, just as the Dirac wave function does [see Eqs. (6.100), (6.101) and (8.11) in Sec. (8.2.2)]. When we expand the gamma matrix factor γ1µ gµν γ2ν = γ10 γ20 − γ⃗1 ⋅ γ⃗2 , we see that the second term connects large to small components and will be small in the nonrelativistic limit. In fact, the spatial Dirac γ matrices typically make contributions on the order of the particle speed, which is O(α) in a nonrelativistic Coulombic bound state. Our second approximation is the neglect of retardation. That is, we will work in the approximation that the potential is instantaneous. This approximation is equivalent to (via Fourier transformation) the neglect of the energy in the photon propagation factor (cf. the treatment of the Breit interaction in Chap. 12). Since the potential can only be instantaneous in one Lorentz frame (modulo spatial rotation), ⃗ and P = γ 0 P 0 . The homogeneous we choose to work in the CM frame where P⃗ = 0, BSEQ, subject to all of these approximations, has the form [cf. Eq. (16.29)] d4 q (−4πiZα)γ10 γ20 i i ] [ ] ∫ Ψnk (q), 4 −(⃗ p − q⃗ )2 ξ1 P + p
− m1 + i 1 ξ2 P − p
− m2 + i 2 (2π) (16.44) where we have defined the CM wave function Ψnk (p) ≡ Ψnk (P⃗ = 0; p). Quite surprisingly, the approximate BSEQ given in Eq. (16.44), while tremendously simplified compared to the full BSEQ, is still too complex to be solved directly. Some of the difficulty is related to the relative energy variable present in the wave function Ψnk (p). The bound-state energy P 0 is independent of relative energy. The relative energy p0 affects the residue of the pole in Eq. (16.27), not the position of the pole. The dependence on the relative energy is most conveniently Ψnk (p) = [

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eliminated by integrating it out and working in terms of a wave function dependent wholly on spatial variables, ∞ dp0 ϕnk (⃗ p) ≡ ∫ Ψnk (p) (16.45) −∞ 2π instead of with Ψnk (p). Integration over the fermion propagation factor is facilitated by separating pole terms according to i Λ+ (⃗ p ) γ0 i Λ− (⃗ p ) γ0 i = + , (16.46) p p ) + i p0 + ω(⃗ p ) − i
− m + i p0 − ω(⃗

where ω(⃗ p ) = (m2 + p⃗ 2 )1/2 and Λ± (⃗ p ) are the positive-energy and negative-energy projectors that we have already encountered in Eq. (14.18), Λ± (⃗ p) ≡

1 1 ω(⃗ p ) ± m ±⃗ σ ⋅ p⃗ [ω(⃗ p ) ± (m − γ⃗ ⋅ p⃗ )γ 0 ] = ( ). ⃗ ±⃗ σ ⋅ p ω(⃗ p )∓m 2ω(⃗ p) 2ω(⃗ p)

(16.47)

The Λ± (⃗ p ) are projection operators satisfying Λ± (⃗ p )Λ± (⃗ p ) = Λ± (⃗ p ),

Λ± (⃗ p )Λ∓ (⃗ p ) = 0,

Λ± (⃗ p ) + Λ∓ (⃗ p ) = 1.

(16.48)

The relative energy integral is done by poles. It has four terms, two of which vanish upon integration (having poles on the same side of the axis), leaving ∫

∞ −∞

∞ dp0 i dp0 i S0 (P ; p) = ∫ [ ] [ ] 2π −∞ 2π ξ1 P + p ξ − m + i P − p − 2 1 1

m2 + i 2 ∞ dp0 i Λ+ (⃗ p ) γ0 i Λ− (⃗ p ) γ0 =∫ [ + ] ξ1 P 0 + p0 − ω1 (⃗ p ) + i ξ1 P 0 + p0 + ω1 (⃗ p ) − i 1 −∞ 2π

×[

P0

iΛ+ (−⃗ p )γ 0 iΛ− (−⃗ p )γ 0 + ] 0 0 0 − p − ω2 (⃗ p ) + i ξ2 P − p + ω2 (⃗ p ) − i 2

ξ2 ⎧ ⎪ ⎪ p )γ 0 ]2 [Λ− (⃗ p )γ 0 ]2 ⎫ p )γ 0 ]1 [Λ+ (−⃗ p )γ 0 ]1 [Λ− (−⃗ ⎪ [Λ+ (⃗ ⎪ = i⎨ − ⎬, 0 0 ⎪ P − ω1 (⃗ p ) − ω2 (⃗ p) P + ω1 (⃗ p ) + ω2 (⃗ p) ⎪ ⎪ ⎪ ⎩ ⎭ (16.49) where ωi (⃗ p ) ≡ (m2i + p⃗ 2 )1/2 . The approximate three-dimensional BSEQ (the integral of (16.44) over relative energy) takes the form Λ1+ (⃗ p )Λ2+ (−⃗ p) Λ1− (⃗ p ) Λ2− (−⃗ p) d3 q −4πZα − } ϕnk (⃗ q ), ∫ P 0 − ω1 (⃗ p ) − ω2 (⃗ p ) P 0 + ω1 (⃗ p ) + ω2 (⃗ p) (2π)3 (⃗ p − q⃗ )2 (16.50) where Λi± (⃗ p ) is defined with ωi (⃗ p ) and mi instead of ω(⃗ p ) and m. This approximate BSEQ can be put into a form resembling a two-particle Dirac equation. We define the single-particle Dirac Hamiltonian ϕnk (⃗ p) = {

⃗ ⋅ p⃗ = (mi − γ⃗ ⋅ p⃗ ) γ 0 = ωi (⃗ Hi (⃗ p ) ≡ βmi + α p ) (Λi+ (⃗ p ) − Λi− (⃗ p )) .

(16.51)

Under action of the operator P −H1 (⃗ p )−H2 (−⃗ p ), the approximate BSEQ becomes 0

(P 0 − H1 (⃗ p ) − H2 (−⃗ p )) ϕnk (⃗ p) = (Λ1+ (⃗ p )Λ2+ (−⃗ p ) − Λ1− (⃗ p )Λ2− (−⃗ p )) ∫

d3 q −4πZα ϕnk (⃗ q). (2π)3 (⃗ p − q⃗ )2

(16.52)

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The approximation to the BSEQ given in Eq. (16.52) is known as the “Salpeter equation” (see Ref. [534]). To wit, Salpeter actually derived a more general equation containing an arbitrary instantaneous kernel. The derivation for such a general kernel is no different from what we just did. The Salpeter equation is similar in form to the Breit equation [574], which can be written in a form much like (16.52), but missing the projection operators on the right-hand side. Just a few additional approximations will bring us to a form amenable to actual solution. We start from (16.50), as (16.52) would require a two-particle Foldy– Wouthuysen expansion of the type discussed in Chaps. 11 and 12. We break ϕnk (⃗ p) up into its large-large, large-small, etc., components using a seemingly trivial identity, ϕnk (⃗ p ) = {Λ1+ (⃗ p ) + Λ1− (⃗ p )}{Λ2+ (−⃗ p ) + Λ2− (−⃗ p )}ϕnk (⃗ p) = {Λ1+ (⃗ p )Λ2+ (−⃗ p ) + Λ1+ (⃗ p )Λ2− (−⃗ p) + Λ1− (⃗ p )Λ2+ (−⃗ p ) + Λ1− (⃗ p )Λ2− (−⃗ p )}ϕnk (⃗ p) = ϕ++ p ) + ϕ+− p ) + ϕ−+ p ) + ϕ−− p), nk (⃗ nk (⃗ nk (⃗ nk (⃗

(16.53)

where ϕ++ p ) = Λ1+ (⃗ p )Λ2+ (−⃗ p )ϕnk (⃗ p ), etc., and multiply both sides of (16.50) by nk (⃗ Λ1+ (⃗ p )Λ2+ (−⃗ p ) to produce an equation for the large-large component of the wave function. We realize that ∣⃗ p ∣/m (and ∣⃗ q ∣/m) are small in a nonrelativistic bound 1 0 0 p ) − ω2 (⃗ p ) can be approximated as state, so Λ± (⃗ p ) → 2 (1 ± γ ) and P − ω1 (⃗ P 0 − ω1 (⃗ p ) − ω2 (⃗ p ) → (m1 + m2 + E) −(m1 +

p⃗ 2 p⃗ 2 p⃗ 2 + ⋯)−(m2 + + ⋯)≈ E − , 2m1 2m2 2µ (16.54)

where E = P 0 −m1 −m2 is the binding energy and µ = (1/m1 +1/m2 )−1 is the reduced mass as before. In this limit, the equation satisfied by the large-large component is p⃗ 2 ++ d3 q (−4πZα) ++ ϕnk (⃗ p) + ∫ ϕ (⃗ q ) = E ϕ++ p). nk (⃗ 2µ (2π)3 (⃗ p − q⃗ )2 nk

(16.55)

This is just the standard nonrelativistic Schr¨odinger–Coulomb equation (in momentum space). We note that the ϕ−− component describes states of two antiparticles, while the ϕ+− and ϕ−+ components vanish in the nonrelativistic limit. 16.2.2

Questions of Gauge and Transformations

The BSEQ gives information about bound-state energies and scattering amplitudes, but our focus in this chapter is on energies. The energies appear as eigenvalues of the homogeneous equation (16.29) or as the positions of poles of the 2-to-2 Green function found by solving the inhomogeneous BSEQ, given in Eq. (16.25). Due to our specific focus on bound state energies, we can make use of a variety of options and transformations of the BSEQ that leave the energies unchanged. As one immediate example, the bound-state binding energy is independent of the Lorentz

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frame. The Lorentz invariance of QED implies that a bound state pole of G must have the covariant form A G(P ; p, q) → 2 + B + ⋯, (16.56) P − Mn2 for P near the four-momentum of a bound state, where Mn is an invariant (A and B are non-singular near this pole). The binding energy En of the bound state is defined through Mn = m1 + m2 + En .

(16.57)

It constitutes an invariant. The equations we wrote down earlier for the Bethe– Salpeter wave function (16.29) and normalization condition (16.41) were specifically for the CM frame, but they could have been written for a general Lorentz frame. Other options and transformations that we will explore in this section are choice of gauge, the possibility of putting conditions on the relative time or energy variables, and the freedom associated with redefinitions of the non-interacting 2-to-2 Green function S. Gauge invariance implies the independence of physical quantities from the choice of gauge (see Secs. 4.6.2 and 8.2.1). It is a fundamental feature of a QED. The basic interaction responsible for atomic structure is the Coulomb force. A physical photon such as one given off in an atomic transition propagates with two possible polarization states that are orthogonal to the direction of motion. These physical properties are captured most directly in the Coulomb gauge, which has the propagator [see Eq. (9.73)] 00 DC =

1 , k⃗ 2

0i i0 DC = DC = 0,

ij DC =

k2

ki kj 1 (δij − ). + i k⃗ 2

(16.58)

(We remember that Greek indices cover the range 0, 1, 2, 3, while Latin indices are spatial and cover the range 1, 2, 3.) On the other hand, calculations that involve integration over loops containing virtual photons can be done much more simply and efficiently in a covariant gauge such as Feynman gauge, with the propagator [see Eq. (9.121)] g µν . (16.59) + i Bethe and Salpeter used Coulomb gauge in their early work, and Salpeter [534] suggested that Feynman gauge was problematic in that unphysical corrections of anomalously low order in α appear that are only canceled when contributions from a number of graphs are considered together. A numerical exploration [575] using Feynman gauge confirms the appearance of energy corrections at order O(µα3 ln(1/α)) as corrections to the Bohr energy levels EnBohr = µα2 /(2n2 ). A systematic exploration [576] reveals contributions at the anomalously low orders O(µα3 ln(1/α)) and O(µα3 ) when using Feynman gauge. Contributions at these orders are unwanted, as the first (relativistic) physical corrections in fact begin at O(µα4 ). In fact, these DFµν = −

k2

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spurious contributions cancel, but only with the inclusion of an infinite number of additional terms in the kernel involving crossed photons. The problem can be seen to be associated with retardation present in the Feynman time-time propagator DF00 = 1/(k⃗2 − k02 ), as distinct from the instantaneous interaction mediated by 00 the Coulomb gauge time-time propagator DC = 1/k⃗ 2 . The moral is that Coulomb gauge should be used for the calculation of properties of nonrelativistic Coulombic bound states. Insistence on the use of Coulomb gauge for bound-state calculations raises a number of issues associated with the fact that radiative corrections, and loop integrals in general, are much easier to evaluate in a covariant gauge. Fortunately, it is often possible to use a covariant gauge such as Feynman gauge for subsets of diagrams (under certain conditions), even when using Coulomb gauge for the calculation as a whole. As an immediate example, the vacuum polarization tensor [see Eq. (10.228)] Πµν (k) = (k 2 gµν − kµ kν ) Π(k 2 )

(16.60)

can be expressed in terms of a scalar function Π(k 2 ) that is independent of gauge. The vacuum polarization tensor also satisfies the transversality condition k µ Πµν (k) = 0, which can be useful in demonstrating a broader type of gauge independence because of the following connection between Coulomb gauge and Feynman gauge photon propagators (see Sec. 9.5.2) µν DC (k) = DFµν (k) +

1 (k µ f ν + f µ k ν ) , 2k 2

f0 =

k0 , k⃗ 2

fi = −

ki . k⃗ 2

(16.61)

In any subgraph where the terms involving f µ can be shown to vanish (to some approximation), Feynman gauge can be used instead of Coulomb gauge for calculations. Many bound-state calculations involve vertex corrections. Since the fermions in a bound state are often nearly at rest, the on-shell version of the vertex function [see Eq. (10.96)] i σµν k ν F2 (k 2 ) (16.62) 2m can often be used. In this expression, k = p2 − p1 , and the Dirac and Pauli form factors F1 (k 2 ) and F2 (k 2 ) are independent of gauge. The identification of gaugeinvariant sets of diagrams that can be evaluated in Feynman gauge is a useful strategy for higher-order bound-state calculations. The BSEQ is an intrinsically four-dimensional equation that, in the CM, depends on the total CM energy and either the relative energy and momentum variables (in momentum space) or the relative time and position variables (in coordinate space). The variables represented by relative energy or relative time are not essential for purposes of determining energy levels and greatly complicate the equations. Moreover, the bound-state energies are independent of the values of the relative coordinates. A number of ways have been developed to eliminate the relative energy (or time) variables and, so, to simplify the equations. One approach is to work with Γµ (p2 , p1 ) = γµ F1 (k 2 ) +

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the wave function or 2-to-2 Green function in which the relative time variables have been set to zero. The corresponding momentum-space version of this condition is to work with the three-dimensional wave function of (16.45) instead of the full fourdimensional ψ(p0 , p⃗ ). A complete three-dimensional formalism and perturbation theory can be developed based on t = 0 quantities [534]. Some other works using t = 0 wave functions and Green functions include Refs. [2, 576–578]. Another line of approach is to put conditions on the relative energy, either setting the relative energy to zero or to a value that places one of the bound particles on its mass shell. A wide variety of bound-state formalisms have been constructed based on the general idea that the relative energy variable can be eliminated, and at the same time a simpler form for the non-interacting propagator can be used. Let us focus first on the second idea, which is that some simpler non-interacting 2-to-2 propagator S0 can be used instead of S. One way to ensure that such a manipulation leaves the bound-state energies unchanged is to arrange matters so that the truncated Green function GT is unchanged by the process. This is because G and GT have bound-state poles at the same positions, as can be seen from (16.35). So, as long as the positions of the poles of GT are unaffected, the energies are unchanged. If GT stays constant, an alteration in S0 must be accompanied by a compensating alteration in the kernel. Given the BSEQ of (16.36) and the new BSEQ for GT in terms of S0 and the new kernel K GT = K + K S0 GT ,

(16.63)

we can eliminate GT between the two using operator manipulations (represented here as simple algebra) to show that the new kernel must have the form −1

K = (1 − K(S − S0 )) K = K + K(S − S0 )K + ⋯ .

(16.64)

The foregoing manipulation has often been used together with an astute choice of S0 inspired by the physics of nonrelativistic bound states to produce a useful bound-state formalism. Several bound-state formalisms have started with the observation that the full non-interacting Green function S that appears in the BSEQ contains far more information than is needed for a description of a nonrelativistic Coulombic bound state (at least to a first approximation). To start with, the radiative corrections found in the self-energy functions contained in S only affect bound-state energies at higher order. Considering now just the free 2-to-2 non-interacting propagator, the large-large (in the bispinor sense) component dominates for purposes of bound-state physics. We see from Eq. (16.49) that the dominant part of S is [i Λ+ (−⃗ p ) γ 0 ]2 [i Λ+ (⃗ p ) γ 0 ]1 . (16.65) S(P 0 ; p) ≈ ξ1 P 0 + p0 − ω1 (⃗ p ) + i ξ2 P 0 − p0 − ω2 (⃗ p ) + i With the choice ξi = mi /(m1 + m2 ) and under the condition that ∣E∣ ≪ ∣⃗ p ∣ ≪ mi , we can see that the denominators are strongly peaked about p0 = 0. The simplified non-interacting propagator −i[iΛ+ (⃗ p )γ 0 ]1 [iΛ+ (−⃗ p )γ 0 ]2 S0 (P 0 ; p) ≡ 2πδ(p0 ) (16.66) 0 P − ω1 (⃗ p ) − ω2 (⃗ p ) + i

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expresses this small-p0 behavior through a Dirac-δ function (as can be demonstrated by integrating S(P 0 ; p) and S0 (P 0 ; p) over the relative energy p0 ). Use of S0 instead of S requires a modification of the kernel, as seen in the preceding paragraph, but the modification just amounts to the addition of some (hopefully small) number of new contributions to the kernel. If we now set the relative energy equal to zero in (16.63), we find GT ∣p0 =q0 =0 = K∣p0 =q0 =0 + (KS0 GT )∣p0 =q0 =0 ,

(16.67)

where all dependence on relative energy has been eliminated, and (16.63) becomes a purely three-dimensional equation for GT (P 0 ; p0 = 0, p⃗, q 0 = 0, q⃗ ). Many alternative formalisms to the one just described have been investigated. One group of alternatives has a S0 similar to that of (16.66), with the same 2πδ(p0 ), an appropriate projection onto the large-large component, and a denominator that reduces to the nonrelativistic denominator E − p⃗ 2 /(2µ) in the nonrelativistic limit, but that differs in details. Another class of formalisms puts one of the particles onto its mass shell and uses the corresponding Dirac-δ function 2πδ(ξ1 P 0 + p0 − ω1 (⃗ p )) in S0 . Many of these formalisms were developed to study nucleon-nucleon scattering, but they are applicable to bound-state QED as well. (There are a number of reviews and comparisons of the many options that have been investigated, for example in [336,579]. The conditions required of a formalism in order that it have the correct relativistic one-body limit (i.e., the Dirac equation) were discussed in [580].) 16.2.3

Reference Kernels and Exact Solutions

The key to performing higher-order calculations of properties of two-body bound states is to find an appropriate lowest order, or reference, problem with an exact solution, and to base a perturbative procedure on that foundation. Without an exactly soluble reference problem, even the lowest order wave function is uncertain, and it is difficult to be sure that all contributions to a physical property such as an energy level at a given perturbative order are accounted for. In this section, we will discuss strategies for obtaining exact solutions to a reference BSEQ and go through the details of one particular example. This will be followed in the next section by the development of a perturbation theory appropriate for extracting corrections to energy levels (and wave functions) from the BSEQ, and in the following section by the use of our sample exact solution in calculations. The first step in finding a solvable reference problem is to choose a simplified non-interacting 2-to-2 propagator to use in place of i i ] [ ] , S(P ; p) = [ ξ1 P + p − m − Σ (ξ P + p) + i ξ P − p − m − Σ 1 1 1 2 2 2 (ξ2 P − p) + i 2 1

(16.68) where Σi (p) is the self-energy function for the ith particle. We will call the simplified propagator S0 , and eliminate G between the BSEQ (16.25) and the modified BSEQ G = S0 + S0 KG ,

(16.69)

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to arrive at the expression for the modified kernel K: K = K + S0−1 − S −1 . (16.70) The presence of the extra terms in K dependent on S0 and S are required to compensate for the use of S0 in (16.69). This manipulation only works if S0−1 exists. An example of a simplified non-interacting propagator that does not have an inverse is Eq. (16.66) above. No inverse exists for the S0 given in Eq. (16.66), because it contains projection operators, and a fundamental characteristic of projection operators is that they cannot be inverted. If a formalism utilizing a non-invertible S0 is desired [such as the one given in Eq. (16.66)], one must work with the truncated Green function GT instead of G. The BSEQ for GT was given in (16.63), and the appropriate modified kernel (16.64) does not require S0 to be invertible. It is not a disadvantage to work with GT instead of G, as GT has bound-state poles at the same positions as G. For the exact solution we choose to display here, however, we will work with G, not GT , and use as our reference non-interacting propagator S0 the free propagator i i ] [ ] , (16.71) S0 (P ; p) = [ ξ1 P + p ξ − m + i P − p − 2 1 1

m2 + i 2 which is invertible. All self-energy corrections have been shifted into the kernel with minimal alteration of the propagator. The job of projection onto the relevant large-large subspace will be accomplished by our choice of lowest-order kernel. We must now choose a lowest-order kernel K0 so that the reference BSEQ G0 = S0 + S0 K0 G0 (16.72) can be solved exactly for G0 . Our solution will be based on the solution to the nonrelativistic Schr¨ odinger-Coulomb (NRSC) equation. K0 must be carefully crafted so that, working together with S0 , Eq. (16.72) reduces to the nonrelativistic equation. Our plan is to use the reference BSEQ in the series form G0 = S0 + S0 K0 S0 + S0 K0 S0 K0 S0 + S0 K0 S0 K0 S0 K0 S0 + . . . = S0 + S0 K0 S0 + S0 [K0 S0 K0 + K0 S0 K0 S0 K0 + . . . ] S0 = S0 + S0 K0 S0 + S0 HS0 , (16.73) and sum the series H ≡ K0 S0 K0 + K0 S0 K0 S0 K0 + ⋯ . (16.74) This sum will be done by transforming it into the form of the corresponding sum for the nonrelativistic problem, which we must examine now. The NRSC Green function was discussed at length in Chap. 4. We review the results here in a slightly different notation useful for our present purposes. The NRSC Green function g(E) satisfies the equation (E − HS )g(E) = 1, which can be written either in coordinate or momentum space: p⃗ 2 ⃗, y⃗ ) = δ (3) (⃗ − V (∣⃗ x∣)) g(E; x x − y⃗ ) , (16.75) (E − 2m (E −

p⃗ 2 d3 ` ⃗ q⃗ ) = (2π)3 δ (3) (⃗ ) g(E; p⃗, q⃗ ) − ∫ V (⃗ p − `⃗) g(E; `, p − q⃗ ) , 2m (2π)3

(16.76)

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with HS given by Eq. (4.1) (except that we use a mass parameter m here instead of the reduced mass µ and will find that m is related to m1 and m2 in a non-trivial way), and V (r) = −Zα/r and V (⃗ p ) = −4πZα/⃗ p 2 . We will use the more flexible operator notation to write the solution to the inhomogeneous NRSC equation as −1 p⃗ 2 −1 − V + i) = (1 − s(E)V ) s(E) g(E) = (E − 2m = s(E) + s(E)V s(E) + s(E)V s(E)V s(E) + ⋯ = s(E) + s(E)V s(E) + s(E)h(E)s(E) , (16.77) where 1 . (16.78) s(E; p⃗, q⃗ ) = s(E; p⃗ )(2π)3 δ (3) (⃗ p − q⃗ ), s(E; p⃗ ) = 2 E − p⃗ /(2m) + i The free propagator is s(E) and h(E) ≡ V s(E)V + V s(E)V s(E)V + ⋯ (16.79) is a known function. (The exact form for h(E) follows from the known form for g(E).) The i is needed to enforce the causal boundary conditions on s(E) and g(E). For future use, we write out the product V s(E)V in full: d3 ` V (⃗ p − `⃗)s(E; `⃗)V (`⃗ − q⃗ ). (16.80) [V s(E)V ] (⃗ p, q⃗ ) = ∫ (2π)3 The bound-state poles of the NRSC Green function can be displayed in + p )ψnk (⃗ q) ∑ ψnk (⃗ g(E; p⃗, q⃗ ) → k , E → En , (16.81) E − En where the index k labels the degeneracy. We now return to our four-dimensional reference BSEQ shown in Eq. (16.72), with the goal of finding a reference kernel K0 for which (16.72) can be solved exactly. As we will show, an appropriate reference kernel is 1 1 q )] [γ 0 Λ+ (−⃗ p ) (1 + γ 0 )Λ+ (−⃗ q )] K0 (P 0 ; p, q) = [γ 0 Λ+ (⃗ p ) (1 + γ 0 )Λ+ (⃗ 2 2 1 2 × N (E; p⃗ 2 )(−i)V (⃗ p − q⃗ )N (E; q⃗ 2 ) , (16.82) where V (⃗ p ) = −4πZα/⃗ p 2 is the usual (momentum-space) Coulomb potential and N is a function to be specified below. The point of the projection operators in K0 and our coming choice of N is to make the sum H look like the corresponding sum for the NRSC Green function (16.79) discussed above. We start by evaluating the first term in H, which is K0 S0 K0 = [K0 S0 K0 ] (E; p, q), d4 ` [K0 S0 K0 ](E; p, q) = ∫ K0 (E; p, `) S0 (E; `) K0 (E; `, q) (2π)4 d4 ` =∫ N (E; p⃗ 2 ) (−i) V (⃗ p − `⃗) N 2 (E; `⃗ 2 ) (−i) V (`⃗ − q⃗ )N (E; q⃗ 2 ) (2π)4 1 i 1 γ 0 Λ+ (`⃗) (1 + γ 0 )Λ+ (⃗ q )] × [γ 0 Λ+ (⃗ p ) (1 + γ 0 ) Λ+ (`⃗) 2 2 ξ1 P +
` − m1 + i 1 1 i 1 × [γ 0 Λ+ (−⃗ p ) (1 + γ 0 ) Λ+ (−`⃗) γ 0 Λ+ (−`⃗) (1 + γ 0 ) Λ+ (−⃗ q )] . 2 2 ξ2 P −
` − m2 + i 2 (16.83)

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We use Eq. (16.49) to perform the integral over `0 (taking into account the imaginary infinitesimals), d`0 i d`0 i S0 (P ; `) = ∫ [ ] [ ] 2π 2π ξ1 P +
` − m1 + i 1 ξ2 P −
` − m2 + i 2 ⎧ ⎪ ⎪ [Λ+ (`⃗)γ 0 ]1 [Λ+ (−`⃗)γ 0 ]2 [Λ− (`⃗)γ 0 ]1 [Λ− (−`⃗)γ 0 ]2 ⎫ ⎪ ⎪ − ⎬ , (16.84) = i⎨ 0 0 ⃗ ⃗ ⃗ ⃗ ⎪ P + ω1 (` ) + ω2 (` ) − i ⎪ ⎪ ⎪ ⎭ ⎩ P − ω1 (` ) − ω2 (` ) + i and note that the Λ− projection operators are eliminated by the Λ+ projections in K0 . A further simplification is achieved using the relation

∫

1 ωi (`⃗) + mi 1 1 ) (1 + γ 0 ) . (1 + γ 0 )Λ+ (`⃗ ) (1 + γ 0 ) = ( 2 2 2 2ωi (`⃗)

(16.85)

We find that K0 S0 K0 reduces to d3 ` (−i)V (⃗ p − `⃗) (2π)3 ω1 (`⃗) + m1 iN 2 (E; `⃗2 ) ω2 (`⃗) + m2 × {( ) ( )} 0 ⃗ ⃗ ⃗ 2ω1 (` ) P − ω1 (` ) − ω2 (` ) + i 2ω2 (`⃗) 1 × (−i)V (`⃗ − q⃗ )N (E; p⃗ 2 ) [γ 0 Λ+ (⃗ p ) (1 + γ 0 )Λ+ (⃗ q )] 2 1 1 0 0 2 × [γ Λ+ (−⃗ p ) (1 + γ )Λ+ (−⃗ q )] N (E; q⃗ ). (16.86) 2 2 An astute choice of N (E; `⃗) can turn the expression in the curly bracket of (16.86) into a nonrelativistic propagator. Following [581, 582], we use [K0 S0 K0 ] (E; p, q) = ∫

N 2 (E; `⃗ 2 ) ≡ (

2ω1 (`⃗) 2P 0 ((P 0 )2 − (m1 − m2 )2 ) ) ω1 (`⃗) + m1 (P 0 + ω1 (`⃗) + ω2 (`⃗))((P 0 )2 − (ω1 (`⃗) − ω2 (`⃗))2 )

×(

2ω2 (`⃗) ). ω2 (`⃗) + m2

(16.87)

After a little bit of algebra, one can confirm that −1 ⃗2 ω1 (`⃗) + m1 N 2 (E; `⃗2 ) ω2 (`⃗) + m2 ˜ − ` + i) ≡ s˜(E; ˜ `⃗) , ) ( ) = (E 2m ˜ 2ω1 (`⃗) P 0 − ω1 (`⃗) − ω2 (`⃗) + i 2ω2 (`⃗) (16.88) which is the free nonrelativistic propagator, but with altered values for its nonrelativistic energy and its mass:

(

0 2 2 ˜ = (P ) − M = E(1 + E/(2M )) , E 0 2P 1 + E/M

m ˜ =

(P 0 )2 − (m1 − m2 )2 4m1 m2 + 2M E + E 2 = , 4P 0 4(M + E)

(16.89a) (16.89b)

where we recall that P 0 = M +E with E the nonrelativistic energy and M = m1 +m2 . (Note that E and M here are to be distinguished from En and Mn , which are

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the nonrelativistic energy and total mass respectively of the state having quantum numbers n and k.) Exactly the same type of simplification occurs for K0 S0 K0 S0 K0 , K0 S0 K0 S0 K0 S0 K0 , etc., allowing us to perform the sum for H [see Eq. (16.74)]. We find that the infinite series for H contains within it the series for h, which is known. One has H(E; p⃗, q⃗ ) = K0 S0 K0 + K0 S0 K0 S0 K0 + ⋯ = {∫

d3 ` ˜ `⃗)(−i)V (`⃗ − q⃗ ) (−i)V (⃗ p − `⃗)i˜ s(E; (2π)3

d3 ` d3 k ˜ `⃗)(−i)V (`⃗ − k⃗ )i˜ ˜ k⃗ )(−i)V (k⃗ − q⃗ ) + ⋯} (−i)V (⃗ p − `⃗)i˜ s(E; s(E; (2π)3 (2π)3 1 1 q )] [γ 0 Λ+ (−⃗ p ) (1 + γ 0 )Λ+ (−⃗ q )] N (E; q⃗ 2 ) × N (E; p⃗ 2 ) [γ 0 Λ+ (⃗ p ) (1 + γ 0 )Λ+ (⃗ 2 2 1 2 1 2 0 0 ˜ E; ˜ p⃗, q⃗ )N (E; p⃗ ) [γ Λ+ (⃗ p ) (1 + γ )Λ+ (⃗ = −ih( q )] 2 1 1 0 0 2 × [γ Λ+ (−⃗ p ) (1 + γ )Λ+ (−⃗ q )] N (E; q⃗ ) . (16.90) 2 2 ˜ E) ˜ is obtained from h(E) by the replacements m → m, ˜ We are Here, h( ˜ E → E. in a position now to write down an exact formula for our reference 2-to-2 Green function G0 (E) in terms of the known nonrelativistic Green function. The relations we use are +∫

G0 = S0 + S0 K0 S0 + S0 H S0

(16.91)

for the reference Green function, and g = s + sV s + shs,

h = s−1 gs−1 − s−1 − V

(16.92)

for the nonrelativistic Green function. In all, we find that G0 (E; p, q) = S0 (E; p, q) + S0 (E; p) K0 (E; p, q) S0 (E; q) ˜ p⃗ )˜ ˜ p⃗, q⃗ )˜ ˜ q⃗ ) − s˜−1 (E; p⃗, q⃗ ) − V (⃗ − iN (E; p⃗ 2 ) {˜ s−1 (E, g (E; s−1 (E; p − q⃗ )} N (E; q⃗ 2 ) 1 1 × S0 (E; p) [γ 0 Λ+ (⃗ p ) (1 + γ 0 )Λ+ (⃗ q )] [γ 0 Λ+ (−⃗ p ) (1 + γ 0 )Λ+ (−⃗ q )] S0 (E; q) . 2 2 1 2 (16.93) The only bound-state poles on the right-hand side of Eq. (16.93) are in the nonrelativistic Green function g˜, and they have the form given by (16.81). The bound ˜ and are located at the Bohr energies for mass m. state poles of G0 (E) live in g˜(E) ˜ That is, the poles are at values of E given by 2 m(Zα) ˜ ˜ = n ∣ E = − ≡ mβ ˜ n, βn ≡ −(Zα)2 /(2n2 ) , (16.94) m→m ˜ 2n2 ˜ and m and n ≡ βn m represents the Bohr energy. We use (16.89) to express E ˜ in terms of the physical binding energy E and masses, and find that the energy poles occur when P 0 → Pn0 (or, equivalently, E → En ) with 1/2

Pn0 = M (1 +

2µβn ) (1 − βn /2)M

(16.95)

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and 1/2

En = Pn0 −M = M (1 +

2µβn ) (1 − βn /2)M

−M = µβn +

µβn2 µ (1 − )+O(α6 ). (16.96) 2 M

˜ p⃗, q⃗ ) is The pole term of g˜(E; ∗ p )ψn`m (⃗ q) ˜ p⃗, q⃗ ) → ∑ ψn`m (⃗ g˜(E; ˜ E −  ∣ n m→m ˜ `m

∗ ∗ ψn`m (⃗ p )ψn`m (⃗ q) ψn`m (⃗ p )ψn`m (⃗ q) , →∑ ˜ E − βn m ˜ `m `m (1 − βn /2)(E − En )

=∑

(16.97)

˜ and m as E → En . We have used the definitions of E ˜ from Eq. (16.89), and the formula for En from Eq. (16.96). The corresponding pole term in G0 (E; p, q) 0

G0 (E; p, q) → i ∑ k

Ψ0nk (p)Ψnk (q) E − En

(16.98)

allows us to define the reference wave function and its conjugate. We equate the pole terms in (16.98) and (16.93), using (16.97), to find ∗ p ) ψn`m (⃗ q) i2 ∑`m ψn`m (⃗ 0 2 −1 ˜ 0 s (En ; p⃗ ) ∑ Ψnk (p)Ψnk (q) = N (En ; p⃗ )˜ 1 − βn /2 k 1 ˜n ; q⃗ )N (En ; q⃗ 2 ) S0 (Pn0 ; p) [γ 0 Λ+ (⃗ × s˜−1 (E p ) (1 + γ 0 )Λ+ (⃗ q )] 2 1 1 0 0 0 q )] S0 (Pn ; q). (16.99) × [γ Λ+ (−⃗ p ) (1 + γ )Λ+ (−⃗ 2 2 In order to complete the factorization of the right-hand side into a sum of products, we note the Fierz-like identity

1 1 χ χ (1 + γ 0 )aa′ (1 + γ 0 )bb′ = ∑ ( s1z ) ( s2z ) (χ+s1z , 0)a′ (χ+s2z , 0)b′ , 0 0 b 2 2 s1z ,s2z a

(16.100)

where the sums are over the possible spin states of the fermions (+1/2 or −1/2) and χm is a fundamental two-component Pauli spinor defined in Eq. (6.42). We see that the degeneracy index k actually represents `, m, s1z , and s2z , and that the reference wave functions are Ψ0nk (p) = η S(Pn0 ; p)ψn`m (⃗ p ) u1,s1z (⃗ p ) u2,s2z (−⃗ p ),

(16.101a)

0 Ψnk (p)

(16.101b)

=

∗ η u1,s1z (⃗ p )u2,s2z (−⃗ p )ψn`m (⃗ p ) S(Pn0 ; p) ,

where η ≡ (1 − βn /2)−1/2 ,

(16.102)

and ω1 (⃗ p ) + m1 S(P ; p) = i ( ) 2ω1 (⃗ p) 0

×

1/2

1/2

ω2 (⃗ p ) + m2 ( ) 2ω2 (⃗ p)

˜− (E

p⃗ 2 ) 2m ˜

N (E; p⃗ 2 )

(ξ1 P 0 + p0 − ω1 (⃗ p ) + i) (ξ2 P 0 − p0 − ω2 (⃗ p ) + i)

,

(16.103)

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and ψn`m are the usual nonrelativistic Schr¨odinger–Coulomb wave functions of Eq. (4.60). We use Dirac spinors defined by 2ωi (⃗ p) ) ui,m (⃗ p) = ( ωi (⃗ p ) + mi

1/2

⎡ ⎤ χm ⎞⎥⎥ ⎢ ⎢⎜ ⎟⎥ ⎢⎜ σ ⎟⎥ ⎢⎝ ⃗ ⋅ p⃗χm ⎠⎥ ⎢ ω (⃗ ⎥ ⎣ i p ) + mi ⎦i (16.104)

1/2 ⎢⎛

ωi (⃗ p ) + mi χ [Λ+ (⃗ p ) ( m )] = ( ) 0 i 2ωi (⃗ p)

and having normalization u+i,sz (⃗ p )ui,s′z (⃗ p ) = δsz ,s′z .

(16.105)

The S function is strongly peaked near p0 = 0 for nonrelativistic momentum and satisfies −1/2

ω1 (⃗ p ) + m1 dp0 S(P 0 ; p) = ( ) ∫ 2π 2ω1 (⃗ p)

(

−1/2

ω2 (⃗ p ) + m2 ) 2ω2 (⃗ p)

N −1 (E; p⃗ 2 ),

(16.106)

which is approximately one when E and the kinetic energies can be neglected compared to the rest masses. One can verify that the reference wave functions satisfy both the homogeneous reference BSEQ (16.29) Ψ0nk = S0 (Pn0 )K0 (Pn0 )Ψ0nk

(16.107)

and the normalization condition (16.41), 0

δrs = iΨnr 16.2.4

0 d d (S −1 (P 0 ) − K0 (P 0 ))∣ Ψ0ns . (16.108) G−1 (P 0 )∣ 0 Ψ0ns = iΨnr Pn0 Pn dP 0 0 dP 0 0

Bound-State Perturbation Theory

The purpose of bound-state perturbation theory is to find a systematic procedure for approximating the exact energy levels and wave functions (and the 2-to-2 Green function) based on the solution of a reference problem. The reference problem contains the dominant physics of the exact problem, but in a simplified form that allows the reference problem to be solved exactly. The difference between the exact and reference problems is in some sense small, and the perturbation theory we will construct will be in the form of a series in increasing powers of the small perturbation. For us, the exact problem is the BSEQ for the full renormalized 2-to-2 Green function G. This BSEQ has the form of (16.69), G = S0 + S0 KG

(16.109)

with the free propagator S0 shown in (16.71)) and a kernel K given by (16.70). The reference problem is a BSEQ of the same form, with the same free 2-to-2 free propagator S0 but having an approximate kernel K0 , G0 = S0 + S0 K0 G0 ,

(16.110)

as in (16.72). We can multiply by S0−1 on the left and G−1 or G−1 0 on the right to rewrite the two BSEQs (16.109) and (16.110) in the forms S0−1 = G−1 + K and S0−1 =

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−1 G−1 + K = G−1 0 + K0 . We then eliminate the common factor S0 to obtain G 0 + K0 , which can be solved for G,

G = G0 + G0 δK G = G0 + G0 δK G0 + G0 δK G0 δK G0 + ⋯ ,

(16.111)

where δK ≡ K − K0 . The full Green function G has poles at the exact bound state energies with residues giving the exact wave functions, while the reference Green function G0 has poles at the reference bound state energies with residues containing the reference wave functions. The task of perturbation theory is to extract from 0 (16.111) explicit relations between the exact E, Ψ, Ψ and the reference E 0 , Ψ0 , Ψ . There are several ways to develop the perturbative expansions for energy levels and wave functions coming from (16.111). We use the procedure outlined in [581]. There are two main differences between the perturbation theory that we will develop and the one used in ordinary nonrelativistic quantum mechanics. In quantum mechanics, Green functions, potentials, and wave functions depend on the coordinates of three-dimensional space (whether in coordinate or momentum representation), while our Green functions, kernels, and wave functions depend on the coordinates of four-dimensional space (the relative position or relative momentum variables). We point out that the dependence on relative time (or relative energy) is not a disadvantage. Rather, it is essential in order to include the effects of retardation. Of more immediate importance is the dependence of the perturbing kernel δK on the unknown CM energy. This is a new feature here that doesn’t typically occur in ordinary quantum mechanics. Our procedure can account for an energydependent kernel in a natural way, but some new terms are introduced that are not present in the usual Rayleigh–Schr¨odinger perturbation theory of ordinary quantum mechanics. Bound state energies appear as the positions of the poles of the 2-to-2 Green function, and we exploit this dependence to develop a perturbation theory. Near a bound-state pole En , G has the form G(E) = i ∑ k

Ψnk Ψnk + R(E), E − En

(16.112)

where n indexes all the distinct energy levels, k labels the degeneracy, and the remainder function R(E) is non-singular in the limit E → En . Note that there will always be some degeneracy because (in the absence of coupling to external fields), the energies can never depend on the z component of the total angular momentum. The reference Green function G0 has a similar expansion in terms of reference quantities G0 (E) = i ∑ k

Ψ0nk Ψ0nk + R0 (E), E − En0

(16.113)

as in (16.98). (When G0 has more degeneracy than G, then the usual techniques of degenerate perturbation theory must be employed, such as using suitable states of G0 in the first place or finding the suitable states by diagonalizing in the degenerate

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subspace. Our presentation here ignores this subtlety for now.) We assume that perturbative expansions exist for En and the wave functions Ψnk , Ψnk of the form En = En0 + En1 + En2 + ⋯, Ψnk = Ψnk =

Ψ0nk Ψ0nk

+ Ψ1nk + Ψ1nk

+ Ψ2nk + Ψ2nk

(16.114a)

+ ⋯,

(16.114b)

+ ⋯,

(16.114c)

where the reference energies and wave functions were found in Sec. 16.2.3 above. The perturbative order is indicated by a superscript. In our calculations here, we will only carry terms of O(δK)2 and ignore any higher-order terms, although it is just a matter of patience to work out higher order contributions. For the left-hand side of (16.111), we have G(E) =

0 1 2 0 1 2 iΨnk Ψnk i (Ψnk + Ψnk + Ψnk + ⋯) (Ψnk + Ψnk + Ψnk + ⋯) = 1 2 E +E +⋯ E − En (E − E 0 ) (1 − n n0 ) n

=

E−En

E 1 iΨ0 Ψ0 i(Ψ1nk Ψ0nk + Ψ0nk Ψ1nk ) iΨ0nk Ψ0nk + { n nk 0 nk + } 2 0 E − En (E − En ) E − En0 (E 1 )2 iΨ0nk Ψ0nk En2 iΨ0nk Ψ0nk + En1 i(Ψ1nk Ψ0nk + Ψ0nk Ψ1nk ) +{ n + (E − En0 )3 (E − En0 )2 iΨ2 Ψ0 + iΨ1nk Ψ1nk + iΨ0nk Ψ2nk + nk nk } + O(δK)3 . (E − En0 )

(16.115)

Summation over the degeneracy index k is implicit. (We have omitted the remainder term because it is non-singular in the limit E → En and doesn’t contribute to the perturbative results for the energies or wave functions.) Our next step is to expand the right-hand side of (16.111). For this expansion we use G0 (E) =

iΨ0nk Ψ0nk ˆ iΨ0nk Ψ0nk ˆ ˆ ′0 + O(E − En0 )2 , (16.116) + G (E) = + G0 + (E − En0 ) G 0 E − En0 E − En0

where 0 0 ˆ 0 (E) = G0 (E) − iΨnk Ψnk , G E − En0 and a similar expansion for δK:

ˆ0 = G ˆ 0 (En0 ), G

δK(E) = δK0 + (E − En0 )

ˆ ′0 = d G ˆ0∣ , G 0 En dE

d δK∣ 0 + O(E − En0 )2 , En dE

(16.117)

(16.118)

where δK0 ≡ δK(En0 ). We find that G(E) = G0 + G0 δKG0 + G0 δKG0 δKG0 + O(δK)3 =(

iΨ0nk Ψ0nk ˆ iΨ0nk Ψ0nk ˆ iΨ0nk Ψ0nk ˆ + G (E)) + ( + G (E)) δK(E) ( + G0 (E)) 0 0 E − En0 E − En0 E − En0

+( ×(

iΨ0nk Ψ0nk ˆ iΨ0nk Ψ0nk ˆ + G (E)) δK(E) ( + G0 (E)) δK(E) 0 E − En0 E − En0

iΨ0nk Ψ0nk ˆ + G0 (E)) + O(δK)3 , E − En0

(16.119)

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ˆ which, upon expanding out G(E) and δK(E), becomes iΨ0nk Ψ0nk iΨ0nk ⟨iδK⟩Ψ0nk } + { E − En0 (E − En0 )2 ˆ 0 + iΨ0 ⟨iδK⟩′ Ψ0 ˆ 0 δK0 iΨ0 Ψ0 + iΨ0 Ψ0 δK0 G G nk nk nk nk nk nk } + (E − En0 )

G(E) = {

+{

iΨ0nk ⟨iδK⟩2 Ψ0nk 1 ˆ 0 δK0 iΨ0nk ⟨iδK⟩Ψ0nk + [G 0 3 (E − En ) (E − En0 )2

ˆ 0 + iΨ0nk ⟨iδK G ˆ 0 δK⟩Ψ0nk + 2iΨ0nk ⟨iδK⟩⟨iδK⟩′ Ψ0nk ] + iΨ0nk ⟨iδK⟩Ψ0nk δK0 G 1 ˆ 0 δK⟩′ } [iΨ0nk Ψ0nk {⟨iδK⟩⟨iδK⟩′′ + (⟨iδK⟩′ )2 + ⟨iδK G E − En0 ˆ 0 δK0 Ψ0nk Ψ0nk δK0 G ˆ 0 + iG ˆ 0 δK0 Ψ0nk Ψ0nk ⟨iδK⟩′ + i[G ˆ 0 δK]′ Ψ0nk Ψ0nk ⟨iδK⟩ + iG

+

ˆ 0 δK0 G ˆ 0 δK0 Ψ0nk Ψ0nk + iΨ0nk Ψ0nk δK0 G ˆ 0 ⟨iδK⟩′ + iΨ0nk Ψ0nk [δK G ˆ 0 ]′ ⟨iδK⟩ + iG ˆ 0 δK0 G ˆ 0 ]} + O(δK)3 + O(E − En0 )0 , + iΨ0nk Ψ0nk δK0 G

(16.120)

where ⟨iX⟩ ≡ iΨ0nk X(En0 ) Ψ0nk , ⟨iX⟩′′ ≡ iΨ0nk

d2 X ∣ Ψ0 , dE 2 E 0 nk n

⟨iX⟩′ ≡ iΨ0nk [Y ]′ ≡

dX ∣ Ψ0 , dE En0 nk

dY ∣ . dE En0

(16.121a) (16.121b)

ˆ 0 δK, which depend on energy and two relative Here, X could be δK or δK G momenta. The quantities ⟨iX⟩, ⟨iX⟩′ , and ⟨iX⟩′′ are numbers, with no dependence ˆ 0 δK or δK G ˆ 0 , which depend on on relative momentum. The quantity Y could be G energy and two relative momenta, while [Y ]′ is a function only of the two relative momenta. Finally, we compare the two expansions (16.115) and (16.120) for G. These expansions are organized into powers of δK and (E − En0 ), and like powers in each of the two forms for G must match. The first correction to the energy is found by comparing the terms of O(δK × (E − En0 )−2 ), with the result that En1 = ⟨iδK⟩. The second correction comes from terms of O(δK 2 × (E − En0 )−2 ), which leads to ˆ 0 δK⟩ + ⟨iδK⟩⟨iδK⟩′ . Corrections to the wave function and the relation En2 = ⟨iδK G conjugate wave function come from comparison of the O(δK × (E − En0 )−1 ) and O(δK 2 × (E − En0 )−1 ) terms. In all, we find that ˆ 0 δK⟩ + ⟨iδK⟩ ⟨iδK⟩′ + O(δK)3 , En = En0 + ⟨iδK⟩ + ⟨iδK G ˆ 0 δKΨ0nk + 1 ⟨iδK⟩′ Ψ0nk + O(δK)2 , Ψnk = Ψ0nk + G 2 1 0 0 ˆ Ψnk = Ψnk + Ψnk δK G0 + ⟨iδK⟩′ Ψ0nk + O(δK)2 . 2

(16.122a) (16.122b) (16.122c)

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We did not write out the O(δK)2 corrections to the wave functions, because we will not be using them. One should note that the corrected wave functions will not automatically be normalized. Terms with powers of (E − En0 ) other than −1 or −2 provide checks of the previous results but do not lead to new relations. Because only the 1/(E − En0 ) and 1/(E − En0 )2 terms in (16.120) are needed in order to work out perturbations to the energies and wave functions, it is possible to proceed in a more elegant manner using the residue theorem to extract from G(E) only the required contributions [583–585]. We choose a small closed contour Cn in the complex energy plane that encloses (in a counter-clockwise way) the pole En of G(E) and En0 of G0 (E) and no other pole of either. Application of the residue theorem to (16.112) gives iΨnk Ψnk = ∮

Cn

En iΨnk Ψnk = ∮

Cn

dE G(E) , (16.123a) 2πi dE dE E G(E) = En0 iΨnk Ψnk + ∮ (E − En0 ) G(E) . 2πi Cn 2πi (16.123b)

By way of illustration, we write out the results coming from (16.123) through O(δK), ˆ 0 δK0 iΨ0nk Ψ0nk + iΨ0nk Ψ0nk δK0 G ˆ0 iΨnk Ψnk = iΨ0nk Ψ0nk + G + ⟨iδK⟩′ iΨ0nk Ψ0nk + O(δK)2 , En iΨnk Ψnk =

En0 iΨnk Ψnk

+ ⟨iδK⟩iΨ0nk Ψ0nk

(16.124a) + O(δK) , 2

(16.124b)

from which the results of (16.122a) follow [through O(δK)]. It is an advantage, especially if higher orders of δK are required, to focus only on the terms in the expansion of (16.120) having orders 1/(E − En0 ) and 1/(E − En0 )2 . 16.2.5

Energy Levels at O(Zα)4 (BSEQ Approach)

In this section, we obtain the corrections to the energy levels of (spin-1/2–spin-1/2) two-body Coulombic bound states at order α4 . These corrections are well known, and have been discussed already in Chaps. 11 and 12. Still, it is instructive to see them again from a different point of view, and coming from a formalism that can be extended in a systematic way to higher orders. These corrections arise from relativistic effects (relativistic kinetic energy, magnetic current-current interaction, and Darwin term) and interactions associated with spin (the spin-orbit interaction and a spin-spin tensor interaction). In the related case of a particle-antiparticle bound state, there is also a correction at this order arising from virtual annihilation to a single photon. The energy levels of a Coulombic system are found by starting with the known solution of a reference problem. We use the solution shown in Sec. 16.2.3 and a perturbative expansion as in the preceding Sec. 16.2.4. The reference energy levels

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were given in Eq. (16.96), with the result En0 = M (1 +

2βn µ ) 1 − βn /2 M

1/2

− M = µβn +

µβn2 µ (1 − ) + O(α6 ) , 2 M

(16.125)

where βn = −(Zα)2 /(2n2 ), while µ = m1 m2 /M is the reduced mass and M = m1 +m2 the total mass. The first of the O(α4 ) energy corrections comes from the O(βn2 ) term in the expansion of the reference energy. Additional energy corrections at O(α4 ) come from first-order perturbation theory. In Eq. (16.122a), this is the correction containing only one factor of δK: ∆E ≈ ⟨iδK⟩ .

