Quantum Electrodynamics

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A.I. Akhiezer V.B. Berestetskii

Quantum Electrodynamics

INTERSCIENCE MONOGRAPHS AND TEXTS IN PHYSICS AND ASTRONOMY Edited by R. E. MARSHAK

Volum e

I:

E. R. Cohen, K. M. Crowe, and J. W. M. DuMond THE FUNDAMENTAL CONSTANTS OF PHYSICS

Volum e

II:

Volum e

III:

N. N. Bogoliubov and D. V. Shirkov INTRODUCTION TO THE THEORY OF QUANTIZED FIELDS

Volum e

IV:

J. B. Marion and J. L. Fowler, Editors FAST NEUTRON PHYSICS In two parts Part I: Techniques Part II: Experiments and Theory

Volum e

V:

D. M. Ritson, Editor TECHNIQUES OF HIGH ENERGY PHYSICS

Volum e

VI:

R. N. Thomas and R. G. Athay PHYSICS OF THE SOLAR CHROMOSPHERE

G. J. Dienes and G. H. Vineyard RADIATION EFFECTS IN SOLIDS

Volum e VII:

L. H. Aller THE ABUNDANCE OF THE ELEMENTS

Volum e VIII:

E. N. Parker INTERPLANETARY DYNAMICAL PROCESSES

Volum e

IX:

C. L. Longmire ELEMENTARY PLASMA PHYSICS

Volum e

X:

Volum e

XI:

R. Brout and P. Carruthers LECTURES ON THE MANY-ELECTRON PROBLEM A. I. Akhiezer and V. B. Berestetskii QUANTUM ELECTRODYNAMICS

Volum e XII:

H. Panofsky and J. Lumley STRUCTURE OF ATMOSPHERIC TURBULENCE

Volum e XIII:

R. D. Heidenreich INTRODUCTION TO TRANSMISSION ELECTRON MICROSCOPY Additional volumes in preparation

INTERSCIENCE MONOGRAPHS AND TEXTS IN PHYSICS AND ASTRONOMY Edited by R. E. MARSHAK University of Rochester, Rochester, New York

VOLUME XI

E d ito r ia l A d v iso ry B o a rd

A. ABRAGAM, College de France, Paris, France H. ALFVEN, Royal Institute of Technology, Stockholm, Sweden L. V. BERKNER, Southwest Research Conference, Dallas, Texas H. J. BHABHA, Tata Institute for Fundamental Research, Bombay, India L. BIERMANN, Max Planck Institut fur Physik und Astrophysik, Munich, Germany N. N. BOGOLIUBOV, J.I.N.R., Dubna, U.S.S.R. A. BOHR, Institute for Theoretical Physics, Copenhagen, Denmark S. CHANDRASEKHAR, Enrico Fermi Institute, Chicago, Illinois J. W. DuMOND, California Institute of Technology, Pasadena, California J. FRIEDEL, University of Paris, Orsay, France L. GOLDBERG, Harvard University, Cambridge, Massachusetts M. GOLDHABER, Brookhaven National Laboratory, Upton, New York H. A. GOVE, Institute for Theoretical Physics, Copenhagen, Denmark C. HERRING, Bell Telephone Laboratories, Murray Hill, New Jersey J. KAPLAN, University of California, Los Angeles, California C. M0LLER, Institute for Theoretical Physics, Copenhagen, Denmark W. K. H. PANOFSKY, Stanford University, Stanford, California R. E. PEIERLS, Oxford University, Oxford, England F. PRESS, California Institute of Technology, Pasadena, California B. ROSSI, Massachusetts Institute of Technology, Cambridge, Massachusetts A. SCHAWLOW, Stanford University, Stanford, California R. A. SMITH, Massachusetts Institute of Technology, Cambridge, Massachusetts L. SPITZER, Jr., Princeton Observatory, Princeton, New Jersey B. STROMGREN, Institute for Advanced Study, Princeton, New Jersey G. TORALDO di FRANCIA, University of Florence, Florence, Italy G. E. UHLENBECK, Rockefeller Institute, New York, New York L. VAN HOVE, C.E.R.N., Geneva, Switzerland V. F. WEISSKOPF, C.E.R.N., Geneva, Switzerland H. YUKAWA, Institute for Theoretical Physics, Kyoto, Japan

QUANTUM ELECTRODYNAMICS

A. I. AKHIEZER Physico-Technical Institute, Academy of Sciences, Khar’kov, U.S.S.R. V. B. BERESTETSKII Institute for Theoretical and Experimental Physics, Academy of Sciences, Moscow, U.S.S.R.