(16.126)

Moreover, only two specific terms in δK are required. Let KC and KT represent the kernels for exchange of a single Coulomb photon and a single transverse photon. It turns out that the only terms in δK that contribute O(α4 ) energy corrections (for the particle-particle case) are KC − K0 , which is the difference between a Coulomb exchange and the reference kernel, and KT . In the remainder of this section we will give the details of how the contributions of KC − K0 and KT can be reduced to an effective perturbing Hamiltonian that can be used with the NRSC wave functions to give the energy level corrections. We begin with the transverse photon contribution ∆ET = ⟨iKT ⟩ = iΨ0 KT Ψ0 . By making explicit the momentum dependence of wave functions and kernel and using the QED Feynman rules for KT , we find the transverse energy contribution to be ˆ iδ ⊥,ij (k) d4 p d4 q 0 i j [−ieγ ] [−i(−Ze)γ ] Ψ0 (q), (16.127) Ψ (p) ∆ET = i ∫ 4 4 2 1 2 (2π) (2π) k where e is the electron charge, −Ze is the charge of the positively charged particle, and k = p − q is the photon momentum. The spatial part of the Coulomb gauge photon propagator given in Eq. (9.73) contains the transverse Dirac-δ function in momentum space [see also Eq. (2.54)], ki kj δ ⊥,ij (k⃗ ) = δ ij − kˆi kˆj = δ ij − . (16.128) k⃗ 2 Inserting the form of the reference wave functions from (16.101), the ∆ET energy correction becomes d3 p d3 q dp0 dq 0 0 S(P ; p) S(Pn0 ; q) ∫ ∫ n (2π)3 (2π)3 2π 2π δ ⊥,ij (k⃗ ) × ψ ∗ (⃗ p ) u1 (⃗ p )γ i u1 (⃗ q) 2 u2 (−⃗ p )γ j u2 (−⃗ q ) ψ(⃗ q ), k − k⃗ 2

∆ET = − 4πZα η 2 ∫

(16.129)

0

−1/2

where η = (1 − βn /2) was defined on Eq. (16.102) and ψ, ψ ∗ are the nonrelativistic Schr¨ odinger wave functions of Eq. (4.60). Subscripts were attached to the Dirac spinors to indicate which particle they belong to (1 for the electron, 2 for the

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positively charged one), but otherwise subscripts labeling the state being considered have been left implicit. The correction ∆ET , simple as it is, is still far to complex to be evaluated exactly. We shall have to make approximations and find in ∆ET the O(α4 ) part. The approximations needed are based on considerations of the physical sizes and frequencies (or equivalently, momenta and energies) that characterize nonrelativistic Coulombic bound states. The physical size is set by the Bohr radius (as appropriate for an atom of reduced mass µ and positive charge number Z), which in natural units is a0 = 1/(Zαµ). The corresponding momentum scale is Zαµ, which characterizes the width of the wave functions in momentum space. The typical atomic energy scale (Zα)2 µ is set by the Bohr energy levels. An examination of the positions of the p0 poles in the reference wave functions [see Eqs. (16.101) and (16.103)], which are found at √ (16.130a) p0 = −ξ1 P 0 + ω1 (⃗ p ) = −m1 − ξ1 E + m21 + p⃗ 2 ≈ −ξ1 E + p⃗ 2 /(2m1 ) , √ p0 = ξ2 P 0 − ω2 (⃗ p ) = m2 + ξ2 E − m22 + p⃗ 2 ≈ ξ2 E − p⃗ 2 /(2m2 ) , (16.130b) confirms the fact that the scale of the Bohr energy levels characterizes the relative energy dependence of the reference wave functions. In passing, we remark that, in deriving (16.130), we used the definition ξi = mi /M . We can now use our knowledge of the scales characterizing the problem to make a number of approximations. The first is to neglect k02 = (p0 − q0 )2 relative to k⃗ 2 in the transverse photon propagator. We have seen that k 2 is of O(α4 ), while k⃗ 2 0

is of order O(α2 ) for typical energies and momenta of the system. So, to a first approximation, k 2 = k02 − k⃗ 2 ≈ −k⃗ 2 . With k0 in the photon propagator out of the way, the relative energies p0 and q0 are decoupled, and we can use (16.106) to perform the energy integrals −1/2

∫

dp0 ω1 (⃗ p ) + m1 S(Pn0 ; p) = ( ) 2π 2ω1 (⃗ p)

(

−1/2

ω2 (⃗ p ) + m2 ) 2ω2 (⃗ p)

N −1 (En ; p⃗ 2 ) 1/2

={

(Pn0 + ω1 (⃗ p ) + ω2 (⃗ p ))((Pn0 )2 − (ω1 (⃗ p ) − ω2 (⃗ p ))2 ) } 2Pn0 ((Pn0 )2 − (m1 − m2 )2 )

=1+(

1 1 1 p⃗ 2 βn µ + 2− ) − +⋯ 2 m1 m2 m1 m2 8 4M

= 1 + O(⃗ p 2 ) + O(α2 ) , (16.131) where we have expanded the result of the integration for small α and small p⃗/m1 and p⃗/m2 . The leading order contributions to the Dirac matrix elements are easy to work out: ⎛ χ ⎞ 0 σi ⃗ ⋅ p⃗ χ+ σ ⎟ u(⃗ p )γ i u(⃗ q ) ≈ (χ+ , − )( i )⎜ ⃗ ⋅ q⃗χ ⎟ −σ 0 ⎜ σ 2m ⎝ ⎠ 2m 1 + i 1 + ⃗ i ] χ, (16.132) ⃗ ⋅ q⃗ + σ ⃗ ⋅ p⃗ σ i ) χ = = χ (σ σ χ [(p + q)i + i (⃗ σ × k) 2m 2m

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p )γ i u(−⃗ q ) is minus the same. The first term in (16.132), proportional to and u(−⃗ p⃗+⃗ q , constitutes the convection term (see Secs. 11.4.2 and 12.5.3), as it has to do with the motion of a charged particle. The second term gives rise to a spin-dependent −1 interaction. We neglect η 2 = (1 − βn /2) , which is 1 + O(α2 ), and obtain 3 3 d p d q ∆ET ≈ −4πZα ∫ (2π)3 (2π)3 i⎤ j⎤ ⎡ ⎡ ⃗ ) ⎥ δ ⊥,ij (k⃗ ) ⎢(p + q)j − i (k⃗ × σ ⃗) ⎥ ⎢(p + q)i − i (k⃗ × σ ⎥ ⎥ χ1 χ2 ψ(⃗ ⎢ × ψ ∗ (⃗ p )χ+1 χ+2 ⎢⎢ q ). ⎥ ⎥ ⃗ 2 ⎢⎢ 2m1 2m2 ⎥ ⎥ ⎢ ⎦1 k ⎦2 ⎣ ⎣ (16.133) If we define an effective Hamiltonian through d3 p d3 q ∗ ψ (⃗ p )χ+1 χ+2 HA χ1 χ2 ψ(⃗ q ), (16.134) ∆EA ≡ ∫ (2π)3 (2π)3 4 then the effective Hamiltonian for transverse photon exchange at O(α ) is πZα 1 −4 2 2 (⃗ ⃗2 ) + (k⃗ × σ ⃗1 ) ⋅ (k⃗ × σ ⃗2 )} . p × q⃗ ) ⋅ (⃗ σ1 + σ p ⋅ q⃗ )2 ) − 2i(⃗ p q⃗ − (⃗ HT = { m1 m2 k⃗ 2 k⃗ 2 (16.135) The first term in HT comes completely from the coupling of convection currents and represents the basic magnetic interaction. The second term contributes part of the spin-orbit interaction, and the third term is a spin-spin tensor interaction. All contribute to the energy at exactly O(α4 ). The Coulomb exchange contribution requires a little more effort to evaluate because ⟨iKC ⟩ and ⟨iK0 ⟩ both come in at O(α2 ) and only the difference contributes at the order O(α4 ) of interest. We start with the Coulomb exchange kernel i i4πZα[γ 0 ]1 [γ 0 ]2 [−i(−Ze)γ 0 ]2 = KC = [−ieγ 0 ]1 , (16.136) k⃗ 2 k⃗ 2 with expectation value d4 p d4 q (16.137) ⟨iKC ⟩ = iΨ0 KC Ψ0 = −4πZαη 2 ∫ (2π)4 (2π)4 1 × ψ ∗ (⃗ p )S(Pn0 ; p) S(Pn0 ; q)ψ(⃗ q )u1 (⃗ p )γ 0 u1 (⃗ q ) u2 (−⃗ p )γ 0 u2 (−⃗ q). k⃗ 2 There is no coupling between p0 and q 0 since the Coulomb photon is instantaneous, and so the relative energy integrals can be done already. The result is given in (16.131). This time, we need to keep terms of O(⃗ p 2 ) and O(α2 ). The required Dirac matrix elements are χ 1/2 1/2 ⎛ ⎞ ⃗ ⋅ p⃗ ⎜ χ+ σ ω(⃗ p) + m ω(⃗ q)+m 0 + ⎟ )⎜ σ u(⃗ p )γ u(⃗ q) ≈( ) ( ) (χ , ⃗ ⋅ q⃗ χ ⎟ ω(⃗ p) + m 2ω(⃗ p) 2ω(⃗ q) ⎝ ω(⃗ q ) + m⎠ 1/2

1/2

⃗ ω(⃗ p) + m ω(⃗ q)+m p⃗ ⋅ q⃗ + i(⃗ p × q⃗ ) ⋅ σ ) ( ) χ+ [1 + ]χ 2ω(⃗ p) 2ω(⃗ q) (ω(⃗ p ) + m)(ω(⃗ q ) + m) k⃗ 2 i ⃗} χ + ⋯ , = χ+ {1 − + (⃗ p × q⃗ ) ⋅ σ (16.138) 8m2 4m2

=(

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where the terms that were left out are of fourth order in the momenta. For ⟨iKC ⟩ we find d3 p d3 q (2π)3 (2π)3

⟨iKC ⟩ = − 4πZα η 2 ∫

(16.139)

1 1 1 (⃗ p 2 + q⃗ 2 ) βn µ 1 ) − } ψ(⃗ q ) {1 + ( 2 + 2 − m1 m2 m1 m2 8 2M k⃗ 2 ⃗1 ⃗2 1 1 k⃗ 2 i σ σ × χ+1 χ+2 {1 − ( 2 + 2 ) + (⃗ p × q⃗) ⋅ ( 2 + 2 )} χ1 χ2 + ⋯ . m1 m2 8 4 m1 m2 × ψ ∗ (⃗ p)

The corresponding contribution to the effective Hamiltonian is HC = − −

k⃗ 2 1 1 (⃗ p 2 + q⃗ 2 ) 1 1 1 4πZα 2 η {1 − ( 2 + 2)+ ( 2+ 2− ) 2 ⃗ 8 m1 m2 8 m1 m2 m1 m2 k ⃗1 ⃗2 βn µ i σ σ + (⃗ p × q⃗) ⋅ ( 2 + 2 )}. 2M 4 m1 m2

(16.140)

This term contributes initially at O(α2 ) with a remainder of O(α4 ). The leading order part will be canceled by the subtraction of the reference kernel. Finally, we need to evaluate the expectation value of the reference kernel ⟨iK0 ⟩. One has ⟨iK0 ⟩ = iΨ0 K0 Ψ0 = iη 2 ∫

d4 p d4 q (2π)4 (2π)4

× ψ ∗ (⃗ p )S(Pn0 ; p)N (En ; p⃗ 2 )(−i)V (⃗ p − q⃗ )N (En ; q⃗ 2 )S(Pn0 ; q)ψ(⃗ q) × u1 (⃗ p )u2 (−⃗ p ) [γ 0 Λ+ (⃗ p)

1 + γ0 1 + γ0 Λ+ (⃗ q )] [γ 0 Λ+ (−⃗ p) Λ+ (−⃗ q )] u1 (⃗ q )u2 (−⃗ q). 2 2 1 2 (16.141)

Using the explicit formulas for the projectors, one obtains ⟨iK0 ⟩ = iη 2 ∫

d4 p d4 q (2π)4 (2π)4

× ψ ∗ (⃗ p )(−i)V (⃗ p − q⃗ )ψ(⃗ q ) S(Pn0 ; p)N (En ; p⃗ 2 )N (En ; q⃗ 2 )S(Pn0 ; q) ×(

ω1 (⃗ p ) + m1 ) 2ω1 (⃗ p)

= η2 ∫

1/2

(

ω1 (⃗ q ) + m1 ) 2ω1 (⃗ q)

1/2

1/2

(

ω2 (⃗ p ) + m2 ) 2ω2 (⃗ p)

d3 p d3 q ψ ∗ (⃗ p )V (⃗ p − q⃗ )ψ(⃗ q), (2π)3 (2π)3

(

ω2 (⃗ q ) + m2 ) 2ω2 (⃗ q)

1/2

(16.142)

where the last step was to perform the relative energy integrals using Eq. (16.106). The corresponding contribution to the effective Hamiltonian is H0 = η 2 V (⃗ p − q⃗ ) = −

4πZα 2 η . k⃗ 2

(16.143)

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The complete effective Hamiltonian for O(α4 ) corrections is the sum of the transverse part HT and the Coulomb part HC − H0 : Heff = V (k⃗ ){ − +

1 1 (⃗ p 2 + q⃗ 2 ) 1 βn µ ( 2+ 2− )− 8 m1 m2 m1 m2 2M

(⃗ p 2 q⃗ 2 − (⃗ p ⋅ q⃗)2 ) 1 k⃗ 2 1 ( 2 + 2)+ 8 m1 m2 m1 m2 k⃗ 2

⃗2 2(⃗ ⃗2 ) ⃗1 ⃗1 ) ⋅ (k⃗ × σ ⃗2 ) σ σ1 + σ σ (k⃗ × σ i (⃗ p × q⃗) ⋅ ( 2 + 2 + )− } 4 m1 m2 m1 m2 4m1 m2

(16.144)

where k⃗ = p⃗ − q⃗. We were able to approximate η by one in the difference HC − H0 because the O(α2 ) contributions cancel. This O(α4 ) effective Hamiltonian agrees in all details with the one obtained in Ref. [582] using a closely related formalism (and, of course, with the results presented in Chap. 12). The first two terms can be seen to arise from the first relativistic correction to the particles’ kinetic energy (as discussed in Chap. 17). The remaining contributions are the Darwin term, the magnetic interaction, the spin-orbit term, and the spin-spin tensor interaction (see also Chap. 12). 16.3 16.3.1

Relativistic Recoil Correction (BSEQ Approach) Exploring Integration Regions

We now supplement the derivation of the relativistic recoil correction (Salpeter correction) outlined in Sec. 15.5 by a derivation based on the Bethe–Salpeter equation for QED. The recoil correction is relevant to a bound system such as hydrogen or muonium at order (Zα)5 . The terms come from interactions involving particle exchange but not radiative corrections to one or the other particle’s interaction (such as would be encoded in a form factor, in a first approximation). Energy corrections coming from a Dirac equation, which includes relativistic corrections, or the Breit equation commence at order O(Zα)4 and go on to order O(Zα)6 and higher even powers of Zα because they are ultimately obtained from expansions in the small quantity v 2 /c2 ∼ (Zα)2 . Terms at order (Zα)5 have their origin in quantum field theory and its description of the bound state as a two-body system. Energy corrections of order (Zα)5 arise naturally from perturbing kernels involving the exchange of two photons, just as the one-photon exchange graphs of Sec. 16.2.5 gave corrections of order (Zα)4 . The power counting that leads to this expectation includes one power of Zα for each exchanged photon and a factor of (Zα)3 coming from the wave function at the origin (or more generally, simply from the wave function). However, power counting for bound states is more subtle, and in fact all of the diagrams of Fig. 16.4 give contributions of order (Zα)5 . The one-photon-exchange (1γE) diagrams contribute initially at order (Zα)4 , as seen in the preceding section, but have higher-order contributions as well. Such behavior is characteristic of bound state graphs. Their effect commences at some initial order

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Fig. 16.4 The recoil graphs that contribute to bound state energies through order (Zα)5 are shown here. The type of exchange photon involved is indicated both graphically and with a label: C for Coulomb, 0 for reference, and T for transverse. These graphs all involve exchange photons only. Contributions with radiative corrections on one fermion line or the other as well as vacuum polarization effects are not considered here. The first six graphs shown are part of δK and appear in the expectation value ⟨iδK⟩. The final four graphs come from the second order ˆ 0 δK⟩ with only the 0-potential part of G ˆ0: G ˆ 0 → S0 . The label perturbation theory term ⟨iδK G C + indicates a sum of one, two, up to an infinite number of Coulomb photon exchanges, all in a ladder configuration with respect to each other and all crossed by the single transverse photon.

but they contribute at higher orders as well. More interesting are the diagrams involving C + where C + denotes a sum of terms involving exchange of one, two, etc., Coulomb photons in a ladder configuration. These diagrams all give contributions at order (Zα)5 , no matter how many Coulomb photons are exchanged. The behavior here is similar to that discussed in Sec. 15.4 for the usual Lamb shift and will be fully explicated in the present context below. As a first step in analyzing the order (Zα)5 recoil corrections, we note that the graphs in Fig. 16.4 with two exchange photons have an internal loop with a loop momentum that must be integrated over. That loop momentum can be either relativistic or nonrelativistic. (In this context, relativistic refers to values of energy and momentum comparable or greater than the smaller of the two masses.) In most cases, a nonrelativistic loop momentum leads to energy corrections of order (Zα)6 for these graphs. As a quick guide to this estimate, we note that a graph with two reference photons in a ladder configuration would give an energy contribution at order (Zα)2 , the same as for a single reference photon exchange, because of the reference BSEQ: Ψ0 = S0 K0 Ψ0 = S0 K0 S0 K0 Ψ0 = . . . . For nonrelativistic momenta, the reference photon and the Coulomb photon are effectively the same and the difference C −0 (Coulomb minus reference photon) involves corrections proportional to the square of the momenta. Similarly, the transverse photon connects to a spatial Dirac γ matrix on each end, which leads to factors of the momenta at each end. The consequence is that the (KC − K0 )S0 (KC − K0 ), (KC − K0 )S0 KT , and KT S0 (KC − K0 ) graphs of Fig. 16.4 contribute at order (Zα)6 for nonrelativistic loop momenta. The reasoning is different for the crossed-Coulomb graph. If

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Fig. 16.5 We depict the recoil graphs that contribute to bound state energies through order (Zα)5 with a nonrelativistic loop momentum for the two-photon-exchange graphs.

Fig. 16.6 We depict the recoil graphs that contribute to bound-state energies through order (Zα)5 . These are the same graphs that were displayed in Fig. 16.4 but with the C − 0 terms expanded out and after use of the reference BSEQ.

we refer to the two poles in (16.46) as the particle and antiparticle poles, then in the integral over the loop energy for the crossed-Coulomb graph, the only nonvanishing contributions have one particle pole and one antiparticle pole, leading to an expression smaller in the numerator by a factor of (Zα)2 and less small in the denominator by (Zα)2 than the two-rung Coulomb ladder, i.e., a contribution of order (Zα)6 . The graph with crossed Coulomb and transverse does contribute at order (Zα)5 in a way that we will discuss at length. Also, the two-transverse graphs contribute through the [Λ− ]1 [Λ− ]2 part of S0 because u(p2 )γ i Λ− (⃗ q )γ 0 γ j u(p1 ) is of order one for nonrelativistic particles, so the power counting gives a total contribution of order (Zα)5 in the same way as for relativistic loop momentum as will be discussed below. The complete set of diagrams with nonrelativistic loop momenta that contribute at order (Zα)5 is shown in Fig. 16.5. When considering relativistic loop momentum, it is perhaps easier to put the order (Zα)5 graphs shown in Fig. 16.4 in the form of Fig. 16.6. The contributions here with relativistic loop momentum are shown in Fig. 16.7. The power counting is now easy. The fact that the loop momentum is relativistic means, by application of the uncertainty principle, that the space-time extent of the loop (roughly equal to 1/µ) is small compared to the size of the atom, which is set by the Bohr radius 1/(Zαµ). So, these contributions factorize into the product of a scattering amplitude calculated for external particles at rest [order (Zα)2 ] times the probability that the two particles making up the bound system will be in contact, the latter

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Fig. 16.7 The recoil graphs that contribute to bound state energies through order (Zα)5 with a relativistic loop momentum for the two-photon-exchange graphs.

being equal to ∣ψ(⃗ r = 0)∣2 = O(Zα)3 . This gives order (Zα)5 in all for the energy correction. Our separation of the calculation into two regions (relativistic and nonrelativistic) brings about a complication: the relativistic and nonrelativistic contributions are separately divergent. The complete energy shift at order (Zα)5 , and in fact each graph by itself, is completely finite and well defined, but the process of separation into regions engenders divergences due to the approximations that will be made appropriate for each region. The generation of divergences is inevitable when graphs are pulled apart into regions and simplified enough as to allow for analytic evaluation of the contribution at each order of Zα. The alternative to this procedure of divide-and-conquer is to evaluate the graphs as a whole using numerical integration. The integrands are typically quite complex, multi-dimensional, and hard to integrate with good precision. So, analysis by regions is the usual approach. We will use dimensional regularization (DR) to regularize the divergent integrals we encounter. We have encountered this method already in Secs. 10.3.6, 10.4.2 and 11.4.4. However, a more thorough discussion might still be useful. In general, among the various approaches to regularization than have been explored, DR does the least violence to the theory. It does not break gauge invariance and leaves the forms and algebraic properties of propagators and vertices unchanged, except for the need to interpret the dimensions of space-time (or energy-momentum space) as d-dimensional instead of four-dimensional. We imagine that d can take a general (possibly complex) value, that will tend to four at the end of the calculation after all divergences have canceled. In practice, we write d = 4 − 2ε [see Eq. (10.172)] and let ε → 0 at the end. It is also useful to define D = d − 1 = 3 − 2 [see Eq. (10.141)] as the dimension of space when dealing, as here, with bound systems at rest in a specific (CM) frame. The regularized space-time dimension alters the spacetime volume elements (dd x and dd p/(2π)d instead of d4 x and d4 p/(2π)4 ), some

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algebraic properties (so that g µ µ = d, δ ii = D, etc.), but is mostly just hidden in the background until the time comes to actually do the integrals. One aspect of DR that we need to account for is the alteration of the units of some physical quantities in d dimensional space-time. While mass, energy, and spatial momentum all have units of mass, distances and times have units of inverse mass. The action is dimensionless in DR, the quantum fields for the fermions and photon have a dimension altered by a quantity of O(ε) from what they had in four dimensions. A consequence is that the quantum of charge is no longer dimensionless in d dimensions but must have units mε where m is a mass. The exact value chosen for m is arbitrary as it disappears in the limit ε → 0, and we use this arbitrariness to our advantage. We define the charge quantum so that the product of the charges for the bound-state constituents is q1 q2 = −Ze2 = −4πZαµ2ε ,

µ2 = µ2

e γE . 4π

(16.145)

This definition is consistent with Eq. (10.173). We recognize γE as the Euler– Mascheroni constant γE = 0.57721 56649 . . . . The reason for this definition, suggested by the modified minimal subtraction (“MS-bar”, MS), is that µ is arbitrary anyway so it doesn’t really matter whether we include the extra factors of eγE and 4π or not, but with the eγE /(4π) factor the divergences arising from loop integrals take a much simpler form. Specifically, the expansion eεγE Γ(ε) =

1 1 1 9 + ζ(2)ε − ζ(3)ε2 + ζ 2 (2)ε3 + O(ε4 ) ε 2 3 40

(16.146)

is much simpler than the expansion of Γ(ε) by itself. The factor of 4π acts to cancel part of the volume element dd p/(2π)d that remains after the momentum integral has been performed — as will be seen below. (The customary notation in field theory for the MS mass parameter is µ instead of µ, but we reserve µ for the reduced mass as before.) The technique we use to isolate the contributions of our bound state integrals coming from various parts of momentum space is called the method of regions. This approach is a powerful technique for the separation of scales and constitutes a generalization of the scale-separation parameter or “overlapping parameter” as described in Sec. 4.5.2. We describe the method of regions in some detail in the next section and apply it in the following sections. 16.3.2

Method of Regions

A central challenge of bound-state physics is the evaluation of integrals that contain important contributions from several different parts of the parameter space. The method of regions [586–590] is a tool that, in conjunction with DR, allows contributions from the various regions to be separated and evaluated. In this section we will describe the method of regions, work through a detailed example of its use, and discuss its application to bound-state physics.

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The method of regions consists of the following steps: (i) Identify the relevant regions that contribute to the integral under consideration. (ii) In each region, expand the integrand in a Taylor series in terms of the quantities that are small in that region. (iii) In each region, integrate the expanded integral, term by term, over the whole integration domain of the integration variables. (iv) Add up the contributions from the various regions to regain the full integral. It is expected that the approximations appropriate for the various regions will lead to new divergences (ones not present in the original integral) that are automatically regulated by DR. New divergences generated in this way will cancel when the contributions of the different regions are combined. As is usual in DR, scaleless integrals are regulated to have the value zero. A particularly clear exposition of this fact, together with further explanatory remarks, is given in Chap. 8 of Ref. [201] [see the text following Eq. (8.52) of Ref. [201]]. We return to the example studied in Sec. 4.5.2, ∞ √ (16.147) F (α) = ∫ dω e−ω α2 + ω 2 , 0

where α ≪ 1 is a small parameter. We saw before how the function F (α) could be (approximately) evaluated by breaking the integration region into parts separated by an overlapping parameter ε where α ≪ ε ≪ 1, and making appropriate approximations in each part. Now, we find the value of F (α) by use of the method of regions. Our first step is to identity the relevant regions that √ contribute to F (α). We see that the integrand is a product of two terms, e−ω and α2 + ω 2 . The exponential clearly has a characteristic scale set by ω ∼ 1, while the square root has a characteristic scale set by ω ∼ α. So for this example there are two regions to deal with, which we call the hard (ω ∼ 1) and soft (ω ∼ α) regions. Since we plan to use DR as a regularization, we need to dimensionally generalize our integral. Usually, for four-dimensional space-time, we generalize by the replacement d4 p → dd p = pd−1 dp dΩp = p3−2ε dp dΩp , where dΩp represents the integral over the d − 1 angles of p when using spherical coordinates. For our one-dimensional integral F (α), we dispense with angles and just work with the simplified dimensional generalization ∞ √ F (α, ε) ≡ ∫ dω ω −2ε e−ω α2 + ω 2 . (16.148) 0

Now, we apply the method of regions (and dimensional regularization). For the hard region, we can expand the square root using √ √ α2 α4 α2 + ω 2 = ω 1 + (α/ω)2 = ω {1 + 2 − 4 + O(α6 )} . (16.149) 2ω 8ω The required integrals can be done immediately using the integral representation of the gamma function: Γ(n + 1) = ∫

∞ 0

dω e−ω ω n .

(16.150)

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For the hard part of F (α, ε), we find the expansion FH (α, ε) = ∫

∞ 0

dω e−ω ω 1−2ε {1 +

α2 α4 − 4 + O(α6 )} 2 2ω 8ω

α2 α4 Γ(−2ε) − Γ(−2 + 2ε) + O(α6 ) 2 8 α2 1 α4 1 3 γE =1+ (− − γE ) − (− + − ) + O(ε) + O(α6 ) . (16.151) 2 2ε 8 4ε 4 2 For the soft region ω is small compared to one and we need to expand the exponential in a Taylor series. The dependence on α is displayed most clearly if we scale the ω variable using ω = αx. We find that = Γ(2 − 2ε) +

FS (α, ε) = ∫

∞ 0

dω ω −2ε {1 − ω +

ω2 ω3 − + ⋯} (α2 + ω 2 )1/2 2 6

= α2−2ε {χ(−2ε, 1/2) − α χ(1 − 2ε, 1/2) + −

α2 χ(2 − 2ε, 1/2) 2

α3 χ(3 − 2ε, 1/2) + . . . } 6

= α2 [

1 1 1 2 α3 1 1 1 2 + + ln ( )] + + α4 [− + − ln ( )] 4ε 4 2 α 3 32ε 64 16 α

α5 + O(ε) + O(α6 ), 45 where we have used the integral formula −

χ(m, n) ≡ ∫

∞ 0

dx xm (1 + x2 )n =

(16.152) ) Γ (− m+1 Γ ( m+1 − n) 2 2 2Γ(−n)

,

(16.153)

which converges when m > −1 and m + 2n < −1. We note in passing that the integral χ(m, n) fails to converge for n = 1/2 no matter what value we take for m. The integral χ(m, n) is defined instead by analytic continuation using Eq. (16.153), which is perfectly well defined when n = 1/2 except for occasional poles. We obtain the full F (α, ε) integral by adding the hard and soft parts: F (α, ε) = FH (α, ε) + FS (α, ε) 1 γE 1 2 α3 = 1 + α2 [ − + ln ( )] + 4 2 2 α 3 5 γE 1 2 α5 + α4 [− + − ln ( )] − + O(ε) + O(α6 ). (16.154) 64 16 16 α 45 Since the original integral F (α) is defined as the limit of F (α, ε) as ε → 0, we have completed our task, at least through terms of O(α5 ). Our expression (16.154) agrees with the earlier result (4.276) that was obtained using an overlapping parameter. It can also be confirmed by analytic integration of (16.147) followed by expansion in α, but the possibility of direct analytic integration of bound state integrals is seldom possible in practical calculations.

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Quantum Electrodynamics in Action: Atoms, Lasers and Gravity Table 16.1 Regions appearing in QED bound state calculations. Here m represents a typical mass — the reduced mass say — and α is the fine structure constant. (In this analysis we are not trying to distinguish between the possibly different scales set by m1 and m2 if they are distinct.) region

notation

typical energy

typical momentum

hard

H

m

m

soft

S

mα

mα

potential

P

mα2

mα

US

mα2

mα2

ultrasoft

The DR regions approach to the evaluation of multi-scale integrals has a number of attractive features compared to the use of overlapping parameters. One is the fact that each term that appears in each region after performing the Taylor expansion has a unique power of the small parameter (modulo possible logarithms). Also, integrals over the whole infinite integration range (even when supplemented by factors like x−2ε ) are often easier to evaluate and have more convenient analytic properties than the integrals over a partial range that would result from overlap parameters. For integrals resulting from quantum loops, the DR regions approach is even more advantageous, as it is awkward to decide just what parameter to have its range restricted by an overlapping parameter: the energy of one of the particles, or of another, or all; or perhaps not the energy at all but the three-momentum magnitude or the Euclidean four-momentum magnitude after Wick rotation. The advantages of DR regions are even more apparent when a single integral has more than just two relevant regions. Experience has shown that there are four regions relevant to the physics of nonrelativistic Coulombic bound states. They are the hard, soft, potential, and ultrasoft regions with characteristic values of energy and spatial momenta as shown in Table 16.1. We take some time to understand something about the origin and nature of each of these regions. We look first at the wave function (16.101): Ψ0nk (p) = η S(Pn0 ; p) ψn`m (⃗ p ) u1 (⃗ p ) u2 (−⃗ p),

(16.155)

and recall the definitions of η from Eq. (16.102) and S(Pn0 ; p) from Eq. (16.103). The relevant part of this function for understanding the regions that are characteristic of a nonrelativistic bound state is 1 ψn`m (⃗ p) (ξ1 Pn0 + p0 − ω1 (⃗ p ) + i)(ξ2 Pn0 − p0 − ω2 (⃗ p ) + i) 1 ≈ ψn`m (⃗ p). 2 ⃗ p p⃗ 2 (ξ1 En + p0 − 2m1 + i) (ξ2 En − p0 − 2m + i) 2

Ψ0nk (p) ∝

(16.156)

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631

Here ωi (⃗ p ) ≡ (m2i + p⃗ 2 )1/2 , while En is the binding energy defined in Eq. (16.96). The expansion in the momenta implied in Eq. (16.156) is relevant for nonrelativistic motion of the constituents. As described in Sec. 4.4.4, the dependence of the wave function on the magnitude of the spatial momentum has the typical behavior 1 , (16.157) ψn`m (⃗ p) ∝ 2 (⃗ p + γn2 )2 where γn = µZα/n. This specific form for the wave function is for the ground state, but all bound states involve powers of the factor (⃗ p 2 + γn2 ) in the denominator, setting the momentum scale ∣⃗ p ∣ ∼ γn ∼ mα. The scale for the relative energy p0 is determined by the positions of the poles in Ψ0 (p). Given that both En and p⃗ 2 /m have the scale γn2 /m, we see that the natural scale for the relative energy set by the wave function is p0 ∼ γn2 /m ∼ mα2 . So, the natural scale for bound-state processes, as set by the wave function, is of the potential kind according to the classification in Table 16.1. This is the region we assumed as the dominant region in the order(Zα)4 calculations of Sec. 16.2.5. Other regions are engendered by the transverse photon propagator 1/(k 2 + i) = [(k 0 − ∣k⃗ ∣ + iε)(k 0 + ∣k⃗ ∣ − i)]−1 when retardation effects (those requiring the presence of k 0 in the propagator) are important. Both poles of the photon propagator set the photon energy magnitude equal to its spatial momentum magnitude. If that spatial momentum is ∼ mα, then the energy will be ∼ mα as well, and the soft region is obtained. In a like manner, if the energy is ∼ mα2 , then the spatial momentum will be ∼ mα2 , giving the ultrasoft region. If we don’t assume that the motion is nonrelativistic, then it is clear from the form of 2 2 the fermion propagator i/(p
− m + i) → 1/(p − m + iε) that relativistic values of 0 0 p and p⃗ (p ∼ ∣⃗ p ∣ ∼ m) can be important. This “hard” region also plays a role in the physics of QED bound states. 16.3.3

Contribution from the Hard Region

We are now ready to tackle the calculation of relativistic recoil corrections at O(α5 ). We already saw that we can separate these into ones involving a nonrelativistic loop momentum as shown in Fig. 16.5 and ones involving a relativistic loop momentum as in Fig. 16.7. The nonrelativistic ones can be further segregated into soft, potential, and ultrasoft contributions, as we will show. The relativistic ones represent hard region contributions. We will discuss the hard region contributions first, then move on to the nonrelativistic regions, and put our results together at the end. The hard-region contribution refers to the part of the total energy shift coming from the region of momentum space where the photons and the two fermions connecting the photons are all hard. The diagrams contributing in this region are shown in Fig. 16.7. We will go through the calculation of these contributions pair by pair, starting with the first two diagrams Fig. 16.7. These are just the ones involving two Coulomb photons. In Fig. 16.8(a), we depict these two diagrams with momentum labels included. The associated wave functions on each side, Ψ0 (p) and

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Fig. 16.8 These are the graphs — ladder plus crossed — involving two Coulomb photons that contribute to order (Zα)5 energy levels from the hard region. Part (a) shows the graphs with ⃗ ) is the CM momentum, ξi = mi /(m1 + m2 ), momentum labels included. Here P = (m1 + m2 + E, 0 and p, k, and q represent the relative momenta. Part (b) shows the same graphs but with hard region approximations indicated for the central loop integral, using ξi P ± k → mi n ± k and ξ1 P + ⃗ ). Wave functions are implicitly present on both sides. p + q − k → m1 n − k with n ≡ (1, 0

Ψ0 (q), are assumed to be present but are not shown. The hard region approximations appropriate for the central loop involve the neglect of p and q compared to the loop momentum k as well as the neglect of the reference bound state energy E relative to m1 + m2 as shown in Fig. 16.8(b). The integrals over p, k, and q factorize in this approximation, and the central part is the two-particle scattering amplitude at rest due to the exchange of two Coulomb photons. For the wave function integrals, we use (16.131) and

∫

dD p ⃗ = ψ(⃗0) . ψ(⃗ p ) = ψ(⃗ r = 0) (2π)D

(16.158)

⃗ will be interpreted in the following as being (A wave function at the origin ψ(0)

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Bethe–Salpeter Equation

taken at the origin of coordinate space, not momentum space.) We see that ∫ ∫

dd p 0 ⃗ 1 (⃗0)u2 (0), ⃗ Ψ (p) ≈ ψ(0)u (2π)d

(16.159a)

dd p 0 ⃗ 2 (0) ⃗ , Ψ (p) ≈ ψ ∗ (⃗0)u1 (0)u (2π)d

(16.159b)

⃗ with i = 1, 2 are evaluated at the origin of where the fundamental spinors ui (0) momentum space. So, the contribution factorizes into the product of a free particle at-rest scattering amplitude times the probability of contact ∣ψ(⃗0)∣2 . Physically, the scattering event happens with relatively high energy and momentum, i.e., effectively instantaneously and at a point relative to the spatial size of the bound state. It remains to calculate the scattering-amplitude part of the diagram. We apply the usual Feynman rules of QED to calculate the scattering amplitude, starting with the ladder (left side) diagram of Fig. 16.8(b): ⃗ 2∫ ∆ECC;L ≈ i∣ψ(0)∣ ×(

i dd k 0 ⃗ ⃗ u1 (0)(−ieγ ) (−ieγ 0 )u1 (0) d (2π) m1n
+ k
− m1

i i 2 0 ⃗ ⃗ ) ) u2 (0)(iZeγ (iZeγ 0 )u2 (0) 2 ⃗ m2n k
− k
− m2

⃗ 2 (4πZαµ2ε ) ∫ = i∣ψ(0)∣ 2

−1 dd k [(k 2 + 2 k⋅n m1 )(k 2 − 2 k⋅n m2 )(k⃗ 2 )2 ] d (2π)

0 0 0 × (u1 γ 0 [m1n
+ k
+ m1 ] γ u1 ) (u2 γ [m2n
− k
+ m2 ] γ u2 )

(16.160)

where n = (1, ⃗0) is the timelike unit vector, and we have used the abbreviation ⃗ = u(⃗ here that u = u(0) p = 0). In terms of the fundamental Pauli spinors χ [see Eq. (6.42)], the Dirac spinor for a particle at rest is u = (χ, 0)T . Since we are working out the spin-independent–that is, the spin averaged–energy shift in this section, we find for the Dirac expectation value 1 1 uXu = Tr[Xuu] → ∑ Tr[Xuu] = Tr [X(γ 0 + 1)] , (16.161) 2 spin 4 µ

because the spin-averaged outer product can be evaluated as follows, 1 1 10 1 1 ∑ χ ⊗ χ+ 0 ) = ( ) = (γ 0 + 1), ∑ u ⊗ u = ( spin 0 0 2 spin 2 2 00 4

(16.162)

where the Dirac matrices are represented in matrix form by giving the large-large, large-small, small-large, and small-small components (see also Chap. 15). We find after performing the spin averaging and the traces that ⃗ 2 (4πZαµ2ε )2 ∫ ∆ECC;L → i∣ψ(0)∣

dd k (2m1 + k 0 )(2m2 − k 0 ) . (2π)d (k 2 + 2 k⋅n m1 )(k 2 − 2 k⋅n m2 )(k⃗ 2 )2 (16.163)

We recall, for convenience, Eq. (16.145). The integral is most easily done using Feynman parameters, completing the square in the denominator, followed by a

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Wick rotation and finally the actual integration as in the Appendix of Ref. [591] (see also Chaps. 11 and 15). Despite the fact that the integral looks like it would have an infrared divergence (because of the (k⃗ 2 )2 factor in the denominator), the dimensionally regularized result is finite in the limit d → 4. Its value is 2 1 2 2 − 2), ∆ECC;L = (Zα)2 ∣ψ(⃗0)∣2 (− 2 − 3 m1 m1 m2 m2

(16.164)

in agreement with (3.7) of Ref. [535], where a different form of regulation was used. (The potential divergence in (16.163) is a Coulomb singularity coming from the ladder configuration of Coulomb photons. Such a singularity is regulated to zero by DR [592].) The crossed-Coulomb calculation is done in the same way, leading to ⃗ 2 (4πZαµ2ε ) ∫ ∆ECC;X = i∣ψ(0)∣ 2

dd k (2m1 − k 0 )(2m2 − k 0 ) (2π)d (k 2 − 2 k⋅n m1 )(k 2 − 2 k⋅n m2 )(k⃗ 2 )2

2 ⃗ 2( 2 − 1 + 2 ) , = (Zα)2 ∣ψ(0)∣ 3 m21 m1 m2 m22

(16.165)

in agreement with (3.9) of Ref. [535]. The total Coulomb contribution is the sum: 2 −2 ∆ECC = ∆ECC;L + ∆ECC;X = (Zα)2 ∣ψ(⃗0)∣2 ( ). (16.166) 3 m1 m2 All graphs with one transverse photon and one Coulomb photon vanish in the hard region, as we now proceed to show. The same momentum assignments as in Fig. 16.8(b) are used, even though now not all photons are Coulomb ones. Consider the first graph on the bottom row of Fig. 16.7. The trace factor for the top fermion line is 1 1 0 0 a 0 0 a 0 a ∑ u1 γ (m1 γ +k
+m1 )γ u1 = Tr [γ (m1 γ + k
+ m1 )γ (γ + 1)] = k , (16.167) 2 spin 4 where a is the spatial index for one end of the transverse photon propagator. Since that propagator carries the transverse delta factor δ ⊥,ab (k⃗ ) = δ ab − kˆa kˆb and k a δ ⊥,ab (k⃗ ) = 0, the contribution of this graph is zero. The remaining three hardregion graphs with one of each kind of photon vanish for the same reason. We complete our hard-region calculation with the evaluation of the double transverse graphs. Again using the momentum assignments of Fig. 16.8(b), the transverse-transverse ladder graph gives i dd k u1 (−ieγ a ) (−ieγ c )u1 d (2π) m1n
+ k
− m1 iδ ⊥,ab (k⃗ ) iδ ⊥,cd (k⃗ ) i × u2 (iZeγ b ) (iZeγ d )u2 k2 k2 m2n
− k
− m2 d ⃗ 2 (4πZαµ2ε )2 ∫ d k [(k 2 + 2 k⋅n m1 )(k 2 − 2 k⋅n m2 )(k 2 )2 ]−1 = i ∣ψ(0)∣ (2π)d c b d × δ ⊥,ab (k⃗ )δ ⊥,cd (k⃗ )(u1 γ a [m1n
+ k
+ m1 ]γ u1 )(u2 γ [m2n
− k
+ m2 ]γ u2 ) .

⃗ 2∫ ∆ETT;L = i ∣ψ(0)∣

(16.168)

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We perform the spin average as before and find that the traces simplify remarkably. The transverse-transverse ladder contribution becomes (2 − d)(k 0 )2 dd k 2 ∆ETT;L = i∣ψ(⃗0)∣2 (4πZαµ2ε ) ∫ . (2π)d (k 2 + 2 k⋅n m1 )(k 2 − 2 k⋅n m2 )(k 2 )2 (16.169) The integral here is a standard covariant loop integral that can be performed with the usual methods, leading to the result ∆ETT;L =

(Zα)2 2ε ⃗ 2 1 1 m1 ln(m2 /µ) + m2 ln(m1 /µ) µ ∣ψ(0)∣ { + − +O(ε)} (16.170) m1 m2 2ε 2 m1 + m2

after setting d = 4 − 2ε and expanding about ε = 0. (Note that only one factor of µ2ε was used in the expansion.) The result has an infrared divergence that is manifest as the pole at ε = 0. Because of the divergence, we cannot just set d = 4 in the wave function. Fortunately, all divergences will cancel before we need to know the value of the wave function in d dimensions, at which point we can go to the usual four-dimensional space-time and use the standard result for ψ(⃗0). An entirely analogous calculation for the transverse-transverse crossed contribution leads to ⃗ 2 (4πZαµ2ε ) ∫ ∆ETT;X = i∣ψ(0)∣ 2

=

dd k (d − 2)(k 0 )2 d 2 (2π) (k + 2 k⋅n m1 )(k 2 + 2 k⋅n m2 )(k 2 )2

(Zα)2 2ε ⃗ 2 1 1 m1 ln(m2 /µ) − m2 ln(m1 /µ) µ ∣ψ(0)∣ { + − +O(ε)} . (16.171) m1 m2 2ε 2 m1 − m2

The total transverse-transverse hard contribution is the sum of the ladder plus crossed terms: ∆ETT = ∆ETT;L + ∆ETT;X =

m2 ln(m2 /µ) − m22 ln(m1 /µ) (Zα)2 2ε ⃗ 2 1 µ ∣ψ(0)∣ { + 1 − 2 1 + O(ε)} . (16.172) m1 m2 ε m21 − m22

We can now combine our Coulomb-Coulomb and transverse-transverse results to get the total contribution from the hard region. Adding the results from (16.166) and (16.172), we find ∆Ehard = ∆ECC + ∆ETT =

m2 ln(m2 /µ) − m22 ln(m1 /µ) (Zα)2 2ε ⃗ 2 1 1 µ ∣ψ(0)∣ { − − 2 1 +O(ε)} . (16.173) m1 m2 ε 3 m21 − m22

We anticipate that the infrared divergence exhibited here will cancel against an ultraviolet divergence in the nonrelativistic region to leave a finite total. Along with that cancellation of divergences, the µ factor in the logarithms will be replaced by one of the nonrelativistic scales to produce a logarithmic O((Zα)5 ln(Zα)) contribution to the overall recoil correction. It is useful to point out that the central part of this calculation, that of the amplitude for scattering of two fermions at rest involving the exchange of two photons, could have been done in Feynman gauge instead of Coulomb gauge. The

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two-photon-exchange diagrams form a gauge-invariant set and any gauge can be used for the calculation. The Feynman gauge calculation is simpler as there is only one kind of photon (with propagator −ig µν /k 2 ) and one kind of vertex (−ieγ µ ), so fewer terms appear and those that do are more compact and covariant. Of course, the result obtained using Feynman gauge is the same as the Coulomb gauge result of (16.173), demonstrating the gauge invariance of the hard region, separately. 16.3.4

Soft Contribution from Two Transverse Photons

We now turn to contributions coming from nonrelativistic regions, and start with the final two graphs of Fig. 16.5 involving the exchange of two transverse photons. We begin by looking at the top fermion line, having Dirac spinors at beginning and end, vertex factors, and the fermion propagator, which we assume carries relative momentum k. This fermion line makes a contribution 1 γ c u1 (⃗ q) ξ1 P + k
− m1 + i Λ− (k⃗ )γ 0 Λ+ (k⃗ )γ 0 + } γ c u1 (⃗ q) = u1 (⃗ p )γ a { ξ1 P 0 + k 0 − ω1 (k⃗ ) + i ξ1 P 0 + k 0 + ω1 (k⃗ ) − i ⎫ ⎧ ⃗ q⃗ ) ⎪ ⎪ σa σc F ac (⃗ p, k, ⎪ ⎪ + ≈ χ+1 ⎨ ⎬ χ1 (16.174) 2 2 ⃗ ⃗ k k ⎪ ⎪ 0 0 ⎪ ⎭ ⎩ ξ1 E + k − 2m1 + i 2m1 + ξ1 E + k + 2m1 + i ⎪

p )γ a N1 ≡ u1 (⃗

⃗ q⃗ ) is quadratic in the small momenta p⃗, k, ⃗ q⃗. The fermion propawhere F ac (⃗ p, k, gator terms coming from Λ+ give small contributions in any of the nonrelativistic regions [of order O(α2 )], and denominator factors are not able to completely reverse this tendency. Only the Λ− terms contribute, and it turns out that these only contribute in the soft region of the loop momentum k. In this soft region, k 0 is O(α) as are all the spatial momenta. The leading approximation to the two-transversephoton graph in the ladder configuration is dD p dD q ∗ χ+1 σ a σ c χ1 χ+2 σ b σ d χ2 ψ (⃗ p )ψ(⃗ q ) (2π)D (2π)D 2m1 2m2 ⊥,ab ⊥,cd ⃗ d ⃗ δ (⃗ p−k) δ (k − q⃗ ) d k . (16.175) ×∫ (2π)d (k 0 )2 − (⃗ p − k⃗ )2 + i (k 0 )2 − (k⃗ − q⃗ )2 + i

soft ∆ETT;L = i(4πZαµ2ε )2 ∫

The spin average takes χ+1 σ a σ c χ1 → δ ac , and similarly χ+2 σ b σ d χ2 → δ bd . We make a shift in the integration variable k⃗ → k⃗ + p⃗ and note that the crossed graph contribution at this order is identical to that of the ladder graph. In all, the spin-averaged recoil correction due to the soft region with two exchanged transverse photons is soft ∆ETT =

(4πZα)2 2ε dD p dD q ∗ µ ∫ ψ (⃗ p )ψ(⃗ q )I(⃗ p, q⃗ ) , 2m1 m2 (2π)D (2π)D

(16.176)

where I(⃗ p, q⃗ ) is a divergent (in the ultraviolet) non-covariant integral that can be

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evaluated by the usual methods, giving I(⃗ p, q⃗ ) = i µ2ε ∫ =

δ ⊥,ab (k⃗ + p⃗ − q⃗ ) dd k δ ⊥,ab (k⃗ ) (2π)d k 2 + i (k 0 )2 − (k⃗ + p⃗ − q⃗ )2 + i

1 1 ∣⃗ p − q⃗ ∣ 8 5 {− + 2 ln ( ) − ln 2 + } . 8π 2  µ 3 3

(16.177)

The resulting energy shift is soft ∆ETT =

(Zα)2 2ε ⃗ 2 [− 1 − 2 ln µ − 8 ln 2 + 5 ] + 2⟨ ln(∣⃗ µ {∣ψ(0)∣ p − q⃗ ∣)⟩p,⃗ } , (16.178) ⃗q m1 m2  3 3

where ⟨ln(∣⃗ p − q⃗ ∣)⟩p,⃗ p − q⃗ ∣), which ⃗ q is the momentum space expectation value of ln(∣⃗ equals the usual expectation value of the Fourier transform of ln k: ⟨ ln(∣⃗ p − q⃗ ∣)⟩p,⃗ =∫ ⃗q

dD p dD q ∗ ψ (⃗ p ) ln(∣⃗ p − q⃗ ∣)ψ(⃗ q) (2π)D (2π)D

= ∫ dD x ψ ∗ (⃗ x ) FT(ln k)(⃗ x ) ψ(⃗ x) = ⟨FT(ln k)⟩

(16.179)

⃗ The result for this expectation value is given in the Appendix of where k = ∣k∣. Ref. [593]: n−1 2µZα ⎧ ⎪ ) + Hn + ln ( ⎪ ⎪ (µZα)3 ⎪ n 2n ⎪ ⟨FT(ln k)⟩ = ( )⎨ 3 1 ⎪ πn ⎪ ⎪ − ⎪ ⎪ ⎩ 4`(` + 1)(` + 1/2)

(` = 0) ,

(16.180)

(` > 0)

where Hn = ∑nj=1 1j is the nth harmonic number. We note that the ultraviolet divergence arising from the two-transverse-exchange soft region just cancels the infrared divergence that was found in the hard region (16.173). The expectation value of the operator given in Eq. (16.180) is related to the Araki–Sucher distribution [see Eq. (15.133) and Refs. [537, 538]. 16.3.5

Soft Contribution from a Single Transverse Photon

There are three contributions to recoil energies at order (Zα)5 involving the exchange of a single nonrelativistic transverse photon, as shown in Fig. 16.5. We consider first the contribution from the second graph in that figure containing an isolated transverse photon. This graph is shown in Fig. 16.9 including specifications for the fermion and photon momenta. If the left and right bound states surrounding that exchange photon in ∆E = iΨ0 δKΨ0 have relative momenta p and q, then the momentum flowing up the exchanged photon is k ≡ p − q. If k is soft, momentum conservation requires at least one of p or q be soft as well. The regions where the ordered set {p, k, q} is {potential, soft, soft} or {soft, soft, potential} give the O(Zα)5 contributions. Consider then the region {potential, soft, soft} in the sense

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Fig. 16.9 Graphs containing a single transverse exchange photon crossing zero, one, or two Coulomb photons. Wave functions are implicitly present on both sides.

of Table 16.1. The single transverse exchange energy shift takes the approximate form dd p dd q 0 iδ ⊥,ab (k⃗ ) P,S,S a [iZeγ b ]2 Ψ0 (q) ∆ET = i∫ Ψ (p) [−ieγ ] 1 (2π)d (2π)d k 2 + i ≈ −4πZαµ2ε i ∫

dD p dD q dq0 ∗ ψ (⃗ p ) u1 (⃗ p )γ a u1 (⃗ q ) u2 (−⃗ p )γ b u2 (−⃗ q ) ψ(⃗ q) (2π)D (2π)D 2π

× δ ⊥,ab (⃗ p − q⃗ )(E −

q⃗ 2 −1 )[(q0 − ∣⃗ p − q⃗ ∣ + i)(q0 + ∣⃗ p − q⃗ ∣ − i)(q0 + i)(−q0 + i)] . 2µ (16.181)

In obtaining this approximation, we performed the p0 integral using ∫

dp0 0 Ψ (p) ≈ u1 (⃗ p )u2 (−⃗ p )ψ(⃗ p) 2π

(16.182)

0

or specifically its analog for Ψ , which hold in the nonrelativistic limit. We also made the soft q 0 approximation that q 0 ∼ α dominates terms of O(α2 ) such as p0 , E, and q⃗ 2 , and implemented a number of general nonrelativistic approximations based on the smallness of E, p⃗ 2 /(2µ), and q⃗ 2 /(2µ) compared to the reduced mass scale µ. Our next step is to perform the q 0 integration, which we do by poles. A subtle point is that the poles at q 0 = 0 are disallowed because they would take q 0 out of the soft region which would be inconsistent with the assumed softness of q. Furthermore, the Dirac expectation value takes the approximate form ⃗ ⋅ q⃗ + σ ⃗ ⋅ p⃗σ a σa σ pa + q a ui (⃗ p )γ a ui (⃗ q ) ≈ χ+ χ→ , (16.183) 2mi 2mi where the last form was achieved by averaging over spin. We also note that (p+q)a = (2p − (p − q))a → 2pa when next to a transverse delta. The energy shift takes the form 4πZαµ2ε dD p dD q ∗ pa pb δ ⊥,ab (⃗ p − q⃗ ) q⃗ 2 P,S,S ∆ET =− ψ (⃗ p) (E − ) ψ(⃗ q ). ∫ D D 3 m1 m2 (2π) (2π) 2∣⃗ p − q⃗ ∣ 2µ (16.184) It is difficult to continue this evaluation in a straightforward way because the q⃗ integration is divergent in the infrared (for ∣⃗ q ∣ ∼ ∣⃗ p∣). In order to proceed, we would need to know the long-distance structure of the wave function in D = 3 − 2ε dimensions. Since we do not know the D-dimensional wave function in that much detail, we will

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instead isolate the wave function from the infrared (IR) divergence by providing a buffer. Specifically, we use the nonrelativistic Schr¨odinger–Coulomb equation (E −

q⃗ 2 dD ` −4πZαµ2ε ) ψ(⃗ q) = ∫ ( ) ψ(`⃗) , 2µ (2π)D (⃗ q − `⃗)2

(16.185)

to insert a Coulomb photon between the soft transverse photon and the wave func⃗ `⃗ → q⃗ to find: tion, and then change integration variables according to q⃗ → p⃗ − k, P,S,S ∆ET =

dD p dD q 8π 2 (Zα)2 µ2ε ∫ m1 m2 (2π)D (2π)D × ψ ∗ (⃗ p ) pa pb (µ2ε ∫

dD k δ ⊥,ab (k⃗ ) ) ψ(⃗ q ). (2π)D ∣k⃗ ∣3 (k⃗ − p⃗ + q⃗ )2

(16.186)

The infrared divergent integral dD k δ ⊥,ab (k⃗ ) (2π)D ∣k⃗ ∣3 (k⃗ − `⃗)2 dD k δ ⊥,ab (k⃗ ) 2k⃗ ⋅ `⃗ 1 + = µ2ε ∫ { } (2π)D ∣k⃗ ∣3 k⃗ 2 + `⃗2 (k⃗ − `⃗)2 (k⃗ 2 + `⃗2 )

I ab (`⃗) ≡ µ2ε ∫

(16.187)

is not too difficult to evaluate in the latter form, since the first term in brackets is relatively simple (if divergent) and the second is finite (so the D → 3 limit may be safely taken). The result is I ab (`⃗) =

1

3π 2 `⃗2

{(−

1 1 2∣`⃗∣ + + ln ( )) δ ab − `ˆa `ˆb } + O(ε). 2ε 2 µ

(16.188)

(We are using `⃗ = p⃗ − q⃗.) If we include the contributions from both soft singletransverse-exchange regions: {potential, soft, soft} and {soft, soft, potential}, the total correction is P,S,S S,S,P soft ∆ET = ∆ET + ∆ET =

×∫

8π 2 (Zα)2 µ2ε m1 m2

dD p dD q ∗ ψ (⃗ p ) (pa pb + q a q b ) I ab (⃗ p − q⃗ ) ψ(⃗ q ). (2π)D (2π)D

(16.189)

We will complete our examination of soft corrections by the evaluation of the crossed transverse-Coulomb graph — the middle one in Fig. 16.9. The relevant part of this graph is the region where the central loop (the photons and connecting fermions) are soft and both wave functions are potential. The full expression for the energy shift coming from this graph is i dd p dd q dd k 0 Ψ (p) [(−ieγ a ) (−ieγ 0 )] d d d (2π) (2π) (2π) ξ1 P + p − k − m + i 1
1
iδ ⊥,ab (k⃗ ) i i × 2 [(iZeγ 0 ) (iZeγ b )] Ψ0 (q). k + i (⃗ ξ2 P −
q − k
− m2 + i p − q⃗ − k⃗ )2 2 (16.190)

∆ETC = i ∫

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The nonrelativistic expansion of the electron propagator, for k soft and p potential, takes the form iΛ+ (⃗ p − k⃗ )γ 0 i = 0 0 0 ξ1 P + p p − k⃗ ) + i
− k
− m1 + i ξ1 P + p − k − ω1 (⃗ iΛ− (⃗ p − k⃗ )γ 0 + ξ1 P 0 + p0 − k 0 + ω1 (⃗ p − k⃗ ) − i

iΛ+ (⃗ p − k⃗ )γ 0 iΛ− (⃗ p − k⃗ )γ 0 + , (16.191) −k 0 + i 2m1 with a similar form for the positive fermion propagator. The first term in each dominates because k 0 /m1 ∼ α. All dependence on p0 and q 0 is confined to the wave functions, so the relative energy p0 and q 0 integrals can be done. We find the approximate soft region energy correction to be dD p dD q dd k ∗ soft ψ (⃗ p ) u1 (⃗ p )γ a Λ+ (⃗ ∆ETC ≈ (4πZαµ2ε )2 i ∫ p − k⃗ )γ 0 γ 0 u1 (⃗ q) (2π)D (2π)D (2π)d 1 1 δ ⊥,ab (k⃗ ) u (−⃗ p )γ 0 Λ+ (−⃗ q − k⃗ )γ 0 γ b u2 (−⃗ q ) ψ(⃗ q). × 2 0 + i)2 2 2 ⃗ k + i (⃗ (−k p − q⃗ − k ) (16.192) ≈

We perform the integration over k 0 using only the poles in the transverse photon propagator — use of the soft fermion poles would be inconsistent with the assumptions that the fermions are soft — and evaluate the spin-averaged Dirac expectation values: dD p dD q dD k soft ∆ETC ≈ (4πZαµ2ε )2 i ∫ (2π)D (2π)D (2π)D (2p − k)a −iδ ⊥,ab (k⃗ ) (−2q − k)b ψ(⃗ q) × ψ ∗ (⃗ p) 2m1 2∣k⃗ ∣3 (k⃗ − p⃗ + q⃗ )2 2m2 8π 2 (Zα)2 µ2ε dD p dD q ∗ p − q⃗ ) ψ(⃗ q ). (16.193) ψ (⃗ p ) (−pa q b ) I ab (⃗ ∫ m1 m2 (2π)D (2π)D The result (16.193) must be doubled to account for the contribution of the graph like the middle one of Fig. 16.9, but reflected right-to-left. The total spin-averaged recoil energy contribution coming from a soft exchanged transverse photon is the sum of (16.189) and twice (16.193). The spatial momentum factors combine according to (pa pb − 2pa q b + q a q b ) → (p − q)a (p − q)b since I ab (⃗ p − q⃗ ) of (16.188) is symmetric in a, b. For the total soft transverse energy contribution (T + TC + CT) we find 8π 2 (Zα)2 µ2ε dD p dD q ∗ soft ψ (⃗ p ) (p − q)a (p − q)b I ab (⃗ p − q⃗ ) ψ(⃗ q) ∆ET,total = ∫ m1 m2 (2π)D (2π)D =

=

1 1 8(Zα)2 µ2ε dD p dD q ∗ 2∣⃗ p − q⃗ ∣ ψ (⃗ p ) {− − + ln ( )} ψ(⃗ q) ∫ D D 3m1 m2 (2π) (2π) 2ε 2 µ

=

8(Zα)2 2ε ⃗ 2 [− 1 − 1 + ln ( 2 )] + ⟨ ln(∣⃗ µ {∣ψ(0)∣ p − q⃗ ∣)⟩p,⃗ }, ⃗q 3m1 m2 2ε 2 µ

(16.194)

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where the expectation value of ln(∣⃗ p − q⃗ ∣) was given in (16.179) and (16.180) above. We anticipate that the infrared divergence found in the soft region here will be canceled against an ultraviolet divergence from the ultrasoft region that remains to be worked out. 16.3.6

Ultrasoft Contribution from a Single Transverse Photon

The ultrasoft region makes a contribution at order (Zα)5 for all three graphs in Fig. 16.9. In fact, there are contributions at this order from graphs with any number of ladder Coulomb photons crossed by a single ultrasoft transverse photon. We will work out the effect of this infinite sum of terms using a procedure analogous to that used in the evaluation of the standard Lamb shift, and will find that the same Bethe logarithm makes its appearance here as it did there. We commence our work with an examination of the single transverse graph with no Coulomb photons and build up from there. The contribution of the first term in Fig. 16.9 to the energy shift is US ∆ET = i∫

iδ ⊥,ab (⃗ p − q⃗ ) dd p dd q 0 a [iZeγ b ]2 Ψ0 (q) , Ψ (p) [−ieγ ] 1 (2π)d (2π)d (p − q)2 + i

(16.195)

where we have to remember that the transverse photon momentum (p−q) is ultrasoft (and p and q are potential), since otherwise this expression is the same as what one would write down for the full graph with no regard to regions. The condition that (p − q) be ultrasoft puts an effective constraint on the part of p–q space that contributes to the integral. The reference wave functions were given in (16.101) with S defined in (16.103). The nonrelativistic approximation to S is S(P ; p) ≈

is−1 (E; p⃗ ) (ξ1 E + p0 −

p⃗ 2

2m1

+ i) (ξ2 E − p0 −

p⃗ 2 2m2

(16.196)

+ i)

−1

where s(E; p⃗ ) = (E − p⃗ 2 /(2µ) + i) is the nonrelativistic propagator. The nonrelativistic approximation to the spin-averaged Dirac expectation values were worked out in (16.183). For the energy shift, we find US ∆ET =−

4πZαµ2ε dD p dD q ∫ 4m1 m2 (2π)D (2π)D

× ψ ∗ (⃗ p )s−1 (E; p⃗ )(p + q)a δ ⊥,ab (⃗ p − q⃗ )(p + q)b s−1 (E; q⃗ )ψ(⃗ q) ×∫

dp0 dq 0 p⃗ 2 p⃗ 2 [ (ξ1 E + p0 − + i1 ) (ξ2 E − p0 − + i2 ) 2π 2π 2m1 2m2

× (p0 − q 0 − ∣⃗ p − q⃗ ∣ + i3 ) (p0 − q 0 + ∣⃗ p − q⃗ ∣ − i3 ) × (ξ1 E + q 0 −

q⃗ 2 q⃗ 2 + i4 ) (ξ2 E − q 0 − + i5 ) ] 2m1 2m2

−1

.

(16.197)

We recall Eq. (16.145) and must now perform the integrals over p0 and q 0 . A new feature that appears here (and in more complicated relative energy integrals) is

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the need to choose an order for the positive infinitesimals. We must also choose which integral to do first and in which half plane to close the p0 and q 0 contours. Any of these choices will do and will give the same result, but choices must be made. We choose to integrate over p0 first and to close p0 in the lower half plane (so, p0 → −ξ1 E + p⃗ 2 /(2m1 ) − i1 and p0 → q 0 + ∣⃗ p − q⃗ ∣ − i3 are the two relevant pole positions). Then, with 1 < 3 , we choose to close q0 in the lower half plane as well (with relevant poles q0 → −ξ1 E + p⃗ 2 /(2m1 ) + ∣⃗ p − q⃗ ∣ − i(1 + 3 ) and q 0 → 2 −ξ1 E + q⃗ /(2m1 ) − i4 ). After some algebra, we find dD p dD q ∗ 4πZαµ2ε ψ (⃗ p )(p + q)a δ ⊥,ab (k⃗ )(p + q)b ψ(⃗ q) ∫ 4m1 m2 (2π)D (2π)D ⎧ ⎫ ⎪ ⎪ −1 ⎪ 1 1 ⎪ × ⎨ + ⎬ , (16.198) 2 2 2 2 p⃗ q⃗ p⃗ q⃗ ⃗ ⃗ ⃗ ⎪ ⎪ 2∣k ∣ ⎪ ⎩ E − ∣k ∣ − 2m1 − 2m2 + i E − ∣k ∣ − 2m2 − 2m1 + i ⎪ ⎭

US ∆ET = −

where k⃗ ≡ p⃗− q⃗. The symmetry between m1 and m2 is apparent in this form. We are free to take (p + q)a = (2p − k)a → 2pa and (p + q)b = (k + 2q)b → 2q b in the numerator because of the transverse delta factor and we note that the first denominator factor can be simplified using q⃗ 2 p⃗ 2 p⃗ 2 − q⃗ 2 −1 p⃗ 2 k⃗ ⋅ (⃗ p + q⃗ ) − + i=(E − ∣k⃗ ∣ − + i)+ =s (E − ∣k⃗ ∣; p⃗ )+ . E − ∣k⃗ ∣ − 2m1 2m2 2µ 2m2 2m2 (16.199) Now every term in s−1 (E − ∣k⃗ ∣; p⃗ ) is O(α2 ) while k⃗ ⋅ (⃗ p + q⃗ ) is O(α3 ) for k⃗ ultrasoft, so we can safely neglect the last term. A similar analysis gives the approximation s−1 (E − ∣k⃗ ∣; q⃗ ) for the second denominator in (16.198), so the ultrasoft contribution from transverse photon exchange becomes US ∆ET ≈

dD p dD q ∗ pa δ ⊥,ab (k⃗ )q b 4πZαµ2ε ψ(⃗ q) ψ (⃗ p ) ∫ 2m1 m2 (2π)D (2π)D ∣k⃗ ∣ × {s(E − ∣k⃗ ∣; p⃗ ) + s(E − ∣k⃗ ∣; q⃗ )} . d k (D) ⃗ and δ (⃗ p − q⃗− k) (2π)D US we can rewrite ∆ET as

If we insert a factor of 1 = ∫ D (D)

s(E; p⃗ )(2π) δ US ∆ET ≈

≈

(⃗ p − q⃗),

D

(16.200)

recall the definition s(E; p⃗, q⃗ ) =

4πZαµ2ε dD p dD q dD k ∗ pa δ ⊥,ab (k⃗ )q b ψ (⃗ p) ψ(⃗ q) ∫ D D D 2m1 m2 (2π) (2π) (2π) ∣k⃗ ∣ ⃗ q⃗ )} × {s(E − ∣k⃗ ∣; p⃗, q⃗ + k⃗ ) + s(E − ∣k⃗ ∣; p⃗ − k, 4πZαµ2ε dD p dD q dD k ∗ pa δ ⊥,ab (k⃗ )q b ψ (⃗ p) ψ(⃗ q )s(E − ∣k⃗ ∣; p⃗, q⃗ ) , ∫ D D D m1 m2 (2π) (2π) (2π) ∣k⃗ ∣ (16.201)

where the final approximation, using ∣k⃗ ∣ ≪ ∣⃗ p ∣, ∣⃗ q ∣, was a consequence of the assumed US ultrasoft nature of k. At this point it is best to pause in our evaluation of ∆ET as we will want to combine it with additional ultrasoft contributions later on.

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We move on to a consideration of the second graph in Fig. 16.9 where the transverse photon is ultrasoft while all other lines and wave functions are potential. The energy correction resulting from this graph, written without any approximations, was given in Eq. (16.190) above. When k⃗ was soft, we used Eq. (16.191) to approximate the electron propagator. Now that k⃗ is ultrasoft, the analogous approximation is iΛ+ (⃗ p )γ 0 iΛ− (⃗ p )γ 0 i ≈ + , 2 ⃗ p 0 0 2m1 (ξ1 P + p
− k
) − m1 + i ξ1 E + p − k − 2m1 + i

(16.202)

with a similar approximation for the other fermion propagator. Clearly the first terms give the dominant contribution. The energy shift from (16.190) with an ultrasoft transverse photon takes the form dd p dd q dd k (2π)d (2π)d (2π)d δ ⊥,ab (k⃗ ) × ψ ∗ (⃗ p )S(P ; p) 2 S(P ; q)ψ(⃗ q) (k + i)(⃗ p − q⃗ )2

US ∆ETC ≈ 2(4πZαµ2ε )2 i ∫

× u1 (⃗ p )γ a Λ+ (⃗ p )γ 0 γ 0 u1 (⃗ q ) u2 (−⃗ p )γ 0 Λ+ (−⃗ q )γ 0 γ b u2 (−⃗ q) −1

p⃗ 2 q⃗ 2 × [(ξ1 E + p − k − + i) (ξ2 E − q 0 − k 0 − + i)] 2m1 2m2 0

0

.

(16.203)

The two spin-averaged Dirac products have the approximate values pa /m1 and −q b /m2 . We have anticipated the fact that the left-right reflected one-transverse one-Coulomb graph will give an equal contribution. We use (16.196) for the nonrelativistic approximation to S, and find (4πZαµ2ε )2 dd p dd q dd k i∫ m1 m2 (2π)d (2π)d (2π)d pa δ ⊥,ab (k⃗ )q b −1 × ψ ∗ (⃗ p ) s−1 (E; p⃗ ) s (E; q⃗ ) ψ(⃗ q) (⃗ p − q⃗ )2

US ∆ETC ≈2

× [(ξ1 E + p0 −

p⃗ 2 p⃗ 2 + i1 ) (ξ2 E − p0 − + i2 ) (k 0 − ∣k⃗ ∣ + i3 ) (k 0 + ∣k⃗ ∣ − i3 ) 2m1 2m2

× (ξ1 E + p0 − k 0 − × (ξ1 E + q 0 −

q⃗ 2 p⃗ 2 + i4 ) (ξ2 E − q 0 − k 0 − + i5 ) 2m1 2m2 −1

q⃗ 2 q⃗ 2 + i6 ) (ξ2 E − q 0 − + i7 )] , 2m1 2m2

(16.204)

where again we have employed a separate infinitesimal imaginary part for each distinct propagator. We perform the relative energy integrals (with any convenient

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choice of order, half-planes to close in, and relative sizes of infinitesimals) to find dD p dD q dD k (4πZαµ2ε )2 i∫ m1 m2 (2π)D (2π)D (2π)D pa δ ⊥,ab (k⃗ )q b −1 s (E; q⃗ )ψ(⃗ q) × ψ ∗ (⃗ p )s−1 (E; p⃗ ) (⃗ p − q⃗ )2

US ∆ETC ≈2

−1

× (−i)3 [s−1 (E; p⃗ ) (2∣k⃗ ∣) s−1 (E − ∣k⃗ ∣; p⃗ )s−1 (E − ∣k⃗ ∣; q⃗ )s−1 (E; q⃗ )] =

4πZαµ2ε dD p dD q dD k ∗ pa δ ⊥,ab (k⃗ )q b ψ(⃗ q) ψ (⃗ p) ∫ D D D m1 m2 (2π) (2π) (2π) ∣k⃗ ∣ × s(E − ∣k⃗ ∣; p⃗ )V (⃗ p − q⃗ )s(E − ∣k⃗ ∣; q⃗ ).

(16.205)

Again, we pause to consider additional ultrasoft graphs that contribute at order (Zα)5 before continuing the evaluation of this contribution. The third graph of Fig. 16.9 shows a pair of ladder Coulomb photons crossed by a transverse photon. When the transverse photon is ultrasoft, we will find that this graphs contributes at the same order (Zα)5 level as the other graphs in Fig. 16.9. Instead of evaluating this diagram from scratch, we focus on the changes between the one- and two-Coulomb graphs. We see that the two-Coulomb contribution can be obtained from the one-Coulomb contribution (16.190) by the replacement (−ieγ 0 )1 (iZeγ 0 )2 1 dd r i →∫ [(−ieγ 0 ) (−ieγ 0 )] d 2 2 (2π) (⃗ p − r⃗ ) ξ1 P +
r − k
− m1 + i (⃗ p − q⃗ − k⃗ ) 1 i i (iZeγ 0 )] , × [(iZeγ 0 ) ξ2 P −
r − m2 + i (⃗ r − q⃗ − k⃗ )2 2

(16.206)

or, with nonrelativistic approximations, 1

(⃗ p − q⃗ − k⃗ )2

→ 4πZαµ2ε (−i) ∫ ×

= −4πZαµ2ε ∫

⃗ )2 (⃗ r −k 2m1

+ i

1

1 ξ2 E − r0 −

dd r 1 1 d 2 (2π) (⃗ p − r⃗ ) ξ1 E + r0 − k 0 −

r⃗ 2 2m2 D

+ i (⃗ r − q⃗ − k⃗ )2

d r 1 D (2π) (⃗ p − r⃗ )2 E − k 0 −

1 ⃗ )2 (⃗ r −k 2m1

1

−

r⃗ 2

2m2

. r − q⃗ − k⃗ )2 + i (⃗ (16.207)

For ultrasoft k, the replacement takes the simple form, 1 dD r 1 1 → −4πZαµ2ε ∫ s(E − k 0 ; r⃗ ) , 2 (⃗ p − q⃗ ) (2π)D (⃗ p − r⃗ )2 (⃗ r − q⃗ )2

(16.208)

or V (⃗ p − q⃗ ) → ∫

dD r V (⃗ p − r⃗ )s(E − k 0 ; r⃗ )V (⃗ r − q⃗ ) = [V sV ] (E − k 0 ; p⃗, q⃗ ) (16.209) (2π)D

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in the compact notation where the internal convolution integrals are implicit. The resulting energy shift is thus dD p dD q dD k ∗ pa δ ⊥,ab (k⃗ )q b 4πZαµ2ε US ψ (⃗ p) ψ(⃗ q) ∆ETCC ≈ ∫ D D D m1 m2 (2π) (2π) (2π) ∣k⃗ ∣ × [sV sV s] (E − ∣k⃗ ∣; p⃗, q⃗ ). (16.210) It is now clear that graphs with additional ladder Coulomb photons can be handled in the same way, each time giving extra factors of V times s to be included in the string seen in (16.210). The total contribution of ultrasoft transverse photon exchange at order (Zα)5 is the sum of (16.201), (16.205), (16.210), plus all diagrams with additional ladder Coulomb photons. The infinite series of terms can be summed using (16.77) to give pa δ ⊥,ab (k⃗ )q b dD p dD q dD k ∗ 4πZαµ2ε US ψ (⃗ p ) ψ(⃗ q) ∆ET(total) ≈ ∫ m1 m2 (2π)D (2π)D (2π)D ∣k⃗ ∣ × [s + sV s + sV sV s + sV sV sV s + ⋯](E − ∣k⃗ ∣; p⃗, q⃗ ) =

4πZαµ2ε dD p dD q dD k ∗ pa δ ⊥,ab (k⃗ )q b ψ(⃗ q )g(E − ∣k⃗ ∣; p⃗, q⃗ ), ψ (⃗ p ) ∫ m1 m2 (2π)D (2π)D (2π)D ∣k⃗ ∣ (16.211)

where g(E, p⃗, q⃗ ) is the NRSC Green function of (16.75) and (16.81). The reader has no doubt noticed the extremely close relationship between this result and the related expression involving the NRSC Green function and an ultrasoft transverse photon that appeared in our study of the Lamb shift (see Chap. 4 for the low-energy part of the self-energy). As for the Lamb shift work, we use the sum-over-states form (16.81) for the Green function. It is useful now to also include an index (which we denote as n), in order to specify what state we are finding the energy shift of, and we will convert to Dirac bra/ket notation when convenient. In addition, we must point out that, although the early work in this chapter, including the definition of the NRSC Green function shown in (16.75)–(16.81) was written down in three dimensional space, we now realize that the dimensionally continued version of that Green function is required. For the ultrasoft energy shift of the nth state, we resort to a derivation akin to the one employed in Secs. 4.6.2 and 15.5.2, but this time, starting from momentum space. We have dD p dD q dD k δ ⊥,ab (k⃗ ) 4πZαµ2ε US ∆ET(total) ≈ ∫ m1 m2 (2π)D (2π)D (2π)D ∣k⃗ ∣ ×∑ n′

=

ψn∗ (⃗ p )pa ψn′ (⃗ p ) ψn∗′ (⃗ q )q b ψn (⃗ q) ⃗ En − ∣k ∣ − En′ + i

a ′ ′ a 4πZαµ2ε D − 1 ΩD ⃗ ∣ ∣k⃗ ∣D−2 ⟨n∣p ∣n ⟩⟨n ∣p ∣n⟩ , ( ) d∣ k ∑ ∫ m1 m2 D (2π)D n′ (En − En′ + i) − ∣k⃗ ∣ (16.212)

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where we appeal to Eq. (10.138) for the definition of the solid angle in D dimensions. One finally arrives at the following result, where we use the shorthand notation X(D) = Γ(D − 1)Γ(2 − D)(D − 1)/D for an intermediate step, US ∆ET(total) = −

= −

ΩD 4πZαµ2ε a ′ ′ a D−2 X(D) ∑⟨n∣p ∣n ⟩⟨n ∣p ∣n⟩ [−(En − En′ + iε)] m1 m2 (2π)D n′ (En − En′ ) 4πZα a ′ ′ a ∑⟨n∣p ∣n ⟩⟨n ∣p ∣n⟩ m1 m2 n′ 6π 2

1 −2(En − En′ + iε) 5 × { − 2 ln ( ) + + O(ε)} . ε µ 3

(16.213)

In the reductions shown above, we integrated the transverse delta function over D − 1 angles using δ ab D − 1 ab ⊥,ab ⃗ (k ) = ∫ dΩD (δ ab − kˆa kˆb ) = ΩD (δ ab − ) = ΩD ( )δ . ∫ dΩD δ D D (16.214) ⃗ The integral over ∣k ∣ was done using the general formula ∫

∞ 0

dx

Γ( a+1 xa = m b −c b c (m + x )

a+1 ) Γ (c − a+1 ) b b

b Γ(c)

.

(16.215)

The logarithm obtained by performing the small-ε expansion has an argument that can be negative (when En > En′ ), leading to an imaginary part for the logarithm in that case. We define the logarithm in the usual way with a cut on the negative real axis, so that ln (−

En − En′ ∣En − En′ ∣ − i) = ln ( ) − iπΘ(En > En′ ) , µ µ

(16.216)

where Θ is the Heaviside step function. The imaginary part gives the correction to the decay rate from state n (cf. Sec. 3.2.3) Γ = −2 Im(∆E) =

8Zα 3m1 m2

∑

⟨n∣pa ∣n′ ⟩⟨n′ ∣pa ∣n⟩(En − En′ ) ,

(16.217)

n′ (En′ y10 , y20 actually implies a more stringent theta function, in the form Θ(min(x01 , x02 ) − max(y10 , y20 )), rather than simply Θ(X 0 − Y 0 ). Show that use

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Bethe–Salpeter Equation

of the more exact theta function changes neither the positions nor the residues of the bound state poles in G′ . Hint: Use min(x01 , x02 ) = X 0 − ξ(x0 )∣x0 ∣ , max(y10 , y20 )

ξ(x0 ) = {

(2)

(3) (4) (5)

(6)

(7)

(16.255)

= Y + [1 − ξ(y )] ∣y ∣ , 0

ξ1 ξ2

0

x0 > 0 . x0 < 0

0

(16.256) (16.257)

The theta function will be Θ(X 0 − Y 0 − ξ(x0 )∣x0 ∣ − [1 − ξ(y 0 )] ∣y 0 ∣) instead of simply Θ(X 0 −Y 0 ). Show that the final form of (16.14) contains an extra factor of exp[i(P 0 − ωn (P⃗ ))(ξ(x0 )∣x0 ∣ + (1 − ξ(y 0 ))∣y 0 ∣)], which changes neither the position of the pole at P 0 = ωn (P⃗ ) nor its residue. Inhomogeneous Bethe–Salpeter Equation. Consider the inhomogeneous BSEQ as shown in the symbolic equation depicted in Fig. 16.2(c). Show that the individual diagrams in G′ implicitly contained in the diagram on the lefthand side of the equation given in Fig. 16.2(c) are in one-to-one correspondence with the sum of two terms. The first of these is the non-interacting 2-to-2 Green function S ′ of Fig. 16.1(b), and the second is the product of S ′ times the two-particle irreducible kernel K ′ of Fig. 16.2(c) times G′ itself. Alternative Form of the Bethe–Salpeter Equation. Show the equivalence of Eqs. (16.22) and (16.23). Series Representation of the Two-Particle Green Function. Show Eq. (16.34). Schr¨ odinger–Coulomb Hamiltonian. Show that Eq. (16.55) is the momentum-space transform of the eigenvalue equation corresponding to the Schr¨ odinger–Coulomb Hamiltonian given in Eq. (4.1). Hyperfine Structure for Equal Masses. Calculate the correction (16.248) in the limit of positronium (m1 = m2 ) and evaluate it numerically for positronium. Two-Time Green Function Method. Compare the matching of scattering amplitudes, outlined in Sec. 12.2.1 and elsewhere in Chap. 12 and Chap. 15 in this treatise with the use of contour integrals for the derivation of expressions pertaining to energy shifts [see Eq. (16.123)]. Then, consult the two-time Green function method introduced by Shabaev1 in Ref. [595] for additional interesting interconnections.

1 Vladimir

Moiseevich Shabaev (b. 1959).

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Chapter 17

NRQED: An Effective Field Theory for Atomic Physics

17.1

Overview

In this chapter, we develop a general approach to the physics of nonrelativistic Coulombic bound states based on an effective nonrelativistic quantum field theory. Specifically, we describe and illustrate the use of Non-Relativistic Quantum Electrodynamics (NRQED). The use of effective theories to describe low energy physics has a long history (for an overview, see Ref. [596]). Effective field theory has been used in atomic physics, at least implicitly, since the earliest days. The explicit construction of an effective nonrelativistic theory was initiated by Caswell1 and Lepage2 in Ref. [597] with the introduction of NRQED, which has become a dominant framework for modern analytic and semi-analytic calculations of higher-order corrections to bound-state properties. Some early uses of NRQED and developments of the approach are given in Refs. [45, 329, 598–603]. NRQED and related work have been reviewed in [588, 598, 604]. There are two general approaches to the development of effective low-energy quantum field theories. In the top down approach, one assumes the knowledge of some more general field theory valid at higher energies and integrates out the high energy field modes, leaving an effective low-energy theory (see also Chap. 18). In principle, using the top down approach, one is able to derive all details of the low-energy theory from the more general theory. In practice, such derivations can be challenging. In the bottom up approach, knowledge of an overarching theory is not assumed. Instead, one simply writes down a Lagrangian for the effective field theory (EFT) consistent with general principles such as Hermiticity, translational and rotational symmetry, gauge invariance, and parity and time reversal invariance. The various terms in the EFT contain some undetermined coefficients, the Wilson3 coefficients, that are determined phenomenologically. Often, an intermediate strategy is employed. The overarching theory is assumed to be known. In our case it is QED, or more generally the Standard Model. The EFT is created as in the 1 William

Edward Caswell (1947–2001). Peter Lepage (b. 1952). 3 Kenneth Geddes Wilson (1936–2013). 2 Gerard

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bottom up approach, as the most general quantum field theory subject to the desired conditions and symmetries. The various terms of the EFT carry coefficients that are determined by the requirement that the EFT and the overarching theory agree when used to calculate some set of processes for which both are valid. The process of determining the coefficients thus is independent of the phenomenological aspects. In this context, the Wilson coefficients are often termed matching coefficients, since their values are found through a matching procedure. Low-energy effective theories such as NRQED have two characteristic and related attributes. First, they are not renormalizable. Renormalizability is an extremely restrictive condition that effective theories do not satisfy. As one simple example of an unrenormalizable term in the NRQED Lagrangian one could mention the operator representing the Darwin or zitterbewegung term, q ⃗ 2 A0 ) ψ . (17.1) LD = − 2 cD ψ + (∇ 8m For the case q = e (electron field), where ψ represents the electron-field operator and m the electron mass, this operator contributes part of the Lamb shift (see Chap. 11). It contains a dimensionless constant cD that is only determined through the matching operation. The presence of the inverse mass squared in the multi⃗ 2 A0 ) ψ is too high plicative factor makes clear that the operator dimension of ψ + (∇ ⃗ 2 A0 ) ψ for this term to be renormalizable. Specifically, the mass dimension of ψ + (∇ ⃗2 is six, consistent with a mass dimension of ψ of 3/2, a mass dimension of the ∇ 0 operator of 2, and a mass dimension of qA (potential energy) of one. A trivial calculation shows that the mass dimension of the operator given in Eq. (17.1) is 2 × 32 + 2 + 1 = 6. Because the mass dimension of LD must be 4, the coefficient multiplying the operators must have dimension −2, consistent with the explicit factor of 1/m2 . This is but one of an infinite number of such operators having higher and higher dimension, and more and more inverse masses in their coefficients. They involve a corresponding infinitude of matching constants that are only known by virtue of the matching procedure. As a general remark, higher-dimension operators give rise to higher-order high-energy divergences in the amplitudes they are involved in, leading to a theory that is not renormalizable. The second and related point is that these terms, with increasingly negative powers of mass in their coefficients, necessarily make smaller and smaller numerical contributions to whatever quantity is being calculated. This is because every explicit mass in the denominator must be balanced by a factor of the characteristic energy or momentum in the numerator. In a nonrelativistic effective theory of Coulombic bound states, the characteristic energies and momenta are small compared to the constituent masses, being typically of order α or α2 times a characteristic mass. So, even though there are infinitely many operators in the effective Lagrangian, only a well-defined and finite number of them contribute at any given power of the fine-structure constant. Under these conditions, lack of renormalizability is not a disadvantage. In this chapter, we show how the matching coefficients can be obtained, and use our results to develop a convenient description of nonrelativistic Coulombic

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two-fermion bound states. We aim to demonstrate that NRQED contains the physics of these bound systems in an extremely natural way. We also develop the Bethe–Salpeter equation (BSEQ) of NRQED. As is to be expected, the NRQED BSEQ is identical to the nonrelativistic Schr¨odinger equation (in leading order). It also provides a framework for including relativistic and radiative corrections. Perturbation theory in NRQED is more convenient than in QED for a number of reasons. There are more NRQED operators, but those operators are simpler to work with and are more directly related to identifiable physical effects. It is easier using NRQED to estimate the order that each term in perturbation theory initially contributes, and much easier to find the higher-order contributions that are present in any given term of the series. These points are illustrated through examples. Specifically, in the following, we use NRQED to derive the effective Hamiltonian that could be used to find all energy levels at O(α4 ). The spin-averaged O(Zα)5 recoil correction, and the leading O(Zα)5 hyperfine corrections for atoms consisting of an electron and an elementary positive constituent are also derived. 17.2 17.2.1

Basics of NRQED NRQED Lagrangian and Feynman Rules

The NRQED Lagrangian is constructed from basic fields representing the nonrelativistic fermions and the photons. The quantum field for the ith spin-1/2 fermion is a (nonrelativistic, two-component) Pauli-spinor valued field ψi (x) having mass mi and charge qi (see Chap. 6). There are no antiparticle degrees of freedom contained in ψi (x) (in contrast to the Dirac formalism described in Chaps. 7 and 8). So, if we want to describe both an electron and a positron, then we need to do this using separate and independent fields for each of them. Typically, we include just two fermion fields. One of them describes a positively charged particle, the other, a negatively charged particle. The terms in the NRQED Lagrangian have various inverse powers of the fermion masses. As discussed earlier, terms with higher negative powers of the masses tend to give smaller effects, so we can organize the terms entering L according to the power of mass. We will focus on recoil corrections of O(Zα)5 . For these contributions, it is sufficient (with one exception) to include terms of mass order 1/m2 (but not higher). We describe the construction of the NRQED Lagrangian following the discussion of Ref. [605] and then use that Lagrangian to derive the corresponding NRQED Feynman rules. The principle of gauge invariance greatly restricts the number of possible terms that can enter the effective Lagrangian L. We assume that L is invariant under the local gauge transformation [see Eq. (8.2) and the surrounding discussion in Chap. 8] ψ(x) → eiqΛ(x) ψ(x) ,

(17.2)

while the Hermitian adjoint transforms as follows, ψ + (x) → ψ ′+ (x) = ψ + (x)e−iqΛ(x) .

(17.3)

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Quantum Electrodynamics in Action: Atoms, Lasers and Gravity Table 17.1 We consider the behavior of the building blocks of the NRQED Lagrangian under the parity and time reversal symmetries. The time variable and the scalar potential are scalars, while the gradient operator and the vector potential are vectors. The magnetic field is an axial vector. For time reversal, one must remember that the time reversal operator is represented as an anti-unitary operator involving complex conjugation and that the vector potential is produced by moving charges. The behavior of ⃗ is determined by noting that ψ + σ ⃗ ψ transforms as an angular momentum σ (odd under time reversal, even under parity). iDt

⃗ iD

A0

⃗ A

⃗ E

⃗ B

⃗ σ

P

+

−

+

−

−

+

+

T

+

−

+

−

+

−

−

This implies the presence of a gauge field Aµ (x) with the corresponding transformation Aµ (x) → A′µ (x) = Aµ (x) − ∂µ Λ(x) .

(17.4)

In place of the usual derivative ∂µ = ∂/∂xµ , L will be constructed using the covariant derivative Dµ = ∂µ +i q Aµ defined in Eq. (8.3). We here denote the charge as q rather than e, inspired by the development of the NRQED formalism, which originated as an effective field theory, describing the low-energy (bound-state) limit of quantum electrodynamics. In the literature on NRQED, the charge of the particle is usually denoted by the general symbol q rather than the more special electron charge e, and we follow this convention in the current discussion. The gauge field can enter the Lagrangian through the Faraday tensor i Fµν = (− ) [Dµ , Dν ] = ∂µ Aν − ∂ν Aµ , q

(17.5)

⃗ and the magnetic or other gauge invariant forms. Specifically, the electric field E, ⃗ induction field B, ∂ A⃗ ⃗ = −∇A ⃗ 0− , E ∂t

⃗=∇ ⃗ × A⃗ , B

(17.6)

are gauge invariant and enter the NRQED Lagrangian. We also require our NRQED Lagrangian to respect the symmetries of rotational invariance, parity, and time reversal, which it inherits from QED. Rotational symmetry is built in if we construct the NRQED Lagrangian from scalar and vector quantities in a rotationally invariant way. The scalars involved will be A0 , iDt , and ⃗ E, ⃗ B, ⃗ D ⃗ and ψ + σ ⃗ = ∇−iq ⃗ ψi+ ψi , while the relevant vectors are A, A, i ⃗ ψi . The behavior of the various building blocks under action of the discrete symmetries is shown in Table 17.1.

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Upon taking the various symmetries into account, we find that the Lagrangian for NRQED adequate for energy levels through O(Zα)5 can be written as L = ∑ ψi+ {iDt + i

⃗2 ⃗4 D qi D ⃗ + cDi qi (D ⃗ −E ⃗ ⋅ D) ⃗ ⋅E ⃗ ⃗i ⋅ B + + cFi σ 3 2mi 8mi 2mi 8m2i

iqi ⃗ −E ⃗ × D) ⃗ ×E ⃗ + ⋯} ψi − 1 Fµν F µν + ⋯ ⃗i ⋅ (D σ 8m2i 4 dS dV + + ⃗ab ⋅ σ ⃗cd } ψ1b ψ2d + ⋯ , + ψ1a ψ2c { δab δcd + σ m1 m2 m1 m2 + cSi

(17.7)

where i is an index used to distinguish the electron (i = 1) from the positive particle (proton or positive muon, i = 2). Of course, the covariant derivative in Eq. (17.7) acts on the argument x of ψi (x), where ψi (x) is a second-quantized (Schr¨odinger– ⃗ = E(x) ⃗ ⃗ = B(x) ⃗ Pauli) field, and E and B are second-quantized field operators. The index i enumerates the particle species that enter the Lagrangian. (In contrast to our treatment of bound systems in Chap. 12, the fermions that make up the atoms here are “second quantized”, i.e. they are quantized fields subject to creation and annihilation.) The parameters qi and mi denote the charge and mass of the ith particle; and a, b, c, d are spinor indices for the Pauli spinors and the Pauli sigma matrices, defined in Eq. (4.343). For example, the term given in Eq. (17.1) is obtained from the full Lagrangian (17.7) according to qi ⃗ −E ⃗ ⋅ D) ⃗ ⋅E ⃗ ψi → −cDi qi ψ + (∇ ⃗ 2 A0 ) ψ i , ψ + (D (17.8) cDi 8m2i i 8m2i i ⃗ → −∇A ⃗ 0 from Eq. (17.6). At this order, we only need to upon picking the term E include terms up to quadratic in denominator masses except for the single 1/m3 term coming from the nonrelativistic expansion of the kinetic energy, √

⃗2 −m = m2 + π

⃗2 ⃗4 π π − + ⋯, 2m 8m3

(17.9)

⃗ which contributes at O(Zα)4 . Values for the Fermi cF , Darwin ⃗ = −iD, with π cD , spin-orbit cS , and scalar dS and vector dV Wilson coefficients will be found by matching, i.e., by requiring agreement between QED and NRQED for a set of low-energy processes that can be readily computed from both theories. The Feynman rules for NRQED are obtained from the Lagrangian in the usual way. There are propagators for the fermions and photons (Coulomb and transverse) found by isolating the terms in L quadratic in the field operators and working in momentum space. The rules for interactions are found by considering the interaction terms in iL one by one and imagining the presence of factors eip⃗1 ⋅⃗x for fields ⃗ are replaced carrying momentum p⃗1 in and e−ip⃗2 ⋅⃗x for outgoing p⃗2 . Derivatives ∇ by i⃗ p1 for incoming and −i⃗ p2 for outgoing momenta. The basic Coulomb interaction is contained in iDt . Temporarily dropping the index i labeling fermion type, the fermion-fermion-photon term coming from iDt is iL → iψ + i(iqA0 )ψ, leading to the Feynman rule −iq for the Coulomb interaction (the field A0 and the fermion

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operators are of course not part of the Feynman rule). The coupling to a transverse ⃗ 2 term. Specifically, photon is contained in the D iL → iψ +

2 q + 1 ⃗ − iq A⃗ ) ψ → ⃗ ⋅ A⃗ + A⃗ ⋅ ∇) ⃗ ψ. (∇ ψ (∇ 2m 2m

(17.10)

The corresponding rule for the interaction of a fermion with a spatial photon of q iq [i(k + p1 )j + i(p1 )j ] = 2m index j is 2m (p2 + p1 )j . This coupling can be called convective because of its dependence on the fermion motion (see Secs. 11.4.2 and 12.5.3). Another example, containing some new features, is the Fermi coupling (proportional to cF ) to a photon carrying spatial index k. The interaction Lagrangian gives q iL → icF 2m ψ + σ i ijk ∇j Ak ψ where the ∇j ≡ ∂j derivative acts only on Ak . The corq q responding Feynman rule is icF 2m σ i ijk (ik j ) = −cF 2m σ i ijk k j , where the incoming photon momentum is k⃗ = p⃗2 − p⃗1 . One subtlety concerns the presence of the antisymmetric symbol ijk in the Feynman rules. The problem is that the definition of ijk is tied to the three dimensional nature of space. When we generalize the number of spatial dimensions to D with D ≠ 3, ijk no longer makes sense. One solution to this difficulty is to replace the product ijk σ k by σ ij where i σ ij ≡ − [σ i , σ j ] . 2

(17.11)

We note that a set of sigma matrices satisfying {σ i , σ j } = 2δ ij exists even in D dimensions (for any positive integer D), and that σ ij as defined generates a representation of the rotation group in D dimensions [606]. In three dimensions we find that σ ij = ijk σ k and σ i = 12 ijk σ jk , so the replacement works when D = 3 and provides a convenient generalization to D ≠ 3. The Feynman rules that we will need for our calculations are displayed in Fig. 17.1. 17.2.2

Matching Coefficients

The NRQED Lagrangian comes with a number of coefficients: cS , cD , cF , etc. Since NRQED is an effective theory for an exactly known more general theory, QED, these coefficients are calculable. We will use the matching technique to work out their values. The idea of matching is to find some appropriate low-energy processes that can in principle be calculated using both QED and NRQED since QED is valid for all energy scales while NRQED is valid for the low energy processes under consideration. The NRQED results will involve the coefficients while the QED results are fully determined, so the values of the various coefficients can be found. These coefficients are often referred to as matching coefficients because of this procedure for finding their values. In order to find all corrections at O(α5 ) we would need values for the spinorbit, Darwin, and Fermi matching coefficients cS , cD , and cF through order α, the vacuum polarization coefficient cVP through order α, and the four-fermion contact terms dS and dV through O(α2 ). The explicit processes that we will focus on in this chapter are the recoil corrections at O(Zα)5 , for which only dS and dV are

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Fig. 17.1 The Feynman rules of NRQED are given here. Solid lines represent fermions, dashed lines stand for Coulomb photons, and wiggly lines for transverse photons. The NRQED vertices are represented by dots. Incoming and outgoing momenta on the fermion line (with mass m) are ⃗ = p⃗2 − p⃗1 is the incoming momentum of the photon. represented by p⃗1 and p⃗2 respectively, and k Indices i, j, k, . . . are spatial indices while a, b, c, . . . are spin indices. In the four-fermion contact term, m1 and m2 represent the masses of the two particles involved. Additional Feynman rules exist for higher order terms in the effective Lagrangian.

required, but in order to illustrate the procedure we will describe the evaluation of the other coefficients as well. The coefficients for terms involving a single fermion (incoming plus outgoing) interacting with a single photon can be found by considering the scattering of an on-shell fermion from an external electromagnetic source at linear order in the electromagnetic field. We considering only low-energy incoming and outgoing fermions so that NRQED is available for performing the calculations. The one-photon matching condition is illustrated in Fig. 17.2(a), where the large solid dots represent all radiative corrections. The fermion momentum labels in this diagram

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Fig. 17.2 Diagrams entering into the matching calculation for NRQED. Part (a) shows the matching condition for interactions involving a single fermion interacting with a photon. As long as all momenta are nonrelativistic, both NRQED and QED describe this interaction equally well. On the NRQED side, this amplitude involves the matching coefficients cF , cD , cS , etc., while on the QED side the full effect of fermion-photon interactions is contained in the Dirac and Pauli form factors. Part (b) shows QED diagrams contributing to the interaction between two fermions. The electron on the top is shown as a fermion, while the positive particle (positron, proton, positive muon, etc.) is shown as an antiparticle on the bottom. The incoming and outgoing relative momenta (p, q) are negligible compared to the exchange momentum k. In the extreme nonrelativistic limit with at-rest or nearly-at-rest fermions, this interaction can be equally well described by QED or by NRQED through the four-fermion contact term involving dS and dV .