Authorized English Edition Revised and Enlarged by the Authors

Translated from the Second Russian Edition by G. M. Volkoff University of British Columbia, Vancouver, B.C., Canada

INTERSCIENCE PUBLISHERS

1965

a division of John Wiley & Sons, New York • London • Sydney

English Translation and All Additions to the Russian Edition Copyright (c) 1965, John Wiley & Sons, Inc. ALL RIGHTS RESERVED. This book or any part thereof must not be reproduced in any form without the written permission of the publisher.

Library of Congress Catalog Card Number 63-17766 Printed in the United States of America

Translator’s Note

This translation follows in the main the second Russian edition of 1959, but in addition incorporates several new sections (in particular §§ 9.4, 26.2, 26.6, 48.4, 48.5, 51.5, 52.5) provided by the authors specially for this edition, as well as a number of minor revisions, and corrections of numerous typographical errors in the formulas. The translator is grateful to the authors for their cooperation. G. V o l k o f f Vancouver, Canada

v

P r e fa c e to th e S eco n d E d itio n

In preparing the second edition we have subjected the book to extensive revision. The prineipal aim and the contents of the book have not been altered; the book is devoted to the systematic presen­ tation of electromagnetic processes only. Only some general theorems and methods go beyond the framework of electrodynamics proper. In the second edition the space devoted to these has been increased (reflection properties, Green’s functions, functional methods, etc.). In the presentation of the principles of quantum electrodynamics the theory of renormalizations has been subjected to the most exten­ sive revision. Without claiming complete mathematical rigor we have attempted to present the concept of renormalization from a single simple physical point of view, avoiding purely prescription-like methods of eliminating divergences, and making maximum use of the general properties of quantum mechanical systems. In connection with this the structure of the book has been somewhat altered. The investigation of the scattering matrix together with the theory of radiation corrections has been segregated into a separate chapter (Chapter VII). The investigation of electrodynamic processes in the first nonvanishing approximation, which involves neither the removal of divergences nor renormalizations, is carried out in Chapters V and VI, while higher order approximations are considered in Chapter VIII. In presenting specific effects we have aimed, within reasonable limits, at the greatest possible degree of completeness of results and of detail of calculations. The number of different electrodynamic phenomena described has been increased. In particular, the theory of processes involving polarized particles, the method of impact param­ eters, etc., have been included. We wish to express our sincere gratitude to V. Aleksin, V. Bar’iakhtar, V. Boldyshev, D. Volkov, S. Peletminskii, R. Polovin, and P. Fomin, who have aided us significantly in the preparation of the second edition of this book. A. I. A k h iezer , V. B. B erestetsk ii vi