0 are interpreted as follows. For NRQED, p√ is the nonrelativistic kinetic energy 0 2 0 4 3 p = p⃗ /(2m) + O (⃗ p /m ). For QED, p = p⃗ 2 + m2 is the total energy. In either case we consider only the nonrelativistic limit ∣⃗ p ∣/m ≪ 1. The same considerations apply to the four-vector q. Radiative corrections do not contribute in the NRQED vertex function [600] making the NRQED side of the matching equality easy to evaluate. A remark is in order. When expanding about p⃗ = 0, p0 = 0, which is appropriate for matching calculations in NRQED, the integrals in the vertex function all vanish in dimensional regularization since there is no nonrelativistic scale. This consideration becomes obvious when one takes into account that the evaluation of NRQED loop integrals proceeds in the momentum range k ∈ (0, ) with  ≪ m, much in the spirit of the evaluation of the low-energy part of the bound-electron self-energy outlined in Chap. 4. Hence, it is always possible to expand in the ratio

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k/m, which renders a number of important integrals scaleless and hence zero in dimensional regularization. Working directly from the Feynman rules given in Fig. 17.1, we find for the 0 and j components of the NRQED scattering amplitude shown in Fig. 17.2(a) the results (k⃗ = p⃗ − q⃗) M0NRQED = −q ηf+ {1 − MjNRQED = −q ηf+ {−

iσ `m p` q m k⃗ 2 c + cS + ⋯} ηi , D 8m2 4m2

(pj + q j ) iσ jk k k + cF + ⋯} ηi , 2m 2m

(17.12) (17.13)

where ηf and ηi are Pauli spinors for the spin states of the outgoing and incoming particles. For the result given in Eq. (17.12), we sum the contributions from the first, fourth and fifth Feynman rule in the first column of Fig. 17.1, while for Eq. (17.13) we include contributions from the second and sixth rule in that first column. The dots represent the contributions of higher-order operators giving terms of O(1/m4 ) in M0 and O(1/m3 ) in Mj . For the QED side, single-photon scattering is a standard process whose result can be expressed in terms of form factors, and these form factors are known to high precision. The QED scattering amplitude is MµQED = −q u(p) {γ µ F1 (k 2 ) +

iσ µν kν F2 (k 2 )} u(q) . 2m

(17.14)

The fermion momenta are on-shell and the photon momentum is k = p − q. We use the nonrelativistic spinor normalization u+ (p) u(p) = 1,

u(p) u(p) = m/E ,

so that ⎛ η ⎞ u(p) = N ⎜ σ ⃗ ⋅ p⃗ ⎟ , ⎝ E + m η⎠ u(p) = N (η + , −η +

√ N=

E+m , 2E

⃗ ⋅ p⃗ σ ). E+m

(17.15)

(17.16a) (17.16b)

Note that the spinors used here are manifestly different from the helicity basis spinors encountered in Sec. 7.3.1. For the nonrelativistic particles being considered here, it is appropriate to use solutions that reduce to the fundamental Pauli spinors χ [see Eq. (6.42)], and are normalized to one particle per unit volume. Because the fermions are nonrelativistic, we will expand everything in the small quantities p⃗ 2 /m2 and q⃗ 2 /m2 . The expansion for the spinors is 4 6 2 ⎛ [1 − p + 11p + O ( p )] η ⎞ ⎟ 8m2 128m4 m6 ⎛ η ⎞ ⎜ ⎜ ⎟ ⎟, ⎜ u(p) = N ⎜ σ ⃗ ⋅ p⃗ ⎟ ≈ ⎜ ⎟ ⎟ 2 4 ⎝ E + m η⎠ ⎜ ⃗ ⋅ p⃗ 3p p σ ⎜ ⎟ ⎝ [1 − 8m2 + O ( m4 )] 2m η ⎠

(17.17)

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with an analogous expansion for u(p). The Dirac and Pauli form factors F1 (k 2 ), F2 (k 2 ) have the expansions 1 Fj (k 2 ) = Fj (0) + k 2 Fj′ (0) + (k 2 )2 Fj′′ (0) + ⋯ , 2 2 ′′ k2 ′ 1 k2 ≡ Fj + 2Fj + ( 2) Fj + ⋯, m 2 m 2 ⃗ ′ k (k 0 )2 ′ k⃗ 4 ′′ = Fj − 2Fj + F + F + ⋯, j m m2 2m4 j

(17.18) (17.19) (17.20)

with an obvious notation for the quantities with bars. Here, we have k0 ≈

p⃗ 2 q⃗ 2 − 2m 2m

(17.21)

for on-shell nonrelativistic incoming and outgoing fermions. Results for the coefficients in these expansions are known (see below). The nonrelativistic expansions of the QED amplitudes give k⃗ 2 iσ ab pa q b k⃗ 2 ′ + + ⋯} η × {F − F + ⋯} i 1 8m2 4m2 m2 1 iσ ab pa q b k⃗ 2 − qηf+ { − + ⋯} ηi × {F 2 + ⋯}, 2m2 4m2

M0QED = − q ηf+ {1 −

MjQED = − q ηf+ {− − qηf+ {

(17.22)

pj + q j iσ ja k a + + ⋯} ηi × {F 1 + ⋯} 2m 2m

iσ ja k a + ⋯} ηi × {F 2 + ⋯} . 2m

(17.23)

A comparison of the NRQED and QED results for the scattering amplitudes gives values for the NRQED coefficients in terms of the form factors: 1 = F 1, cD =

(17.24a)

′ F 1 + 2F 2 + 8F 1 ,

(17.24b)

cS = F 1 + 2F 2 ,

(17.24c)

cF = F 1 + F 2 .

(17.24d)

The form factors themselves are power series in α, for which the first several terms are known. We worked out their values through one-loop order in Sec. 10.3.6 and found the results [see Eq. (10.190)] F 1 = 1,

(17.25a)

α 1 1 µ 1 {− − ln ( 2 ) − } + O(α2 ), π 6 6 m 8 α 2 F2 = + O(α ) . 2π ′

F1 =

2

(17.25b) (17.25c)

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For the matching coefficients, we obtain 4α 1 µ2 { + ln ( 2 )} + O(α2 ), (17.26a) 3π  m α (17.26b) cS = 1 + + O(α2 ), π α cF = 1 + + O(α2 ). (17.26c) 2π In a similar way, the vacuum polarization coefficient cVP can be obtained. We compare the NRQED and QED results for the static potential between two charges at small values of k⃗ 2 /m2 , as corrected by vacuum polarization. From NRQED we find i i −ik⃗ 4 i + (cVP 2 ) + ⋯} (−iq2 ) V NRQED → i(−iq1 ) { m k⃗ 2 k⃗ 2 k⃗ 2 q1 q2 k⃗ 2 = (17.27) {1 + cVP 2 + ⋯} , m k⃗ 2 while from standard QED (see Sec. 10.4.3), we have q1 q2 α k⃗ 2 V QED → {1 + + ⋯} . (17.28) 15π m2 k⃗ 2 cD = 1 −

Comparison of the NRQED and QED forms yields the value α cVP = . (17.29) 15π The two-photon-exchange contact term requires more work. The relevant diagrams are shown in Fig. 17.2(b). The exchanged photons and fermions in the inner loop are high energy, with momenta of the order of the fermion masses. So, as far as atomic distance scales are concerned, these interactions are point-like. The NRQED representation of these hard corrections is contained in the four-fermion contact term, with amplitude dV dS + + ⃗ab ⋅ σ ⃗cd } η1ib η2id . δab δcd + σ (17.30) iMNRQED = iη1f a η2f c { m1 m2 m1 m2 Here η1ib , for example, is the bth component of the Pauli spinor describing the incoming spin state of fermion 1. We find it convenient to calculate the amplitudes directly for spin-0 and spin-1 combinations of the two spin-1/2 fermion spin states. (Since this is a point-like interaction, S states are the only ones that will contribute, so the total angular momentum is just due to the spins of the two fermionic constituents.) The relativistic (QED) two-particle spin states for a particle-antiparticle system were found in Sec. 16.4. Here, we need the nonrelativistic (NRQED) twoparticle spin states (no antiparticle involved). Initially, we work in three spatial ⃗i dimensions, where the total spin operator is the usual S⃗tot = S⃗1 + S⃗2 with S⃗i = 12 σ being the spin operator for the ith particle. Let ξ by the desired two-particle spin state, and take the action of a general operator A1 B2 on ξ to be [A(1) B (2) ξ]ab = Aaa′ Bbb′ ξa′ b′ = [AξB T ]ab .

(17.31)

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We here introduce the ξ matrices as auxiliary devices in the calculation of the twoparticle matrix elements, in a distant analogy to Casimir’s trick for the calculation of Dirac traces. The introduction of the ξ matrices, together with the calculational prescription (17.31), eliminates the need for a separate consideration of the projection operators onto the two-particle states. The explicit forms for the two-particle ξ states of spin S = 0 and S = 1 are 1 01 1 0 1 10 00 ) , ξ 11 = ( ) , ξ 10 = √ ( ) , ξ 1−1 = ( ) . ξ 00 = √ ( 00 01 2 −1 0 2 10

(17.32)

One can verify that the spin eigenvalues of these states have been identified properly by making some quick calculations: 1 1 z Stot ξ SM = (S1z + S2z ) ξ SM = σ 3 ξ SM + ξ SM (σ 3 )T = M ξ SM , (17.33) 2 2 2 1 3 3 (S⃗tot )2 ξ SM = (S⃗1 + S⃗2 ) ξ SM = ( ) ξ SM + σ i ξ SM (σ i )T + ξ SM ( ) 4 2 4 = S(S + 1) ξ SM . (17.34) The multiplet structure can be verified by showing 1 x 1 1 T y ± SM Stot ξ = (σtot ± iσtot ) ξ SM = (σ x ± iσ y ) ξ SM + ξ SM (σ x ± iσ y ) 2 2 2 √ = (S ∓ M )(S ± M + 1) ξ S(M ±1) . (17.35) These two-particle spin states are orthonormal and complete: +

Tr [(ξ SM ) ξ S M ] = δSS ′ δM M ′ ′

′

,

∗

SM SM ∑ ξab (ξcd ) = δac δbd .

(17.36)

SM

A more convenient representation of the two-particle spin states can be obtained by action of the 2 × 2 “charge conjugation” matrix c = −iσ2 on the indices of the second (positively charged) particle. We define this new representation as χSM = ξ SM c.

(17.37)

SM

The χ matrices are just the ones found in Eq. (16.234) for the relativistic twoparticle spin states consisting of particle and antiparticle: 1 1 ⃗ ⋅ e⃗M , χ1M = √ σ χ00 = √ 12×2 , (17.38) 2 2 ∓1 where the polarization vectors in the spherical basis are e⃗M = √ (1, ±i, 0), and 2 e⃗0 = (0, 0, 1). The expectation value of a spin operator A1 B2 takes the convenient form

̃ , ξ + A1 B2 ξ = Tr [ξ + A ξB T ] = Tr [χ+ Aχc−1 B T c] = Tr [χ+ AχB]

(17.39)

̃ is obtained from B by reversing the order of all σ matrices in B and where B ̃ and B follows from the changing the sign of each. The connection between B + −1 T properties of the charge conjugation matrix c = c = c = −c and the relationship c−1 (σ i )T c = −σ i .

(17.40)

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At this stage we promote our result to D dimensions and take (17.39) to give the definition of an NRQED two-particle spin expectation value for general D. We are now in a position to exploit the matching conditions to obtain coefficients dS and dV . We start with the NRQED calculation. For the scalar expectation value we find ∗

+

SM SM ) δab δcd ξbd (ξac = Tr [(ξ SM ) ξ SM ] = 1,

(17.41)

while the vector expectation value is ⎧ ⎪ (S = 0) + ⎪−D ∗SM SM ξac (σ k )ab (σ k )cd ξbd = tr [(χSM ) σ k χSM (−σ k )] = ⎨ . (17.42) ⎪ D − 2 (S = 1) ⎪ ⎩ We have used (17.39), ̃ σ k = −σ k and the D-dimensional sigma matrix relations {σ a , σ b } = 2δ ab , and σ k σ k = D, as well as σ k σ n σ k = −(D − 2)σ n with an understood D-dimensional summation over k. The D-dimensional trace formulas are tr[12×2 ] = 2, tr[σ a ] = 0, and tr[σ a σ b ] = 2δ ab . For the complete NRQED expectation value, we find dS dV SM SM ∗ ⃗ab ⋅ σ ⃗cd } ξbd δab δcd + σ MSM NRQED = (ξac ) { m1 m2 m1 m2 ⎧ if S = 0 dS dV ⎪ ⎪−D = + ⎨ , (17.43) m1 m2 m1 m2 ⎪ ⎪ ⎩D − 2 if S = 1 which implies m1 m2 S=1 {(D − 2)MS=0 (17.44a) dS = NRQED + DMNRQED }, 2(D − 1) m1 m2 S=1 { − MS=0 (17.44b) dV = NRQED + MNRQED }. 2(D − 1) On the QED side, the graphs of Fig. 17.2(b) give the scattering amplitude dd k i −i 2 µ ν u (−iq γ ) ) (−iq γ ) u ( 1f 1 1 1i (2π)d k2 m1n
+ k
− m1 i × {v 2i (−iq2 γν ) (−iq2 γµ ) v2f −m2n
+ k
− m2

iMQED = (−1) ∫

+ v 2i (−iq2 γµ )

i (−iq2 γν ) v2f }, −m2n
− k
− m2

(17.45)

which contains a Fermi antisymmetry factor (−1). The imaginary infinitesimals in propagator denominators required to enforce Feynman boundary conditions are implicit. The amplitude can be written as MQED = i(q1 q2 )2 ∫ × (u1i v 2i ){

dd k 2 −1 ν [(k 2 + 2k ⋅ n m1 ) (k 2 ) ] Tr[γ µ (m1n
+ k
+ m1 ) γ d (2π) γν (−m2n
+ k
+ m2 ) γµ k2

− 2k ⋅ n m2

+

γµ (−m2n
− k
+ m2 ) γν k 2 + 2k ⋅ n m2

}(v2f u1f )] . (17.46)

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In this case, we have q1 q2 = +4πZαµ2 [see Eq. (16.145)]. We have converted the product of two spinor bilinears into a single trace. Also, we make use of (16.233) written as 1 1 1 1 0 χSM γ γ5 (n ) = √ { 5 } (−n } (u1 v 2 )SM = (
+ 1) = √
+ 1) { 0 0 − e − 2 2 2
M 2
eM

(17.47)

0 12×2 ). The specific definition 12×2 0

for S = 0 (above) and S = 1 (below), where γ5 = (

of γ5 used here satisfies γ52 = 1 and {γ5 , γ µ } = 0. Other definitions of γ5 are more useful for theories involving chiral symmetries, but for our purposes, the present definition is convenient (see also Sec. 10.7). One also has (u1 v 2 )+ = γ 0 v2 u1 γ 0 , so that 1 −γ 1 1 1 0 0 −γ5 (−n ) = √ { ∗5 } (n
+ 1) = √
+ 1) { ∗ } . (17.48) − e −χ+SM 0 − 2
M 2 2 2
eM We use the fact that eM is a spatial unit vector: e∗M ⋅ eM = −1 (with no sum on M ), and e∗M ⋅ n = n ⋅ eM = 0. The two diagrams of Fig. 17.2(b) will be evaluated in turn. The first (ladder) graph contributes (v2 u1 )SM = (

ML;QED = i(4πZαµ2 )2 ∫

dd k −1 NL [(k 2 + 2 k ⋅ n m1 )(k 2 − 2 k ⋅ n m2 )(k 2 )2 ] , (2π)d (17.49)

where

⎧ ⎫ ⎪
− k
+ m2 )γµ ⎪ ⎪ ⎪ 1 ⎪ −γν (m2n ⎪ µ ν NL = Tr[γ (m1 n ⎬ (n
+ k
+ m1 )γ (n
+ 1) ⎨
+ 1)] ⎪ ⎪ ∗ 8 ⎪ ⎪ ⎪
+ k
+ m2 )γµ
eM ⎪ ⎩
eM γν (−m2n ⎭ ⎧ ⎫ ⎪ ⎪ 2(d − 1)k 2 + (d − 2)2 (k ⋅ n)2 + 2(m1 − m2 )k ⋅ n − 4m1 m2 ⎪ ⎪ ⎪ ⎪ =⎨ ⎬, ⎪ ∗ 2 2 2 ⎪(d − 3)[2(em ⋅ k)(k ⋅ em ) + k ] + (d − 4) (k ⋅ n) + 2(m1 − m2 )k ⋅ n − 4m1 m2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (17.50) valid for

⎧ ⎪ ⎪ ⎪S ⎨ ⎪ ⎪ ⎪ ⎩S

= 0⎫⎪⎪⎪ ⎬. Similarly, the crossed contribution is = 1⎪⎪⎪⎭

MX;QED = i(4πZαµ2 )2 ∫ with

dd k −1 NX [(k 2 + 2k ⋅ nm1 )(k 2 + 2k ⋅ nm2 )(k 2 )2 ] (2π)d (17.51)

⎧ ⎫ ⎪
+ k
+ m2 )γν ⎪ ⎪ ⎪ 1 ⎪ −γµ (m2 n ⎪ µ ν NX = Tr[γ (m1 n ⎬ (n
+ k
+ m1 )γ (n
+ 1) ⎨
+ 1)] ⎪ ⎪ ∗ 8 ⎪ ⎪ ⎪
− k
+ m2 )γν
eM ⎪ ⎩
eM γµ (−m2n ⎭ ⎧ ⎫ ⎪ ⎪ 2(d − 3)k 2 +(d − 2)(d − 4)(k ⋅ n)2 − 2(m1 + m2 )k ⋅ n − 4m1 m2 ⎪ ⎪ ⎪ ⎪ =⎨ ⎬ ⎪ 2 ∗ ⎪2(d − 5)k +R(d, k, n) + 4(d − 3)(eM ⋅ k)(k ⋅ eM ) − 2(m1 + m2 )k ⋅ n − 4m1 m2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (17.52)

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where R(d, k, n) = (d2 − 10d + 20)(k ⋅ n)2 . The d-dimensional momentum space integrals can be done using Feynman parameters, to give MQED =

(Zα)2 µ2 m1 m2 ⎧ ⎫ 1 1 m2 ln(m22 /µ2 ) − m22 ln(m21 /µ2 ) − 3m1 m2 ln(m21 /m22 ) ⎪ ⎪ ⎪ ⎪ ⎪ + O()⎪ − + + 1 ⎪ ⎪ 2 2 ⎪ ⎪ ⎪  3 ⎪ m1 − m2 ×⎨ ⎬, 2 2 2 2 ⎪ ⎪ 1 1 m ln(m /µ ) + m ln(m /µ ) ⎪ ⎪ 1 2 2 1 ⎪ ⎪ ⎪ ⎪ − + + + O() ⎪ ⎪ ⎪ ⎪  3 m1 + m2 ⎩ ⎭ (17.53)

⎧ ⎪ ⎪ ⎪S ⎨ ⎪ ⎪ ⎪ ⎩S

= 0⎫⎪⎪⎪ ⎬ as the total of the ladder plus crossed contributions. These results are = 1⎪⎪⎪⎭ consistent with the hard region results of the previous chapter, in Eq. (16.251). We note that the energy correction is related to the expectation value of the amplitude by ∆E = −∣ψ(0)∣2 M. Finally, we can obtain dS and dV from the QED amplitudes using (17.44). We find that [see Eq. (16.145) for the relation of µ and µ] ⎧ m2 m2 ⎫ ⎪ ⎪ ⎪ m21 ln ( 22 ) − m22 ln ( 21 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ µ µ ⎪ 1 ⎪ 2 2 ⎪ 1 dS = (Zα) µ ⎨− + + ⎬, 2 2 ⎪ ⎪  3 m − m ⎪ ⎪ 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

for

m21 ) m22 . (17.54) dV = (Zα)2 µ2 m21 − m22 Our results for dS and dV are consistent with those of Ref. [592]. The difference in the sign of dS is due to a difference in how the positively charged particle fields are defined. m1 m2 ln (

17.2.3

Bethe–Salpeter Equation of NRQED

As in standard QED, the Bethe–Salpeter equation provides a convenient means for working out the energy levels of two-body bound systems in the context of NRQED. In Chap. 16, we have discussed, for the ordinary Bethe–Salpeter equation, the 2to-2 Green functions, the two-body irreducible kernel, the inhomogeneous BSEQ, the presence of bound-state poles, the homogeneous BSEQ, the normalization conditions, a solvable reference problem, and a perturbation theory built upon that solution. These are present in analogous forms for NRQED. Here, we describe the NRQED approach. The 2-to-2 Green function of NRQED is defined as for QED: G(x1 , x2 ; y1 , y2 ) = ⟨0∣Tψ1 (x1 )ψ2 (x2 )ψ 1 (y1 )ψ 2 (y2 )∣0⟩,

(17.55)

except that there is no need to renormalize G as hard-region effects do not exist in NRQED by construction, and the rest energies have been extracted from the fields: ̃i (x) = e−imi t ψi (x). ψ (17.56)

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̃i (x) is a Pauli fermion field with the standard time dependence, while ψi (x) is Here ψ a field with the dominant time dependence factored out. The practical consequence of this redefinition is that the Feynman rules for the NRQED propagator of a fermion, say fermion 1, carrying energy ξ1 P0 + p0 = m1 + ξ1 E + p0 , and spatial momentum p⃗ (in the CM), is i/(ξ1 E + p0 − p⃗ 2 /(2m1 ) + i). The transformation to CM and relative coordinates goes through as before, as does the proof that the 2to-2 Green function contains poles at the positions of the bound-state energies. The bound state pole in the 2-to-2 Green function (in the CM) takes a form analogous to Eq. (16.15), Ψnk (p)Ψnk (q) + ⋯. n,k E − En + i

G(E; p, q) = i ∑

(17.57)

The inhomogeneous BSEQ of NRQED is G = S ′ + S ′ K ′ G. ′

(17.58)

′

The graphical definitions of G, S , and K are shown in Fig. 17.3. As was the case for QED, each of G, S ′ , K ′ has an infinite number of terms. As in the previous chapter, we choose to rewrite the BSEQ in terms of a simpler non-interacting 2-to-2 reference propagator S, where for S we use the uncorrected product S(E; p) = iS1 (E; p) × iS2 (E; −p) =

i ξ1 E + p0 −

p⃗ 2 2m1

+ i

×

i ξ2 E − p0 −

p⃗ 2 2m2

. + i (17.59)

This reference propagator has the crucial property that ∫

dp0 i dp0 S(E; p) = ∫ 2π 2π ξ1 E + p0 −

p⃗ 2 2m1

+ i

×

i ξ2 E − p 0 −

p⃗ 2 2m2

+ i

= i s(E; p⃗ ), (17.60)

where s(E; p⃗ ) is the nonrelativistic propagator s(E; p⃗ ) =

1 E−

p⃗ 2 2µ

+ i

.

(17.61)

So, the integral over the relative energy of the reference noninteracting 2-to-2 propagator is just equal to the imaginary unit times the standard free nonrelativistic Green’s function for an effective particle with reduced mass µ. We simplify the BSEQ by replacing the fully-corrected propagator S ′ by the reference propagator S so that all self-energy corrections are moved to the modified kernel K, G = S + S K G,

(17.62)

K = K ′ + S −1 − (S ′ )−1 .

(17.63)

where now

The reference propagator S and modified kernel K ′ are depicted in parts (d) and (e) of Fig. 17.3.

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Fig. 17.3 Green functions and kernel entering into the NRQED Bethe–Salpeter equation. Part (a) represents the full 2-to-2 Green function G of NRQED. The electron line is shown at the top and the proton at the bottom. Dashed and wiggly lines represent Coulomb and transverse photons, respectively. The various vertices of NRQED are represented collectively by the dots. Part (b) shows S ′ , the product of a fully corrected electron propagator with a fully corrected proton propagator. Part (c) shows K ′ , the two-particle irreducible kernel of the NRQED BSEQ. As before, the external fermion lines are shown with vertical hashes to indicate that they are amputated. These lines are shown only for clarity. Part (d) shows the free 2-to-2 propagator S, and part (e) shows the modified kernel K for use in a BSEQ with propagator S. Only a suggestive subset of diagrams is shown.

Our next goal is to find an approximation to the NRQED BSEQ that can be solved exactly for use as the basis of a perturbative expansion. The NRQED reference equation is analogous to the one obtained in full QED in the previous chapter, but significantly simpler. As a reference kernel we can make the simplest possible choice, namely, an unmodified Coulomb interaction between the two charged particles. That is, we choose the reference kernel as follows, i = −iV (⃗ p − q⃗ ) . (17.64) K0 (E; p, q) = (−iq1 )(−iq2 ) (⃗ p − q⃗ )2 Here, V (k⃗ ) = −4πZαµ2 /k⃗ 2 is the usual momentum-space Coulomb potential (in

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D dimensions) using Eq. (16.145). As we will show, this reference BSEQ G0 = S + SK0 G0

(17.65)

can be solved, at least for D = 3. As in the previous chapter, we build a solution for the reference 2-to-2 propagator using the series G0 = (1 − SK0 )

−1

S = S + SK0 S + SK0 SK0 S + ⋯

= S + SK0 S + S (K0 SK0 + K0 SK0 SK0 + ⋯) S = S + SK0 S + SHS ,

(17.66)

where H = K0 SK0 + K0 SK0 SK0 + ⋯. The terms in H implicitly involve integrals over the full d-dimensional space-time, but because of (17.60) and the fact that the reference potential is energy-independent, they can all be reduced to purely spatial integrals of D = d − 1 dimensions. The first term of H is dd k [K0 S K0 ](E; p, q) = ∫ K0 (E; p, k)S(E; k)K0 (E; k, q) (2π)d dD k (−i)V (⃗ p − k⃗ )s(E; k⃗ )V (k⃗ − q⃗ ) = −i[V s V ](E; p⃗, q⃗ ) . (2π)D (17.67)

=∫

Likewise, the second term of H reduces as [K0 SK0 SK0 ] (E; p, q) = −i [V sV sV ] (E; p⃗, q⃗ ) ,

(17.68)

and so on. Comparison with (16.79) and (16.77) shows that H(E) is essentially the corresponding sum that enters the nonrelativistic Schr¨odinger-Coulomb Green function. One has H(E) = −i h(E) where g(E) = s(E) + s(E)V s(E) + s(E)h(E)s(E)

(17.69)

is the Schr¨ odinger equation of standard quantum mechanics. Given the known solution for g(E), we also have the solution to our NRQED reference problem: G0 = S + SK0 S + S(−ih)S.

(17.70)

The bound-state poles of g have the form p ) ψn+ (⃗ q) ∑ ψn (⃗ g(E; p⃗, q⃗ ) → , (17.71) 0 E − En where the sum is over the degenerate quantum numbers labeling orbital angular momentum and spin. There are no poles in s or sV s, so Eq. (17.69) implies that the poles of h are s(En0 ; p⃗ )−1 ψn (⃗ p )ψn+ (⃗ q )s(En0 ; q⃗ )−1 h(E; p⃗, q⃗ ) → ∑ . (17.72) E − En0 n These same bound-state poles are present in the reference Green’s function G0 as seen by (17.70), so that G0 (E; p, q) → − i ∑ n

S(En0 ; p) s(En0 ; p⃗ )−1 ψn (⃗ p ) ψn+ (⃗ q ) s(En0 ; q⃗ )−1 S(En0 ; q) E − En0 0

≡ i∑ n

Ψ0n (p)Ψn (q) . E − En0

(17.73)

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This allows us to reduce the problem of the calculation of the two-particle wave function to the calculation of a reference wave function ψn (⃗ p ), which is a function of the relative momentum p⃗ only, Ψ0n (p) = −i S(En0 ; p) s(En0 ; p⃗ )−1 ψn (⃗ p ),

(17.74a)

0 Ψn (p)

(17.74b)

= −i ψn+ (⃗ p ) s(En0 ; p⃗ )−1 S(En0 ; p) .

0 Ψn (p)

Here, Ψ0n (p) and correspond to the nonrelativistic limit of the wave functions which had been calculated in Eq. (16.101) for the relativistic Bethe–Salpeter 0 formalism. Note, however, that Ψn (p) in Eq. (17.74) does not contain a γ 0 matrix [cf. Eq. (7.34)]. The γ 0 matrix enters the relativistic (four-component) bispinor formalism, but is absent in the nonrelativistic (two-component) spinor formalism that is used in NRQED. The reference wave functions contain the nonrelativistic wave functions in a straightforward (multiplicative) way, and satisfy the integral identities dp0 0 dp0 Ψn (p) = −i ∫ S(En0 ; p)s(En0 ; p⃗ )−1 ψn (⃗ p) ∫ 2π 2π = −i [is(En0 ; p⃗ )] s(En0 ; p⃗ )−1 ψn (⃗ p ) = ψn (⃗ p), (17.75) and dp0 0 Ψ (p) = ψn+ (⃗ p ). (17.76) 2π n The reader should recognize how much more naturally the nonrelativistic Coulomb problem fits into the formalism of the quantum field theory NRQED compared to the contortions required to adapt the Coulomb problem to the relativistic theory of QED. Additional aspects of this simplicity are illustrated below. The bound-state equation satisfied by the reference wave functions can be obtained from the pole part of ∫

1 = G−1 0 G0 ,

(17.77)

which gives the reference homogeneous Bethe–Salpeter equations: 0

0 0 −1 0 G−1 0 (En ) Ψn = Ψn G0 (En ) = 0 .

(17.78) −1

The reference Green’s function has the value G0 = (1 − SK0 ) G−1 0

=S

−1

(1 − SK0 ) = S −1

−1

S, so that

− K0 .

(17.79)

The reference Bethe–Salpeter equation (S − = 0, multiplied by S, takes the form dd q Ψ0n (p) = S(En0 ; p) ∫ K0 (En0 ; p, q) Ψ0n (q) . (17.80) (2π)d We use the fact that the reference kernel is energy independent, so we can perform the q 0 integral on the wave function with Eq. (17.75), and integrate both sides over relative energy p0 , to find ψn (⃗ p ) = is(En0 ; p⃗ ) ∫

K0 )Ψ0n

dD q (−i) V (⃗ p − q⃗ ) ψn (⃗ q), (2π)D

(17.81)

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or

p⃗ 2 dD q ) ψn (⃗ p ) = s(En0 ; p⃗ )−1 ψn (⃗ p) = ∫ V (⃗ p − q⃗ )ψn (⃗ q ). (17.82) 2µ (2π)D So, the reference BSEQ in NRQED reduces exactly to the usual Schr¨odingerCoulomb equation. Finally, we consider the normalization condition for the reference wave functions. It can be obtained by expanding the identity G0 = G0 G−1 (17.83) 0 G0 about the bound-state pole. Near the pole, one has 0 0 ⎛ Ψ0 Ψ ⎞ Ψ0 Ψ i n n0 + ⋯ = i n n0 + ⋯ E − En ⎝ E − En ⎠ (En0 −

0

−1 ⎞ ⎞ ⎛ Ψ0 Ψ ⎛ 0 0 dG0 ∣ + ⋯ i n n0 + ⋯ . (17.84) × G−1 0 (En ) + (E − En ) dE E=E 0 ⎠ ⎠ ⎝ E − En ⎝ n The term in the expansion with a single pole gives the normalization condition: −1 0 dG0 dd p dd q 0 dG−1 0 (E; p, q) 1 = iΨn Ψ0n = i ∫ Ψ0n (q). ∣ ∣ Ψn (p) d d dE E=E 0 (2π) (2π) dE 0 E=En n (17.85) −1 Since G−1 − K0 and K0 is independent of E, only the S −1 term needs to be 0 = S kept in the energy derivative. The S −1 term satisfies S −1 (E; p, q) = S(E; p)−1 (2π)d δ (d) (p − q) , (17.86)

S −1 (E; p) = [iS1 (E; p) iS2 (E; −p)] = − (ξ1 E + p0 −

−1

p⃗ 2 p⃗ 2 + i) (ξ2 E − p0 − + i) . 2m1 2m2

(17.87)

So, one has dG−1 0 (E; p, q) = −(2π)d δ (d) (p−q) (ξ1 S2 (En0 ; −p)−1 +S1 (En0 ; p)−1 ξ2 ), (17.88) ∣ dE 0 E=En and the normalization condition becomes dd p dd q 0 dG−1 0 (E; p, q) 1 = i∫ Ψ (p) Ψ0n (q) ∣ n d d (2π) (2π) dE E=E 0 n

dd p 0 = − i∫ Ψ (p)(ξ1 S2 (En0 ; −p)−1 + S1 (En0 ; p)−1 ξ2 )Ψ0n (p) . (17.89) (2π)d n With the help of Eq. (17.74), we can write this as dd p + 1 = i∫ ψ (⃗ p )s(En0 ; p⃗ )−1 (ξ1 S1 (En0 ; p)2 S2 (En0 ; −p) + ξ2 S1 (En0 ; p)S2 (En0 ; p)2 ) (2π)d n × s(En0 ; p⃗ )−1 ψn (⃗ p) = i∫

dD p + ψ (⃗ p )s(En0 ; p⃗ )−1 (−iξ1 s(En0 ; p⃗ )2 − iξ2 s(En0 ; p⃗ )2 )s(En0 ; p⃗ )−1 ψn (⃗ p) (2π)D n

=∫

dD p + ψ (⃗ p )ψn (⃗ p ), (2π)D n

(17.90)

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which is the usual normalization. This agreement provides a check of the procedure. In the reduction shown above we have used the integral formula ∫

dp0 dp0 [S1 (E; p)]2 S2 (E, −p) = ∫ S1 (E; p) [S2 (E; −p)]2 = −i s(E; p⃗ )2 . (17.91) 2π 2π

17.3 17.3.1

Applications of NRQED Energy Levels at O(Zα)4 (NRQED Approach)

In this section, we consider energy level corrections to two-fermion bound states such as hydrogen or muonium using NRQED. In Chap. 16, we obtained the same result using a QED-based formalism in Eq. (16.144), but seeing the NRQED derivation allows for a direct comparison of the two approaches. All diagrams that contribute to the energy levels of nonrelativistic two-fermion Coulombic bound states through O(α5 ) are shown in Fig. 17.4. Only a subset of these enter at O(α4 ). These are, specifically, the diagrams in Figs. 17.4(a)–(f). (There is a contribution at O(α4 ) to the energy levels of fermion-antifermion bound states such as positronium from the contact term Fig. 17.4(f) accounting for virtual annihilation into a single photon, but we will not consider this contribution here.) We aim to discuss the O(α4 ) energy contributions one by one, first noting that the fermion momenta are all from the potential region (in the sense of Table 16.1), as enforced by the wave function. (Graphs with virtual fermions that are not part of the wave function do not contribute at this order.) Also, the exchanged photons are exclusively potential at this order. We will explore the contributions of nonpotential fermions and photons when we move on to O(α5 ). The relativistic kinetic energy correction of Fig. 17.4(a) is the only one that requires a bit of special handling as the corresponding kernel is produced by the S −1 − (S ′ )−1 term of (17.63) instead of coming directly from an interaction diagram in K ′ . In order to sort this out, we consider the corrected single-particle propagator Si (E; p), Si (E; p) =

1 ξi E + p 0 −

p⃗ 2 2mi

− Σi (E; p) + i

,

(17.92)

p⃗ where Σi (E; p) = − 8m The fully 3 + ⋯ is the NRQED self-energy of particle i. i corrected non-interacting 2-to-2 propagator has the form 4

S ′ (E; p, q) =

i2 (2π)4 δ (4) (p − q) [ξ1 E + p0 −

p⃗ 2 2m1

− Σ1 (E; p) + i] [ξ2 E − p0 −

p⃗ 2 2m2

− Σ2 (E; −p) + i]

,

(17.93) so that the contribution to K from the O(⃗ p 4 ) part (fourth-order correction to the

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Fig. 17.4 Diagrams contributing to NRQED energy levels at orders α4 [diagrams (a)–(f)] and α5 [diagrams (b)–(k)]. (Diagram k also contributes at O(α4 ) for particle-antiparticle bound states.) The various contributions are (a) the first relativistic correction to the kinetic energy, (b) spin-orbit, (c) Darwin, (d) magnetic interaction due to transverse photon exchange, (e) a spin-orbit contribution from the Fermi interaction, (f) Fermi-Fermi tensor interaction, (g) vacuum polarization, (h) transverse photon spanning zero, one, two, etc. Coulomb photons (which are represented collectively as a vertical double line), (i) transverse photon exchange crossing one, two, etc. Coulomb photons, (j) seagull interaction, and (k) four-fermion contact interaction.

kinetic energy, appropriately named K4 ) is [S −1 − (S ′ )−1 ](E; p, q) → KK4 ≡{

p⃗ 4 p⃗ 4 S2 (E; −p)−1 + S1 (E; p)−1 } (2π)4 δ (4) (p − q) . 3 8m1 8m32 (17.94)

For the corresponding energy shifts, we use ∆E = ⟨iδK⟩ = i ∫

0 d4 p d4 q Ψ (p) δK(E; p, q) Ψ0 (q) , 4 4 (2π) (2π)

(17.95)

with wave functions given by Eq. (17.74) as Ψ0 (p) = iS1 (E 0 ; p)S2 (E 0 ; −p)s(E 0 ; p⃗ )−1 ψ(⃗ p ), 0

Ψ (p) = iψ + (⃗ p )s(E 0 ; p⃗ )−1 S1 (E 0 ; p)S2 (E 0 ; −p) ,

(17.96a) (17.96b)

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to find ∆EK4 = i ∫ = i∫

d4 p d4 q 0 Ψ (p)KK4 (E 0 ; p, q)Ψ0 (q) (2π)4 (2π)4 d4 p p⃗ 4 + 0 −1 ⃗ S1 (E 0 ; p)2 S2 (E 0 ; −p) iψ (⃗ p )s(E ; p ) { (2π)4 8m31

+ S1 (E 0 ; p)S2 (E 0 ; −p)2 =∫

p⃗ 4 } is(E 0 ; p⃗ )−1 ψ(⃗ p) 8m32

p⃗ 4 d3 p + p⃗ 4 } ψ(⃗ p). ψ (⃗ p ) {− 3 − 3 (2π) 8m1 8m32

(17.97)

The relative energy integrals were done by way of (17.91). It follows that the effective Hamiltonian coming from the relativistic kinetic energy correction is p⃗ 4 p⃗ 4 Heff (K4 ) = (− 3 − ) (2π)3 δ (3) (⃗ p − q⃗) . (17.98) 8m1 8m32 The remaining contributions to the effective Hamiltonian arise more directly from the corresponding diagrams of Fig. 17.4. The kernels involved are all independent of relative energy, so the energy integrals in ⟨iδK⟩ can be done trivially using Eqs. (17.75) and (17.76). In general, for kernel contributions independent of relative energy, we find that d4 p d4 q 0 Ψ (p) δK(⃗ p, q⃗ ) Ψ0 (q) ⟨iδK⟩ = i ∫ (2π)4 (2π)4 = i∫

d3 p d3 q + ψ (⃗ p ) δK(⃗ p, q⃗ ) ψ(⃗ q), (2π)3 (2π)3

(17.99)

and we identify δHeff = iδK

(17.100)

as the corresponding contribution to the effective Hamiltonian. The kernel for the first spin-orbit correction, from the first part of Fig. 17.4(b), is ⃗1 q1 i p⃗ × q⃗ σ ⃗1 K(SO) = (⃗ p × q⃗ ) ⋅ σ (−iq2 ) = −πZα ⋅ , (17.101) 4m21 (⃗ p − q⃗ )2 (⃗ p − q⃗ )2 m21 and the contribution from the second part of Fig. 17.4(b) is analogous. The total spin-orbit effective Hamiltonian is ⃗1 ⃗2 p⃗ × q⃗ σ σ Heff (SO) = −iπZα ⋅( + ). (17.102) (⃗ p − q⃗ )2 m21 m22 We note that where there are no divergences, we can work with D = 3 and take ⃗ ⋅ (⃗ σ `m p` q m → σ p × q⃗ ), and to the order of interest we can take cS = cD = cF = 1. The Darwin correction, from Fig. 17.4(c), has the form Heff (D) = i × { =

iq1 i i iq2 (⃗ p − q⃗ )2 (−iq2 ) + (−iq1 ) (−⃗ p + q⃗ )2 } 2 2 2 8m1 (⃗ p − q⃗ ) (⃗ p − q⃗ ) 8m22

πZα 1 1 ( 2 + 2). 2 m1 m2

(17.103)

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The kernel for the magnetic interaction of Fig. 17.4(d) is iq1 iδ ⊥,ij (⃗ p − q⃗ ) iq2 K(M) = (p + q)i (−p − q)j . (17.104) 2m1 (p − q)2 + i 2m2 We can use the transversality of δ ⊥,ij (⃗ p − q⃗ ) and the identity k⃗ = p⃗ − q⃗ to simplify the convection factors as follows, (p + q)i → (2p − k)i → 2pi , and (−p − q)j → (−k − 2q)j → −2q j . The transverse photon propagator contains the four-vector photon momentum squared as opposed to the three-momentum squared of the Coulomb photon. This photon denominator gives rise to soft and ultrasoft contributions in addition to the usual potential-region contribution, but the potential region still dominates and is the only one that enters at O(α4 ). The same neglect of retardation holds for the remaining O(α4 ) contributions. Consequently, at O(α4 ), we replace (p − q)2 by its spatial contribution −(⃗ p − q⃗ )2 . The magnetic effective Hamiltonian takes the form p ⋅ q⃗ )2 4πZα p⃗ 2 q⃗ 2 − (⃗ . (17.105) Heff (M) = − m1 m2 (⃗ p − q⃗ )4 The Fermi spin-orbit contribution of Fig. 17.4(e) involves a convection interaction with one fermion combined with a Fermi (spin-dependent) interaction on the other and exchange of a transverse photon. For the corresponding kernel, we find q1 ik iδ ⊥,ij (⃗ p − q⃗ ) iq2 K(FSO) = σ1 (p − q)k (−p − q)j 2m1 −(⃗ p − q⃗ )2 2m2

iq1 iδ ⊥,ij (⃗ p − q⃗ ) q2 jk (p + q)i σ (−p + q)k . (17.106) 2m1 −(⃗ p − q⃗ )2 2m2 2 The transverse delta function reduces to the usual Kronecker delta because of the antisymmetry of σ ij . Because this contribution is finite we can use the threedimensional reduction σ ij = − 2i [σ i , σ j ] → ijk σ k . The corresponding effective Hamiltonian is ⃗2 ) p⃗ × q⃗ (⃗ σ1 + σ Heff (FSO) = −2iπZα ⋅ . (17.107) 2 (⃗ p − q⃗ ) m1 m2 Finally, the Fermi-Fermi, or “tensor”, contribution of Fig. 17.4(f) is ⃗1 ] ⋅ [(⃗ ⃗2 ] πZα [(⃗ p − q⃗ ) × σ p − q⃗ ) × σ Heff (FF) = . (17.108) m1 m2 (⃗ p − q⃗ )2 We have now obtained all terms that contribute to energy levels at O(α4 ) (subject to the exception for particle-antiparticle systems that have an additional contact contribution at this order). The complete (momentum space) effective Hamiltonian at O(α4 ) is p⃗ 4 p⃗ 4 πZα 1 1 ) (2π)3 δ (3) (k⃗ ) + ( 2 + 2) Heff = (− 3 − 8m1 8m32 2 m1 m2 +

⃗1 ⃗2 2(⃗ ⃗2 ) 4πZα p⃗ 2 q⃗ 2 − (⃗ p ⋅ q⃗ )2 p⃗ × q⃗ σ σ σ1 + σ − iπZα ⋅( 2 + 2 + ) 4 2 ⃗ ⃗ m1 m2 m1 m2 m1 m2 k k ⃗1 ) ⋅ (k⃗ × σ ⃗2 ) πZα (k⃗ × σ + . (17.109) 2 ⃗ m1 m2 k −

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All terms that enter here can be immediately identified as identical to the corresponding terms in the QED version of the effective Hamiltonian (16.144) except for the first term in (17.109), coming from the relativistic kinetic energy, which is not obvious. We note that on the QED side, the correction to the Bohr energy level comes from the reference energy (16.125) as well from the QED effective Hamiltoµβ 2 µ ) coming from nian. The comparison must include the extra contribution 2n (1 − M 2 2 the reference energy where βn = −(Zα) /(2n ) as defined in Eq. (16.94). In other words, agreement between the QED and NRQED versions of the O(α4 ) effective Hamiltonian requires µβn2 µ d3 p d3 q + (1 − )+∫ ψ (⃗ p )V (k⃗ ) 2 M (2π)3 (2π)3 ×{ =∫

(⃗ p 2 + q⃗ 2 ) 1 1 1 βn µ ( 2+ 2− )− }ψ(⃗ q) 8 m1 m2 m1 m2 2M p⃗ 4 p⃗ 4 d3 p + ψ (⃗ p ) (− − ) ψ(⃗ p ). (2π)3 8m31 8m32

(17.110)

This condition can be verified using En0 = µβn , ⟨V ⟩ = 2En0 , and ⟨⃗ p 2 ⟩ = −2µEn0 , 1 1 1 1 3 + − = − , m21 m22 m1 m2 µ2 m1 m2

1 1 1 3 + = − , m31 m32 µ3 µ2 M

(17.111)

and the momentum-space Schr¨ odinger equation p⃗ 2 ψ(⃗ p ) = 2µ {En0 ψ(⃗ p) − ∫

d3 q V (⃗ p − q⃗ )ψ(⃗ q )} , (2π)3

(17.112)

applied to one factor of p⃗ 2 in each term on the right-hand side. As in Chap. 16, the actual energy levels at O(Zα)4 can be found by evaluating some standard Coulomb expectation values. 17.3.2

Relativistic Recoil Correction (NRQED Approach)

Here we will obtain the spin-averaged relativistic recoil corrections to energy levels at O(Zα)5 using NRQED. In the end, we will compare the QED (BSEQ) and NRQED calculations. The spin averaging will be performed quite differently in NRQED compared to the QED method of Chap. 16. Before, in our QED calculation, we averaged the spins of the two particles separately. Here, we average the combined spins using X avg ≡

X 00 + ∑M X 1M D+1

(17.113)

where X SM is the expectation value of an operator X evaluated in a state of spin S and z-component M . We note that there are D (the number of spatial dimensions) possible values for the z-component M in a D-dimensional space when S = 1. The recoil corrections of interest here all involve a single inverse power of m2 . We

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p

q

p

p-k

k

p-q

q

p

p-q-k

p-k

r-k

q

r-q-k

k p-r

p

q (a)

-p

-q-k (b)

-q

-p

-r (c)

-q-k

-q

Fig. 17.5 NRQED kernels containing a single transverse exchange photon crossing zero, one, or two Coulomb photons, showing momentum assignments. The particle i fermion line implicitly caries an energy ξi E in addition to the relative momentum shown.

consider the ratio m1 /m2 to be small and ignore terms containing higher powers of m2 in the denominator. The kernels that contribute to the spin-averaged recoil energies at O(Zα)5 are shown in Figs. 17.4(d), (i), (j), and (k). The relativistic kinetic energy from the p⃗ 4 /m3 term is pure O(Zα)4 , while the next relativistic correction involves the operator p⃗ 6 /m5 , which is pure O(Zα)6 . None of these contribute at O(Zα)5 nor do they have the right power of m2 for our current interest. The spin-orbit and Darwin kernels of Figs. 17.4(b) and (c) have α5 contributions, because the matching coefficients cS and cD contain terms which are O(α/π), but the powers of m2 are either (m2 )0 = 1 (when the spin-orbit or Darwin vertex affects particle 1) or (m2 )−2 (when the spin-orbit or Darwin vertex affects particle 2). In neither case do we find the desired 1/m2 dependence. The kernels of Figs. 17.4(e) and (f) do give corrections of O(Zα)5 with 1/m2 mass dependence, but their spin average vanishes. For, say, the second graph of Fig. 17.4(e), the spin average in+ + volves the trace Tr [(χSM ) χSM σ ij ], which vanishes because (χ00 ) χ00 = 1/2,

∑M χ+1M χ1M = D/2, and Tr[σ ij ] = 0. Similar considerations show that all spin averages for Figs. 17.4(e) and (f) vanish. Neither the vacuum polarization kernel of Fig. 17.4(g) nor either of the Lamb shift terms of Fig. 17.4(h) have the right m2 dependence to be of interest here. We now focus our attention on the kernels of Figs. 17.4(d), (i), (j), and (k) that do contribute spin-averaged recoil corrections at O(Zα)5 . We begin our calculation with the magnetic interaction of Fig. 17.4(d), shown with momentum assignments in Fig. 17.5(a). Application of the NRQED Feynman rules leads to the energy correction iδ ⊥,ij (⃗ p − q⃗ ) iq2 dd p dd q 0 iq1 Ψ (p) (p + q)i (−p − q)j Ψ0 (q). ∆ET = i ∫ d d (2π) (2π) 2m1 (p − q)2 + i 2m2 (17.114) When all the momenta are potential, this contribution is O(Zα)4 as seen in the previous section. Higher order corrections from this region are O(Zα)6 . For our present calculation of (Zα)5 recoil corrections, the relevant regions for {p, p−q, q} are {P,S,S}, {S,S,P}, and {P,US,P} (using the notation introduced in Table 16.1).

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We start with the soft correction from region {P,S,S} and make the appropriate soft corrections to Ψ0 (q) and k 2 : P,S,S ∆ET =

i4πZαµ2 dd p dd q ∫ 4m1 m2 (2π)d (2π)d 0

× Ψ (p)

(2pi ) δ ⊥,ij (⃗ p − q⃗ ) (2pj ) s−1 (E 0 ; q⃗ ) ψ(⃗ q ), q02 − (⃗ p − q⃗ )2 + i (q0 + i)(−q0 + i)

(17.115)

where we have used (p + q)i = (2p − (p − q))i → (2p)i to simplify the convection factors. We use (17.76) to perform the p0 integral and recall that the q 0 = 0 poles are disallowed in the q 0 integral. After performing the relative energy integrals, we find the contribution 4πZαµ2 dD p dD q + pi pj δ ⊥,ij (⃗ p − q⃗ ) q⃗ 2 ψ (⃗ p) (E 0 − ) ψ(⃗ q ), ∫ D D 3 m1 m2 (2π) (2π) 2∣⃗ p − q⃗ ∣ 2µ (17.116) which is identical to the corresponding contribution (16.184) found in the previous chapter using the BSEQ of QED after a number of approximations. The manipulations described following Eq. (16.184) go through just as before, and we find the total result ({P,S,S}+{S,S,P}) from single transverse photon exchange to be given by (16.189) as before, P,S,S ∆ET =−

dD p dD q + 8π 2 (Zα)2 µ2 ψ (⃗ p ) (pi pj + q i q j ) I ij (⃗ p − q⃗ ) ψ(⃗ q ), ∫ m1 m2 (2π)D (2π)D (17.117) where the integral I ab (k⃗ ) was given in Eq. (16.188). For the ultrasoft contribution from the {P,US,P} region, we have directly that Eq. (17.114) takes the form of Eq. (16.197). We perform the relative energy integrals and insert an extra integration with corresponding Dirac-δ function (contained within s(E − ∣k⃗ ∣; p⃗, q⃗ )), as in the previous chapter to find [see Eq. (16.201)], soft ∆ET =

4πZαµ2 dD p dD q dD k + pi δ ⊥,ij (k⃗ )q j ψ(⃗ q )s(E − ∣k⃗ ∣; p⃗, q⃗ ). ψ (⃗ p ) ∫ m1 m2 (2π)D (2π)D (2π)D ∣k⃗ ∣ (17.118) This completes, for now, our reduction of the magnetic interaction of Fig. 17.4(d). We move on to consider the kernels represented by Fig. 17.4(i) containing one, two, etc., ladder Coulomb photons crossed by a single transverse photon. All of these kernels contribute at the order of interest when the crossing transverse photon is ultrasoft. This gives rise to a term that is closely related to the usual Lamb shift. There is also a contribution from the first of these kernels, involving a single Coulomb photon, when the inner loop is soft. We begin our analysis with this soft contribution. The relevant kernel is represented by the diagram shown in Fig. 17.5(b). Working with the NRQED Feynman rules, we write down the US ∆ET =

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corresponding energy contribution soft ∆ETC = i∫

iq1 i dd p dd k dd q 0 Ψ (q) (2p − k)i (2π)d (2π)d (2π)d 2m1 ξ1 E + p 0 − k 0 −

iδ ⊥,ij (k⃗ ) i (−iq2 ) k 2 + i (⃗ p − q⃗ − k⃗ )2 i iq2 × (−2q − k)j Ψ0 (q). ⃗ )2 (−⃗ q − k 2m 2 ξ2 E − q 0 − k 0 − 2m2 + i

⃗ )2 ⃗ k (p− 2m1

+ i

× (−iq1 )

(17.119)

We make the appropriate “soft-k” approximations and perform the p0 and q0 relative energy integrals using (17.75) and (17.76). The k 0 integral is done using poles, remembering that the k 0 = 0 poles are disallowed. We find soft ∆ETC =

dD p dd k dD q −i(4πZαµ2 )2 ∫ m1 m2 (2π)D (2π)d (2π)D

pi q j δ ⊥,ij (k⃗ ) 1 1 ψ(⃗ q) (−k 0 + i)(−k 0 + i) (k 0 )2 − k⃗ 2 + i (⃗ p − q⃗ − k⃗ )2 (4πZαµ2 )2 dD p dD k dD q + pi q j δ ⊥,ij (k⃗ ) ψ(⃗ q) = − ψ (⃗ p ) ∫ m1 m2 (2π)D (2π)D (2π)D 2∣k⃗ ∣3 (⃗ p − q⃗ − k⃗ )2 × ψ + (⃗ p)

=

8π 2 (Zα)2 µ2 dD p dD q + ψ (⃗ p ) (−pi q j ) I ij (⃗ p − q⃗ ) ψ(⃗ q ), (17.120) ∫ m1 m2 (2π)D (2π)D

which is identical to (16.193) of the previous chapter. As before, this contribution must be doubled to account for the additional kernels like those of Fig. 17.4(i), but with the transverse photon slanting down to the left. The soft single transverse contribution of (17.117) combined with the crossed transverse-Coulomb contribution of (17.120) here gives a total soft single-transverse contribution of soft ∆ET(total) =

1 1 2 8(Zα)2 2 µ {∣ψ(0)∣2 [− − + ln ( )] + ⟨ ln(∣⃗ p − q⃗ ∣)⟩p,⃗ } , (17.121) ⃗q 3m1 m2 2 2 µ

analogous to the derivation of Eq. (16.194). Continuing our evaluation of the kernels represented by Fig. 17.4(i), we consider the case where the transverse photon is ultrasoft. We will start with the contribution involving the exchange of a single Coulomb photon crossed by the ultrasoft transverse photon. Using the momentum assignments of diagram in Fig. 17.5(b) and the approximations appropriate to an ultrasoft k, we find the contribution US ∆ETC = i∫

×

iq1 i dd p dd k dd q 0 Ψ (p) (2p)i 0 (2π)d (2π)d (2π)d 2m1 ξ1 E + p − k 0 − iδ ⊥,ij (k⃗ ) i i (−iq2 ) 2 2 0 k + i (⃗ p − q⃗ ) ξ2 E − q − k 0 −

q⃗ 2 2m2

p⃗ 2 2m1

+ i

(−iq1 )

iq2 (−2q j ) Ψ0 (q). + i 2m2 (17.122)

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This integral, once doubled to account for the reflected kernels, is precisely the integral displayed in (16.204). The result after integration over relative energies is the same as was shown in (16.205), US ∆ETC =

4πZαµ2 dD p dD q dD k + pi δ ⊥,ij (k⃗ ) q j ψ (⃗ p) ψ(⃗ q) ∫ D D D m1 m2 (2π) (2π) (2π) ∣k⃗ ∣ × s(E − ∣k⃗ ∣; p⃗ )V (⃗ p − q⃗ )s(E − ∣k⃗ ∣; q⃗ ) . (17.123)

For the two-Coulomb case of Fig. 17.5(c), we follow the same procedure as in the previous chapter, not re-calculating the whole graph but rather just analyzing the changes between the one-Coulomb graph of Fig. 17.5(b) and the two-Coulomb graph of Fig. 17.5(c). The only difference is that the Coulomb propagator in Fig. 17.5(b) is replaced by a loop integration containing two fermions and two Coulomb photons (remembering that k is ultrasoft): 1

(⃗ p − q⃗ − k⃗ )2

→∫ ×

1 i dd r (2π)d (⃗ p − r⃗ )2 ξ1 E + r0 − k 0 − i i 2 0 (⃗ r − q⃗ ) ξ2 E − r −

= −4πZαµ2 ∫

r⃗ 2 2m2

+ i

r⃗ 2 2m1

+ i

(−iq1 )

(−iq2 )

dD r 1 1 s(E − k 0 ; r⃗ ) , D 2 (2π) (⃗ p − r⃗ ) (⃗ r − q⃗ )2

(17.124)

or, equivalently, V (⃗ p − q⃗ ) → [V s V ] (E − k 0 ; p⃗, q⃗ ).

(17.125)

When we combine results for the kernels with no Coulomb photons (17.118), one Coulomb photon (17.123), and two and more Coulomb photons using (17.125) and its generalization to the many-Coulomb case, we find US ∆ET(total) =

=

4πZαµ2 dD p dD q dD k + pi δ ⊥,ij (k⃗ )q j ψ (⃗ p) ψ(⃗ q) ∫ D D D m1 m2 (2π) (2π) (2π) ∣k⃗ ∣ × [s + sV s + sV sV s + sV sV sV s + ⋯] (E − ∣k⃗ ∣; p⃗, q⃗ )

4πZαµ2 dD p dD q dD k + pi δ ⊥,ij (k⃗ )q j ψ(⃗ q )g(E − ∣k⃗ ∣; p⃗, q⃗ ). ψ (⃗ p) ∫ D D D m1 m2 (2π) (2π) (2π) ∣k⃗ ∣ (17.126)

The analysis now proceeds just as in the previous chapter, leading to the same result as (16.222): 8(Zα)2 2 1 5 µ 8µ3 (Zα)5 )} − ln k0 (n, `). µ ∣ψ(0)∣2 { + + ln ( 3m1 m2 2 6 µ(Zα)2 3πm1 m2 n3 (17.127) At this point, we recall the difference between the reduced mass µ and the renormalization scales µ and µ, according to the discussion surrounding Eq. (16.145). We have now completed the evaluation of all spin-averaged recoil corrections of O(Zα)5 US ∆ET(total) =

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involving the exchange of a single transverse photon. There are two classes of corrections: the soft-region terms of (17.121) and the ultrasoft region terms of (17.127). The infrared divergence of the soft term exactly cancels the ultraviolet divergence of the ultrasoft term, leading to a finite total. The sum of these two contributions is ∆ET =

2 1 8(Zα)2 {∣ψ(0)∣2 [ln ( ) + − ln k0 (n, `)] + ⟨ln(∣⃗ p − q⃗ ∣)⟩} , (17.128) 2 3m1 m2 µ(Zα) 3

where k 0 (n, `) is the Bethe log discussed at length in Chap. 4, and the expectation value ⟨ln(∣⃗ p − q⃗ ∣)⟩ is given in (16.180). The total single-transverse-photon contribution of Figs. 17.4(d) and (i) reduces to ∆ET =

8µ3 (Zα)5 4 1 5 {[ln ( ) + Hn − + ] δ`=0 3 3πm1 m2 n Zαn 2n 6 − ln k0 (n, `) −

δ`≠0 }. 4`(` + 1)(` + 1/2)

(17.129)

We recall from Eq. (16.180) that Hn = ∑nj=1 1j is the nth harmonic number. Graphs involving the exchange of two transverse photons also contribute to the spin-averaged recoil corrections at O(Zα)5 . These are contained in the seagull term Fig. 17.4(j) and the contact term Fig. 17.4(k). The contact term includes the effect of the exchange of two photons of all varieties, transverse and Coulomb, but only in the hard region. Our calculation of the matching coefficients for the contact term was done in Feynman gauge, but the values of these coefficients are independent of gauge, so the calculation could have been done in Coulomb gauge if desired, with the same result. The seagull graph of Fig. 17.4(j) makes the following contribution: dd p dd k dd q 1 ∆Eseagull = i ( ) ∫ 2 (2π)d (2π)d (2π)d 0

× Ψ (p) (

−iq12 −iq 2 iδ ⊥,ij (k⃗ ) iδ ⊥,ij (k⃗ + p⃗ − q⃗ ) 0 )( 2 ) 2 Ψ (q) . m1 m2 k + i (k + p − q)2 + i

(17.130)

The initial factor of 1/2 is a symmetry factor needed because the two vertices in the seagull diagram are connected by identical propagators. This term is clearly a recoil correction because of the explicit factor of m2 in the denominator. The leading order is (Zα)5 when the two wave functions are potential but the inner loop is soft. For this region, we must approximate (k + p − q)2 → (k 0 )2 − (k⃗ + p⃗ − q⃗ )2 , so the relative energy integrals factorize. We use Eqs. (17.75) and (17.76) to do the p0 and q 0 integrals, arriving at ∆Eseagull =

dD p dD q + (4πZα)2 2 µ ∫ ψ (⃗ p ) I(⃗ p, q⃗ ) ψ(⃗ q), 2m1 m2 (2π)D (2π)D

(17.131)

where I(⃗ p, q⃗ ) contains the k integration and was defined in Eq. (16.177). We see that the NRQED seagull graph contains exactly the same information, and has the

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same value (16.178), as the soft two-transverse-photon contribution discussed in Sec. 16.3.4. For the seagull contribution, we find (Zα)2 2 1 8 5 µ {∣ψ(0)∣2 [− − 2 ln µ − ln 2 + ] + 2⟨ ln(∣⃗ p − q⃗ ∣)⟩p,⃗ }. ⃗q m1 m2  3 3 (17.132) The contact term shown in Fig. 17.4(k) makes an energy contribution ∆Eseagull =

∆Econtact = i ∫

idS idV dd p dd q 0 ⃗1 ⋅ σ ⃗2 } Ψ0 (q). Ψ (p) { + σ d d (2π) (2π) m1 m2 m1 m2

(17.133)

The reference wave functions contain the spin state factors ξ SM shown in Eq. (17.32), so the expectation value for ∆Econtact reduces to the spin expectation value of the contact interaction (17.43), proportional to MSM NRQED , times the wave function at the spatial origin, ∆Econtact = −∣ψ(0)∣2 MSM NRQED = −

∣ψ(0)∣2 {dS +dV [−DδS=0 +(D−2)δS=1 ]} . (17.134) m1 m2

Values for dS and dV were given in Eq. (17.54). It follows that the spin-averaged energy shift due to the contact interaction is 1 S=0 S=1 {∆Econtact + D∆Econtact } D+1 ∣ψ(0)∣2 D(D − 3) = − {dS + dV } m1 m2 D+1

avg ∆Econtact =

=

(Zα)2 2ε 1 1 m2 ln(m2 /µ) − m22 ln(m1 /µ) µ ∣ψ(0)∣2 { − − 2 1 + O()} . m1 m2  3 m21 − m22 (17.135)

(We note that dV doesn’t contribute.) A glance back at Eq. (16.173) shows that this result is identical to the QED hard contribution computed using Coulomb gauge in the previous chapter. We have now completed the calculation of all contributions to the spin-averaged recoil correction at O(Zα)5 coming from the exchange of two soft transverse photons (seagull term) plus the exchange of two hard photons of any sort (contact term). We note that the ultraviolet −1/ from the soft region cancels the infrared 1/ from the hard region, leaving a finite result. We use (16.180) for the logarithmic expectation value. The sum of the two-photon-exchange terms is ∆Eγγ =

Zα 2 1 7 µ3 (Zα)5 {[2 ln ( ) − ln 2 + 2Hn − + πm1 m2 n3 n 3 n 3 −2

m21 ln(m2 /µ) − m22 ln(m1 /µ) δ`≠0 ] δ`=0 − }. 2 2 m1 − m2 2`(` + 1)(` + 1/2)

(17.136)

The grand total for the spin-averaged recoil correction at O(Zα)5 is the sum of the one-transverse exchange term ∆ET of (17.129) and the two-photon exchange

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term ∆Eγγ of (17.136), spin-average ∆Erecoil =

−2

2 µ3 (Zα)5 1 14 2 1 41 {[ ln ( )+ (ln ( ) + Hn − )+ 3 πm1 m2 n 3 Zα 3 n 2n 9

8 m21 ln(m2 /µ) − m22 ln(m1 /µ) 7δ`≠0 ]δ`=0 − ln k0 (n, `) − }. m21 − m22 3 6`(` + 1)(` + 1/2)

This result is in complete agreement with (16.226) found using QED. 17.3.3

Hyperfine Structure of S States (NRQED Approach)

The leading O(Zα)5 corrections to the hfs come exclusively from the exchange of two hard-region photons that are represented in NRQED by the contact term. This leading effect includes a factor of ln(m2 /m1 ), which is large when m2 ≫ m1 . Such logarithms arise from integration over a range spanning m1 to m2 , all of which is in the hard region. The contact-term energy correction has already been obtained in (17.134). The hfs correction is hfs ∆Econtact =−

ln(m1 /m2 ) ∣ψ(0)∣2 2(D − 1)dV → −8∣ψ(0)∣2 (Zα)2 , m1 m2 m21 − m22

(17.137)

in agreement with the result (16.248) obtained using QED and Coulomb gauge in the previous chapter and with known results, found, e.g., in Sec. 10.1.1 of Ref. [27]. 17.4

Discussion

In Chaps. 16 and 17, we have displayed two versions of a bound-state formalism, both based on quantum field theory and both available for the calculation of relativistic, recoil, and other field-theory corrections to the properties of two-body Coulombic bound states. In each case, we have used the formalism to calculate an effective Hamiltonian that could be used to find the energy levels at O(Zα)4 , the spin-averaged recoil corrections at O(Zα)5 , and the leading O(Zα)5 correction to the hyperfine structure of S states. These results were worked out in detail to illustrate the types of considerations needed to derive results for these corrections. We have two main observations to make about the QED and NRQED versions of the Bethe–Salpeter formalism. First, we notice that the two formalisms have many common features. The two approaches are instances of the same basic development and are based on the same solvable nonrelativistic Coulomb problem. The perturbative expansions have the same structure. The reference wave functions consist of a spin state, the same nonrelativistic hydrogen-like wave function, and features that accentuate the two-body nature of the bound states, viz., [(ξ1 E + p0 − p⃗ 2 /(2m1 ) + i) (ξ2 E − p0 − p⃗ 2 /(2m2 ) + i)]−1 for NRQED and [(ξ1 P 0 + p0 − ω1 (⃗ p ) + i)(ξ2 P 0 − p0 − ω2 (⃗ p ) + i)]−1 for QED. The calculations of energy contributions boil down to the analogous manipulations eventually because the underlying physics is the same in both cases; yet,

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the NRQED approach allows us to formulate the results in a more systematic fashion, one of the advantages being that results obtained for high-energy contributions to one process are summarized in an effective operator and can be used for the calculation of other processes without having to go through the calculations again. Second, and strikingly, we notice the huge disparity in the complexity of the formalisms. In both cases we are required to employ the nonrelativistic Schr¨odinger– Coulomb problem as the heart of a two-body quantum field theory formalism, but the fact that all relativistic effects have been stripped off from NRQED to be included as perturbations makes the fit for NRQED direct and natural, as opposed to the exceedingly unnatural nature of the fit for QED for the study of nonrelativistic bound systems (where we are allowed to expand in the coupling parameter Zα). To begin with, we note the difference in the fundamental variables used to construct the theories: two-component Pauli spinor fields for NRQED versus four-component Dirac spinor fields for QED. The NRQED fermions are simply second-quantized versions of the nonrelativistic wave functions, while the QED fermions contain antiparticle components that are only remotely relevant for the physics of nonrelativistic bound states. The core reason for NRQED’s relative simplicity is that the hard, i.e., relativistic, aspects of particle interactions are isolated from the low-energy binding aspects by the encapsulation of the hard effects in matching coefficients and the corresponding effective interactions (spin-orbit, Darwin, Fermi, contact, etc.). The actual calculation of the matching coefficients can be challenging, but it is no more challenging than absolutely necessary, given that the matching coefficients can be obtained from standard gauge-independent scattering calculations involving free external particles. The NRQED Lagrangian contains many more terms than that for QED, but those terms represent physical effects that are in a form that leads to relatively straightforward evaluation instead of being hidden inside of a much more complicated QED bound state expectation value that must be teased apart before actual evaluation. The relative simplicity of the NRQED calculations has been illustrated by the calculations of spin-averaged recoil corrections and hyperfine contributions given above. Finally, there is an aspect that did not come up in the calculations shown: each region of each diagram (for instance, the {P,US,P} region of the single transverse exchange of Fig. 17.5(a)) has not only a leading contribution that was calculated here, but higher order contributions as well. It is relatively straightforward to extract these higher-order contributions from NRQED, but extremely difficult in a QED-based formalism. For all these reasons, NRQED in one form or another has been the basis of most modern calculations of higher-order corrections to the properties of two-particle Coulombic bound states.

17.5

Further Thoughts

Here are some suggestions for further thought.

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(1) Spin and Rotations. One particular entry in Table 17.1 concerns the symmetries of the spin current under parity (+) and time reversal (−). It is a good exercise to derive the behavior of the spin current under parity and under rotations, within full QED, noting the transformation of the Dirac wave function under parity, ψ P (x) = P ψ(xP ), with P = γ 0 (see Sec. 7.2.3). Another good exercise is to derive the spinor representation of the parity transformation, which is γ 0 , to apply it to the wave function, and to rederive this result again. To this end, show that γ⃗ → −⃗ γ under parity, while γ 0 → γ 0 under parity. Consider Sec. 7.2.2 and especially Eq. (7.30). Derive the Lorentz generator for a full rotation about the z axis, which has ω12 = ω3 = θ = 2π. Show that the spinor representation of the rotation by 360○ about the z axis is i S = exp [− ωµν σ µν ] = exp [−iπΣ3 ] . 4

(17.138)

In view of exp [−iπ (

(2)

(3)

(4)

(5)

σ3 0 1 )] = − ( 2×2 0 σ3 0

0

12×2

) = −14×4 ,

(17.139)

justify why spinors change sign upon a rotation by a full 360○ . Now, consider the effect of a parity transformation in the space of spin matrices, according to the formalism introduced in Sec. 7.2.3, and show that the spin does not change sign. Darwin and Spin-Orbit Feynman Rules. The Feynman rule for the Fermi vertex was derived in the text. Work out the Feynman rules for the Darwin and spin-orbit vertices in an analogous way, starting from the NRQED Lagrangian. Determination of Matching Coefficients. Consider the scattering of a low-energy electron in an external electromagnetic field. By comparing the QED and NRQED expressions for the scattering amplitude, express the Fermi, Darwin, and spin-orbit matching coefficients cF , cD , and cS in terms of the Dirac and Pauli form factors F1 and F2 . Verify the results in Eq. (17.24). NRQED and Higher-Order Corrections. Consider self-energy corrections to the energy levels of a single-electron atom like hydrogen. Study the terms in the NRQED Lagrangian given in Eq. (17.7), and investigate how these will affect the corrections to the low-energy part of the self-energy, discussed in the velocity gauge in Sec. 4.6.2. Consider Refs. [163, 170, 173, 315], and verify the consistency of the results obtained in the cited references with those that would otherwise be derived from the Lagrangian in Eq. (17.7), and from the formalism introduced in Chap. 11. Also, compare to the results reported in Ref. [326]. Araki–Sucher Distribution. Consider Eq. (17.121) and clarify the role of the matrix element of ln(∣⃗ p − q⃗ ∣) in comparison to the Araki–Sucher distribution given in Eq. (15.133).

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Chapter 18

Fermionic Determinants and Effective Lagrangians

18.1

Overview

The title of Ref. [607] reads as “Effective Interactions are Effective Interactions”. With this wisdom, which follows the general principle that underlines trivial statements like “a soccer ball is spherical” (in German: “Der Ball ist rund!”), we start the discussion of effective Lagrangians. The rationale behind the statement is as follows: Once you have a Lagrangian, use it in any way imaginable! Put it to the test! Plug into it all kinds of fields that you can imagine, provided the field configuration is within the physical regime covered by the effective Lagrangian, i.e., in the regime for which the Lagrangian is valid. The general principle which underlies the calculation of all “effective” Lagrangians is as follows: In general, quantum processes happen at different length scales. In general, one strives to “integrate out” the processes that happen on short length scales (high-energy scales) and to formulate an effective interaction which describes the low-energy degrees of freedom, i.e., the processes at low energy scales. The quantum perturbations due to the high-energy degrees of freedom are summarized in an “effective Lagrangian” which describes the modification of the long-distance physics, due to the presence of the short-range interactions. We had already followed this principle in Chap. 17, in the derivation of the effective NRQED Lagrangian, after the integration of the short-distance physics. The secret then is that the fields appearing in the effective Lagrangian can be interpreted as classical fields, or quantum fields (field operators), or sums of both. This is because in deriving the effective interaction, one actually ignores the quantum fluctuations of the “low-energy” degrees of freedom and only considers the fluctuations of the “high-energy” degrees of freedom. Within field theory, the effective Lagrangian can generally be calculated in the following steps [130, 608]. (i) One writes the complete Lagrangian of the system, including the free Lagrangian for the light, and the free Lagrangian for the heavy degrees of freedom, and the interaction terms. (ii) The path integral representing the generating function is typically dominated by the zero-field configuration for both light and heavy degrees of freedom. (iii) One identifies the field for which 691

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one would like to calculate the effective interaction as a classical field. (iv) The path integral needs to be taken over the configurations of the “heavy” field, while the “light” degrees of freedom are kept as classical “sources”. This path integral can be calculated by a systematic expansion about the saddle point (in general, the Gaussian saddle point where the fields vanish), or in some cases, an all-order calculation is possible. Taking the logarithm of the path integral over the generating functional, one then obtains the effective Lagrangian for the “light” degrees of freedom. Here, we shall attempt to illustrate this program on the basis of the Heisenberg– Euler effective Lagrangian, where the “light” degrees of freedom are the (massless) photons, and the “heavy” degrees of freedom are the electrons and positrons. We shall do so in a rather ad hoc approach, without recourse to the path integral formalism. In Sec. 18.2, we discuss the derivation of the Heisenberg–Euler effective Lagrangian of quantum electrodynamics [29, 87, 88, 609], as well as convenient representations of the expansions in both perturbative as well as nonperturbative regimes, and in Sec. 18.3, we present applications of the formulas, in the context of a modified speed of light in strong background fields, as well as in the context of bound-state theory (long-range asymptotic tail of the Wichmann–Kroll potential). Other applications include simplified calculation of various cross sections using the effective Lagrangian, which can be obtained by simply using the negative of the Lagrangian as the Hamiltonian density, in order to calculate the S matrix of a scattering process, with essentially arbitrary photon fields in the initial and final states. These latter applications are left as an exercise for the reader. 18.2 18.2.1

Derivation of the Heisenberg–Euler Effective Lagrangian Heisenberg–Euler Lagrangian for Electric Fields

In our route toward the derivation of the effective action [87, 88, 610, 611], we first take one step back and recall the general rationale behind the derivation of the photon emission statistics for a given classical current distribution, as discussed in Sec. 9.6.2. If we wish to calculate the photon emission, then we first need to know the current J µ (x), and then we calculate the creation of the quanta of the electromagnetic field. The S matrix must be analyzed for the case where J µ is classical, but Aµ is quantized. In the evaluation of the QED effective action, the situation will be opposite: The classical electromagnetic field configuration Aµ (x) is given, and we calculate the influence of the quantized fermionic fields onto the equations of motion of the photon field, i.e., the modification of the Maxwell equations. In the former case, we assume the classical currents to be strong, and we generate a few quanta of the radiation field. In the latter case, we inherently assume the electromagnetic fields to be strong, and calculate the influence of the virtual quanta of the fermion field onto the classical electromagnetic fields. For our purposes, we write the QED interaction Lagrangian, introduced in

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Fermionic Determinants and Effective Lagrangians

Sec. 10.1, as follows, µ µ cl cl LI (x) = −e Jin (x)Acl µ (x) = −e ψ in (x)γ ψin (x) Aµ (x) = −e ψ in (x) A (x) ψin (x) . (18.1) µ µ Here, Jin (x) = e ψ in (x)γ ψin (x) is the four-vector current, while ψin (x) is the fermionic field operator. The in fields ensure that we match the free fermion fields with the non-interacting field operators in the infinite past, where the vector potential Acl µ (x) is supposed to be adiabatically switched off. The quantity cl

A (x) = γ µ Acl µ (x) is the Feynman slash of the classical vector potential. The interaction Lagrangian is equal to the negative of the interaction Hamiltonian, LI (x) = −HI (x). All field operators are formulated in the interaction picture. Following the discussion in Sec. 9.6.2, we can write down the time-ordered expression for the S matrix, S[ Acl ] = T exp [−i e ∫ d4 x ψ in (x) Acl (x) ψin (x)] .

(18.2)

The amplitude for emitting no pair (vacuum-persistence amplitude) is given by ∞

(−ie)n 4 4 ∫ d x1 ⋯ ∫ d xn n! n=0

⟨0 in ∣S∣ 0 in⟩ = ∑

cl

cl

× ⟨0 ∣T [ψ in (x1 )A (x1 )ψin (x1 ) ⋯ ψ in (xn )A (xn )ψin (xn )]∣ 0⟩ . (18.3) From here on, we leave out the classical (cl) and interaction picture, in-field (in) labels. In order to proceed further, we have a careful look at the contribution to the vacuum persistence amplitude ⟨0 in ∣S∣ 0 in⟩ from the term with definite n in Eq. (18.3). We have interactions at points (x1 , . . . , xn ). For the Wick theorem (see Sec. 9.6.1) to lead to a nonvanishing contribution, we need an operator ψ(x` ) to be contracted with an operator ψ(xk ). In addition, the field operator ψ(xk ) gets multiplied by some A(xk ). The summation over spinor indices yields an expression ∑α Aµ (xk )γαµk α ψα (xk ). In the fully quantized framework of the Wick theorem with the fermionic field operators, the contraction has to proceed via some ψ(x` ), to yield some expressions like −ie ⟨0 ∣T [A(xk )ψ(xk )ψ(x` )]∣ 0⟩. We now define a matrix whose form is inspired by the necessity to contract the expression ∑α Aµ (xk )γαµk α ψα (xk ), taken at spinor index αk and at the space-time coordinate xk , with a field operator, taken at spinor index α` and at space-time coordinate x` , C(αk , xk , α` , x` ) = −ie ∑ ⟨0 ∣T [Aµ (xk ) γαµk α ψα (xk )ψ α` (x` )]∣ 0⟩ .

(18.4)

α

Now, applying the fermionic Wick theorem, S0 [A] can be written ∞

1 4 4 ∫ d x1 ⋯ ∫ d xn n=0 n!

S0 [A] = ∑

× ∑(−1)n εP P

∑

α1 ,...,αn

C(α1 , x1 , αP1 , xP1 )⋯C(αn , xn , αPn , xPn ) ,

(18.5)

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where P is a permutation of the indices 1, . . . , n, and εP is the sign of the permutation. One might wonder about the origin of the factor (−1)n . The −ie from the expansion of the exponential in Eq. (18.2) is already absorbed in the prefactor of the matrix C. So, one might assume that there should be no factor (−1)n , and also, the origin of the factor εP is unclear. Both factors are of combinatorial origin. The reason for its occurrence lies in the sign of the permutations εP , which depends on n in quite a subtle fashion. Roughly speaking, one may illustrate the occurrence by considering the prefactor of a “standard term” C(α1 , x1 , α2 , x2 ) C(α2 , x2 , α3 , x3 )⋯C(αn , xn , α1 , x1 ), which has a direction of indices 1 → 2 → 3 → ⋅ ⋅ ⋅ → n → 1, and corresponds to a “standard” cyclic perturbation where all numbers advance one place, while the last one flips back to unity; on other words, one has P1 = 2, P2 = 3, P3 = 4, . . . , Pn = 1. The permutation P connected with the “standard term” fulfills εP = (−1)n+1 , i.e., it is odd if n is even, and vice versa. When calculating the prefactor of the “standard term” C(α1 , x1 , α2 , x2 ) C(α2 , x2 , α3 , x3 )⋯C(αn , xn , α1 , x1 ), we have to take into that according to the Wick theorem, this term acquires an additional negative prefactor −1, because it describes a closed fermion loop, or in other words, the “last” fermion operator has to be commuted with an odd number (2n + 1) of other fermion operators, and this always generates an additional factor (−1). The “standard term” must obtain a prefactor (−1), however, εP = (−1)n+1 for the “standard permutation”. This works provided we supply an addition prefactor of (−1)n in the order n. For permutations other than the “standard one”, the additional minus signs generated by the Wick theorem compensate against the additional minus signs due to the permutations, and the prefactor (−1)n is retained. It is very instructive to illustrate the prefactor on the basis of an example calculation, say, for n = 2 and n = 3, by way of example. We shall investigate the contraction leading to a “standard term” which is given by a “cyclic contraction” of the fermionic field operators, with the with the “first” field operator becoming the “last” after the permutation. For n = 2, this means that ⟨0∣T [ψ(x1 )A(x1 )ψ(x1 )ψ(x2 )A(x2 )ψ(x2 )]∣0⟩ ˆ 1 )ψ(x1 )ψ(x2 )]∣ 0⟩ ⟨0 ∣T [A(x ˆ 2 )ψ(x2 )ψ(x1 )]∣ 0⟩ , (18.6) = (−1)2 εP2 ⟨0 ∣T [A(x because (−1)2 εP2 = −1 where P2 is the “standard permutation” for n = 2, P2 = (

12 ), 21

εP2 = −1 .

(18.7)

We reemphasize that the prefactor in Eq. (18.6) can otherwise be understood as a “standard prefactor (−1) resulting from a closed fermion loop”, which, in our example case, is written as the product of (−1)2 and εP2 = −1. For n = 3, the

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prefactor is obtained in the form (−1)n εPn with n = 3, i.e. ˆ 1 )ψ(x1 )ψ(x2 )A(x ˆ 2 )ψ(x2 )ψ(x3 )A(x ˆ 3 )ψ(x3 )]∣0⟩ ⟨0∣T [ψ(x1 )A(x ˆ 1 )ψ(x1 )ψ(x2 )]∣ 0⟩ = (−1)3 εP3 ⟨0 ∣T [A(x ˆ 2 )ψ(x2 )ψ(x3 )]∣ 0⟩ ⟨0 ∣T [A(x ˆ 3 )ψ(x3 )ψ(x1 )]∣ 0⟩ . × ⟨0 ∣T [A(x

(18.8)

Here, we have (−1)3 εP3 = −1 where the “standard permutation” is P3 = (

123 ), 231

εP3 = +1 .

(18.9)

For general n, we define the “standard permutation” as Pn = (

12 3 ⋯n ), 23⋯ n 1

εPn = (−1)n+1 .

(18.10)

The prefactors given in Eqs. (18.6) and (18.8) are manifestations of the general prefactor (−1)n εP in Eq. (18.5). They allow us to identify S0 [A] with a determinant. Indeed, for any matrix, one has the identity det(1 − Γ) = exp [tr ln(1 − Γ)] = 1 − ∑ Γaa

(18.11)

a

+

(−1)2 ∑ (Γaa Γbb − Γab Γba ) + . . . 2 ab

+

(−1)n n!

∑

∑ εP Γα1 αP1 ⋯Γαn αPn + . . . .

α1 ,...,αn P

So, if we associate the integrals over the dxi in Eq. (18.5) with sums over a discretized space ∫ dxk → ∆x ∑` xk,` , in the limit of an infinite number of integration points (Riemannian definition of an integral), then we have S0 [A] = det(1 − Γ) ,

(18.12)

where Γ is a matrix with two spinor and two space-time coordinate indices, Γαk xk α` x` = ⟨αk , xk ∣Γ∣ α` , x` ⟩ = C(αk , xk , α` , x` ) .

(18.13)

We have four spinor indices for every coordinate index. Now we can even write Γ as a matrix, bearing in mind that a matrix acting on an infinite-dimensional Hilbert space in fact is an operator. So, we need to find the operator representation of Γ, bearing in mind that the determinant of a matrix acting in an infinite-dimensional space is a Fredholm determinant. The Hilbert space is spanned by the fundamental eigenkets ∣α, x⟩, which represent the spinor indices and the coordinates. The operators Xµ and Pµ , which are denoted here by capital letters, act on the eigenkets as follows, Xµ ∣α, x⟩ = xµ ∣α, x⟩ ,

⟨α, x ∣Pµ ∣ ϕ⟩ = i

∂ ⟨α, x∣ϕ⟩ , ∂xµ

(18.14)

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where ∣ϕ⟩ is a state in the Hilbert space. The fundamental commutator is [X µ , P ν ] = −i g µν . We normalize the eigenkets for the coordinate and the momentum space in such a way that we can directly associate the integrations with summations if we replace the Kronecker deltas with Dirac delta functions. This is accomplished as follows, ⟨α, x∣α′ , x′ ⟩ = δαα′ δ (4) (x − x′ ) ,

⟨α, p∣α′ , p′ ⟩ = δαα′ δ (4) (p − p′ ) ,

δαα′ ei p ⋅x . (2π)2 ′

⟨α, x∣α′ , p′ ⟩ =

(18.15)

The transformation from coordinate to momentum space (corresponding to a definition of the wave functions) is given by the last of the above equations. The orthogonality relations in Eq. (18.15) hold for an integration measure obtained from the limiting process of replacing sums by integrals, without any additional factors of 1/(2π)4 , in either coordinate or momentum space, ⟨α, x∣α′ , x′ ⟩ = ∑ ∫ d4 p′′ ⟨α, x∣α′′ , p′′ ⟩ ⟨α′′ , p′′ ∣α′ , x′ ⟩ = δαα′ δ (4) (x − x′ ) ,

(18.16a)

⟨α, p∣α′ , p′ ⟩ = ∑ ∫ d4 x′′ ⟨α, p∣α′′ , x′′ ⟩ ⟨α′′ , x′′ ∣α′ , p′ ⟩ = δαα′ δ (4) (p − p′ ) .

(18.16b)

α′′

α′′

With these considerations, we can now easily write down the expression for the vacuum persistence amplitude, having identified integrations as the limiting process of summations over the coordinates, 1 1 ] = det [(P − e A(X) − m + i) ] S0 [A] = det [1 − e A(X) P − m + i P − m + i 1 = exp [Tr ln ((P − e A(X) − m + i) )] . (18.17) P − m + i Here, the classical field A(X) is a function of the space-time coordinate X, which constitutes an operator. It could appear that the expression for S0 [A] from Eq. (18.17) is not gauge invariant. Indeed, under a gauge transformation, one transforms Aµ (X) → Aµ (X)− ∂µ Λ(X) (see Sec. 8.2.1). The operator P − eA(X) − m + i transforms as P − eA(X) − m + i → eie Λ(X) (P − e A(X) − m + i) e−ie Λ(X) .

(18.18)

A gauge transformation, however, is not complete unless one also transforms the wave functions, in the sense that ′ ′ 1 1 δαα′ e−i p ⋅x → δαα′ e−i p ⋅x eie Λ(x) . (18.19) ⟨α, x∣α′ , p′ ⟩ = (2π)2 (2π)2 The eigenkets of the position operator thus acquire a phase under the gauge transformation, ∣α, x⟩ → ∣α, x⟩ e−ie Λ(x) .

(18.20)

However, one can argue that the operators in (18.17) do not know anything about the gauge transformation of the wave functions, and therefore, it remains to show

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that Eq. (18.17) is gauge invariant without the transformation of the wave functions. In the sense of a matrix transformation, we observe that the transformation ∣x⟩ → e−ie Λ(x) ∣x⟩ is unitary, because it leaves the scalar products of the fundamental eigenkets invariant, ⟨α, x ∣eie (Λ(x )−Λ(x)) ∣ α′ , x′ ⟩ = eie (Λ(x )−Λ(x)) ⟨α, x∣α′ , x′ ⟩ = ⟨α, x∣α′ , x′ ⟩ . ′

′

(18.21)

Any unitary transformation leaves the determinant of a matrix invariant, and the same applies to the Fredholm determinant (determinant over an infinite-dimensional Hilbert space) from Eq. (18.17). We conclude that the expression (18.17) is gauge invariant without any additional gauge transformation of the wave functions or eigenkets. In order to better understand the role of the unitary character of the gauge transformation, we write the logarithm of the effective action as follows, ln S0 [A] = ln det (P − e A(X) − m + i) − ln det (P − m + i) .

(18.22)

The gauge transformation then acts only on the first term, and it is unitary in the Hilbert space, which leaves the determinant unaltered. In order to evaluate S0 [A], we are facing the problem of getting rid of the spinor structures, which make it impossible to separate the numerator P − e A(X) − m + i from the denominator P − m + i. We should at least try to change the denominator into a scalar quantity, so that it becomes possible to separate numerator and denominator by an integral representation, which in turns makes it possible to evaluate the traces explicitly, e.g., in coordinate space. We are doing this with a trick, which entails the charge conjugation operation. It turns out to be advantageous to use charge conjugation invariance of the vacuum persistence amplitude (the vacuum should be as likely to return to its state no matter what the sign of the interacting charges is), and to add the charge conjugated vacuum persistence amplitude to its original expression. The physical reason is that the vacuum persistence amplitude cannot depend on the sign of the charge of the electron, or in other words, in a fictitious world in which the electron charge changes sign, just as many electron-positron pairs would be produced as with the physical electron charge. We recall the defining property of the charge conjugation (7.49), which reads as Cγ µ C −1 = −(γ µ )T . So, the charge conjugation invariance implies that ln S0 [A] = Tr ln [(P − eA − m + i)

1 ] P − m + i

= Tr ln [C (P − eA − m + i) C −1

1 ] = ln S0 [A]C . (18.23) C(P − m + i)C −1

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Let us transform the charge-conjugated expression,

ln S0 [A]C = Tr ln [C (P − eA − m + i) C −1

1 ] C(P − m + i)C −1

1 ] − m + i) ⎡ T⎤ ⎢ ⎥ 1 (P − eA + m − i)) ⎥⎥ = Tr ln ⎢⎢( ⎢ (P + m − i) ⎥ ⎣ ⎦ 1 (P − eA + m − i)] . = Tr ln [ (P + m − i) = Tr ln [(−P T + eAT − m + i)

(−P T

(18.24)

Here, we have used the fact that the matrix exponential fulfills the condition exp(AT ) = [exp(A)]T , and that the trace of a matrix is equal to the trace of its transpose. The matrix logarithm is the inverse operation of the matrix exponential. Thus, the logarithm of the transpose of a matrix is equal to the transpose of the logarithm of the same matrix. We now add the vacuum persistence amplitude and its charge conjugate,

1 (ln S0 [A] + ln S0 [A]C ) 2 1 1 1 (P − eA + m − i)] = Tr ln [(P − eA − m + i) 2 P − m + i P + m − i 1 1 (P − eA + m − i) ] . (18.25) = Tr ln [(P − eA − m + i) 2 2 P − m2 + i

ln S0 [A] =

There might be a question regarding the transformation. Namely, in general, the matrix exponential exp(A) exp(B) is only equal to exp(A + B) if A and B commute. Hence, we cannot claim that ln(A B) is equal to ln(A) + ln(B). Setting A = S0 [A] and B = S0 [A]C , we might encounter a problem, in the transformations leading to Eq. (18.25). However, the product of Fredholm determinants fulfills det(A B) = det(A) det(B). Taking the logarithm and using the fact that ln(det A) = Tr ln(A), one can then show Eq. (18.25). Otherwise, one has to be extremely careful with traces in infinite-dimensional vector space. E.g., the relation ∗ Tr(A B) = {Tr(B + A+ )} holds, but we cannot simply assert that Tr(AB) = Tr(B A). Consider, e.g., in one-dimensional quantum mechanics, with a coordinate q and a momentum operator p, the trace Tr(qp − pq) = Tr(i 1), where 1 is the unit operator. We now insert a γ 0 matrix, use the identities ln(det A) = Tr ln(A), as well as det(A+ ) = (det A)∗ and det(A B) = det(A) det(B), and γ 0 γ µ γ 0 = (γ µ )+ ,

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repeatedly. The net result is to transform ln S0 [A] into the form 1 1 γ 0 (P − eA + m − i) γ 0 ] ln S0 [A] = Tr ln [γ 0 (P − eA − m + i) γ 0 2 2 P − m2 + i + 1 1 + + (P − eA + m + i) ) = ln det ((P − eA − m − i) ( 2 ) 2 P − m2 − i

∗ 1 1 ln det ((P − eA + m + i) (P − eA − m − i) 2 ) 2 P − m2 − i 1 1 2 = Tr ln [((P − eA) − m2 + i) 2 ]. (18.26) 2 P − m2 + i We have not completely eliminated the spinor structure, but in view of the identity e 2 (P − eA) = (P − e A)2 + σµν F µν , F µν = ∂ µ F ν − ∂ ν F µ , (18.27) 2 it has been significantly simplified. Here, the spin matrices are given as σµν = i [γµ , γν ], according to Eq. (7.30). 2 Up to now, we have only considered the vacuum persistence amplitude S0 [A], which is a functional of the electromagnetic field configuration A = A(X), where X = (X µ ) is an operator, but the stated aim of the current derivation is to calculate the effective action of the electromagnetic field. Let us go back to the definition of the vacuum persistence amplitude as the time-ordered product,

=

S0 [A] = T ⟨0 ∣exp [i ∫ d4 x LI (x)]∣ 0⟩ .

(18.28)

The time evolution operator is calculated in the interaction picture. We assume that the electromagnetic field is adequately described as a classical field, whereas the quantum fluctuations of the fermionic field are described by the fermionic determinant. The vacuum persistence amplitude finds a natural interpretation as an exponentiated integral over the effective action of the classical electromagnetic field which generates the interaction, S0 [A] = exp [i ∫ d4 x Leff (x)] .