F ro m th e P r e fa c e to th e F irst E d itio n

At the present time a number of particles is known which corre­ spond to various quantum fields interacting with each other. However, of the many types of physical interactions existing in nature the only one, apart from gravitation, that has been studied in sufficient detail is the electromagnetic interaction. The theory of the latter interaction is the subject of quantum electrodynamics, to which the systematic exposition of this book is devoted. Since the electromagnetic interaction is the fundamental one in the case of electrons and photons, quantum electrodynamics enables us to explain and to predict a wide range of phenomena related to the behavior of these particles. As for the application of quantum electrodynamics to other particles (nucleons and mesons), it is con­ siderably restricted because of the essential role played by other types of interactions (nuclear or meson interactions) in the case of those particles. Therefore, problems relating to mesons are not treated in this book, and the interaction of nucleons with the electromagnetic field is treated only in the limit of low velocities. The formulation of the fundamental equations of quantum electro­ dynamics, and even the very possibility of separating the interacting fields into the electromagnetic and electron-positron fields, is based on the fact that the interaction between these fields is a weak one. This circumstance finds its expression in the smallness of the constant a = e1/h e which characterizes the interaction. Therefore the inter­ action between the fields is treated in quantum electrodynamics as a small perturbation, and the mathematical method employed in quan­ tum electrodynamics is perturbation theory in which all quantitative results are expressed in terms of power series in a. Since both the electromagnetic and the electron-positron fields are systems with an unlimited number of degrees of freedom, the ap­ plication of perturbation theory gives rise to divergent expressions characteristic of the present theory, which are absent only in the first nonvanishing approximation of perturbation theory. The development vii

Vlll

FROM THE PREFACE TO THE FIRST EDITION

of quantum electrodynamics in recent years has permitted the estab­ lishment of principles for regularizing divergent expressions, so that it has then become possible to calculate higher order approximations (the so-called radiation corrections). This progress is to a great extent associated with the new invariant formulation of perturbation theory. Invariant perturbation theory has made it possible to present the results in a compact and relativistically invariant form, and this has allowed the rules for regularization to be formulated. On the other hand, the use of invariant perturbation theory has significant practical advantages over the earlier methods even in the case of first-order calculations. Therefore the whole pre­ sentation in this book is based on invariant perturbation theory. Although it is a completely satisfactory theory within a definite field of physical phenomena, modern quantum electrodynamics has the important drawback that in order to remove the divergences which arise in the theory additional concepts must be introduced which are neither contained in the fundamental formulation of the theory, nor reflected in its basic equations. This state of affairs is apparently due to profound causes. They are implicit in the fact that it is frequently impossible to construct a closed theory of a limited set of phenomena (in the present case, the purely electromagnetic ones) without taking into account a wider class of interactions existing in nature. We wish to express our gratitude to Academician L. D. Landau and Professor I. Ia. Pomeranchuk and to the participants in the seminars directed by them for discussions of a number of problems presented in this book.

Contents

CH A PTER I

QUANTUM MECHANICS OF THE PHOTON § 1.

The Photon Wave Function 1. Introduction. 2. The Photon Wave Function in k-Space. 3. Energy. 4. Normalization of the Photon Wave Function.

1

§ 2.

Photon States of Definite Momentum 1. Photon Momentum Operator. 2. Impossibility of In­ troducing a Photon Wave Function in the Coordinate Representation. 3. Plane Waves. 4. Polarization Density Matrix for the Photon.

9

§ 3.

Angular Momentum. Photon Spin . 1. Angular Momentum Operator. 2. Photon Spin Oper­ ator. 3. Photon Spin Wave Functions.

17

§ 4.

Photon States of Definite Angular Momentum and Parity . . . . 1. Eigenfunctions of the Photon Angular Momentum Operator. 2. Longitudinal and Transverse Vector Spher­ ical Harmonics. 3. Parity of Photon States. 4. Expan­ sion in Spherical Waves. 5. Expressions for the Electric and Magnetic Fields.

24

§ 5.

Scattering of Photons by a System of Charges 1. Incoming and Outgoing Waves. 2. Effective Scattering Cross Section. 3. The Optical Theorem. 4. Dispersion Relations.

36

§ 6.

The Photon Field Potentials . . . 1. Transverse, Longitudinal, and Scalar Potentials. 2. Longitudinally Polarized “Photon.” 3. Potentials for Plane and Spherical Waves. ix

46

CONTENTS

X

§ 7.

System of Photons 1. Wave Function for a System of Two Photons. 2. Even and Odd States of Two Photons. 3. Classification of the States of Two Photons of Definite Angular Momentum. 4. Wave Function for a System of an Arbitrary Number of Photons.

52

§ 8.

L-Vectors and Spherical Harmonics 1. Irreducible Tensors. 2. The Algebra of L-Vectors. 3. Spherical Harmonics.