(18.29)

In other words, having kept the field configuration as a variable entity in our derivation, we are now in the position to interpret the result of our calculation as an effective Lagrangian for the remaining (classical) electromagnetic fields. We can now use the integral representation of the logarithm, ∞ ds a [eis(a+i) − eis(b+i) ] , (18.30) ln ( ) = − ∫ b s 0 together with the integral (18.30) and the matching (18.29), to write the effective action as ∞ ds 2 2 i 4 4 P −e A)2 e−is(m −i) tr (⟨x ∣eis ( ∣ x⟩ − ⟨x ∣eis P ∣ x⟩) ∫ d x Leff (x) = ∫ d x ∫ 2 s 0 ∞ ds 2 i 4 = ∫ d x∫ e−is(m −i) [M (A) − M (0)] . (18.31) 2 s 0

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Here, tr is now reduced to the trace over the (4 × 4)-spinor space, but not anymore to the trace over the coordinate space; we have replaced Tr → tr ∫ d4 x. For slowly varying fields, the integral over d4 x can be dropped on both sides of Eq. (18.31). The central object of the next considerations therefore is the matrix element e P −e A)2 M (A) = tr ⟨x ∣eis ( ∣ x⟩ = tr ⟨x ∣exp (is (P − e A)2 + σµν F µν )∣ x⟩ , 2

(18.32)

which leaves only the expression containing σµν F µν within spinor algebra. We evaluate it for a spatially constant electric field. A3 (x) = −E t ,

F 03 = −F 30 = −E ,

(18.33)

while all other components of the field-strength tensor vanish, and the electric field ⃗ = (0, 0, E). Furthermore, vector points in the positive z-direction, E σ03 =

i [γ0 , γ3 ] = −iγ 0 γ 3 = −iα3 , 2

(18.34)

where αi = γ 0 γ i [see Eqs. (7.14) and (7.18)]. The spinor trace thus leads to e σµν F µν )] = tr [exp (−s e α3 E)] 2 14×4 1 = tr [(14×4 − s e α3 E + (seE)2 − (seE)3 α3 + ⋯)] 2! 3! ∞ 1 (seE)2n = 4 (1 + (seE)2 + ⋯) = 4 ∑ = 4 cosh(seE) . 2! n=0 (2n)!

tr [exp (−s

(18.35)

The result 4 cosh(seE) commutes with (P − eA)2 , and therefore M (A) = 4 cosh(seE) ⟨x ∣exp (is (P − e A)2 )∣ x⟩ .

(18.36)

We can write the scalar quantity (P − eA)2 as (P − eA)2 = (P 0 )2 − (P 1 )2 − (P 2 )2 − (P 3 + eEt)2 = e−iP

0

P 3 /(eE)

[(P 0 )2 − (P 1 )2 − (P 2 )2 − (eEt)2 ] eiP

0

P 3 /(eE)

,

(18.37)

where we have used the notation P = (P 0 , P x ≡ P 1 , P y ≡ P 2 , P z ≡ P 3 ). The identity (18.37) follows from the Campbell–Baker–Hausdorff formula (9.149), which we use in the form 1 exp(A) B exp(−A) = B + [A, B] + [A, [A, B]] + ⋯ , (18.38) 2! so that, with a as a = P 3 /(eE), e−iaP t2 eiaP = ea∂t t2 e−a∂t 0

0

= t2 + [a ∂t , t2 ] +

1 [a∂t , [a∂t , t2 ]] = t2 + 2 a t + a2 = (t + a)2 . (18.39) 2

Now, U = exp[−iP 0 P 3 /(eE)] mediates a unitary transformation of Hilbert space of momentum eigenstates just like the previously discussed unitarity transformation

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of the coordinate-space eigenkets. Setting p1 = (ω1 , p⃗1 ), p2 = (ω2 , p⃗2 ) and p⃗⊥ = (px , py , 0), one can insert a basis of momentum eigenkets as follows, M (A) = ⟨x ∣exp {is [(P 0 )2 − (P 1 )2 − (P 2 )2 − (P 3 + eEt)2 ]]∣ x⟩ 4 cosh(seE) = ⟨x ∣exp [is e−iP

0

P 3 /(eE)

[(P 0 )2 − (P 1 )2 − (P 2 )2 − (eEt)2 ] eiP

=∫ d4 p1∫ d4 p2 ⟨x∣p1 ⟩ ⟨p1 ∣eisU [(P

) −(P ) −(P ) −(eEt) ]U

0 2

1 2

2 2

2

0

−1

P 3 /(eE)

]∣ x⟩

∣ p2 ⟩ ⟨p2 ∣x⟩. (18.40)

We have carefully distinguished momentum operators (written in capital letters) from momentum eigenvalues (written in lowercase letters) and used the fact that ⟨x ∣p1 ⟩ ⟨p2 ∣ x⟩ =

1 exp[i(p2 − p1 ) ⋅ x] , (2π)4

(18.41)

which is compatible with Eq. (18.15). Then, with U = exp[−iP 0 P 3 /(eE)], the vector structure in the matrix element in the integrand of Eq. (18.40) can be disentangled as follows, M = ⟨p1 ∣eisU [(P

) −(P 1 )2 −(P 2 )2 −(eEt)2 ]U −1

0 2

∣ p2 ⟩

= ⟨p1 ∣exp [i s exp[−iP 0 pz2 /(eE)] [(P 0 )2 − (px2 )2 − (py2 )2 − (eEt)2 ] × exp[iP 0 pz2 /(eE)]]∣ p2 ⟩ = e−isp⃗⊥ δ (3) (⃗ p1 − p⃗2 ) ⟨ω1 ∣exp [i s exp[−iP 0 pz2 /(eE)] [(P 0 )2 − (eEt)2 ] 2

× exp[iP 0 pz2 /(eE)]]∣ ω2 ⟩ .

(18.42)

Here, p⃗⊥ = px2 ˆex + py2 ˆey . The Dirac-δ function δ (3) (⃗ p1 − p⃗2 ) ensures that both momenta are equal and can be written as p⃗ after one of the integrations has been performed. Furthermore, we have identified the matrix element as being evaluated between eigenstates of the four-momentum ∣pi ⟩ = ∣ωi , p⃗i ⟩, with i = 1, 2. Then, ⟨ω1 ∣ω2 ⟩ = δ(ω1 − ω2 ) and ⟨⃗ p1 ∣⃗ p2 ⟩ = δ (3) (⃗ p1 − p⃗2 ). After all the algebra corresponding to the momentum operators has been performed, we can finally use the representation P 0 = −i∂/∂t and write M as follows, M = e−isp⃗⊥ δ (3) (⃗ p1 − p⃗2 ) ⟨ω1 ∣exp [i s exp [ 2

× exp [−

pz2 ∂ ] [(P 0 )2 − (eEt)2 ] eE ∂t

pz2 ∂ ] ]∣ ω2 ⟩ eE ∂t

= e−isp⃗⊥ δ (3) (⃗ p1 − p⃗2 ) ⟨ω1 ∣exp [i s [(P 0 )2 − (pz2 + eEt)2 ] ]∣ ω2 ⟩ . 2

(18.43)

Inserting this result into Eq. (18.40), and rewriting the remaining integration

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variable in momentum space as p⃗ ≡ p⃗1 , one obtains M (A) 1 3 = ∫ d p ∫ dω1 ∫ dω2 exp [i(ω2 − ω1 ) t] 4 cosh(seE) (2π)4 × e−isp⃗⊥ ⟨ω1 ∣eis[(P 2

=

) −(pz +eEt)2 ]

0 2

∣ ω2 ⟩ ,

pz 1 3 )] ∫ d p ∫ dω1 ∫ dω2 exp [i(ω2 − ω1 ) (t − 4 (2π) eE × e−isp⃗⊥ ⟨ω1 ∣eis[(P 2

) −(eEt)2 ]

0 2

∣ ω2 ⟩ ,

(18.44)

where the time variable has been shifted by the replacement t → t − pz /(eE), which is permissible because we assume a constant electric field (in time). The d3 p integral can easily be carried out, splitting the integration into a transverse integral over d2 p⊥ = dpx dpy and an integral over dpz . 2 2 pz d2 p⊥ p3 dp3 d3 p exp [i(ω1 − ω2 ) ] e−isp⃗⊥ = ∫ exp [i(ω1 − ω2 ) ] e−isp⃗⊥ ∫ 4 3 (2π) eE (2π) 2π eE ∞ dp p 2 eE ⊥ ⊥ =∫ eE δ(ω1 − ω2 ) e−isp⊥ = 2 δ(ω1 − ω2 ) . (18.45) (2π)2 8π i s 0

∫

In view of (P 0 )2 = −∂t2 , we have M (A) = cosh(seE)

eE is[−∂t2 −(eEt)2 ] ∣ ω⟩ . ∫ dω ⟨ω ∣e 2 2π i s

(18.46)

We can now identify is[−∂t −(eEt) ] ∣ ω⟩ = Tr (eisHosc ) = ∫ dt ⟨t ∣eis[−∂t −(eEt) ] ∣ t⟩ , ∫ dω ⟨ω ∣e 2

2

2

2

(18.47)

simply as the trace of a quantum-mechanical operator, which can be taken in the frequency or the time representation. The fact that there are no additional prefactors to consider, can be traced to our normalization of the four-momentum eigenstates given in Eqs. (18.15) and (18.41). The operator Hosc is that of a harmonic oscillator whose “coordinate variable” is the time Hosc = −

1 1 ∂2 + m0 ω02 ξ 2 , 2m0 ∂ξ 2 2

ξ ≡ t,

(18.48)

and which has an imaginary eigenfrequency, m0 = 1/2 and ω0 = 2 i eE. Now, the trace can be taken either in the frequency, or the time representation, or in the representation of energy eigenfunctions, which as we know display a well-known discrete spectrum of eigenfrequencies (n + 1/2)ω0 . The trace therefore is equal to ∞

∞

n=0

n=0

Tr (eisHosc ) = ∑ eis(n+1/2)ω0 = ∑ e−2seE(n+1/2) =

1 . 2 sinh(seE)

(18.49)

Finally, we obtain a simple and compact result for M (A), as defined in Eq. (18.32), M (A) = coth(seE)

eE , 4π 2 i s

M (0) =

1 4π 2 i s2

,

(18.50)

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where the latter result is obtained in the limit E → 0. Inserting this compact result for M (A) into Eq. (18.31), one obtains ∞ ds 2 i 4 e−is(m −i) [M (A) − M (0)] ∫ d x∫ 2 s 0 ∞ ds 2 1 1 4 =∫ d x 2 ∫ e−is(m −i) (eE coth(seE) − ) . 8π s2 s 0

4 ∫ d x Leff (x) =

(18.51)

We can now drop the d4 x integration and obtain ∞ ds 2 1 1 e−is(m −i) (eE coth(seE) − ) . (18.52) ∫ 8π 2 0 s2 s The leading term in the expansion of the hyperbolic cotangent in the integrand for s → 0 leads to a 1/s3 divergence; this divergence is exactly compensated by the term M (0). However, there is a second divergence, in view of the expansion 1 1 e E coth(se E) = + s(e E)2 + O(s2 ) , (18.53) s 3 for small proper time s. We recall that the unperturbed Maxwell Lagrangian reads ⃗ = 0⃗ reduces to the expression E ⃗ 2 /2. The loga⃗2 − B ⃗ 2 ), which for B as L[0] = 21 (E rithmic divergence generated by the 1/s term, for small proper time s, multiplies the ⃗ 2 . According to the renormalization program discussed square of the electric field E in Sec. 10.6, it necessitates a Z3 renormalization. Let us study this in more detail. The Maxwell Lagrangian L[0] , for an electric field, plus the effective Lagrangian (18.52), produces the following term, propor⃗2, tional to E

Leff =

∞ ds 2 e2 1 ⃗2 . e−is(m −i) ) E L[0] + Leff ∼ ( + 2 ∫ 2 8π 3s 

(18.54)

Here,  is a small proper time cutoff. The renormalization program dictates that a counter term has to be added to the original Maxwell Lagrangian, in terms of a Z3 renormalization, so that the complete theory, formulated in terms of the renormalized, finite, physical fields, produces finite Green functions and in our case, a finite effective Lagrangian. This is fulfilled if we set, on the one-loop level, ∞ ds 2 e2 e−is(m −i) . (18.55) ∫ 2 8π 3s  The counter term is perturbatively small, of order e2 ∝ α, but logarithmically divergent for small . The unrenormalized and renormalized electric fields are related by 2 ∞ ds √ 2 ⃗ 0 = Z3 E ⃗ ≈ (1 − 1 e ∫ ⃗. E e−is(m −i) ) E (18.56) 2 2 8π 3s 

(Z3 − 1)

[1]

=−

⃗ in Eq. (18.52) as the unrenormalized, bare electric field One may now interpret E ⃗ E0 , use Eq. (18.56), and then write the result in terms of the physical electric field strength. If this is done, to one-loop order, i.e., cutting off the expansion at order

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e2 ∝ α, then it has the same effect as a simple subtraction of the logarithmically divergent term under the integral sign in Eq. (18.52). The result of this operation is Leff =

∞ ds e2 E 2 s 1 1 −is(m2 −i) e (eE coth(seE) − − ). ∫ 8π 2 0 s2 3 s

(18.57)

The renormalization counter term given in Eq. (18.55) restores the Maxwell Lagrangian in the low-energy (weak-field) limit without additional, finite renormalizations; the Lagrangian L[0] + Leff , with Leff given in Eq. (18.57), has the leading terms 12 E 2 + O(E 4 ), with the prefactor of the quadratic term being exactly 12 . This corresponds to the on-mass-shell renormalization; the renormalization-group invariance of the QED effective action is analyzed in greater detail in Sec. 19.4.1. The QED interaction Lagrangian does not contain derivative terms; therefore, the QED interaction Hamiltonian is equal to the negative of the QED interaction Lagrangian. In Sec. 3.3.1, we have learned that a decay width can be described as a complex resonance energy, according to the identification E = Re E − 2i Γ, where Γ is the width. The QED effective action as written in Eq. (18.57) has a manifestly nonvanishing imaginary part, which can thus be interpreted according to Leff = Re [Leff ] +

i w, 2

(18.58)

where w = w(x) is the pair-production amplitude per space-time volume, i.e., the probability of producing an electron-positron pair in a very intense, static, electric field. The vacuum persistence probability, which is the absolute square of the vacuum persistence amplitude, then decays according to a time evolution proportional to exp[− ∫V d3 r w(t, r⃗)], where we have written w(x) = w(t, r⃗), and V is the reference volume. We can alternatively write the imaginary part of Leff as the real part of the following expression, w=−

∞ ds 2 1 e2 E 2 s 1 Re [i e−is(m −i) ] (eE coth(seE) − − ). ∫ 2 2 4π s 3 s 0

(18.59)

Both factors in the integrand change sign under the transformation s → −s, in the limit of  → 0. Therefore, the integrand is even in s, and we may write w=−

∞ ds 1 e2 E 2 s 1 −ism2 −∣s∣ Re i e (eE coth(seE) − − ), ∫ 8π 2 3 s −∞ s2

(18.60)

where of course, ∣s∣ in the infinitesimal suppression factor in the exponential denotes the modulus of s. From now on, we assume eE > 0. Because of the structure of the argument of the exponential, the integration contour for s must be closed in the lower complex half plane; the singularities are at s = −i s′ , with s′ = nπ/(eE), with positive integer n > 0. The singularity at n = 0 is canceled by the counterterm −1/s. Near the singularities, the hyperbolic cotangent can be expanded as follows, coth(seE) =

1 1 2 + (seE + inπ) + O [(seE + inπ) ] , seE + inπ 3

(18.61)

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so that Res coth(seE) =

s=− inπ eE

1 . eE

(18.62)

Collecting, carefully, all the prefactors, one finally obtains the expression w= −

1 ∞ nπ eE 2 ) exp (−m2 ) i (−2πi) (−i ∑ 8π 2 n=1 nπ eE

∞

α E2 nπm2 exp (− ), 2 2 eE n=1 n π

= ∑

(18.63)

where we have used e2 = 4πα. This is the nonperturbative (in the QED coupling e) result for the pair production probability in a vacuum dressed by an intensive electric field of magnitude E. It is known as the Schwinger1 mechanism [89]. 18.2.2

Heisenberg–Euler Lagrangian for General Fields

The result (18.57) needs to be generalized to cases where both the electric as well as the magnetic fields are nonvanishing [87, 88]. It is intuitively clear that the general result must be expressible in terms of Lorentz invariants of the fields. From Chap. 8 of Ref. [81], we recall the field-strength tensor F µν

x y z ⎛ 0 −E −E −E ⎞ x z y ⎜E 0 −cB cB ⎟ ⎟, =⎜ ⎜ E y cB z 0 −cB x ⎟ ⎜ ⎟ ⎝ E z −cB y cB x 0 ⎠

(18.64)

and its dual 1 F̃µν = µνρδ Fρδ . 2

(18.65)

It reads, in components, F̃µν

x y z ⎛ 0 −cB −cB −cB ⎞ z y ⎟ ⎜ cB x 0 E −E ⎟ =⎜ . ⎜ cB y −E z 0 Ex ⎟ ⎜ ⎟ ⎝ cB z E y −E x 0 ⎠

(18.66)

The general result for the QED effective action can be expressed in term of the eigenvalues (“secular invariants”) of the field strength tensor. As a (4 × 4) matrix, F µν has four eigenvalues, expressible as ±a, ±b. They are expressed as follows, √√ √√ a= F 2 + G2 + F , b= F 2 + G2 − F , (18.67) in terms of the Lorentz invariants F and G, 1 1 ⃗2 ⃗2 1 F = Fµν F µν = (B − E ) = (a2 − b2 ) , 4 2 2 1 ⃗ ⋅B ⃗ = ±ab . G = Fµν F̃µν = −E 4 1 Julian

Seymour Schwinger (1918–1994).

(18.68a) (18.68b)

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⃗ = ⃗0, we have b = 0 and a = ∣B∣, ⃗ whereas for B ⃗ = 0⃗ we have a = 0 and b = ∣E∣. ⃗ For E ⃗ ⃗ ⃗ ⃗ More generally, we have b = ∣E∣, and a = ∣B∣, if and only if B is (anti-)parallel to E. ⃗ ∥ −B. ⃗ For G < 0, it is For G > 0, it is possible to choose a Lorentz frame with E ⃗ ⃗ possible to choose a Lorentz frame with E ∥ B. For a plane-wave, monochromatic, traveling laser wave, we have G = 0, and therefore b = 0. As we shall soon see, this implies that a plane, monochromatic laser wave cannot produce electron-positron ⃗ ⋅ B∣ ⃗ > 0, pairs. In any case, because a and b are positive definite, we have a b = ∣E which clarifies the sign ambiguity in (18.68b). The Maxwell Lagrangian is given by 1 1 ⃗2 ⃗2 1 − B ) = (b2 − a2 ) . (18.69) L[0] = −F = − Fµν F µν = (E 4 2 2 The QED effective action Leff is conveniently expressed as follows, Leff =

∞ ds e2 a2 − b2 1 −is (m2 −i) e [ab coth(eas) cot(ebs) − − ] (18.70) ∫ 2 8π s 3 (e s)2 0

i∞+η ds 2 e2 a2 − b2 1 lim + ∫ e−(m −i)s [ab coth(eas) cot(ebs) − − ], 2 8π ,η→0 η s 3 (es)2 where in the second step, the rotation of the contour gives rise to an overall minus sign in front of the integral. An expansion in powers of α gives rise to the following leading terms,

=−

2 α2 ⃗ 2 ⃗ 2 2 14 α2 ⃗ ⃗ 2 (E − B ) + (E ⋅ B) , (18.71) 45m4 45m4 where L[0] has been given in Eq. (18.69). We should note that an expansion of Eq. (18.70) in powers of α gives rise to a divergent series [612]. This is an expansion in interactions with external fields, which can be visualized by Feynman diagrams. Borel summability of the divergent series, in the sense of Eq. (3.4), has been studied and shown, both within the ordinary [612] as well as within the distributional sense for the case of an electric field [126,613]. The divergent character of the perturbative expansions does not limit their predictive power [614]. For a number of applications, it is useful to express the nonperturbative expression (18.70) in terms of special functions [615, 616]. In particular, the real and imaginary parts can be expressed as L = L[0] + L[1] ,

Re Leff = − an =

Im Leff =

∞ e2 [an + dn ] , ab ∑ 4π 3 n=1

) coth ( nπb a n

dn = − 2

L[1] =

[Ci (

) coth ( nπa b 2n ∞

(18.72a)

nπm2 nπm2 nπm2 nπm2 ) cos ( ) + si ( ) sin ( )] , ea ea ea ea

[enπm

2

/(eb)

Ei (−

2 nπm2 nπm2 ) + e−nπm /(eb) Ei ( )] , eb eb

e ab 1 nπa nπm2 coth ( ) exp (− ). ∑ 8π 2 n=1 n b eb

(18.72b)

On the occasion, we recall that m in these formulas denotes the electron mass, not an extraneous summation index. The “asymmetry” in these formulas regarding

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cosine and sine integrals is not arbitrary; the generally accepted definitions for these integrals [159] are indeed asymmetric: Ci(z) = − ∫

∞ z

dt

cos(t) t

si(z) = − ∫

∞ z

dt

sin(t) π = Si(z) − , t 2

(18.73)

whereas Si(z) = ∫0 dt t−1 sin(t) and we assume that z > 0. The exponential integral is given as a principal-value expression, z

Ei(u) = −(P.V.) ∫

∞ −u

e−t dt , t

u ∈ R.

(18.74)

Under an appropriate deformation of the contour, a unified representation for both the real and the imaginary parts is obtained [617], Leff = lim+ − →0

bn = −

∞ e2 ab ∑ (bn + cn ) , 3 4π n=1

) coth ( nπb a 2n

+ exp (i cn =

[exp (−i

nπm2 nπm2 ) Γ (0, −i ) ea ea

nπm2 nπm2 ) Γ (0, i )] , ea ea

) coth ( nπa b 2n + exp (−

[exp (

(18.75a)

(18.75b)

nπm2 nπm2 ) Γ (0, ) eb eb

nπm2 nπm2 ) Γ (0, − + i)] . eb eb

(18.75c)

Here, we make extensive use of the incomplete Gamma function defined as [see Eq. (6.5.3) of [159]] Γ(a, z) = ∫

∞ z

dt e−t ta−1 .

(18.76)

For a = 0, the quantity Γ(0, z) as a function of z has a branch cut along the negative real z axis, and we assume that lim→0+ Im Γ(0, −x + i ) = −π, for x > 0. The effective action has branch cuts along the positive and negative b-axis as well as along the positive and negative imaginary a-axis. For more details, the reader is referred to [618]. The incomplete Gamma function Γ(0, z) is defined in the entire complex plane with a cut along the negative real axis [159, 618]. 18.3 18.3.1

Application of the Heisenberg–Euler Lagrangian Modification of the Speed of Light in Background Fields

Now that we have evaluated the QED effective action, and especially the phenomenologically important perturbative terms from Eq. (18.71), the question is how to employ the concept in experimentally relevant calculations. From Ref. [81] and other textbooks on classical electrodynamics, we know that the homogeneous

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Maxwell equations are an inherent property of the field strength tensor and thus do not change if we take into account the quantum corrections, ∂ ⃗ ⃗ = 0. ⃗ ⋅B B = 0, ∇ (18.77) ∂t In curved space-times, the homogeneous Maxwell equations can be traced to socalled Bianchi identities fulfilled by the field-strength tensor [see Eq. (8.262) of Ref. [81]]. However, the inhomogeneous Maxwell equations, which couple the sources to the fields, are derived from an action principle. This is true in flat as well as curved space-times [see Eq. (8.258) of Ref. [81]]. The inhomogeneous Maxwell equations thus receive a quantum correction. For convenience, we recall the Maxwell Lagrangian, the one-loop correction, and the modified Lagrangian which couples to the sources, as follows, ⃗+ ⃗ ×E ∇

L = L[0] + L[1] − J µ Aµ ,

L[0] =

1 2

⃗2 − B ⃗ 2) , (E

2 α2 ⃗ 2 ⃗ 2 2 14 α2 ⃗ ⃗ 2 (E − B ) + (E ⋅ B) , 45m4 45m4 ∂ (L[0] + L(1) ) ∂L[0] , F µν = . = ∂(∂µ Aν ) ∂(∂µ Aν )

(18.78a)

L[1] =

(18.78b)

F µν

(18.78c)

The variational equations become [0] [1] ∂ ∂ ∂ (L + L ) µν F = = Jν . ∂xµ ∂xµ ∂(∂µ Aν )

(18.79)

⃗ (in place of the The tensor F µν is composed of the dielectric displacement field D ⃗ ⃗ electric field E) and of the magnetic field H (in place of the magnetic induction ⃗ These are the effective fields in the presence of the background field. The field B). ⃗ and B ⃗ fields, and we obtain field-strength tensor Fµν = ∂µ Aν − ∂ν Aµ contains the E ⃗ ⃗ by differentiating the dielectric displacement D and the magnetic field strength H the effective Lagrangian. In Cartesian components, we have, for spatial indices i, j = 1, 2, 3, ∂ j (L[0] + L[1] ) = εij (18.80a) r E , ∂E i ∂ ij j (L[0] + L[1] ) = (µ−1 Hi = − (18.80b) r ) B . ∂B i We recall that  = µ0 = 1 in our units. The radiatively corrected Maxwell equations assume the form ⃗ = ρ, ⃗− ∂D ⃗ = J⃗ . ⃗ ⋅D ⃗ ×H ∇ ∇ (18.81) ∂t Di =

j We have formulated the dielectric displacement Di = εij r E and the magnetic field i −1 ij j ⃗ and the magnetic induction B, ⃗ and H = (µr ) B in terms of the electric field E ij the vacuum-induced relative permittivity tensor εr and vacuum-induced permeij ability tensor (µ−1 r ) . The Cartesian components of the (relative) permittivity and

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permeability tensors are given as follows, 4α2 ⃗2 − B ⃗ 2 ) δ ij + 7 B i B j ] , [2(E 45 m4 4α2 ij ij ⃗2 − B ⃗ 2 ) δ ij − 7 E i E j ] , (µ−1 [2(E r ) =δ + 45 m4 4α2 ij ⃗2 − B ⃗ 2 ) δ ij − 7 E i E j ] . [2(E µij r =δ − 45 m4 ij εij r =δ +

(18.82a) (18.82b) (18.82c)

In the last step, we have calculated the matrix inverse to the leading order in α2 . Terms of order α4 and higher are neglected in Eq. (18.82). It is very instructive to derive the effective wave equations for an electromagnetic wave propagating under the presence of a large magnetic background field. To this end, we replace, in the field equations, ⃗ → δE ⃗, E

⃗ → δD ⃗, D

⃗→B ⃗ + δB ⃗, B

⃗ →H ⃗ + δH ⃗, H

(18.83)

where the fields denoted by capital letters are the background fields, and the fields denoted by the lowercase letters denote a small perturbation. From Eqs. (18.80a) and (18.82a), we obtain the dielectric displacement, δDi = (1 −

⃗2 28 α2 B i B j 8 α2 B i ) δE + δE j , 45 m4 45 m4

(18.84)

whereas Eqs. (18.80b) and (18.82b) imply that there is an additional term when we ⃗ expand to first order in the δ H, ⃗ + δ B) ⃗ 2 ij 8 α 2 (B δ ) (B j + δB j ) (18.85) 45 m4 ⃗2 ⃗2 8 α2 B 8 α2 B 16 α2 B i B j i i ≈ (1 − ) B + {(1 − ) δB − δB j } . 45 m4 45 m4 45 m4

H i + δH i = (δ ij −

The term in curly brackets in Eq. (18.85) is identified as δH i . We thus have the following relative, effective permittivity and permeability tensors (which we denote as εij and µij ). These are relevant to the perturbative fields, δDi = εij δE j ,

ij

δH i = (µ−1 )

δB j ,

⃗2 28 α2 B i B j 8 α2 B )+ , 4 45 m 45 m4 ⃗2 8 α2 B 16 α2 B i B j ij (µ−1 ) = δ ij (1 − ) − , 45 m4 45 m4 ⃗2 8 α2 B 16 α2 B i B j ) + . µij = δ ij (1 + 45 m4 45 m4 εij = δ ij (1 −

(18.86a) (18.86b) (18.86c) (18.86d)

⃗ δ D, ⃗ δB ⃗ and δ H ⃗ describe a propagating wave with wave Let us assume that δ E, ⃗ ⃗ In a vector k and angular frequency ω, in a strong homogeneous background field B.

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source-free region of space, the homogeneous and inhomogeneous Maxwell equations ⃗ then imply that for a wave, propagating with a wave vector k, ⃗ − ω δB ⃗ = 0⃗ , ⃗ = 0, ⃗ = 0, ⃗ + ω δD ⃗ = 0⃗ . (18.87) k⃗ × δ E k⃗ ⋅ δ B k⃗ ⋅ δ D k⃗ × δ H From this point onwards, we can treat the wave propagation problem just like in ⃗ and the propagation vector k⃗ span a reference wave optics. The background field B ⃗ plane. The δ E field, which defines the polarization vector of the electric field (unit ⃗ can either lie in this plane or it can be perpendicular. vector in the direction of δ E), ⃗ If δ E is perpendicular to the reference plane, then we speak of the transverse (⊥) ⃗ is perpendicular to the reference plane, we have a longitudinal (∥) mode. In case δ H mode. The strong background magnetic field defines a susceptibility tensor, and we would like to find a solution to the effective wave equations as listed in Eq. (18.87). For the transverse mode (where the electric field is perpendicular to the background magnetic field), this solution reads as follows, ⃗ = δE ⃗⊥ = E0 kˆ × B ⃗ = E0 χ (n⊥ (kˆ ⋅ B) ˆ, ˆ kˆ − n−1 ˆ δE δH (18.88a) ⊥ B) ,

⃗2 ⃗2 8 α2 B 8 α2 B 2 ⃗ , ⃗ k) sin θ , θ = ∠( B, χ = 1 − . (18.88b) 45 m4 45 m4 Here, E0 is a field amplitude, which is multiplied by the vector product of the unit ⃗ k∣ ⃗ denotes the unit vector. Let us verify that indeed, ˆ As usual, kˆ = k/∣ vectors kˆ × B. n⊥ has the physical meaning of a refractive index. To this end, we use the equation ⃗ = −ω δ D ⃗ and observe that, by virtue of the definition of the perpendicular k⃗ × δ H mode, ˆ j = E0 χ δ ij (kˆ × B) ˆ j = E0 χ (kˆ × B) ˆ i. δDi = εij δE j = E0 εij (kˆ × B) (18.89) n⊥ = 1 +

⃗ either using Eq. (18.88a), or using We can now write the vector product k⃗ × δ H Eq. (18.87), and compare, ⃗ = −E0 k n−1 χ (kˆ × B) ⃗ = −E0 ω χ (kˆ × B) ˆ = −ω δ D ˆ . k⃗ × δ H (18.90) ⊥ So, indeed, ˆ ˆ ˆ ˆ − k n−1 ⊥ χ (k × B) = −ω χ (k × B) .

(18.91)

We therefore have, for the phase velocity, ω = n−1 (18.92) ⊥ = c⊥ , k where c⊥ is the speed of light in the prepared medium, along the specific polarization axis (we recall that in our unit system, we have c = 1 where c is the speed of light in vacuum). For the parallel mode (∥ mode, with the magnetic field of the traveling wave being perpendicular to the background magnetic field), the solution reads as follows, ⃗ = δH ⃗ ⊥ = E0 kˆ × B ˆ, δH n∥ = 1 +

⃗2 14 α2 B sin2 θ , 45 m4

⃗= δE

E0 ˆ kˆ + 1 B) ˆ , (−n∥ (kˆ ⋅ B) χ n∥

⃗ , ⃗ k) θ = ∠(B,

χ=1−

⃗2 8 α2 B . 45 m4

(18.93a) (18.93b)

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In the presence of the strong background field, the vacuum becomes birefringent (n⊥ ≠ n∥ ), and the two different refractive indices are n⊥ and n∥ , with the results given in Eqs. (18.88b) and (18.93b). The speed of light along the ordinary axis (the “fast” axis) of the vacuum is c/n⊥ . Along the extraordinary (“slow”) axis, the speed of light is c/n∥ . Very strong magnetic fields can be found in so-called neutron stars, and some current high-precision experiments currently attempt to detect the vacuum birefringence effects caused by the excitation of virtual electron-positron pairs, as described by the effective Lagrangian [619, 620].

e e

e

e

(a)

(b)

Fig. 18.1 The left Feynman diagram (a) is a light-by-light scattering diagram which gives rise to the perturbative terms of fourth order in the electromagnetic fields, as listed in Eq. (18.71). The diagram (b) on the right is the Wichmann–Kroll correction to the Coulomb interaction, which is described by a quantum correction to the Coulomb photon, due to three (or more) interactions of the electron-positron pair in the loop, with the external Coulomb field.

18.3.2

Effective Lagrangian and Wichmann–Kroll Potential

In Sec. 10.4, we have analyzed the one-loop correction to the photon propagator, due to a vacuum-polarization loop, as well as the concomitant correction to the Coulomb potential (Uehling potential), and we had mentioned that there exist further corrections to the Coulomb interaction, which are described by diagrams with more Coulomb vertices; these are known as the Wichmann–Kroll corrections [90,91]. The result for the fourth-order terms in the Heisenberg–Euler effective Lagrangian given in Eq. (18.71) corresponds to the light-by-light scattering Feynman diagram of Fig. 18.1. We should thus clarify the connection of the Heisenberg–Euler Lagrangian to the Wichmann–Kroll potential, which in leading order also entails the same diagram, but in a different kinematic region. This is very important, because in the spirit of effective Lagrangians, we should be able to use the Heisenberg–Euler Lagrangian in order to minimize the effective action for the quantum-corrected Coulomb field, obtaining at least an asymptotic form of the Wichmann–Kroll correction.

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⃗ = −∇Φ. ⃗ The Coulomb potential Φ generates a static electric field E We should ⃗ thus start from the effective action for a pure electric field E, generated by a radially symmetric potential Φ, so that the Lagrangian (18.71) with the source term (18.78) reads as 2 α2 1 ⃗ 2 2 α2 ⃗ 4 1 ⃗ 2+ ⃗ 4 − ρΦ. (∇Φ) E + (18.94) E − ρ Φ = (∇Φ) 4 2 45m 2 45m4 The variational principle demands that the Coulomb potential, including the quantum correction, should minimize the action, which is the space-time integral of the Lagrangian. Because the charge configuration is static, the action S is obtained as the integral of L over all space, L=

S[Φ] = ∫ d3 r ( = 4π ∫

∞ 0

1 2 α2 ⃗ 2+ ⃗ 4 − ρ Φ) (∇Φ) (∇Φ) 2 45m4

dr r2 (

2 α2 1 2 4 (∂r Φ) + (∂r Φ) − ρ Φ) , 2 45m4

(18.95)

where we assume that Φ(⃗ r) = Φ(r) is radially symmetric, ∂r Φ =

∂ ⃗ r)∣ . Φ(r) = ∣∇Φ(⃗ ∂r

(18.96)

We now calculate the variation S[Φ + δΦ] − S[Φ] = ∫

∞ 0

dr r2 δΦ [(−∂r2 −

2 8 α2 2 3 ∂r ) Φ − (−∂r − ) (∂r Φ) − ρ] . r 45m4 r (18.97)

Here, we have used the following identity which is valid for arbitrary, radially symmetric functions A(r) and B(r), ∫

0

∞

dr r2 (∂r A(r)) B(r) = ∫

∞ 0

2 dr r2 A(r) (−∂r − ) B(r) . r

(18.98)

The variational equation thus reads as (−∂r2 −

2 8 α2 2 3 ∂r ) Φ + (−∂r − ) (∂r Φ) = ρ . r 45m4 r

(18.99)

⃗ 2 A(⃗ For a radially symmetric function A(⃗ r) = A(r), we have ∇ r) = 2 [∂r + (2/r) ∂r ] A(r), and thus, the variational equation can be reformulated as follows, ⃗ 2Φ + ∇

8 α2 2 3 (∂r + ) (∂r Φ) = −ρ . 4 45m r

(18.100)

We now assume that Φ = ΦC + Ξ is given by the sum of the Coulomb potential ΦC and a quantum correction Ξ. The charge density of the nucleus and the Coulomb potential are given by ρ(⃗ r) = Z ∣e∣ δ (3) (⃗ r) ,

ΦC (⃗ r) =

Z ∣e∣ , 4π r

⃗ 2 ΦC (⃗ ∇ r) = −ρ(⃗ r) .

(18.101)

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We note that in view of the nonlinearity of Eq. (18.100), the absolute normalization of the potential matters in the current calculation; it would thus be a mistake to replace Φ → e Φ = −Zα/r, where e is the electron charge. Now, we enter into Eq. (18.100) with an ansatz 8 α2 2 2 3 ∂r ) Ξ + (∂r + ) (∂r ΦC ) = 0 . (18.102) 4 r 45m r Here, we have inserted the first-order solution ΦC into the nonlinear corrective quantum term, whereas Ξ is used in the leading, nonlinear term. We have also used the fact that (∂r + 2/r) (1/r) = 0. In order to justify this approximation, we recall that no further variation is necessary; we simply have to solve Eq. (18.100) with a suitable ansatz Φ = ΦC + Ξ, where Ξ is the correction to be determined. It is straightforward to observe that Eq. (18.102) is solved by a potential proportional to r−5 , and we finally obtain Φ = ΦC + Ξ ,

(∂r2 +

Φ = ΦC + Ξ =

Z ∣e∣ 2 α (Zα)2 (1 − ), 4π r 225 π (m r)4

(18.103a)

Zα 2 α (Zα)3 + . (18.103b) r 225 π m4 r5 The latter term in the above expression is equal to the long-distance tail of the Wichmann–Kroll potential [90, 91]. We note that a tail of the form 1/r5 for the Wichmann–Kroll potential is different in its functional dependence from the Uehling potential, which decays exponentially at large distances [379]. The term of order (α/π) (Zα)3 /(m4 r5 ) generates an energy shift proportional to α (Zα)8 m. We should note that this term does not constitute the “leading” term in the Wichmann– Kroll correction for S states in hydrogen and other, simple atomic systems. The leading Wichmann–Kroll term is otherwise known to be proportional to a Dirac-δ function and generates an energy shift of the order α (Zα)6 m. Our variational calculation shows, at the same time, the usefulness of the effective Lagrangian for bound-state calculations, but also, its limitations: Namely, near the atomic nucleus, the Coulomb potential and its derivative, the Coulomb field, vary considerably on the length scale of an electron Compton wavelength, which exceeds the operational parameters for the Heisenberg–Euler Lagrangian. So, it is impossible to use the QED effective action in order to obtain the Dirac–δ term of the Wichmann–Kroll potential [90, 91], due to the very starting point used in the calculation of the effective Lagrangian. In bound-state calculations, the usefulness of effective field-theory methods typically is limited. They are useful in order to obtain the asymptotic limits of corrections, but they can seldomly be used in order to obtain complete results. At some risk to oversimplification, we can say that bound-state theory is a “Rallye Monte Carlo”, which puts many more refined, and more ramified demands on theory than the plain “Formula One” course of loop correction in the field theory of free quantum states. The latter simply entail “high-profile specifications” without the need to deal with the “narrow curves” of bound-state theory, encountered near the singularities of the Coulomb potential, or V = eΦ = −

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the “narrow curves” due to the electron-electron repulsive force. Still, our treatment serves to convince ourselves that the non-exponential decay of the Wichmann–Kroll potential at long distances is fully consistent with the principles of field theory. We reemphasize that this asymptotic behavior is in sharp contrast to the exponential asymptotic behavior of the (one-loop) Uehling potential for large distances [379]. 18.4

Further Thoughts

Here are some suggestions for further thought. (1) Matrix Exponential and Logarithm. Show that the matrix exponential and matrix logarithm have the properties exp(C A C −1 ) = C exp(A) C −1 ,

ln(C A C −1 ) = C ln(A) C −1 .

(18.104)

On account of these identities, ramify the transformation (18.26). (2) Effective Action and Divergent Series. Calculate the first few terms, or even, the general terms (accompanying the nth power of α) in the asymptotic expansion of the effective Lagrangian (18.57) in powers of the electric field, and convince yourself of the divergent character of the expansion. Try to find a graphical representation for the terms in the expansion, conceivably in terms of Feynman diagrams. ⃗ (3) Effective Action and Electric Field. Find the limit as a → 0, b = ∣E∣, of the general expressions of the real part, and the imaginary part, of the effective action as given in Eqs. (18.72a) and (18.72b). Write the result as a ⃗ For the real part, convince yourself of function of the field strength E = ∣E∣. the consistency of your result with the original expression given in the integral representation (18.57), conceivably comparing to a numerical evaluation of the integral given in Eq. (18.57). For the imaginary part, you should obtain a result consistent with Eq. (18.63). (4) Effective Action and Vacuum Birefringence. Derive both Eqs. (18.88) and (18.93) corresponding to vacuum birefringence in a strong magnetic field. (5) Effective Action and Wichmann–Kroll Potential. Attempt to derive Eq. (18.102) by examining the Euler–Lagrange equations corresponding to the Lagrangian (18.95). (6) Applications of the Light-by-Light Scattering Tensor. Study a few applications (and historical notes on the possible observation of photon splitting) recorded in Ref. [621]. In particular, investigate whether or not, in some applications related to photon production in heavy-ion collisions, virtual light-by-light scattering processes can dominate over tree-level amplitudes [622, 623].

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Chapter 19

Renormalization-Group Equations

19.1 19.1.1

Orientation and Motivation Renormalization on Different Scales

Quantum electrodynamics is a quantum field theory, and one of the paradigmatic developments in this area, continued over the last couple of decades, concerns the so-called renormalization group which establishes connections between the behavior of quantum electrodynamics at varying distance, momentum, or energy scales. As a final part of this treatise, an overview is to be given regarding the applications of related concepts to precision physics. Indeed, we shall attempt, below, to give an illustrative overview of the main ideas that have been pursued over the last couple of decades, in the field of renormalization-group transformations in the context of gauge theories. This discussion will be illustrated by a few concrete calculations in the following. The renormalization program implies that in the canonical approach, we start from the original Lagrangian and do perturbation theory. The perturbative corrections yield further terms which need to be taken into account. Thus, we identify the original Lagrangian as the bare Lagrangian. The multiplicative renormalization implies that the original bare Lagrangian, written in terms of the bare quantities, can be rewritten as the same Lagrangian, but in terms of the renormalized quantities. If L stands for the mapping of the parameters of the theory to the Lagrangian, then, for a scalar theory with a bare field φ0 (renormalized field φ), and a bare coupling g0 (renormalized coupling g), we would have the schematic relationship L0 = L(φ0 , g0 , . . . ),

LR = L(φ, g, . . . ) + LCT (φ, g, . . . ) .

(19.1)

Here, L0 is the bare Lagrangian, LR is the renormalized Lagrangian, and the counter terms LCT are obtained from the loop corrections, i.e., they represents those contributions which need to be reabsorbed into the physical parameters of the theory. The multiplicative nature of the renormalization is manifest in the following relations, φ0 = Z φ φ ,

g0 = Zg g ,

(19.2)

which relate the bare parameters of the theory to the physical, renormalized ones. 715

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A different interpretation of the renormalization program is as follows. We start from the Lagrangian of the theory, and, evaluating the loop corrections, we see that we cannot maintain the desired interpretation of the physical parameters in the Lagrangian as the physical, renormalized fields. However, we would like to keep this interpretation. We can thus simply replace the parameters in the original Lagrangian of the theory by the physical parameters, provided we renormalize them accordingly. Here, Eq. (19.1) is recalled as LR = L(Zφ φ, Zg g, . . . ) = L(φ, g, . . . ) + LCT (φ, g, . . . ) .

(19.3)

The counter terms shift the parameters back to the original, physical ones by the additional perturbations created. The new terms which need to be added to the Lagrangian are proportional to the ones that were there before, and therefore, after the renormalization, we end up with the same Lagrangian that we had before, but under the replacement (φ0 → Zφ φ, g0 → Zg g, . . . ). The identification of the physical parameters is fixed by a set of renormalization conditions. In contrast to our discussion in Chap. 10, we here leave out the subscript R for the renormalized quantities. The question now is whether some additional insight can be drawn from the process of renormalization. The most obvious route to additional insight, originally taken by Gell-Mann and Low [624], and later explained in greater detail by Akhiezer and Berestestkii [4], starts from the following consideration. We have already realized that the electron charge is “screened” by a dynamically induced “virtual charge” of opposite sign, i.e., the bare electron charge, visible at large momenta or very close approach, is larger than the electron charge seen from afar. The increase of the electron charge with the momentum scale is encoded in the vacuum polarization. In Sec. 10.4, we had evaluated the modified Coulomb potential VC (k 2 ) with k 2 = −k⃗2 as follows (e0 is the bare charge), VC (k 2 ) =

e2 e2 e2 = = , k 2 (Z3 − Π(k 2 )) k 2 (1 + Π(0) − Π(k 2 )) k 2 (1 − ΠR (k 2 ))

e2 = Z3 e20 ,

Z3 = (1 + Π(0)) .

(19.4)

Here, Z3 is a renormalization constant (see Chap. 10) which depends on the cutoff scale Λ; for Λ → ∞, formally, we have Z3 → 0. However, we know that physically, the cutoff scale Λ cannot be sent to infinity. The renormalized vacuum polarization function Π(k 2 ) = Π(k 2 ) − Π(0) only diverges logarithmically for k 2 → ∞; we can thus write down the “effective” interaction at the scale k 2 as e2eff =

e2phys 1 − Π(0)

,

(19.5)

where e2phys is the physical electron charge, renormalized so that it represents the elementary charge that governs the cross section of low-energy Thomson scattering (we have previously denoted this charge as e = ephys ). Yet, at the same time, we know that Λ only acts as a cutoff parameter for the logarithmic divergence of

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the vacuum polarization function ω(k 2 ), i.e., the leading logarithmic divergence of ω(0) as a function of Λ, and of ω(k 2 ) as a function of k 2 are not independent. This consideration suggests that, at least in the asymptotic regime k 2 ≫ m2 , the dependence of Z3 on Λ and of e2eff on k 2 must be interrelated. Furthermore, for k 2 ≫ m2 , one might conjecture that the “running” of the charge with the momentum scale must be independent of m2 ; the concrete numerical value of m2 cannot be important in the asymptotic, high-energy regime. Indeed, as we shall see in the following, it is possible to express the “running” of the charge of high energy with the momentum scale exclusively as a function of the effective charge at a reference scale. Let us formulate this statement in terms of a “running” of the QED coupling α = e2 /(4π). The QED β function expresses the fact that the “running” of the coupling with the scale is a function of the coupling at the reference momentum scale itself, k2

∂ α(k 2 ) = β (α(k 2 )) . ∂k 2

(19.6)

Because the behavior of the β function is determined by the dependence of the Z3 renormalization constant on the cutoff parameter, the renormalization-group (RG) analysis assigns “additional physical significance” to the process of renormalization, beyond the simple statement that the “infinities are reabsorbed” into the physical parameters of the theory observed at a specific scale. The renormalization group is called a “group” because the β function actually allows us to evolve the coupling, say, from the scale k12 to the scale k22 to the scale k32 , by integrating Eq. (19.6) with respect to k 2 , starting from a reference scale k12 , where the value of α(k12 ) needs to be “matched” against observations (e.g., QED at the Z pole). For QED, the theory is canonically matched against the physical theory at long distances and on the mass shell of the electron (on-mass-shell renormalization). This means that we have measured the electron charge at long distances (fine-structure constant), and the free electron mass, and then, we construct QED theory and match the theory against the physically observed parameters on mass shell. However, we could also have measured the electron charge at a momentum scale equal to the weak interaction scale (Z scale) near k 2 ≈ m2Z ≈ 91 GeV2 , and the electron mass at a different scale, and matched the QED Lagrangian against the different set of observed parameters at the Z scale. This would be a no less fundamental matching. In dimensional regularization, one naturally introduces a momentum scale µ into the theory, as a parameter into the counterterms. The counterterms therefore depend on the renormalization scale µ, and one can describe the running of the coupling etc. by a change in the renormalization scale µ. Irrespective of where we carry out the matching, the perturbative evolution of the coupling with the scale must still be described, at least approximately, by the same multiplicative factors that we originally invoked in order to describe the evolution of the electron charge with momentum, starting from zero momentum. We shall see that this is universally the case in the limit where the distance scale is much shorter than the electron Compton

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wavelength, i.e., in a regime where we can ignore the electron mass compared to the momentum transfer. QED in that sense is a set of possible theories, with the physical QED theory picked out by nature through the somewhat arbitrary selection of the value of the fine-structure constant α by a mood of nature. The actual value of α is not fixed by any more fundamental consideration. Again, the running of the coupling due to the vacuum polarization therefore is not more or less fundamental than a conceivable change of α, that is, the fine-structure constant in the limit of zero momentum transfer. As one approaches the fundamental, UV, scale by carrying out the renormalization at ever shorter distance scales, one approaches the microscopic distance of the bare theory, and the renormalized parameters become more and more equal to the bare parameters. In QED, since the coupling is ever growing with the scale, the bare coupling is supposed to be very large, or even infinite if the renormalized coupling has a finite, nonzero value (which it has, since the fine-structure constant has a nonvanishing value α ≠ 0). There might have been hope that the running of the charge is a fundamental process that eventually leads us to a potential understanding of the unrenormalized, bare charge at very small distance scales. However, that is not the case: rather, we are taught by the running of the coupling, that the evolution of the charge at smaller distance needs to be described by a more fundamental theory. In that sense, QED theory is the effective theory, in the infrared, of a more fundamental theory that acts on very short distance scales (the Planck scale!?!), and whose details are hidden from us. We cannot hope to learn anything about the more fundamental interactions that govern the Planck scale, from the infrared limit of the theory, which is QED. These consideration eventually lead to an additional interpretation of a quantum field theory itself: Namely, a field theory actually has the value of the coupling as a free parameter. Through the β function, a field theory predicts how the effective coupling evolves with the momentum scale at which processes are observed; it does not fix the value of the coupling at a reference scale; this latter observation must still be made independently, and necessitates an experiment. For QED, we are in the lucky position that the couplings remain finite in the infrared, so that we can match the electron charge with the observed electron charge for large-distance interactions. For other gauge theories, like quantum chromodynamics (QCD), the effective couplings actually diverge in the infrared (at least within a perturbative formalism), and it is thus impossible to fix the QCD coupling constant by observing, say, the interaction of two quarks in the limit in which they are far apart (this limit otherwise serves as a valid ground for fixing the value of the QED coupling). The question then is whether one can generalize the scale transformation argument, originally presented above, to a more general setting.

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Renormalization-Group Equations

The formulation of RG equations may proceed along three different lines. (i) The original approach of Gell-Mann and Low was to formulate a specific function that characterizes the RG evolution of a physical quantity of interest (in their case, the running of the QED coupling, which is formulated in terms of the logarithmic derivative of the coupling with respect to the momentum scale), and to show that it is independent of any other physical parameters in a specified, asymptotic limit (in their case, independent of the electron mass for very short distance scales). The RG equation for the quantity of interest then follows immediately. (ii) The approach of Callan and Symanzik is based on the observation that differentiations of generating vertex functions and other “inverted” Green functions with respect to mass scales generate mass insertions. Paradigmatically, we can formulate this observation as ∂ (n) Γ ∣ = Γ(n,1) , ∂m20 g0 =const.

(19.7)

where Γ(n) is an n-vertex function, and Γ(n,1) is a vertex function with a mass insertion. The mass insertion, for QED as well as for other theories (N -vector model), necessitates the introduction of yet another renormalization constant, called Z0 or Zψψ in the case of QED. The application of the operator mR ∂/∂mR , ∂ ∂m20 ∂ (n) ∂m20 (n) (n,1) Γ0 = mR Γ = (m ) Γ0 , R ∂mR ∂mR ∂m20 0 ∂mR (19.8) then leads to equations of the Callan–Symanzik type, where upon renormalization, both sides of the equation are finally expressed in terms of renormalized quantities. The paradigm is that one compares theories at different length scales, while keeping the bare coupling constant and while varying the bare mass of the theory in such a way that a meaningful equation relating vertex functions is obtained. In appropriate asymptotic limits, one may then discard the right-hand side on dimensional grounds. (n,1) Typically, the mass insertion implies that for very large momenta, the function Γ0 (n) falls off more rapidly than the function Γ0 , which in turn implies that one can ignore the inhomogeneous term in the Callan–Symanzik equation in an appropriate asymptotic limit, which in some cases enables us to recover the approach (i) from approach (ii). Again, the Callan–Symanzik equations compares two theories valid at different renormalization scales while keeping the bare theory invariant (up to a variation of the mass). The approach (iii) of Zinn-Justin [130] is based on the introduction of a cutoff Λ, and upon the comparison of bare (!) theories at the cutoff scale. The Zinn-Justin equation expresses the following fact: Near the scale Λ of the bare theory, it is possible to vary the cutoff scale Λ, the value of the coupling g0 , and the normalization of the field as given by the anomalous dimension η, while keeping the physics at the phenomenological scale of the theory, i.e., in the infrared, invariant. This is expressed in Eq. (13.13) of Ref. [130]. One then expresses the fact that the bare theory, at scale Λ/λ, can be identified with the physical, renormalized (n)

Γ0

(n)

= Z −1 ΓR ,

mR

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theory at scale Λ/λ, provided one applies a scale-dependent renormalization Z = Z(λ) to the “bare” theories which are being compared to each other. The “bare” theory at scale Λ/λ is then being identified with the renormalized theory at that scale, modulo the renormalizations. The latter consideration allows us to identify the β functions of the theory, initially formulated in terms of logarithmic derivatives with respect to the cutoff Λ, in terms of derivatives with respect to the scale λ, of the renormalized coupling, thus restoring the intuitive picture of the β function (description of the RG evolution of the coupling with the scale). As a last remark, we briefly comment on so-called functional RG equations which express the evolution of the effective Hamiltonian of a theory with the cutoff scale. Specifically, the Polchinski [625] and Wegner–Houghton [626] equations are based on the general idea that if one integrates out the fluctuations of the field φ1 in the path integral, one obtains an equation for the field φ2 . In QED, this approach, where φ1 plays the role of the fermion field, leads to an effective interaction for the electromagnetic field (see Chap. 18). Now, Polchinski’s realization is based on the observation that this approach can be applied to the field φ1 itself if one splits the field fluctuations in the following way: Namely, one first isolates those fluctuations that happen at momentum scales k ≥ Λ (field φ2 ) and integrates them out. As a function of Λ, one then obtains an equation for the evolution of the effective potential V = VΛ as a function of the scale Λ (obtaining an equation for the field φ1 ). One can then solve this equation using “trial functions” which represent nothing but an ansatz for the “running effective potential” as a function of the cutoff scale Λ. In this way, one can even study the evolution of the effective potential with distance (or momentum) scales for theories which are not renormalizable in the perturbative sense, because the ansatz that one chooses for the effective potential may contain, e.g., periodic terms in the fields like cos φ which are not renormalizable perturbatively. In a typical practical application, the functional RG equations are excellent tools for the study of the phase structure of perturbatively nonrenormalizable, field-theoretical models. One interesting question to ask is: Does one of the couplings (expansion coefficients for the trial function for the effective potential) reverse its asymptotic behavior in the infrared, as a function of the value chosen for this coupling in the ultraviolet? The Wetterich equation [130] is obtained from the Polchinski equation by a Legendre transformation. It expresses the running of the effective action Γk with the scale k in terms of the derivative of a regulator function ∂k Rk , and of the complete (2) generating functional for the two-point function Γk . A summation (trace) over the quantum fluctuations beyond the regulator scale and a supertrace over the bosonic and fermionic degrees of freedom of the model is implied. Of course, in the perturbative limit, the functional equations reproduce the findings of the perturbative expansions, and it is hard to obtain more accurate predictions for, e.g., the scale-dependent wave function renormalization constant from a nonperturbative equation as compared to the perturbative approach, because

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the Polchinski equation nevertheless introduces an ambiguity into the calculation (exact shape of the smooth cutoff function). The Wegner–Houghton equation is difficult to solve while taking into account all the renormalizations of the couplings and fields during the evolution. In principle, the full knowledge of the effective potential at a given distance scale, obtained by a solution of the Polchinski or Wegner–Houghton equations, would allow us to obtain the RG evolution of all correlation functions. However, this program has not yet been completed to greater accuracy than by a perturbative approach for any nontrivial field theory of practical interest. We proceed as follows. In Sec. 19.2, we first take a step back and discuss scale transformations and their effect on solutions of the Schr¨odinger–Coulomb and Dirac–Coulomb problems. Indeed, the scaling equations will be very similar in form to the Callan–Symanzik RG equations discussed further down. We are then in the position to treat actual renormalization-group transformations in Sec. 19.3, starting first from a demonstration of the “existence” of the QED β function, and then, continuing to the calculation of a few terms in both the Gell-Mann–Low approach, and in the Callan–Symanzik approach. The connection of the Gell-Mann–Low and Callan–Symanzik approaches amounts to a change in the renormalization scheme; this statement will be cast into mathematical formulas. Finally, in Sec. 19.4, we discuss a number of applications of the developed formalism, first, in the context of the RG-invariance of the QED effective action, then, in terms of RG-optimized perturbation theory, before concluding with a discussion of conceivable connections of the physical value of the fine-structure constant with the RG analysis of QED. 19.2 19.2.1

Scale Transformations From Poor Man’s Scaling to Nontrivial Scaling

Scaling transformations are not renormalization-group transformations; however, there is a connection. By a scaling transformation, one can understand how a physical system develops when one of the parameters undergoes a scale change, which may result in a different hierarchy of terms: namely, on different scales, “small”, “perturbative” terms may suddenly become dominant. E.g., for a quartic anharmonic oscillator, the harmonic term x2 in the potential is dominant for small x, whereas the “perturbative” term g x4 is dominant for large x, or, large coupling constant g ≫ 1. In the context of bound-state problems, one can derive scaling laws for the solutions of the Schr¨ odinger–Coulomb, and Dirac–Coulomb problems, which describe the evolution of bound-state solutions as a function of the nuclear charge number Z and the electron mass m. These relations will be used in the following, in order to derive compact formulas for a number of matrix elements, evaluated on relativistic bound states. Renormalization-group transformations are scale transformations which connect the physical parameters of a field theory on different length or momentum scales, due to the dependence of the renormalization constants on

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the reference scale at which the parameters are matched against observations; i.e., renormalization-group transformations constitute a special, “distinguished” class of scale transformations. Let us start with a brief digression on “poor man’s scaling”, which is a scale transformation relevant to the analysis of quantum anharmonic oscillators. From the theory of anharmonic oscillators generalized to the anharmonic potentials, we are familiar with so-called poor man’s scaling which mediates the transition from the weak-coupling perturbative regime, where the main contribution to the energy is from the unperturbed Hamiltonian, to the strong-coupling regime, where the energy is dominated by the anharmonic term. To this end, we consider the Hamiltonian operator of the one-dimensional quartic anharmonic oscillator, 1 1 1 ∂2 1 ∂2 + x 2 + g x4 = − + x2 + O(g) , (19.9) H =− 2 2 2 ∂x 2 2 ∂x 2 where x is the coordinate. For g → 0, the eigenvalues approximate those of the unperturbed harmonic oscillator Hamiltonian − 12 ∂x2 + 21 x2 , and are thus approximate by the formula En ∼ n + 1/2 + O(g), where n is a nonnegative integer. However, for large coupling g → ∞, the functional form of the eigenvalues is totally different, the “perturbative” term g x4 becomes dominant, and a scaling transformation must be used in order to describe the large-coupling asymptotics. We scale the coordinate in Eq. (19.9) as x → g −1/6 x, to obtain a Hamiltonian whose eigenvalues are still the same as those of the original Hamiltonian (19.9), but the functional form is very different, x2 1 ∂2 1 ∂2 4 1/3 + x + + x4 + O(g −2/3 )) . (19.10) ) = g (− H = g 1/3 (− 2 ∂x2 2 ∂x2 2 g 2/3 Here, the last term in round brackets acts as a perturbation that vanishes for g → ∞. The energy levels of H, for g → ∞, are well approximated by the formula En ∼ g 1/3 n , where n is the nth level of the Hamiltonian operator − 12 ∂x2 + x4 . The scale transformation x → g −1/6 x, which reveals the large-coupling asymptotics of the energy eigenvalues, is otherwise known as poor man’s scaling. 19.2.2

Scale Transformation and Schr¨ odinger Hamiltonian

Using the idea of “poor man’s scaling”, we aim to derive a set of identities, based on scale transformations of the Schr¨odinger–Coulomb, and Dirac–Coulomb Hamiltonians, to gain some additional insight into the physics and the scaling properties of the physical system. The systems under investigation are more complex than a simple anharmonic oscillator. We indicate the scaling relations for the Schr¨odinger– Coulomb, and the Dirac–Coulomb case; they have not yet appeared in the literature up to this point, to the best of our knowledge. Let us start from the Schr¨odinger– Coulomb Hamiltonian with an eigenvector ∣φ⟩, p⃗ 2 Zα − , HS ∣φ⟩ = ES ∣φ⟩ , (19.11) HS = 2µ r

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where µ is the reduced mass of the system. We can factor out the reduced mass as follows, 1 2 1 ⃗ − ), HS = (Zα)2 µ (− ∇ 2 ρ ρ

ρ⃗ = Zαµ r⃗ .

(19.12)

The first-order perturbations of the eigenstate wave function under an infinitesimal variation of the mass µ and the nuclear charge number Z read as follows, ′ ∂ 1 p⃗ 2 ∣φ⟩ = ( ) (− ) ∣φ⟩ , ∂µ ES − HS 2µ ′ 1 Zα ∂ ∣φ⟩ = ( ) (− ) ∣φ⟩ . Z ∂Z ES − HS r

µ

(19.13a) (19.13b)

Here, the reduced Green function [see Eq. (5.7)] is denoted with a prime. While the identities are intuitively obvious, the derivation is somewhat nontrivial in the details. Let us write the nonrelativistic kinetic term as H0 = p⃗ 2 /(2µ) and consider the Schr¨ odinger Hamiltonian HS = p⃗ 2 /(2µ) + V = H0 + V . For the Coulomb potential V = −Zα/r, we have Z∂V /∂Z = V . The Schr¨odinger equation can be written as (ES − H0 ) ∣φ⟩ = V ∣φ⟩ .

(19.14)

Let us differentiate this equation with respect to Z, taking into account that the free Hamiltonian H0 is independent of Z. A careful consideration of both sides of the equation leads to the identity, ∂ES ∂ ∂V ∂ ∂ ∣φ⟩ +(ES − H0 ) Z ∣φ⟩ = Z ∣φ⟩ + V Z ∣φ⟩ = V ∣φ⟩ + V Z ∣φ⟩. (19.15) ∂Z ∂Z ∂Z ∂Z ∂Z This can be written as ∂ES ∂ Z ∣φ⟩ + (ES − H0 − V ) Z ∣φ⟩ = V ∣φ⟩ . (19.16) ∂Z ∂Z We project this equation onto the Hilbert space orthogonal to the eigenstate ∣φ⟩. To this end, we first observe that (Z ∂ES /∂Z)∣φ⟩ is proportional to ∣φ⟩, while ∂ Z ∂Z ∣φ⟩ is orthogonal to ∣φ⟩. The latter observation can be made by recognizing ∂ that Z ∂Z ⟨φ∣φ⟩ = 0, due to conservation of the norm. We can now obtain a com∂ plete result for Z ∂Z ∣φ⟩ by simply inverting the operator (ES − H0 − V ) in the space orthogonal to ∣φ⟩, i.e., by applying the reduced Green function to the right-hand side. Thus, Z

Z

′ ′ ∂ 1 1 ∣φ⟩ = ( ) V ∣φ⟩ = ( ) V ∣φ⟩ . ∂Z ES − H0 − V ES − HS

(19.17)

This proves Eq. (19.13b). Analogous relations can be used in order to show Eq. (19.13a). Adding the two perturbations given in Eqs. (19.13a) and (19.13b), one obtains (

′ ′ 1 p⃗ 2 Zα 1 ) (− + ) ∣φ⟩ = −ES ( ) ∣φ⟩ = 0 , ES − HS 2µ r E S − HS

(19.18)

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because the reference state is excluded from the sum over virtual states, by virtue of the definition of the reduced Green function. We thus obtain the scaling equation ∂ ∂ −Z ) ∣φ⟩ = 0 . (19.19) (µ ∂µ ∂Z For the ground state of hydrogen, the wave function in coordinate space is given as φ(⃗ r) =

(Zαµ)3/2 √ exp(−Zαµr) . π

(19.20)

The quantities Z and µ enter the formula for φ(⃗ r) only via the combination Zαµ, it is evident that indeed, the scaling equation (19.19) is fulfilled. 19.2.3

Scale Transformation of the Dirac Hamiltonian

Let us now turn our attention to the Dirac Hamiltonian with an eigenstate ∣ψ⟩, Zα ⃗ ⋅ p⃗ + β m − HD = α , HD ∣ψ⟩ = ED ∣ψ⟩ , (19.21) r where the Dirac energy is denoted as ED . The Dirac equation is a one-particle equation, and we thus work with the electron mass m, not with the reduced mass µ. The first-order perturbations under a change of the mass and the charge number are given as follows, ′ ∂ 1 ∣ψ⟩ = ( ) β m∣ψ⟩ , ∂m ED − HD ′ ∂ 1 Zα Z ∣ψ⟩ = ( ) (− ) ∣ψ⟩ . ∂Z ED − HD r From the eigenvector property we have, immediately,

m

(19.22a) (19.22b)

′ 1 Zα ) (⃗ α ⋅ p⃗ + β m − ) ∣ψ⟩ = 0 , (19.23) ED − HD r if ∣ψ⟩ is an eigenvector, HD ∣ψ⟩ = ED ∣ψ⟩. Combining Eqs. (19.22) and (19.23), and (8.95), we obtain

(

(m

′ ∂ ∂ 1 +Z ) ∣ψ⟩ = ( ) (−⃗ α ⋅ p⃗) ∣ψ⟩ ∂m ∂Z ED − HD ′ 1 =( ) (−i [HD − ED , r⃗] ⋅ p⃗) ∣ψ⟩ ED − HD ′ 1 ∂ ∣ψ⟩ − ∣ψ⟩ i ⟨ψ + ∣⃗ r ⋅ p⃗∣ψ⟩+i( ) r⃗ [HD − ED , p⃗] ∣ψ⟩. = r⃗ ⋅ ∂ r⃗ ED − HD (19.24) ′

We have used the identity [1/(ED − HD )] (ED − HD ) = 1 − ∣ψ⟩ ⟨ψ + ∣ (for absolute clarity, the Hermitian adjoint ψ + rather than the Dirac adjoint ψ + γ 0 is being indicated. The matrix element i ⟨ψ∣⃗ r ⋅ p⃗∣ψ⟩ = ⟨ψ∣⃗ r ⋅ ∂∂r⃗ ∣ψ⟩ can be evaluated for a general eigenvector ∣ψ⟩, as follows, ∂ ∂ ∂ ∂ ⟨ψ + ∣⃗ r⋅ ∣ ψ⟩ = ⟨ψ + ∣[⃗ r, ]∣ ψ⟩ + ⟨ψ + ∣ ⋅ r⃗∣ ψ⟩ = −3 − ⟨ψ + ∣⃗ r⋅ ∣ ψ⟩ , (19.25) ∂ r⃗ ∂ r⃗ ∂ r⃗ ∂ r⃗

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where we have used an integration by parts. This gives immediately 3 i ⟨ψ + ∣⃗ r ⋅ p⃗∣ψ⟩ = − . 2

(19.26)

The commutator on the right-hand side of Eq. (19.24) is easily evaluated, i r⃗ ⋅ [HD − E, p⃗] = i r⃗ ⋅ [−

Zα ∂ Zα Zα , p⃗] = r⃗ ⋅ =− . r ∂ r⃗ r r

(19.27)

We thus have the relation, (m

′ ∂ ∂ 3 1 Zα ∂ +Z ) ∣ψ⟩ = r⃗ ⋅ ∣ψ⟩ + ∣ψ⟩ + ( ) (− ) ∣ψ⟩ . ∂m ∂Z ∂ r⃗ 2 ED − HD r

(19.28)

Finally, using Eq. (19.22b) in reverse, we obtain the result (m

∂ ∂ 3 − r⃗ ⋅ − ) ∣ψ⟩ = 0 . ∂m ∂ r⃗ 2

(19.29)

This equation constitutes the analogue of the relation (19.19) within the fully relativistic formalism. 19.3 19.3.1

Renormalization Group Transformations Functional Equations in the Asymptotic Regime

From Eq. (10.111), using the Ward–Takahashi identity Z1 = Z2 , we infer that the physical charge e of the electron and the bare charge e0 are related by the renormalization constant Z3 , as follows, eR = Z 1/2 e0 ,

Z = Z3 = Z(λ)

eR = e(λ) = eλ .

(19.30)

Let us now compare finite, renormalized quantities at three different scales, denoted here by the symbols 1, 2 and 3 in a somewhat lax notation, e2λ3 =

Z(λ3 ) 2 Z(λ3 ) Z(λ2 ) 2 e = e = F (λ2 , λ2 ) F (λ2 , λ2 ) e2 , Z(λ2 ) λ2 Z(λ2 ) Z(λ1 ) λ1 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹3¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹2¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹2¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹1¹ ¹ ¹¶ λ1

(19.31)

=F (λ23 ,λ21 )

where the F (λ2a , λ2b ) = Z3 (λa )/Z3 (λb ) denotes the ratio of renormalization constants. Let us assume that we have matched the parameter e2λ at the momentum scale λ against the observed, physical electron charge square at the same momentum scale. Then, the F function can be written in terms of all of its arguments as follows, F = F (k 2 , λ2 ) = F (e2λ ,

k 2 m2 , ), λ2 λ2

(19.32)

i.e., it describes the evolution of the charge from the scale λ2 to the scale k 2 . We observe that since F is itself dimensionless, it can only depend on dimensionless arguments, and furthermore, we explicitly indicate the value of e2λ as a third argument,

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because this value is determined by observation and represents “experimental” input. Up to a prefactor −gµν , the renormalized photon propagator at the momentum scale k 2 has the structure (see Chap. 9) Gγ (k 2 ) =

k 2 m2 F (k 2 ) 1 F (e2λ , 2 , 2 ) = , 2 k λ λ k2

(19.33)

where the superscript γ denotes the photon. The function F (k 2 ) describes the running of the charge but it is not the running charge itself; once the reference scale λ and the effective charge square e2λ at the reference scale are fixed, the theory becomes predictive. After the fixing of the reference scale, F is a function of only one variable, namely, k 2 . In order to obtain the actual value of the running charge at a specific momentum scale, one must multiply F (k 2 ) by the value of the running charge at a reference momentum scale, to obtain the expression e2k Gγ (k 2 ). Now, for k 2 = λ2 , we have fixed the physical charge square to be e2λ , and so F (k 2 = λ2 ) = 1 ,

e2λ Gγ (k 2 = λ2 ) =

e2λ F (λ2 ) e2λ = 2, λ2 λ

(19.34)

and the photon propagator at the scale k 2 = λ2 has the desired form. When the three arguments of F are fully written out, the condition F (k 2 = λ2 ) = 1 implies that F (e2λ , 1,

m2 ) = 1. λ2

(19.35)

Before we now explore the properties of the renormalization group function F in the asymptotic regime, we illustrate its properties by a number of further example calculations. First, we observe the we can obtain the invariant charge square e2k at the scale k both using scale transformations λ → k as well as λ′ → k, a fact which is expressed as follows, e2k = e2λ F (e2λ ,

k 2 m2 k 2 m2 , 2 ) = e2λ′ F (e2λ′ , ′2 , ′2 ) . 2 λ λ λ λ

(19.36)

The effective charge e2k thus defined is called the invariant charge: We can either start the RG evolution at scale λ and transform to scale k via the action of the F function, or we can start from the scale λ′ and go over to scale k by the action of an F function of different arguments. In any case, since e2k is dimensionless, it must be expressed as a function of dimensionless arguments. Equation (19.36) also is consistent with Eq. (19.31), in the form e2λ3 = F (λ23 , λ21 ) e2λ1 , upon the identification 1 → λ, λ′ and 3 → k 2 . The group structure, evident from Eq. (19.31), F (λ23 , λ22 ) F (λ22 , λ21 ) = F (λ23 , λ21 ) ,

(19.37)

is expressed as follows in terms of the F functions of three dimensionless arguments, F (e2λ′ ,

k 2 m2 λ′2 m2 k 2 m2 , ′2 ) F (e2λ , 2 , 2 ) = F (e2λ , 2 , 2 ) . ′2 λ λ λ λ λ λ

(19.38)

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Renormalization-Group Equations

This relation exhibits the group structure of the RG transformation in a particularly clear way. We start from e2λ at scale λ and transform to scale λ′ and then to scale k 2 (left-hand side) or directly from scale λ to the scale k 2 (right-hand side). One more alternative point of view is as follows. From the invariance of the bare charge e0 under the RG transformation, e20 = [Z3 (λ′ )]

−1

e20 = [Z3 (λ)]

e2λ′ ,

−1

e2λ ,

(19.39)

we have e2k [Z3 (λ′ )] ⇒

Z3 (λ′ )F (e2λ′ ,

−1

e2λ′

=

e2k [Z3 (λ)]

−1

e2λ

,

2 k 2 m2 m2 2 k , ) = Z (λ) F (e , , ). 3 λ λ′2 λ′2 λ2 λ2

(19.40)

This is the another form of Eq. (19.36). The function F describes the evolution of the renormalization constant Z3 , and therefore, of the electron charge square, with the scale, and therefore we also have the relation ′2 k 2 m2 Z3 (λ′ ) k 2 m2 m2 k 2 m2 2 λ 2 2 , ) = F (e , , ) , ) = F (e , ) F (e ′, ′, λ λ λ λ2 λ2 Z3 (λ) λ2 λ2 λ′ 2 λ′ 2 λ′ 2 λ′ 2 (19.41) ′ ′2 2 2 λ′2 m2 since Z3 (λ )/Z3 (λ) = F (λ , λ ) = F (eλ , λ2 , λ2 ) according to Eq. (19.32), which illustrates once more the group structure given in Eq. (19.38). We now want to describe the flow of the effective QED coupling with the momentum scale, starting from the invariant charge described by Eq. (19.36) and the group structure (19.38) of the RG transformations. Interchanging λ ↔ λ′ , the group structure (19.38) can be written as follows,

F (e2λ ,

F (e2λ ,

2 2 m2 m2 k 2 m2 2 λ 2 k , ) F (e , ) = F (e , ). ′, ′, λ λ λ2 λ2 λ′2 λ′2 λ′2 λ′2

(19.42)

With k2 m2 λ2 , y = , t = , (19.43) λ′2 λ′2 λ′2 we can write Eq. (19.42) as follows, after a multiplication of both sides by a factor e2λ′ , x=

x y F (e2λ , , ) e2λ′ F (e2λ′ , t, y) = e2λ′ F (e2λ′ , x, y) . t t

(19.44)

We can eliminate the quantity e2λ in the argument of the first function, using the invariant charge formula (19.36), e2λ = e2λ′ F (e2λ′ ,

λ2 m2 , ) = e2λ′ F (e2λ′ , t, y) . λ′2 λ′2

(19.45)

So, Eq. (19.44) is rewritten as follows, x y e2λ′ F (e2λ′ , t, y) F (e2λ′ F (e2λ′ , t, y) , , ) = e2λ′ F (e2λ′ , x, y) . t t

(19.46)

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Now, the parameter e2 ≡ e2λ′ is fixed by an observation, and therefore x y e2 F (e2 , t, y) F (e2 F (e2 , t, y) , , ) = e2 F (e2 , x, y) . t t

(19.47)

We now consider the asymptotic regime y → 0, which in view of y = m2 /λ′2 is either the massless limit or the limit λ′ → ∞, i.e., the limit of large momenta. Endowing the functions in the asymptotic regime with a superscript “as”, we can drop the last argument y → 0 in the formulas, and we also replace the first argument by α = e2 /(4π), F as (α, x) ≡ F (e2 , x, 0) ,

(19.48a)

x ). (19.48b) t Equation (19.48b) is equivalent to Eq. (19.47) in the asymptotic region, and we define the effective coupling Gas in the asymptotic region as indicated. Then, a recursive application of the definition (19.48b) leads to the result Gas (α, x) ≡ αF as (α, x) = αF as (α, t) F as (α F as (α, t) ,

Gas (Gas (α, t) ,

x x x ) = Gas (α F as (α, t) , ) = α F as (α, t) F as (α F as (α, t) , ) , t t t (19.49)

and the latter expression is easily identified as the right-hand side of Eq. (19.48b), so that x x Gas (α, x) = Gas (Gas (α, t), ) , αx = Gas (αt , ) , (19.50) t t where in the latter step we have identified αx = Gas (α, x) as the effective coupling at the scale x. The solution to the RG equation (19.50) needs to be found, now. Indeed, Eq. (19.50) is solved by any function of the form Gas (α, x) = f (x f −1 (α)) ,

(19.51)

but this is not obvious. Let us first apply f −1 to both sides of the defining Eq. (19.51). We get f −1 (Gas (α, x)) = x f −1 (α) . We now go back to the original RG equation (19.50) and also apply f

(19.52) −1

,

x )) . (19.53) t Using Eq. (19.52) for both sides of Eq. (19.53) (twice, recursively, to the right-hand side) yields the relation x x x f −1 (α) = f −1 (Gas (α, t)) = t f −1 (α) = x f −1 (α) , (19.54) t t which verifies the solution of the RG equation (19.50). We now look at the derivative of the asymptotic effective coupling Gas with the scale x, f −1 (Gas (α, x)) = f −1 (Gas (Gas (α, t),

∂ as ∂ G (α, x) = f (x f −1 (α)) = f −1 (α) f ′ (x f −1 (α)) . ∂x ∂x

(19.55)

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If we supply an additional prefactor x, then the logarithmic derivative of the coupling can be expressed as a function of a single variable, namely, the coupling as given by Eq. (19.51), ∂ as G (α, x) = (x f −1 (α)) f ′ (x f −1 (α)) = β (f (x f −1 (α))) = β (Gas (α, x)) . x ∂x (19.56) Evidently, this expression is a function, named the β function, of a single variable, namely, x f −1 (α), or alternatively f (x f −1 (α)), which is αx = Gas (α, x). We have derived the result that the logarithmic derivative of the effective coupling with the (square of the) momentum scale, for large momenta, is described by a function of the value of the effective coupling at that scale (and nothing else), ∂ as ∂ αx = x G (α, x) = β (Gas (α, x)) = β (αx ) . (19.57) αx = Gas (α, x) , x ∂x ∂x This is one central result of the renormalization group when applied to gauge theories. The result (19.56) is the basis for the definition of the β function of QED, which describes the running of the effective charge with the momentum scale. We now assume that β is positive, based on the one-loop value from Eq. (19.60) below, and verify consistency with the different possible kinematic regions (time-like momentum transfer k 2 > 0 and space-like exchange momenta k 2 = −⃗ q 2 < 0. Indeed, we shall assume that the β function is positive for both (time-like as well as spacelike) physically relevant domains of the asymptotic, effective coupling Gas (α, x). We first recall the definition of the x variable from Eq. (19.43), i.e., x = k 2 /λ′2 . Here, k 2 is the “final” momentum of the RG evolution step. The most relevant physical domain for k 2 is the deep Euclidean region k 2 = −⃗ q 2 , with q⃗2 ≫ m2 , i.e., the region of a “strong Coulomb potential” with a “hard Coulomb photon exchange”. In this kinematic region, pair production is impossible. For negative x, the value of the effective coupling grows with the modulus of x, i.e., it grows with x as x becomes more negative, and both x as well as ∂x αx are negative. This implies that the product x ∂x αx is positive for both relevant kinematic regions. Let us now fix λ′2 > 0 as a fixed reference scale for the momentum square. Then, we can identify the logarithmic differential operator x ∂/∂x with the operator k 2 ∂/∂(k 2 ). Let us also convince ourselves that the same β function can be used in order to describe the evolution of the coupling in the manifestly Euclidean region, setting the reference scale λ′2 = m2 . Then, x = k 2 /m2 = −⃗ q 2 /m2 , and the RG evolution can 2 2 be expressed in terms of ξ = q⃗ /m as an independent variable, x → k 2 = −⃗ q 2 = −ξ , αx = α−ξ ≡ α(ξ) , ∂ ∂ ∂ αx = β (α−ξ ) = β (α(ξ)) , ξ α(ξ) = ξ α−ξ = x ∂ξ ∂ξ ∂x

(19.58)

so that the RG evolution with the reference momentum scale q⃗2 is indeed given by the β function, q⃗2

∂ α(⃗ q 2 ) = β (α(⃗ q 2 )) , ∂ q⃗2

α(⃗ q 2) = α (

q⃗2 ). m2

(19.59)

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This implies that we can alternatively interpret the RG evolution operator x ∂x in the equation x ∂x α = β(αx ) as a differential operator in x = k 2 /m2 or in x = q⃗2 /m2 , provided we use an appropriate definition of αx , namely, the value of αx at x = k 2 /m2 or x = −⃗ q 2 /m2 . In the above considerations, we have implicitly assumed that the value of α from which we start the RG transformation at zero momentum transfer is the physical value of the fine-structure constant. This is known as the MOM (momentum) renormalization scheme. (We are somewhat using the RG method in a region out of its initial specification, i.e., in the non-asymptotic region. For more remarks on this issue, we refer to the text following Eq. (19.74).) The result for the β function has been given to three loops by Baker and Johnson in Ref. [627], and the four-loop result has been indicated by Gorishny, Kataev, Larin and Surguladze in Ref. [628], α3 α4 1 101 αx2 + x2 + x3 ( ζ(3) − ) 3π 4π π 3 288 α5 5 1 93 + x4 (− ζ(5) + ζ(3) + ) + O(αx6 ) . π 3 3 128

β(αx ) = βMOM (αx ) =

(19.60)

Note that our definition of the β function differs by an overall factor 4π from the one given in Eq. (2.5) of Ref. [628]. The RG evolution equation can be solved as follows, x

∂ αx = β(αx ) , ∂x

∫

Gas (α,x2 ) Gas (α,x

1)

dz x2 = ln ( ) , β(z) x1

(19.61)

where αx = Gas (α, x). Here, we come to an understanding of the often used jargon of “matching and running”: namely, when the coupling Gas (α, x1 ) is “matched” against the observed coupling strength at that scale, then the evolution of the coupling to Gas (α, x2 ) at the scale x2 is given by the QED beta function. We should stress, here, that the RG flow is formulated only for the beta function of the coupling; the flow of the mass is being ignored. The role of this approximation will be clarified below. Let us indicate a method of solving the RG evolution, perturbatively. With the β function begin held constant, we have approximately G (α,x2 ) dz x2 Gas (α, x2 ) − Gas (α, x1 ) )=∫ ≈ , x1 β(Gas (α, x1 )) Gas (α,x1 ) β(z) as

ln (

(19.62)

if x2 and x1 are not too far apart. Then, we have Gas (α, x2 ) ≈ Gas (α, x1 ) + ln (

x2 ) β(Gas (α, x1 )) . x1

(19.63)

A better approximation for the function in the integrand of Eq. (19.62) is 1 β ′ (z0 ) 1 = − (z − z0 ) + ⋯, β(z) β(z0 ) [β(z0 )]2

z0 = Gas (α, x1 ) .

(19.64)

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Here, terms of order (z − z0 )2 and higher have been ignored. Integrating the RG evolution to second order in z − z0 , we find that Gas (α, x2 ) − Gas (α, x1 ) =

x2 1 ln ( ) β(Gas (α, x1 )) 1! x1 1 x2 2 + [ln ( )] β(Gas (α, x1 )) β ′ (Gas (α, x1 )) + ⋯ . 2! x1 (19.65)

This result easily generalizes to all orders, Gas (α, x2 ) − Gas (α, x1 ) =

x2 1 ∂ ln ( ) {β(z) z}∣ 1! x1 ∂z z=Gas (α,x1 ) +

1 x2 2 ∂ ∂ [ln ( )] {β(z) β(z) z}∣ +⋯ 2! x1 ∂z ∂z z=Gas (α,x1 )

∞

x2 n ∂ n 1 [ln ( )] {[β(z) ] z}∣ . (19.66) x1 ∂z n=1 n! z=Gas (α,x1 )

= ∑

This equation defines Gas (α, x2 ) as a power series in the coupling Gas (α, x1 ) = αx1 . 19.3.2

Renormalization Group of Callan and Symanzik

In contrast to the approach of Gell-Mann and Low, and Callan–Symanzik equation also implies a variation of the mass parameter of the physical theory. The Gell-Mann low β function can only be applied in the asymptotic region of large momenta, and it describes only the RG flow of the coupling. If we wish to use the QED β function in the non-asymptotic region, we also have to include the running of the mass parameter. This is done in the approach of Callan and Symanzik [130], which is based on a variation of the renormalized and/or bare masses of the bare theory, while keeping the bare coupling and cutoff scale Λ constant. According to the principle of renormalization, the effective charge at the scale x = −⃗ q 2 /m2 can be written in terms of a renormalization constant Z3 , which in turn can be expressed in terms of the renormalized value αx and the ratio Λ2 /m2 , i.e., in terms of the renormalized quantities, αx = Z3 (

Λ2 , αx ) α0 . m2

(19.67)

As Λ → ∞, the bare coupling which is the effective, high-energy coupling, grows indefinitely, and to keep αx finite, we have to demand that Z3 → 0 for Λ → ∞. In order to fix ideas, we observe that this implies that αx actually increases when we increase the mass parameter m2 in the denominator of the Z3 renormalization constant; this otherwise corresponds to a decrease of Λ and a corresponding increase in Z3 and αx . Callan and Symanzik observed that the logarithmic derivative with respect to the mass of the effective charge at scale x can be expressed in terms of αx alone,

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at least in the limit Λ → ∞. The derivative is to be taken at constant cutoff Λ and constant bare coupling α0 , but varying renormalized mass m (and, as a consequence, varying bare mass m0 ), m2

Λ2 ∂ 2 ∂ α ∣ = m (Z ( , αx ) α0 )∣ x 3 ∂m2 ∂m2 m2 α0 ,Λ=const. α0 ,Λ=const. ∂ Λ2 (Z ( , αx ))∣ 3 ∂m2 m2 α0 ,Λ=const. RRR αx Λ2 RRR 2 ∂ = (Z ( , α )) m 3 x Λ2 RRRR ∂m2 m2 Z3 ( m 2 , αx ) Rα0 ,Λ=const. 2 Λ ∂ = αx m2 ln (Z3 ( 2 , αx ))∣ ∂m2 m α0 ,Λ=const. = α0 m2

= − αx Λ2

Λ2 ∂ Λ→∞ ̃ ln (Z3 ( 2 , αx ))∣ = β(αx ) . 2 ∂Λ m α0 ,m=const. (19.68)

The relation of β̃ to the β function defined in Sec. 19.3.1 will be clarified below. ̃ x ) still is a As already noted, αx shrinks with growing Λ, which explains why β(α positive quantity if x describes a space-like momentum transfers. The derivation of the Callan–Symanzik equation starts from the basic relation fulfilled by the effective charge, which we write as follows, G(α, q 2 , m2 ) = G (Z3 (

Λ2 , α) α0 , q 2 , m2 ) = lim G(α0 , q 2 , m20 , Λ2 ) . Λ→∞ m2

(19.69)

This equation might raise some questions, because the right-hand side appears to be independent of Λ, while the expression in the middle appears to be manifestly Λdependent. The rationale is that, once the cutoff Λ and the physical coupling α are fixed, the renormalization constant Z3 which relates the bare and the renormalized coupling is also determined, by the relation α = Z3 (

Λ2 , α) α0 , m2

(19.70)

where α0 is the bare coupling, i.e., the coupling at scale Λ. The same relation must hold in the limit Λ → ∞, and this justifies the functional form of Eq. (19.69). We now vary the mass while keeping the bare coupling and the scale Λ constant. There is an additional ingredient to the formulation of the Callan–Symanzik equation which we have to discuss. Without derivation, we here indicate that the differentiation of the inverse 1/G of the effective charge, as given by vacuum polarization diagrams, generates mass insertions in the Feynman diagrams, often denoted by a cross in the diagrammatic language. We can define an inhomogeneity function Z0−1 ∆ (α,

q2 ∂ ) = lim m0 G−1 (α0 , q 2 , m20 , Λ2 ) , Λ→∞ m2 ∂m0

(19.71)

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where again we take account of the running of the mass from its bare value m0 to its renormalized value m. The right-hand side can only depend on the parameters that the left-hand side depends upon. Note that the mass insertion does not correspond to any available term in the original QED Lagrangian. A term that generates a mass insertion can be formulated, if desired, by an infinitesimal addition to the QED Lagrangian (or generating functional), but it necessitates the introduction of a further renormalization constant in addition to the vertex renormalization Z1 , the wave function renormalization Z2 , and the charge renormalization Z3 . The new, additional renormalization constant is conveniently referred to as Z0 . We now vary the mass parameter while keeping the bare coupling and the scale Λ constant. A change in the mass scale of the left-hand side of the coupling given in Eq. (19.69) results in the relation ∂ −1 Λ2 ∂ −1 G ∣ =m G (Z3 ( 2 , α) α0 , q 2 , m2 ) m ∂m ∂m m α0 ,Λ=const. = α0 (2 m2

∂ −1 ∂ Λ2 Z ( , α))∣ G (α, q 2 , m2 ) 3 2 2 ∂m m ∂α α0 ,Λ

∂ −1 G (α, q 2 , m2 ) ∂m ∂ −1 ∂ Λ2 2 α m2 Z3 ( 2 , α)∣ G (α, q 2 , m2 ) = Λ2 2 m ∂α Z3 ( m2 , α) ∂m α0 ,Λ +m

∂ −1 G (α, q 2 , m2 ) ∂m ∂ ∂ ̃ +m ) G−1 (α, q 2 , m2 ) . = (2 β(α) ∂α ∂m ̃ Our definition of the β(α) function, α m2 ∂ Λ2 ̃ β(α) = Z3 ( 2 , α)∣ , Λ2 2 m Z3 ( m2 , α) ∂m α0 ,Λ +m

(19.72)

(19.73)

differs by a factor two from the definition in Eq. (13.25) of Ref. [2]. As we shall see later, our definition ensures that the first two terms of the β function are renormalization-scheme independent. For the left-hand side of Eq. (19.72), we can rewrite the equation in terms of differentiations with respect to the bare mass, and, using Eq. (19.71), ∂ −1 lim m G (α0 , q 2 , m20 , Λ2 )∣ Λ→∞ ∂m α0 ,Λ m ∂m0 ∂ ( m0 G−1 (α0 , q 2 , m20 , Λ2 )∣ ) Λ→∞ m0 ∂m ∂m0 α0 ,Λ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

= lim

=Z0−1 ∆(α,q 2 /m2 )

m ∂m0 −1 q2 q2 Z0 ) ∆ (α, 2 ) = (1 + δ(α)) ∆ (α, 2 ) , Λ→∞ m0 ∂m m m ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

= lim (

=1+δ(α)

(19.74)

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where we have used Eq. (19.71). The Callan–Symanzik renormalization-group equation thus reads as ∂ ∂ q2 q2 ̃ (2 β(α) +m ) G−1 (α, 2 ) = (1 + δ(α)) ∆ (α, 2 ) , (19.75) ∂α ∂m m m where the right-hand side describes higher-order mass insertions. The validity is not restricted to the asymptotic region; the price to pay is an additional mass insertion (nonvanishing right-hand side), and the necessity to calculate the additional renormalization constant Z0 for the field due to the mass insertion. Meaningful results can be inferred from Eq. (19.75) in the asymptotic regime of ∣q 2 ∣ ≫ m2 , where the right-hand side of Eq. (19.75) is suppressed by at least one power of q 2 /m2 as compared to the terms on the left-hand side. 19.3.3

Connection of Callan–Symanzik and Gell-Mann–Low

In the asymptotic regime of large momenta q 2 → ±∞, the inhomogeneous term in the Callan–Symanzik RG equation (19.75) is suppressed, and the coupling assumes its asymptotic behavior, G → Gas . Then, the Callan–Symanzik equation reduces to the following statement, ̃ (2 β(α)

∂ ∂ q2 −1 +m ) (Gas ) (α, 2 ) = 0 . ∂α ∂m m

(19.76)

As before, the physically most interesting limit concerns the deep Euclidean regime, ∂ q 2 = −⃗ q 2 → −∞. In view of the chain rule, ∂x [1/f (x)] = −f ′ (x)/[f (x)]2 , this implies that ∂ q2 ∂ ̃ + 2 β(α) ) Gas (α, 2 ) = 0 . (19.77) (m ∂m ∂α m Using the identity m

∂ q2 q2 ( 2 ) = −2 2 , ∂m m m

this can be reformulated as q2 ∂ ∂ q2 as ̃ (−2 ( 2 ) + 2 β(α) ) G (α, ) = 0. m ∂(q 2 /m2 ) ∂α m2

(19.78)

(19.79)

Defining x = q 2 /m2 , this relation translates into the RG equation for the flow of the QED coupling in the asymptotic regime, ∂ as ∂ as ̃ x G (α, x) = β(α) G (α, x) . (19.80) ∂x ∂α Now, this needs to be compared to the Gell-Mann–Low RG equation (19.57), ∂ x Gas (α, x) = β(Gas (α, x)) . (19.81) ∂x Using αx = Gas (α, x), Eqs. (19.80) and (19.81) imply the relation ∂αx ̃ β(αx ) = β(α) . (19.82) ∂α

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Intuitively, one would assume that since both Eqs. (19.80) and (19.81) describe the RG flow of the coupling, the connection between the equations should be found in a Jacobian factor, and this is indeed the case, ∂αx ∂α ∂αx ∂αx ̃ β(αx ) = x = (x )( ) = β(α) . (19.83) ∂x ∂x ∂α ∂α A change in the definition of the coupling amounts to a change in the renormalization scheme. Up to the Jacobian which describes the transition among different renormalization schemes, we see that the Callan–Symanzik formalism and the GellMann Low formalism are actually equivalent in the regime of large momentum transfers. Conversely, the Callan–Symanzik equation clarifies how to augment the RG flow in the non-asymptotic regime, taking into account the running of the mass and the inhomogeneous terms from the Callan–Symanzik equation, generated by the mass insertions. We briefly indicate the available results for the β function in various renormalization schemes, without going into any further detail. In the on-shell scheme, the three-loop result for the QED beta function has been given by de Rafael and Rosner in Ref. [629], and the result has been generalized to four loops by Gorishny, Kataev, Larin and Surguladze in Refs. [628, 630], α3 121 α4 α2 + 2− β̃OS (α) = 3π 4π 288 π 3 5 α 5561 23 4 7 + 4 ( − ζ(3) + ln 2 ζ(2) − ζ(3)) + O(α6 ) . (19.84) π 10368 18 3 16 It is perhaps useful to clarify that the authors of Ref. [628] refer to the β function as the “Gell-Mann–Low function”, in a slight departure from the usual naming conventions. In the minimal subtraction (MS) scheme, we have from Ref. [628] the four-loop result α2 α3 31 α4 α5 13 2785 βMS (α) = + 2− + 4 (− ζ(3) − ) + O(α6 ) . (19.85) 3 3π 4π 288 π π 36 31104 Another result of particular interest concerns so-called “quenched” QED. In the case of “perturbative quenching”, the related Feynman diagrams with internal photon vacuum polarization diagrams are not considered and therefore the coupling constant of QED is renormalized only by the subset of vacuum polarization subgraphs with one external fermion loop and 0 ≤ n ≤ ∞ internal photon lines. It can be shown, that within this approximation the QED vacuum polarization function and the related contribution to the QED β-function do not depend on the choice of the renormalization scheme [631]. A partial five-loop result for the β function βQ (α) subgraphs in quenched QED has been given in Refs. [632–634], α2 α3 1 α4 23 α5 α6 4157 1 + − − + ( + ζ(3)) + O(α7 ) . (19.86) 2 3 3π 4π 32 π 128 π 4 π 5 6144 8 The analytic structure of the 5-loop result of Ref. [11] differs from the previously known terms; it contains no ζ(3)-term in the 4-loop term, but ζ(3) does appear in the 5-loop coefficient. βQ (α) =

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Surprisingly, an inspection of Eqs. (19.60), (19.84), (19.85) and (19.86) shows that the terms of order α2 and α3 in the otherwise scheme-dependent results for the QED β function, in fact, are the same. This is not accidental. We had previously noted the Jacobian character of the scheme dependence in Eq. (19.83), which is manifest in the relations ∂αx ∂α ∂αx ̃ ̃ , β(αx ) = x , β(α) =x . (19.87) β(αx ) = β(α) ∂α ∂x ∂x We can set, by perturbative expansion, ∂αx = 1 + 2 a1 α + 3 a2 α2 + O(α3 ) . (19.88) ∂α The Taylor expansion can of course be inverted to read αx = α (1 + a1 α + a2 α2 + O(α3 )) ,

α = αx (1 − a1 αx + (2a21 − a2 ) αx2 + O(αx3 )) .

(19.89)

The β̃ function in the transformed scheme is given as ̃ β(α) = b2 α2 + b3 α3 + b4 α4 + O(α5 ) ,

(19.90)

where b2 is the one-loop, and b3 is the two-loop coefficient. The scheme independence of the first two perturbative coefficients of the QED β function can be verified as follows, ∂αx = (b2 α2 + b3 α3 + O(α4 )) (1 + 2 a1 α + 3 a2 α2 + O(α3 )) ∂α = b2 α2 + (b3 + 2 b2 a1 ) α3 + O(α4 )

̃ β(αx ) = β(α)

= b2 (αx − a1 αx2 )2 + (b3 + 2 b2 a1 ) αx3 + O(αx4 ) = b2 αx2 + b3 αx3 + O(αx4 ) .