62

CH A PTER II

RELATIVISTIC QUANTUM MECHANICS OF THE ELECTRON § 9.

The Dirac Equation 1. Spinors. Pauli Matrices. 2. Dirac Equations. Dirac Matrices. 3. Unitary Transformations of Bispinors. 4. The Necessity for Four-Component Electron Wave Functions. 5. Symmetric Form of the Dirac Equation. Equation of Continuity. 6. Invariance of the Dirac Equation. 7. Bilinear Combinations of the Components of the Wave Function.

§ 10.

Electron and Positron States. States of Definite Momentum and Polarization 1. Solutions with Positive and Negative Frequencies. 2. The Charge Conjugation Transformation. 3. The Posi­ tron Wave Function. 4. Plane Waves. 5. Polarization of a Plane Wave. 6. Polarization Density Matrix for the Electron. 7. Averaging over Polarization States.

§ 11.

§ 12.

Electron States of Definite Angular M omentum and P a rity ...................................... 1. Orbital and Spin Functions. Spherical Spinors. 2. Wave Function of a State of Definite Angular Momentum. 3. Parity of a State. 4. Expansion in Spherical Waves. Electron in an External Field . 1. The Dirac Equation with an External Field. 2. Sepa­ ration of Variables in a Central Field. 3. Asymptotic Be­ havior of the Radial Functions. 4. Behavior of Energy Levels as Functions of the Potential Well Depth. 5. Elec­ tron in a Constant Homogeneous Magnetic Field.

73

86

105

112

CONTENTS

§ 13.

Motion of an Electron in the Field of a Nucleus 1. Solution of the Radial Equations for the Coulomb Field. 2. Wave Functions for the Continuous Spectrum. 3. Isotopic Level Shift. 4. General Investigation of the Effect of Finite Nuclear Size.

§ 14.

Electron Scattering 1. Spinor Scattering Amplitude. 2. Expression for the Cross Section in Terms of Phases. 3. Polarization and Azimuthal Asymmetry. 4. Scattering by a Coulomb Field. 5. Small Angle Scattering.

§ 15.

Nonrelativistic Approximation 1. Transition to the Pauli Equation. 2. Second Ap­ proximation. 3. Application of the Dirac Equation to Nucleons. CH A PTER III

QUANTIZED ELECTROMAGNETIC AND ELECTRON-POSITRON FIELDS § 16.

Quantization of the Electromagnetic Field 1. Four-Dimensional Form of the Field Equations. 2. Variational Principle. Energy-Momentum Tensor of the Electromagnetic Field. 3. Expansion of the Potentials into Plane Waves. 4. Quantization of the Electromag­ netic Field. 5. Use of the Indefinite Metric.

§ 17.

Commutators of the Electromagnetic Field 1. Commutation Relations for the Potentials and the Field Components. 2. Chronological and Normal Products of Components of the Potential. 3. Singular Functions As­ sociated with the Operators □ and (□ — m 2).

§ 18.

Quantization of the Electron-Positron Field . 1. Variational Principle for the Dirac Equation. EnergyMomentum Tensor of the Electron-Positron Field. 2. Quantization Rules for the Electron-Positron Field.

§ 19.

Anticom mutators of the Electron-Positron Field. Chronological and Normal Products of Field Compo­ nents. Current Density 1. Commutation Relations for Field Components. 2. Chronological and Normal Products of Operators of the Electron-Positron Field. 3. Electric Current Density.

CONTENTS

xii §

20.

G eneral P ro p e rtie s of W ave Fields . 1. Wave Functions of a Field and the Lorentz Group. 2. Irreducible Finite-Dimensional Representations of the Lorentz Group. 3. Energy-Momentum Tensor and An­ gular Momentum Tensor. 4. Current Density Vector. 5. Relativistically Invariant Field Equations. 6. Wave Equations for Particles of Spin Zero and Unity.

§

21.

Q uan tization of Fields. C onnection betw een Spin an d Statistics .237 1. Nondefiniteness of the Charge in the Case of Integral Spin and of the Energy in the Case of Half-Integral Spin. 2. Quantization of Fields for Integral and Half-Integral Spin. Pauli’s Theorem. 3. Inversion of Coordinates and Time Reversal.

214

CH A PTER IV

FUNDAMENTAL EQUATIONS OF QUANTUM ELECTRODYNAMICS §

22.

§23.

§24.

In te ra c tin g E lectrom agnetic an d E lectro n -P o sitro n Fields . . . . 1. System of Equations for Interacting Fields. 2. Lagrangian. Energy-Momentum Tensor. 3. Field Equations in Poisson Bracket Form. 4. Invariance Properties of the Equations of Quantum Electrodynamics. E quations of Q u an tu m E lectrodynam ics in th e I n te r ­ action P ictu re. In v a ria n t P e rtu rb a tio n T h eo ry 1. Heisenberg and Schrodinger Pictures. Interaction Pic­ ture. 2. Transition to the Interaction Picture in Quan­ tum Electrodynamics. 3. Charge Conjugation Operator. 4. Perturbation Theory. T he S catterin g M atrix 1. The Scattering Problem and the Definition of the Scattering Matrix. 2. Matrix Elements of Field Opera­ tors. 3. Representation of the Scattering Matrix as a Sum of Normal Products. 4. General Relation between Tand iV-Orderings. 5. Symmetry of the Scattering Matrix under Time Reversal.

253

268

290

CONTENTS

§ 25.

Graphical R epresentation of the Elements of the Scat­ tering Matrix. The Scattering Matrix in Momentum Space 1. Graphical Representation of Normal Products. 2. Var­ ious Interaction Processes between Fields. 3. Transition to Momentum Space. 4. Closed Electron Loops with an Odd Number of Vertices. 5. Rules for Writing Down Matrix Elements.

§ 26.

Probabilities of Various Processes . 1. General Formula for the Probability. 2. Effective Cross Section. 3. Summation and Averaging over Polari­ zation States of Electrons and Photons. 4. Probabilities of Processes Involving Polarized Particles. 5. Probabili­ ties of Processes in the Presence of an External Field. 6. Feynman’s Notation. v INTERACTION OF ELECTRONS WITH PHOTONS ch a pter

§ 27.

Emission and Absorption of a Photon 1. General Expression for the Matrix Element. 2. Elec­ tric Multipole Radiation. 3. Magnetic Multipole Radia­ tion. 4. Selection Rules. 5. Angular Distribution and Polarization of the Radiation.

§ 28.

Scattering of a Photon by a Free Electron 1. Scattering Matrix Element. 2. Application of Conser­ vation Laws. 3. Differential Cross Section for Unpolar­ ized Particles. 4. Angular Distribution and Total Cross Section. 5. Distribution of Recoil Electrons. 6. Scatter­ ing of Polarized Photons. 7. Scattering of Photons by Polarized Electrons.

§ 29.

Brem sstrahlung . . . 1. Perturbation Theory for an Electron Wave Function in the Continuum. Incoming and Outgoing Waves. 2. Effec­ tive Cross Section for Bremsstrahlung. 3. Angular Dis­ tribution of the Radiation in a Coulomb Field. 4. Polari­ zation of the Radiation. 5. Spectrum of the Radiation. 6. Screening. 7. Radiative Energy Losses. 8. Exact Theory of Bremsstrahlung in the Nonrelativistic Domain. 9. Exact Theory of Bremsstrahlung in the Extreme Rela-

CONTENTS

xiv

tivistic Domain. 10. Radiation Emitted in ElectronElectron and Electron-Positron Collisions. § 30.

Em ission of P hotons of Long W avelength 1. “The Infrared Catastrophe.” 2. Investigation of the Divergence in the Low Frequency Domain by Means of the Scattering Matrix. 3. Relation between the Photon “Mass” and the Minimum Frequency.

413

§31.

P hotoeffect . . . . 1. Photoeffect in the Nonrelativistic Domain. effect in the Relativistic Domain.

429 2. Photo­

§32.