(19.91)

This shows that the first two coefficients of the QED β function are schemeindependent, which is what we wanted to show. The next higher-order term is β(αx ) = b2 αx2 + b3 αx3 + b4 αx4 + (−a21 b2 + a2 b2 − a1 b3 ) αx4 + O(αx5 ) ̃ x ) + (−a2 b2 + a2 b2 − a1 b3 ) α4 + O(α5 ) . = β(α 1 x x

(19.92)

In order to fix ideas, we should mention that an obvious requirement on the β function is that the mapping α → αx must be infinitely differentiable, and it must be strictly monotonous so that no artificial zeros or zeros of the derivatives of the β functions are being introduced. 19.4 19.4.1

RG Equations and Applications Effective Action and Renormalization Group

The QED effective action is a Lagrangian, written in terms of the electromagnetic fields, which describes a quantum-corrected Maxwell Lagrangian. In the strong-field limit, the mass of the electron becomes irrelevant, and one may thus ask the question to which extent the effective charge regime can be related to the analytic form of the

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Renormalization-Group Equations

QED effective Lagrangian, at least in the asymptotic regime. Indeed, connections between the strong-field and short-distance limits of quantum electrodynamics, and the effective Lagrangian, have been studied in Refs. [635, 636]. The first aspect to notice is that the product Aµ0 1/2 = e0 Aµ0 (19.93) e Aµ = (Z3 e0 ) 1/2 Z3 actually is invariant under RG transformations. We can intuitively understand that, in the asymptotic region, the Maxwell Lagrangian must contain the effective charge α(t) of the renormalization group “time” t, via the renormalization of the field and the coupling e2 = 4 π α, 1 1 1 α L0 = − F0µν F0,µν → − F µν Fµν = − F µν Fµν . (19.94) 4 4Z3 4 α(t) This structure is rigorously shown in Ref. [637] (see also the references therein). The parameter t has to be defined to be consistent with the strong-field asymptotic behavior of the QED effective Lagrangian, t=

1 α ∣F 2 ∣ ln ( 4 ) 4 µ

(19.95)

⃗2 − E ⃗ 2 ). Because t is proportional to the logarithm of the where F 2 = F µν Fµν = 2(B renormalization scale µ, and in view of the identity d ln(µ) = µ−1 dµ, the β function can be expressed as 2β(α(t)) =

d α(t) , dt

t=∫

α(t) α

dα′ . 2β(α′ )

(19.96)

The factor two is a consequence of our definition (19.59). We recall, from Eq. (19.60), the first two perturbative terms of the β function as β(α) = b2 α2 + b3 α3 + ⋯, and obtain from the RG evolution, 1 1 = − 2 β2 t − 2 α β3 t + O(t2 ) , (19.97) α(t) α so that the structure of the QED effective Lagrangian (including the leading Maxwell term) reads as follows, 1 α β2 2 α ∣F 2 ∣ α 2 β3 2 α ∣F 2 ∣ L = − F2 + F ln ( 4 ) + F ln ( 4 ) + O(α3 ) . 4 8 µ 8 µ

(19.98)

For QED, the asymptotic behavior of the one-loop, and two-loop Lagrangian is known from the calculations of Ritus [635, 636]. For a constant, strong, magnetic background field, we have ⃗2 ⃗2 ⃗2 ⃗2 αB αB α2 B αB L(1) = ln ( 4 ) , L(2) = ln ( 4 ) . (19.99) 2 12π m 16π m ⃗ 2 , which is valid for a purely magnetic background field, Using the relation F 2 = 2 B we can read off the coefficients β2 =

1 , 3π

β3 =

1 , 4π 2

(19.100)

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in agreement with the result indicated in Eq. (19.60). (We recall that the first two coefficients of the QED β function actually are scheme-independent.) We here identify the electron mass with the renormalization scale µ. One might thus think that the QED effective action would be an ideal vehicle for studying the RG evolution of the effective charge, possibly even superior to the QED β function. However, that is not the case. First, we recall that the RG evolution discussed above relies on the asymptotic regime of strong fields, which is equivalent to the approximation made previously in deriving the Gell-Mann–Low β function. Second, and this point is less obvious, the RG evolution of α from the QED effective action, and from the Gell-Mann–Low β function is not quite as universally matched as one would otherwise expect. In Ref. [637], one considers ̃µν , where self-dual (sd) background fields with Fµν = Fµν = F ̃µν = 1 µνρδ F ρδ , F 2 ≡ F µν Fµν . (19.101) Fµν = F 2 These fields are still constant, and strong, but self-dual, so that the dual field strength tensor equals the field strength tensor. In a self-dual background field [see Eqs. (2.30) and (2.31) of Ref. [637]], the strong-field asymptotics do not match the coefficients of the β functions at all, α ∣F 2 ∣ α2 F 2 α ∣F 2 ∣ α F2 (2) (1) ln ( ) , L = − ln ( ). (19.102) Lsd = − sd 48π m4 16π 2 m4 The explanation for the apparent mismatch of the strong-field behavior of the QED effective Lagrangian in a self-dual background, given in Ref. [637], is subtle and depends on the loop order. At one loop, one can argue that in a self-dual background, as opposed to a constant magnetic field, so-called zero modes contribute to the effective action. These zero modes are virtual states of the Dirac field with a specific helicity, and they contribute to the fermionic determinant, but only in the self-dual background field, which has both electric and magnetic field components. The zero modes come from the states of a massless Dirac field, which always is in a specific helicity state. For the two-loop, self-dual term in Eq. (19.102), a different explanation is offered in Ref. [637]. Namely, it is argued that the RG equation fulfilled by the Lagrangian in the self-dual background should read as follows, ∂ ∂ ∂ + β(α) − γm m ) L(α, m, µ, e Fµν ) = 0 , (19.103) (µ ∂µ ∂α ∂m in contrast to the magnetic-field case, where the mass does not run, ∂ ∂ ⃗ = 0. (µ + βm (α) ) L(α, m, µ, eB) (19.104) ∂µ ∂α ⃗ act as spectators In both Eqs. (19.103) and (19.104), the quantities e Fµν viz. eB under the RG evolution. Furthermore, µ is the renormalization scale, which also enters the asymptotic form of the one-loop Lagrangian, F α F2 α ∣F 2 ∣ α F2 m2 (1) Lsd = − − )+ (19.105) ln ( ln ( 2 ) . 4 4 48π m 12π µ

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One can now read off the anomalous mass dimension as follows, γm = −

3α µ ∂m =− + ⋯. m ∂µ 2 π

(19.106)

This implies that at two-loop order, one can then trade the anomalous dimension term in Eq. (19.103) for a mass-dependent β function, 2 α2 1 α3 α2 + + γ + O(α4 ) m 3 π 2 π2 π 2 α2 α3 2 α2 1 α3 3 α3 = + − + O(α4 ) = − 2 + O(α4 ) (19.107) 2 2 3 π 2 π 2 π 3 π π which perfectly matches the two-loop coefficient of the strong-field limit of the selfdual spinor case given in Eq. (19.102). The correspondence between the strong-field limit and the β function in the self-dual spinor case at two-loop only applies to the β function βm of an implicitly electron-mass-dependent regularization scheme. The mass-dependent scheme is indicated for the self-dual spinor case because of the presence of the zero modes which inhibit a direct massless limit. Via the running one-loop electron mass in the Dirac propagator insertion of the two-loop effective action, the anomalous mass dimension creeps into the renormalization-group analysis of the QED effective action, in the case of a self-dual background field. While the matching of the QED β function with the strong-field limit of the QED effective action does not universally hold for all types of background fields, one can find a very straightforward connection for the case of a magnetic background, and a rather subtle connection for the self-dual background field case. βm (α) =

19.4.2

Brodsky–Lepage–Mackenzie Scale Setting

We start from the two-loop result for the anomalous magnetic moment of the electron, which has been obtained in Refs. [54,639–641]. Combining the one-loop vertex calculation of the electron g factor [2] with the two-loop result, we have α α ge − 2 = (1 + C1 + O(α2 )) , 2 2π π where the two-loop coefficient C1 has the numerical value 197 3 C1 = + ζ(2) − 6ζ(2) ln(2) + ζ(3) = −0.657 . 72 2 The muon anomalous magnetic moment is different, gµ − 2 α α aµ = = (1 + D1 + O(α2 )) , 2 2π π where m2µ 1 25 D1 ≈ C1 + [ln ( 2 ) − ] . 3 me 18 ae =

(19.108)

(19.109)

(19.110)

(19.111)

This result holds if we assume that the two-loop difference of the muon anomalous magnetic moment and the electron anomalous magnetic moment is mostly due to

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e

e µ

µ

(a)

µ

µ

(b)

Fig. 19.1 Insertion of the vacuum-polarization loop [Fig. (b)] into the vertex correction [Fig. (a)] relevant to the muon gµ − 2. This effect can be described, to good accuracy, by an optimally fixed momentum scale in the asymptotic form of the one-loop vacuum polarization. The interaction with the external magnetic field is denoted by the external photon line which ends in a cross.

(electron) vacuum polarization loops. We thus ignore a numerically small term, which is obtained when the electron loop is replaced by a muon loop. From the RG analysis of the running charge, we know that the formula α (19.112) α(Q2 ) = 5 α 1 Q2 1 − ( ln ( 2 ) − ) + ⋯ π 3 me 3 describes the running coupling constant at the scale Q2 = −⃗ q 2 , in the deep Euclidean region. The paradigm of scale setting now is as follows: Let us consider the difference of the muon anomalous magnetic moment and the electron anomalous magnetic moment. The question is whether we can find an “optimum” scale Q2∗ so that the difference in the two-loop terms in the muon versus electron anomalous magnetic moments can be obtained by inserting the running charge at the optimum scale Q2∗ , into the one-loop term of the electron anomalous magnetic moment, i.e., the matching criterion for Q2∗ is as follows, α2 α(Q2∗ ) α − + O(α3 ) = 2 (D1 − C1 ) + O(α3 ) . (19.113) aµ − ae = 2π 2π 2π The notion is that the running charge α(Q2∗ ) describes the two-loop vacuumpolarization insertions indirectly, by a change in the effective coupling used for the calculation of the one-loop effect, with a scale Q2∗ matched so that upon reexpansion of α(Q2∗ ), the two-loop vacuum polarization diagram in Fig. 19.1 is described. This is why the remainder term in Eq. (19.113) is assumed to be of order α3 , not α2 . Expanding the running charge (19.112) to first order in α, the criterion then is that m2µ α 1 Q2 5 α1 25 α(Q2∗ ) − α(Q2 ) = ( ln ( ∗2 ) − ) = [ln ( 2 ) − ] , (19.114) π 3 me 3 π 3 me 18 which is solved by 5 Q2∗ = m2µ exp ( ) ≫ m2e . (19.115) 6

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The hope then is that, inserting the running charge at the optimum scale Q2∗ into the two-loop result for the anomalous magnetic moment of the electron, 1 1 [α(Q2∗ ) − α] + 2 C1 [α2 (Q2∗ ) − α2 ] + ⋯ (19.116) 2π 2π one obtains an approximation to the three-loop effects. In an ideal scenario, one might thus assume that the terms denoted by the ellipsis (. . . ) in Eq. (19.116) are of order O(α4 ), not O(α3 ). The perturbation series for difference of the anomalous magnetic moments of the muon and electron is given as aµ − ae ≈

α 2 α 3 α 4 α 5 aQED − aQED = 1.09 ( ) + 22.87 ( ) + 127.00 ( ) + Cµ ( ) + O(α6 ) , (19.117) µ e π π π π with explicit numerical result for the coefficients through four-loop order. It has been verified by Kataev and Starshenko that the higher-order terms are well represented by inserting the running charge into lower-order diagram [34, 642]. The coefficients in the series (19.117) grow drastically in numerical magnitude with an increase of the loop order. This is a manifestation of the general divergence of perturbation theory in QED [132], which is especially relevant to the current situation because the muon anomalous magnetic moment is obtained from an optimum momentum scale which is much larger than the electron mass scale [see Eq. (19.115)], and therefore the effect of higher-order vacuum-polarization insertions is considerably enhanced. RG improvement is relevant to the tenth-order (in e) graphs, which are of fifth order in α, and contribute to the next higher-order analytic term not included in Eq. (19.117). In Ref. [643] a group led by Kinoshita1 has obtained an estimate of Cµ ≈ 570(14) for the term of order (α/π)5 to be included into Eq. (19.117). Karshenboim [638] has shown that the previous analysis was somewhat incomplete; he analyses the graph in Fig. 19.2(c) and obtains an additional contribution of Cµ ≈ 176(35) for the coefficient. It is worth noting that not all graphs which contribute in fifth order in α can be analyzed by RG-improved perturbation theory. The topologically new graphs in Figs. 19.2(e) and (f), whose contribution has been estimated in Ref. [638], give additional contributions of Cµ ≈ 85 ± 85 and Cµ ≈ ±40, respectively. The total result obtained by adding all of the mentioned contributions reads as Cµ = 930(170) ,

(19.118)

which is an impressive magnitude for a perturbative coefficient; with a twinkle in the eye and with some humor, we can state that the tenth-order contribution to the anomalous magnetic moment of the muon currently holds the “world record” for the largest (in numerical magnitude) perturbative coefficient obtained for any process in field theory, when the perturbation series is expressed in terms of a “reasonable” expansion parameter, namely, in our case, the QED coupling α. Very recently, a 1 Toichiro

Kinoshita (b. 1925).

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e

e

e

e

e

e

µ

µ

e

µ

µ

µ

µ

(b)

(a)

e

e

e

e

(c)

e

e

e

e e e µ

µ (d)

µ

µ (e)

µ

µ (f)

Fig. 19.2 Representative diagrams illustrating the calculation of five-loop contributions (order α5 ) to the anomalous magnetic moment of the muon. Diagram (a) is a three-loop “skeleton” diagram. The number of internal lines is I = 9, the number of vertices is V = 7, and the loop order is I − V + 1 = 3. The four-loop diagram (b) results from a one-loop vacuum-polarization insertion into diagram (a), which can be estimated based on the running charge, inserted into diagram (a). Diagrams (c) and (d) can be estimated based on the two-loop running of the effective charge, inserted into diagram (a). Finally, diagrams (e) and (f) cannot be estimated based on the renormalization-group equations for the running charge. Expressed in terms of the coefficient Cµ [see Eq. (19.117)], their contributions have been estimated (see Ref. [638]) as 185 ± 15 and ±40, respectively. The electron lines and muon lines are denoted by e and µ, respectively.

complete calculation (without estimation) of the coefficient has been presented in Ref. [644] with the result Cµ = 753.29(1.04) , (19.119) for the QED contribution to Cµ (contributions beyond QED are numerically small). This result is in fair numerical agreement with the estimate (19.118). The BLM scale setting is a form of an RG improved perturbation theory. We notice that the expression (19.112) for the running coupling has the form of a socalled [1/1]-Pad´e approximant (for an explanation of the notation, see Ref. [124]), where the running coupling is replaced by a rational function of a different expansion parameter. Furthermore, one knows that if the running were perfect, we could represent e.g. the anomalous magnetic moment of the muon as α(Q∗ )/(2π) where Q∗ is just the value of α at the perfectly matched scale Q∗ . Of course, in their original work [92], Brodsky, Lepage and MacKenzie explicitly state that the matching only makes sense for the vacuum-polarization diagram insertions. They do not

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include topologically new graphs which can contribute in each higher order, such as those shown in Fig. 19.2. Yet, we could be a little bold. Using the effective coupling, we effectively replace the perturbation series by a Pad´e approximant to itself, which upon reexpansion yields back the perturbation series whose convergence was to be improved. This is known an accuracy-through-order relation [124]. As an open problem, we mention here the conceivable replacement of the rational function fixing the effective charge (19.112), by a Pad´e approximant or even a Weniger transformation [645], which might improve the scale setting procedure slightly. We also notice that the continuous readjustment of the optimum scale for the RG improvement, order-by-order in perturbation theory, is reminiscent of the celebrated order-dependent mappings originally introduced by Seznec and Zinn-Justin [646]. 19.5

Further Thoughts

Here are some suggestions for further thought. (1) Predictive Limits. In view of the considerations reported in Sec. 3.4.2, extrapolate some numerical coefficients for the anomalous magnetic moment of the muon and try to determine the predictive limits of quantum electrodynamics. Also, consult Ref. [130]. Note that estimates contain the factor Γ(k/2) in the kth loop order in QED as opposed to Γ(k) (see Sec. 3.4.2). (2) Beta Function at Large Coupling. Consult Refs. [647, 648] for quite recent work on the large-order behavior of the coefficients of the QED β function. It is claimed in Ref. [648] that the perturbative expansion of the β function has the asymptotic form β(α) ∼ C ∑ A−K Γ(N /2 + 6) αK , K

A = 4.886 ,

(19.120)

in accordance with Eq. (3.105). Based on this conjecture, and on the numerically observed trend in the large-coupling behavior of the summations (using order-dependent mappings [130]) of the first few partial sum of the QED β function, it has been conjectured [647] that the entire β function has the asymptotic behavior β(α) ∼ β∞ αλ ,

β∞ = 1.0 ± 0.3 ,

λ = 1.0 ± 0.1 .

(19.121)

The exact equality β∞ = λ = 1 has been conjectured in Ref. [648]. A confirmation of this result by independent methods would otherwise imply a linear growth of the effective QED coupling at large momenta and be consistent with what is expected for Grand Unification. (3) Toward Grand Unification. Study the RG evolution of QED, and the electroweak sector of the Standard Mode, and of quantum chromodynamics (QCD). Why do we conjecture the confluence of all coupling parameters in nature at very high energy scales, and how is this related to Grand Unification?

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(4) Novel Interactions. Explore weak and other beyond-Standard-Model interaction effects in atomic systems. According to Eq. (6.100), the atomic wave function can be represented as follows, ⎛ φ ⎞ ψ=⎜σ ⃗ ⋅ p⃗ ⎟ , ⎝ 2m φ ⎠

⎛ φ+ ⎞ ψ =⎜ +σ ⃗ ⋅ p⃗ ⎟ , ⎝ φ 2m ⎠ +

ψ = ψ+ γ 0 .

(19.122)

Here, m is the bound particle mass (for us, the electron) and φ is the Schr¨ odinger–Pauli wave function, while ψ is the relativistic Dirac wave function. The representation is valid for essentially nonrelativistic systems, with first relativistic corrections in the lower components. Consider a hypothetical novel particle X1 with mass MX1 which has an axial coupling to the electron and a vector coupling to the quarks in the nucleus. Convince yourself that one can obtain a parity-violating (PV) interaction Hamiltonian by combining the µ = 0 axial component of the electron current with the µ = 0 vector component of the quark current, as follows, (1)

HPV = ψ γ 0 geA γ 5 ψ

1 ψ q γ 0 gqVii ψqi , MX1 2 i

(19.123)

with suitable identifications of the parameters. The γ 5 matrix in atomic theory is almost exclusively used in the Dirac representation, and is given in Eq. (7.15). Derive, in the nonrelativistic limit, the relevant effective Hamiltonian, (1)

HPV =

geA gqVii 2 MX

⟨ψ qi γ 0 ψqi ⟩ (

⃗ ⋅ p⃗ (3) ⃗ ⋅ p⃗ σ σ δ (⃗ r) + δ (3) (⃗ r) ) m m

(19.124)

to be evaluated on the atomic wave function. Convince yourself that the corresponding term for the weak interaction is ⃗ ⋅ p⃗ (3) ⃗ ⋅ p⃗ GF 1 (1) σ σ (1) HPV = √ QW (Z, N ) ( δ (⃗ r) + δ (3) (⃗ r) ). m m 4 2m

(19.125)

(1)

Here, QW (Z, N ) is the weak charge of the nucleus. Compare with Refs. [649, 650]. One can also obtain a parity-violating (PV) interaction Hamiltonian by combining the µ = 1, 2, 3 vector component of the electron current, with the µ = 1, 2, 3 axial vector component of the quark current, as follows, (2)

HPV = ψ γ 0 glVaa γ⃗ ψ

1 MX1 2

ψ qi γ 0 gqAii γ⃗ γ 5 ψqi ,

(19.126)

again, with a suitable identifications of the parameters. Convince yourself that one can assume that (q) ⃗ I, I3 ⟩ , ⟨ψ qi γ 0 gqAii γ⃗ γ 5 ψqi ⟩ = ∑ GA (Z, N ) δ (3) (⃗ r) ⟨I, I3′ ∣I∣ q

(19.127)

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where I⃗ is the nuclear spin vector. In view of ψ γ 0 = ψ + , this leads to (2) HPV

(q)

geV ∑q GA (Z, N ) σ ⃗ ⋅ p⃗ ⃗ ⋅ p⃗ ⃗ σ ⃗ ⋅ I⃗ δ (3) (⃗ ⃗ ⋅ I) , = ( σ r) + δ (3) (⃗ r) σ 2 MX m m 1

(19.128)

to be evaluated on the atomic wave function. In fact, the corresponding term for the weak interaction is ⃗ ⋅ p⃗ (3) ⃗ ⋅ p⃗ GF 1 (2) σ σ (2) HPV = √ QW (Z, N ) ( δ (⃗ r) + δ (3) (⃗ r) ), m m m 4 2 (2)

(19.129)

where QW (Z, N ) is an anapole moment of the nucleus. Be inspired by Ref. [651]. (5) Higher-Order Corrections. Generalizing results reported in Refs. [56–59] and Ref. [55], evaluate the electron anomalous magnetic moment to such high loop orders that you are able to check about the asymptotic large-order limit, where the series should divergence factorially. This task might last a lifetime!

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[19] B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules. Benjamin Cummings, New York (2000). [20] M. Weissbluth, Atoms and Molecules. Academic Press, New York (1978). [21] W. R. Johnson, Atomic Structure Theory. Springer, Heidelberg (2010). [22] I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules. Springer, Heidelberg (2007). [23] I. Lindgren and J. Morrison, Atomic Many-Body Theory. Springer, Heidelberg (1986). [24] L. N. Labzowsky, G. L. Klimchirskaya and Y. Y. Dmitriev, Relativistic Effects in the Spectra of Atomic Systems. Taylor and Francis, London (1993). [25] J. J. Sakurai, Modern Quantum Mechanics. Addison-Wesley, Reading, MA (1994). [26] J. J. Sakurai, Advanced Quantum Mechanics. Addison-Wesley, Reading, MA (1967). [27] M. I. Eides, H. Grotch and V. A. Shelyuto, Theory of Light Hydrogenic Bound States (Springer Tracts in Modern Physics 222). Springer, Berlin, Heidelberg, New York (2007). [28] P. W. Milonni, The Quantum Vacuum. Academic Press, San Diego (1994). [29] W. Dittrich and H. Gies, Probing the Quantum Vacuum (Springer Tracts in Modern Physics 166). Springer, Berlin, Heidelberg, New York (2000). [30] P. J. Mohr, B. N. Taylor and D. B. Newell, CODATA recommended values of the fundamental physical constants: 2010, Rev. Mod. Phys. 84, pp. 1527–1605 (2012). [31] See the URL http://physics.nist.gov/cuu/constants (2021). [32] L. Essen, R. W. Donaldson, M. J. Bangham and E. G. Hope, Frequency of the Hydrogen Maser, Nature (London) 229, pp. 110–111 (1971). [33] L. Essen, R. W. Donaldson, E. G. Hope and M. J. Bangham, Hydrogen Maser Work at the National Physical Laboratory, Metrologia 9, pp. 128–137 (1973). [34] S. G. Karshenboim, Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants, Phys. Rep. 422, pp. 1–63 (2005). [35] N. Kolachevsky, M. Fischer, S. G. Karshenboim and T. W. H¨ ansch, High-Precision Optical Measurement of the 2S Hyperfine Interval in Atomic Hydrogen, Phys. Rev. Lett. 92, p. 033003 (2004). [36] S. G. Karshenboim and V. G. Ivanov, Hyperfine structure of the ground state and first excited states in light hydrogen-like atoms and high-precision tests of QED, Eur. Phys. J. D 19, pp. 13–23 (2002). [37] U. D. Jentschura and V. A. Yerokhin, Quantum electrodynamic corrections to the hyperfine structure of excited S states, Phys. Rev. A 73, p. 062503 (2006). [38] W. Liu, M. G. Boshier, S. Dhawan, O. van Dyck, P. Egan, X. Fei, M. Grosse Perdekamp, V. W. Hughes, M. Janousch, K. Jungmann, D. Kawall, F. G. Mariam, C. Pillai, R. Prigl, G. zu Putlitz, I. Reinhard, W. Schwarz, P. A. Thompson and K. A. Woodle, High Precision Measurements of the Ground State Hyperfine Structure Interval of Muonium and of the Muon Magnetic Moment, Phys. Rev. Lett. 82, pp. 711–714 (1999). [39] V. A. Yerokhin and U. D. Jentschura, Electron Self-Energy in the Presence of a Magnetic Field: Hyperfine Splitting and g Factor, Phys. Rev. Lett. 100, p. 163001 (2008). [40] V. A. Yerokhin and U. D. Jentschura, Self-energy correction to the hyperfine splitting and the electron g factor in hydrogenlike ions, Phys. Rev. A 81, p. 012502 (2010).

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Index

3j Symbol, 200, 219 6j Symbol, 200 9j Symbol, 200 F1 Form Factor, 381 F2 Form Factor, 381, 449 LS Coupling, 11, 508 S Matrix, 189, 342, 354, 360, 367, 426, 462 S States, 131 SO(3, 1) Symmetry, 78 SO(4) Symmetry, 76, 104 α(Zα)5 Correction, 561 CPT Invariance, 237 g Factor, 1, 2, 439 Subtlety, 439 jj Coupling, 11, 508 21 cm Line, 491

Arbitrary Spin, 479, 483 Atom-Surface Interaction, 176 Atomic Decay, 8, 142 Atomic Decay Rate, 55 Atomic Units, 22, 27 Auxiliary Heavy Fermions, 403 Bare Charge, 383 Bare Lagrangian, 402 Barker–Glover Correction, 487 Bender–Wu Formulas, 72 Bessel Function, 103, 246, 286 Spherical, 244 Beta Decay, 53 Beta Function, 717, 730 Bethe Logarithm, 8, 75, 116, 129, 133, 519 P States, 136 S States, 135 Helium, 519 Relativistic, 572 Bethe–Salpeter Equation, 11, 593, 659 Exact Solutions, 608 Kernel, 595, 598, 671, 673 NRQED Approach, 671 Orthonormality, 601 Perturbation Theory, 614 Recoil Correction, 623 Reference Kernel, 608 Wave Function, 601 Bethe–Salpeter Wave Function, 596 Bhabha Scattering, 370 Binding Parameter, 11 Bispinor, 209, 227, 230, 239, 425 Black Hole, 308 Bohr Radius, 4, 20, 27, 112

ac Stark Shift, 57, 212 Action at a Distance, 329 Addition Theorem, 110 Adiabatic Damping, 58, 160, 350 Advanced Green Function, 324, 328 Advanced Propagator, 248 Ampere–Maxwell Law, 316 Analytic Function, 376 Anapole Moment, 381 Angular Algebra, 221 Angular Momentum Operator, 192 Angular–Momentum Decomposition, 260 Annihilation Operator, 33 Anomalous Magnetic Moment, vii, 2, 399, 432, 437, 439, 491 Antiparticle, 248, 249, 261, 309, 311, 367, 434, 538 Araki–Sucher Distribution, 579 781

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Borel Plane, 70 Borel–Pad´e Summation, 68 Borel-Type Integrals, 67 Boson, 20 Bound-State Calculations, 21 Bound-State Formalism, 607 Bound-State Pole, 594 Bound-State Wave Function, 266 Breit Hamiltonian, 10, 137, 461, 474, 475, 479, 483, 486, 513, 557, 618 Bremsstrahlung, 382 Brodsky–Lepage–Mackenzie Scale Setting, 739 Brown–Ravenhall Disease, 538 Brownian Motion, 19 Buckingham π Theorem, 25 Callan–Symanzik Equation, 719, 731 Campbell–Baker–Hausdorff Formula, 346, 458, 700 Canonical Momentum, 328, 443 Cartesian Coordinates, 212 Casimir Effect, 116 Casimir Operator, 78 Casimir–Polder Interaction, viii, 20, 40, 149, 159, 182, 463 Cauchy Residue Theorem, 407 Center-of-mass Frame, 76, 461, 489, 492, 494, 497, 510, 512, 513, 576, 580, 594 Charge Conjugation, 227, 234, 311, 697 Charge Renormalization, 384, 400 Charge Screening, 716 Chirality, 241 Cholesky Decomposition, 524 Christoffel Symbol, 293 CI, 542 Clebsch–Gordan Coefficients, 8, 191, 195, 198, 203 Clifford Algebra, 302 CODATA, 21 Complex Plane, 376 Compton Wavelength, 20, 23 Concatenation Approach, 90 Configuration-Interaction Code, 542 Connected Diagram, 417 Contiguous Relations, 97, 99, 277 Continuum State, 51, 86, 89 Continuum-State Wave Functions, 274 Convection Current, 450, 488, 621, 662 Correlated Basis, 10, 520

Coulomb Field, 436 Coulomb Gauge, 9, 29, 322, 328, 359, 432 Coulomb Potential, 8, 26, 79, 85, 391, 392, 406, 443, 689 Non-Integer Dimension, 392 Repulsive, 89 Counter Term, 385 Covariant Derivative, 265, 266, 293, 301, 304 Creation Operator, 33 Current Operator, 442 Fourier Transformation, 447 Darwin Term, 27, 436 Darwin–Foldy Term, 482 Decay Rate Magnetic, 142 Radiative, 140 Density Of States, 52 Deuterium, 503 Dielectric Function, 187 Dielectric Surface, 187 Dimensional Regularization, 9, 360, 390, 394, 399, 402, 403, 419, 423, 454, 626 Dipole Polarizability, 60 Dipole Transition, 218 Dirac Adjoint, 240, 310, 368 Dirac Angular Momentum Operator, 206 Dirac Angular Quantum Number, 138, 206, 241, 267, 545 Dirac Bra-Ket Notation, 199 Dirac Equation, 8, 136, 227, 245 Brown–Ravenhall Disease, 538 Free, 246 Massless, 243 Radiatively Corrected, 137 Dirac Field, 248, 359 Bound-State Propagator, 260 Charge Conjugation, 311 Curved Space-Time, 310 Dirac Adjoint, 240, 310, 368 Dirac–Coulomb Propagator, 281 Dirac–Volkov Propagator, 289 Field Determinant, 692 Free Particle, 427 Free Propagator, 248, 257 Gravitational Coupling, 293 Lagrangian, 238, 298, 311 Operator, 248, 265 Propagator, 248, 368

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Index

Dirac Form Factor, 381, 448, 455, 666 Dirac Green Function, 257 Dirac Hamiltonian, 240, 243, 246, 257, 377, 425 Free, 425, 428 Radiatively Corrected, 429 Dirac Matrix, 229, 232, 365, 371 Covariant Derivative, 304 Curved Space-Time, 297 Dirac Particle, 9, 427 Dirac’s Large Number Hypothesis, 6 Dirac-δ Function, 28, 52 Transverse, 32 Dirac-δ Potential, 137 Dirac–Coulomb Equation, 9 Dirac–Coulomb Green Function, 258 Dirac–Coulomb Hamiltonian, 266, 281, 429, 724 Dirac–Coulomb Propagator, 265, 281 Dirac–Coulomb Wave Function Bound State, 266 Continuum State, 274 Dirac–Mohr Matrices, 316 Dirac–Schwarzschild Hamiltonian, 308, 457 Dirac–Volkov Equation, 9 Dirac–Volkov Propagator, 265, 289 Dirac–Volkov Wave Function, 284 Discretized Field Operator, 40 Dispersion Relation, 566 Distribution, 325 Doppler Shifts, 513 Drachmanization, 438, 528 Dynamic Polarizability, 60, 96, 145 Matrix Element, 100 Static Limit, 100 Dynamic Stark Effect, 57, 128, 212 Dynamic Stark Shift, 128 Effective Action, 736 Effective Hamiltonian, 462, 468 Breit, 475 Single-Particle, 478 Two-Particle, 474, 477, 485 Effective Lagrangian, 691, 711 Wichmann–Kroll Potential, 711 Einstein Equivalence Principle, 5 Einstein Summation Convention, 34, 165, 192 Einstein–Hilbert Action, 294

Einstein–Hilbert Equation, 295 Electric Field Longitudinal, 30, 328 Transverse, 330 Electric Field Operator, 36 Electric-Dipole Interaction, 62 Electromagnetic Waves, 17 Electron Compton Wavelength, 25 Electron Mass, 1 Electron Rest Mass, 27 Electron Spin, 9 Energy Normalization, 88, 89 Energy Variable, 97 Euclidean Four–Momentum, 104 Euler Beta Function, 72 Euler–Lagrange Equation, 18, 296 Euler–Mascheroni Constant, 118, 396 Exponential Integral, 68 External Field, 443, 504 Faraday Law, 179, 189, 316 Faraday Tensor, 660 Fermi’s Golden Rule, 49, 89, 363 Fermion Field, 13 Fermionic Determinant, 691 Feynman Gauge, 342, 371, 412, 419 Feynman Green Function, 248, 324 Feynman Parameter, 392, 394, 404 Feynman Propagator, 159, 248, 253, 321 Dirac Field, 252 Photon Field, 341 Feynman Rules, 361, 367, 659 Furry Picture, 561 Phase Factor, 370 Feynman Slash, 238, 693 Field Anticommutator, 262 Field Commutator, 9, 340 Field Momentum, 253, 416, 558 Field Operator, 159, 248, 265, 321 Discretized, 40 Electric, 36 Magnetic, 36 Field Propagators, 9 Field Quantization, 29, 62, 248 Field Renormalization, 385 Field Strength Tensor, 359, 705 Fifth Current, 423 Fine Structure, 485, 618, 677 Fine–Structure Constant, 3, 13, 20 Finite Rotations, 204

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Quantum Electrodynamics in Action: Atoms, Lasers and Gravity

Finite-Size Correction, 155, 502, 547 First Quantization, 19 Fock Space, 40, 338 Fokker Precession, 457 Foldy–Wouthuysen Transformation, 10, 425, 429, 457, 593 Arbitrary Spin, 479 Eighth Order, 432 Form Factor, 448 Free Particle, 427 General Case, 429 General Electromagnetic Coupling, 432 General Particle Hamiltonian, 434 Gravity, 457 Iterative, 428 Magnetic Field Coupling, 439 Transition Current, 442, 457, 472 Forest Formula, 9, 420 Form Factor, 380, 399, 448, 557, 666 Four-Dimensional Spherical Coordinates, 107 Four-Vector Potential, 328, 359 Fourier Backtransformation, 263 Fourier Transformation, 107, 111, 155, 263 Fredholm Determinant, 695, 697 Friar Moment, 149, 158 Fried–Yennie Gauge, 343 Fundamental Commutators, 35 Furry Picture, ix, 11, 558 Gamma Function, 81, 378 Gamma Traces, 423 Gauge Condition, 322 Lorenz Gauge, 323 Gauge Fixing, 322, 335, 387 Gauge Invariance, 605 Gauge Transformation, 127, 334, 604 Gaunt Coefficients, 203 Gauss’s Law, 316 Gaussian Units, 29 Gegenbauer Polynomial, 109 Gell-Mann–Low–Sucher Theorem, 64, 367 Generalized Bessel Function, 286 GRASP, 544 Gravitational Constant, 308 Green Function, 63, 105, 248, 253, 323, 361 Advanced, 324, 328 Feynman, 324 Helmholtz, 147

Momentum Space, 102, 116 Perturbative, 253 Radial, 91, 93, 96, 98 Retarded, 324, 328 Schr¨ odinger–Coulomb, 8, 75, 90, 99, 102, 609 Time Evolution, 254 Green function Radial, 92 Gupta–Bleuler Condition, 36, 241, 323, 329, 344 Gyromagnetic Ratio, 2 Hamilton–Jacobi Formalism, 18 Hamiltonian Operator, 38 Hartree, 22, 27, 66 Hartree–Fock Approximation, 507, 533 Relativistic, 542 Hartree–Fock Equations, 535 Heaviside Step Function, 251, 393 Heaviside–Lorentz Units, 21, 410 Heisenberg Equation Of Motion, 44 Heisenberg Picture, 43 Heisenberg Uncertainty Relation, 37 Heisenberg–Euler Lagrangian, 12, 323, 692, 705, 711, 736 Speed of Light, 707 Wichmann–Kroll Potential, 711 Helicity, 227, 230, 239, 240 Helium, 10, 507 Correlated Basis, 507, 520 Hylleraas Basis Set, 522 Kato Cusp Condition, 528 Korobov Basis Set, 522 Lamb Shift, 515, 557 Mass Polarization, 511 Numerical Calculations, 525 Ortho-Helium, 508, 509 Para-Helium, 508, 509 Spectrum, 520 Hellmann–Feynman Theorem, 280 Helmholtz Green Function, 147 Hermitian Gravitational Hamiltonian, 308 Hermitian Operator, 36, 189 Hermiticity, 341, 426 Higgs Field, 6 Higgs Mechanism, 6 Hilbert Space, 149, 337, 696 Hydrogen Atom, 76 Hydrogenlike Ion, 61, 75, 450, 460

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Index

Hylleraas Basis Set, 522 Hyperfine Centroid, 3 Hyperfine Structure, 1, 3, 12, 215, 485, 487, 489, 491, 503, 648, 688 Hypergeometric Function, 276 U Function, 81, 82 1 F1 Function, 81, 86, 89 2 F1 Function, 99 2 F0 Function, 82 Confluent, 81 In Field, 350 Infrared Regulator, 398 Interaction Hamiltonian, 121, 163, 359, 477 Interaction Lagrangian, 359 Interaction Picture, 42 Interatomic Interactions, 149, 159, 462 Ion–Atom Interaction, 224 Ionization Cross Section, 88 Jacobi Polynomial, 205 Jacobi–Anger Expansion, 287 Josephson Effect, 26 Josephson Voltage System, 26 Kaluza–Klein Theories, 6 Kato Criterion, 71 Kato Cusp Condition, 528 Kernel, 595, 598, 671, 673 King Nonlinearity, 547 Klein–Gordon Equation, 8, 227 Korobov Basis Set, 522 Kramers–Kronig Relation, 188 Kronecker Delta, 337 Lagrange Multiplier, 535 Lagrangian, 225, 295 Bare, 402, 413 NRQED, 659 QED, 359 Renormalized, 413 Lagrangian Density, 238 Laguerre Polynomial, 75, 83, 92, 94, 267, 271 Associated, 83 Lamb Shift, ix, 11, 119, 425, 456, 486 Definition, 486 Helium, 515 Laser-Dressed, 42

Landau Gauge, 343, 401, 416 Laplace Operator, 79, 110 Large-Order Perturbation Theory, 67 Laser-Dressed Lamb Shift, 42 Legendre Polynomial, 80, 110, 205 Legendre Transformation, 328 Length Gauge, 121, 218 Lepton Magnetic Moment, vii Lerch Transcendent, 99, 135 Levi-Civit` a Tensor, 38, 192 Lie Algebra, 439 Lie Group, 78 Light Shift, 57, 212 Linearization, 227 Lippmann–Schwinger Equation, 11 Local Basis, 293 Local Lorentz Invariance, 5 Local Position Invariance, 5 Longitudinal Component, 30 Longitudinal Electric Field, 30, 328 Longitudinal Photon, 35, 340, 345 Loop Correction, 388 Loop Diagrams, 9 Lorentz Covariance, 227 Lorentz Index, 230, 301, 400 Lorentz Invariance Local, 5 Lorentz Transformation, 227, 513 Lorenz Gauge, 9, 322, 323, 334, 359 Møller Scattering, 370 Magnetic Decay, 142 Magnetic Field Operator, 36 Magnetic Interaction, 142, 143 Magnetic Susceptibility, 499 Many-Electron Code, 542 Many-Particle System, 461, 477, 497, 507, 531 Hartree–Fock Approximation, 533 Mass Counter Term, 125 Mass Dimension, 396 Mass Polarization, 511, 513 Matching, 462, 469, 559, 662 Matter Wave, 18 Maxwell Equations, 29, 296 Maxwell Lagrangian, 12, 703 Maxwell–Dirac–Mohr Equation, 316 MCDHF, 542 Mehler’s Formula, 148 Method of Regions, 627

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Metric, 293 Minimal Subtraction, 396, 420 Minkowski Space, 323, 376 Modified Minimal Subtraction, 396, 420 Momentum Normalization, 88 Momentum Operator, 18, 125 Momentum Space Euclidean, 104 Multi-Configuration Dirac–Fock Code, 542 Multipole Interactions, 183 Muonic Deuterium, 503 Muonic Hydrogen, 503 Muonium, 2 Natural Units, 21, 23, 28, 410 Newton Gravitational Constant, 308 NMR Spectroscopy, 499 Non-Recoil Limit, 479 Nonperturbative Calculations, 11, 572 Nonperturbative Effects, 73 Normal Ordering, 40, 41, 44 NRQED, 12, 657 Bethe–Salpeter Equation, 671 Feynman Rules, 659 Lagrangian, 659 Matching, 662 Matching Coefficients, 662 Nuclear Finite-Size Correction, 155, 502, 547 Nuclear Magnetic Resonance, 499 Nuclear Magnetic Shielding, 499 On-Shell Renormalization, 9 Orthohelium, 508 Out Field, 350 Overlapping Parameter, 118, 410 Pad´e Approximants, 70 Pair Production, 705 Para-Helium, 508 Parity, 227, 233, 235 Pauli Exclusion Principle, 533 Pauli Form Factor, 381, 448, 449, 455, 666 Pauli Matrix, 439, 513 Pauli Spin Matrices, 137 Pauli–Villars Regularization, 370, 390, 419 Perturbation Theory, 8, 149, 160, 614 First Order, 150 Higher Orders, 152

Second Order, 150 Photon Field, 321 Operator, 321 Propagator, 321, 328, 333, 334, 342, 444 Weyl Gauge, 343 Photon Mass, 334, 372, 419, 420, 451 Photon Polarization, 33 Photon Propagator, 328, 334, 342, 401 Coulomb Gauge, 333, 444 Feynman Gauge, 342, 371, 412, 419 PJVS (Programmable Josephson Voltage System), 26 Plane-Wave Solutions, 238 Pochhammer Symbol, 81 Poisson Statistics, 356 Polarizability Dipole, 60 Dynamic, 60, 96, 100 Multipole, 185, 186 Polarization, 21 Polchinski Equation, 720 Poor Man’s Scaling, 721 Positronium, 2, 503 Power–Zienau Transformation, 128, 492 Principal Value, 171 Product State, 161 Programmable Josephson Voltage System, 26 Propagator, vii, ix Proper Time, 461 Proton Radius, 158 QED Effective Lagrangian, 13 Quabla Operator, 323 Quantization, vii First, 19 Second, 19 Quantized Electromagnetic Field, 9 Quantized Fermion Field, 9 Quantized Fields, 62 Quantum Electrodynamics, 1 Quantum Mechanics, 17 Quantum SI Units, 25 Racah–Wigner Algebra, 8, 191, 203 Radial Green function, 92 Radiative Correction, 380 Recoil Correction, 11, 575, 593, 623, 681 Contact Term, 687 Final Result, 590

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Index

Hard Region, 631, 687 High-Energy Term, 587 Low-Energy Term, 577 Middle-Energy Term, 577 NRQED, 681 Seagull Term, 584, 686 Soft Region, 636, 637, 683 Ultrasoft Region, 641, 684 Recoil Corrections, 11 Recoil Operator, 547 Recoupling Coefficients, 201 Recursion Relation for Radial Powers, 137 Reduced Mass Corrections, 453 Reduced Quantity, 25 Reduced-Mass Correction, 450, 486, 488, 496, 497 Regularization Dimensional, 9, 360, 390, 394, 399, 402, 403, 419, 423, 454, 626 Pauli–Villars, 370, 390, 419 Photon Mass, 334, 372, 419, 420, 451 Regulator Infrared, 398 Ultraviolet, 398 Reinterpretation Principle, 249 Relativistic Bethe Logarithms, 572 Relativistic Correction, 27, 513, 531 Relativistic Hartree–Fock Approximation, 542 Relativistic Interaction, 461 Renormalization, 116, 380, 384, 715 Asymptotic Regime, 725 Beta Function, 717 Lagrangian, 413 Multiplicative, 385 Of Charge, 384, 400, 716, 717 Of Electromagnetic Field, 385 Of Vertex, 387 Of Wave Function, 384 On-Shell, 9 Renormalization Constant, 384, 419 Renormalization Group, 13, 388, 715, 717, 719, 721, 731, 736, 739 Asymptotic Regime, 725 Callan–Symanzik Approach, 719, 731 Evolution, 388 Gell-Mann–Low Approach, 719 Renormalized Parameters, 385 Resolvent Operator, 152 Resummation, 69, 70

Resurgent Expansions, 72 Retardation, 11, 159, 175 Retarded Green Function, 324, 328 Ricci Rotation Coefficients, 300, 306 Ricci Tensor, 294 Riemann Curvature Tensor, 294 Riemann Zeta Function, 99, 135 Rotation Matrix, 192, 204 Runge–Lentz Vector, 78 Runge–Lenz Operator, 76 Rutherford Scattering, 361 Rydberg Constant, 1, 4 Rydberg Electron, 219 Rydberg State, 219, 220 Rydberg–Core Interaction, 224 Sachs Form Factor, 158 Salpeter Correction, 11, 12, 517, 575 Salpeter Equation, 604 Sargent’s Law, 54 Scalar Photon, 35, 340, 345 Scalar Potential, 30, 328 Scale Transformation, 721 Dirac–Coulomb Hamiltonian, 724 Schr¨ odinger–Coulomb Hamiltonian, 722 Scattering Amplitude, 558, 561 Schr¨ odinger Equation, 270 Schr¨ odinger Equation, 17 Schr¨ odinger Hamiltonian, 18, 51, 76 Schr¨ odinger Picture, 42, 44, 50, 159 Schr¨ odinger Spectrum, 79 Schr¨ odinger Wave Function, 208 Schr¨ odinger–Coulomb Green Function, 8, 75, 90, 96 Schr¨ odinger–Coulomb Hamiltonian, 76, 77, 83, 85, 91, 92, 97, 510, 722 Schr¨ odinger–Coulomb Propagator, 101, 133 Schr¨ odinger–Coulomb Spectrum, 131 Schr¨ odinger–Pauli Wave Function, 137, 209 Schwarzschild Radius, 308 Schwinger Mechanism, 705 Seagull Term, 126, 584 Second Quantization, 19, 62, 359 Secular Invariant, 705 Self Energy, 121 Self-Energy, 140, 142, 411, 414, 448, 449, 454, 456, 559 High-Energy Part, 139, 449, 456

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Higher-Order Corrections, 561 Higher-Order Terms, 144, 572 Imaginary Part, 141 Low-Energy Part, 121, 132 Magnetic–Photon Exchange, 143 Relativistic Bethe Logarithms, 572 Self-Energy Density, 331 Shapiro Time Delay, 5 SI mksA System, 21 Single-Particle Hamiltonian, 478, 479 Slater Determinant, 535 Sochocki–Plemelj Prescription, 140 Solid Angle in d Dimensions, 378 Speed of Light, 707 Spherical Basis, 194 Spherical Bessel Function, 246 Spherical Biharmonic, 211 Spherical Coordinates, 79 Spherical Harmonic, 79, 107 Four-Dimensional, 107, 109 Spin Connection, 305 Spin Matrix, 199, 232, 439, 513 Spin Wave Functions, 507 Spin-Angular Function, 206 Spin-Orbit Interaction, 514 Spin-Other-Orbit Interaction, 514 Spin-Spin Interaction, 514 Spinor, 9, 206, 513, 533, 545 Covariant Derivative, 301 Lorentz Transformation, 231 Spin Connection, 305 Wave Function, 206 Stark Effect, 8, 57, 68 Dynamic, 57, 212 Static, 65 Static Stark Effect, 65 Sturmian Decomposition, 75 Sturmian Function, 94 Temporal Gauge, 343, 464 Tensor Decomposition, 212, 213, 218 Tetrad Basis, 297 Ricci Rotation Coefficients, 300, 306 Three-Body Bound System, 511 Time Ordering, 346 Time Reversal, 227, 235 Time-Dependent Perturbation Theory, 8, 57 Time-Ordered Perturbation Theory, 49 Trans-Series, 72

Transition Current, 442, 448, 472 Transition Rates, 89 Transverse Component, 30 Transverse Dirac-δ Function, 32, 619 Transverse Electric Field, 330 Transverse Electric Modes, 178 Transverse Magnetic Modes, 178 Tree-Level Diagrams, 9 Triangularity Condition, 201 Two-Body Bound System, 484, 510, 575, 593, 594 Two-Particle Interaction Hamiltonian, 474, 477, 484, 485 Two-Particle Irreducible Kernel, 595, 598, 671, 673 Uehling Potential, 409, 456, 711 Ultraviolet Regulator, 398 Unit System Atomic, 22, 27 Gaussian, 29 Heaviside–Lorentz, 21, 410 Natural, 21, 23, 410 Quantum SI, 25 SI mksA, 21 Unitarity Correction, 356 Unitary Transformation, 426 Vacuum Fluctuation, 21 Vacuum Polarization, 20, 388, 400, 406, 456, 667 Vacuum-Polarization Tensor, 401 Van der Waals Interaction, 149, 159, 169 Vector Addition Coefficients, 195, 198 Vector Potential, 30, 33, 447, 469 Operator, 33, 447 Vector Recoupling Coefficients, 201 Vector Spherical Harmonic, 209 Velocity Gauge, 121, 126 Vertex Correction, 370, 379, 394, 414 Vertex Function, 380, 399 Vertex Renormalization, 387 Virial Theorem, 85, 279 Ward–Takahashi Identity, 384, 389, 395, 415 Watt Balance, 26 Wave Equation Schr¨ odinger–Coulomb, 8 Wave Function Collapse, 18

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Index

Wave Function Renormalization, 384 Wave Vector, 21 Weak Equivalence Principle, 5 Wegner–Houghton Equation, 720 Wentzel–Kramers–Brioullin Expansion, 19 West-Coast Convention, 293, 337 Weyl Gauge, 343, 464 Whittaker Differential Equation, 80, 267 Whittaker Function, 75, 89, 92, 267 M Function, 81, 82, 86, 92, 271 W Function, 81, 82, 92 Wichmann–Kroll Correction, 13, 692, 711 Wick Rotation, 168, 182, 397, 405, 588 Wick Theorem, 323, 346

Wigner 3j Symbol, 200 Wigner 6j Symbol, 200 Wigner 9j Symbol, 200 Wigner Matrices, 8 Wigner–Brioullin Formula, 151, 154 Wilson Coefficient, 657 Zeeman Effect, 497 Relativistic Correction, 499 Zemach Moment, 158 Zeta Function, 99 Zinn-Justin Equation, 719 Zitterbewegung Term, 27, 436, 475

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