P ro d u ctio n of E lectro n -P o sitro n P a irs 1. Production of an Electron-Positron Pair by a Photon in the Field of a Nucleus. 2. Exact Theory of Pair Pro­ duction by a Photon in the Field of a Nucleus in the Nonrelativistic and Extreme Relativistic Cases. 3. Pair Production by Two Photons. 4. Pair Production in a Photon-Electron Collision. 5. Pair Production in a Col­ lision of Two Fast Charged Particles.

438

§33.

A n n ih ilation of E lectron-P ositro n P a irs in to P h o to n s 457 1. Annihilation of a Pair into Two Photons. 2. Polariza­ tion Effects in the Two-Photon Annihilation of a Pair. 3. Annihilation of a Pair into One Photon. 4. Positronium Decay. 5. Three-Photon Decay of Orthopositronium. 6. Multiple Photon Production Accompanying the Annihilation of a Pair.

§34.

T he M ethod of E q u iv alen t P h o to n s 1. The Number of Equivalent Photons. 2. Bremsstrahlung from a Fast Electron in the Field of a Nucleus. 3. Radiation Emitted in an Electron-Electron Collision. 4. Pair Production by a Photon in the Field of a Nucleus. 5. Pair Production in a Collision of Two Fast Particles.

§35.

S catterin g of a P h o to n by a B ound E lectro n , Em ission of Two P hotons . . . 484 1. The Dispersion Formula. 2. Resonance Scattering. 3. Compton Scattering by Bound Electrons. 4. Emission of Two Photons. The Metastable 2 State of the Hy­ drogen Atom.

473

CONTENTS

XV CH A PTER VI

RETARDED INTERACTION BETWEEN TWO CHARGES §36.

Electron-Electron and Positron-Electron Scattering 1. Electron-Electron Scattering. 2. Positron-Electron Scattering. 3. Scattering of Polarized Electrons and Posi­ trons. 4. Annihilation of an Electron-Positron Pair into a /i-Meson Pair.

499

§37.

Retarded Potentials . . . 1. Interaction Function for Two Charges. 2. General Form of the Matrix Element. 3. Retarded Potentials and Transition Currents.

509

§38.

Interaction Energy of Two Electrons to Terms of Order v2/ c 2 1. The Breit Formula. 2. Schrodinger Equation for a Two-Electron System. 3. Interaction between an Elec­ tron and a Positron. 4. Exchange Interaction between an Electron and a Positron.

517

§39.

Positronium . . . 1. Hamiltonian Operator and the Unperturbed Equation. 2. Perturbation Operator. 3. Fine Structure. 4. Zeeman Effect.

527

§40.

Interna] Conversion of Gamma-Rays 1. Expansion of Retarded Potentials in Spherical Waves. 2. Conversion Coefficient. 3. Conversion in the E-Shell. 4. Effect of Finite Nuclear Size. 5. Effect of Electron Shells on Radiation from the Nucleus.

537

§41.

Conversion Accompanied by Pair Production. Excita­ tion o f Nuclei by Electrons 1. Conversion of Magnetic Multipole Radiation. 2. Con­ version of Electric Multipole Radiation. 3. Excitation of Nuclei by Electrons. 4. Monoenergetic Positrons.

§42.

Coulomb (M onopole) Transitions 1. Reduction to the Static Interaction. 2. Conversion and Nuclear Excitation in the Case of an EO-Transition.

554

565

xvi

CONTENTS CH A PTER VII

INVESTIGATION OF THE SCATTERING MATRIX § 43.

Properties of Exact Solutions of the Equations of Quantum Electrodynamics. Propagators 1. Stationary States of a System of Interacting Fields. 2. Propagators and Their Spectral Representation. 3. Connection between Propagators and the Scattering Matrix. Integral Equations for Propagators. 4. Electro­ magnetic Mass of the Electron.

§ 44. Structure of the Scattering Matrix . 1. Self-Energy Parts of Diagrams. 2. Vertex Parts of Diagrams. 3. Renormalization of Electron Mass.

571

593

§ 45. Renorm alization of Electron Charge 605 1. Physical Charge of the Electron. 2. Renormalization of Propagators and Vertex Parts. 3. Three-Photon Vertex Parts. 4. Renormalization of Matrix Elements. 5. Formulation of Perturbation Theory as an Expansion of Powers of ec. § 46.

Divergences in the Scattering Matrix and their Removal . . . .619 1. Divergences in Irreducible Diagrams. 2. Introduction of a Cut-Off Momentum. 3. Convergence of Regularized Expressions for Irreducible Vertex Parts and Self-Energy Parts. 4. Convergence of Regularized Quantities in the Case of Reducible Diagrams.

§ 47.

Evaluation of Self-Energy and Vertex Parts . 1. Evaluation of Integrals over Four-Dimensional Re­ gions. 2. Second Order Electron Self-Energy Part. 3. Second Order Photon Self-Energy Part. 4. Third Order Vertex Part in the Case of External Electron Lines. 5. Third Order Vertex Part in the Case of One External Electron Line.

§ 48.

Functional Properties of Green’s Functions. Limits of Applicability of Quantum Electrodynamics 657 1. Expansion Parameters of Perturbation Theory. 2. Zero Order Approximation in the Expansion in Powers of e2. 3. Integral Equations for the Zero Order Approximation.

631

CONTENTS 4. The Renormalization Group. 5. Derivation of Asymp­ totic Expressions for the Green’s Functions with the Aid of Differential Equations of the Renormalization Group. 6. The Problem of Closure of Quantum Electrodynamics. § 49.

Generalized Green’s Functions 1. Green’s Functions in the Presence of External Fields. 2. Green’s Function for Two Electrons. Equation for Bound States of the Electron-Positron System. 3. Equa­ tions for Green’s Functions in Terms of Variational De­ rivatives. 4. Expressions for Green’s Functions in Terms of Functional Integrals. C H A PT ER V III

RADIATION CORRECTIONS TO ELECTRO­ MAGNETIC PROCESSES § 50.

Effective Potential Energy of the Electron. Radiation Corrections to the Electron Magnetic Moment and to Coulomb’s Law 1. Energy of Interaction of the Electron with the Elec­ tromagnetic Field Taking into Account Corrections of Order «. 2. Radiation Corrections to the Electron Mag­ netic Moment. 3. Radiation Corrections to Coulomb’s Law.

§ 51.

Radiation Corrections to Electron Scattering 1. Electron Scattering by the Coulomb Field of a Nucleus in the Second Born Approximation. 2. Differential Cross Section for the Scattering of an Electron by the Coulomb Field of a Nucleus taking into Account Radiation Correc­ tions of Order a ■ 3. Elimination of the Photon “Mass” from the Scattering Cross Section. 4. Removal of the Infrared Divergence for an Arbitrary Scattering Process. 5. Scattering of High Energy Electrons by an External Field. 6. Radiation Corrections to Electron-Electron and Electron-Positron Scattering.

§ 52.

Radiation Corrections to Photon-Electron Scattering, to Pair Creation and Annihilation, and to Bremsstrahlung 1. Radiation Corrections to the Compton Effect. 2. Lim­ iting Cases of Low and High Energies. 3. Radiation

xviii

CONTENTS Corrections to Two-Photon Pair Annihilations. 4. Radia­ tion Corrections to Bremsstrahlung. 5. Radiation Correc­ tions to Photon Production and Single Photon Annihila­ tion of Pairs.

§ 53.

R ad iatio n C orrections to A tom ic Levels .751 1. Radiation Shift of Atomic Levels. 2. Radiation Shift of the Levels of ^-Mesohydrogen. 3. Natural Line Widths. 4. Photon Scattering near Resonance.

§ 54.

P h o to n -P h o to n S catterin g an d th e L a g ra n g ia n fo r th e E lectrom agnetic Field . 1. Photon-Photon Scattering Tensor of the Fourth Rank. 2. Photon-Photon Scattering. 3. Connection between the Photon-Photon Scattering Cross Section and the Radiation Corrections to the Lagrangian of the Electromagnetic Field. 4. Exact Expressions for the Lagrangian of the Electromagnetic Field.

§ 55.

P h o to n S catterin g by th e C oulom b Field o f a N ucleus 1. General Expression for the Cross Section for Photon Scattering by a Constant Electromagnetic Field. 2. Rela­ tion between the Forward Scattering Amplitude for a Photon and Pair-Production by a Photon in the Field of a Nucleus. 3. Momentum Distribution of Recoil Nuclei Accompanying Pair Production by a Photon in the Field of a Nucleus. 4. Angular Distribution of Recoil Nuclei and Total Cross Section for Pair Production by a Photon in the Coulomb Field of a Nucleus. 5. Small Angle Co­ herent Scattering of Photons by the Field of a Nucleus. C H A PT ER

764

792

ix

ELECTRODYNAMICS OF PARTICLES OF SPIN ZERO § 56.

Field E quations fo r S calar P a rticle s 1. First Order Equations. 2. Quantization of the Free Scalar Field. 3. Commutators of the Field. Vacuum Ex­ pectation Values of Products of Field Components.

819

§57.

T he S catterin g M atrix in S calar E lectrodynam ics 1. The Interaction Picture. 2. Rules for Calculating Ele­ ments of the Scattering Matrix. 3. Divergences of the Scattering Matrix.

827

CONTENTS

xix

§ 58.

Scattering of Scalar Particles . 1. Scattering of Scalar Particles by the Coulomb Field of a Nucleus. 2. Scattering of a Charged Scalar Particle by a Scalar Particle.

835

§ 59.

Scattering of a Photon by a Scalar Particle. Bremsstrahlung Photons from a Scalar Particle 1. Scattering of a Photon by a Scalar Particle. 2. Bremsstrahlung from Scalar Particles.

§ 60.

§ 61.

Production and A nnihilation of Pairs of Scalar Particles . . . 1. Production of Pairs of Scalar Particles by a Photon in the Coulomb Field of a Nucleus. 2. Production of a Pair of Scalar Particles by Two Photons. 3. Two-Photon An­ nihilation of a Pair of Scalar Particles. 4. Annihilation of Pairs of Scalar Particles into Electron-Positron Pairs and the Inverse Process. Polarization of the Vacuum in the Case of Charged Scalar Particles 1. Vacuum Polarization Tensor for Scalar Particles. 2. Correction to Coulomb’s Law. 3. Photon-Photon Scat­ tering. Radiation Corrections to the Lagrangian of the Electromagnetic Field.

838

842

847

Concluding Remarks

852

R eferences

855

Subject Index

863

CHAPTER I

Quantum Mechanics of the Photon

§ 1.

The Photon Wave Function

1.1. Introduction The corpuscular properties of light were historically the first fun­ damental fact which served as a basis for the development of quantum theory. The Planck-Einstein relation between the energy w of a quantum of light, the photon, and the frequency co of the electromagnetic field corresponding to it is expressed as w = tuo and is historically the first relation containing the quantum constant ti. The systematic quantum mechanics of the atom was developed be­ fore that of the photon and this situation has a profound physical reason. Atomic particles, the electrons and the nuclei, have rest masses different from zero. For them there exists a range of energies, small compared to their rest energy, in which relativistic effects may be neglected. Since the rest mass of the photon is zero, there exists for it no nonrelativistic energy region, and the quantum mechanics of the photon must neces­ sarily be relativistic from the outset. As the counterpart of a particle, quantum mechanics introduces a field of one or more wave functions which determine the probability distribution and the expectation values of the various physical quanti­ ties related to the partticle. The wave functions satisfy a certain system of differential equations which determine the nature of the motion of the particle. In order to transpose this description into the relativistic domain we must, first of all, take into account the requirements of the principle of relativity. They amount to the requirement that the field equations must be invariant under Lorentz transformations. However, this by itself is not sufficient for a unique determination of the form of the field equations which would give expression to the individual properties of a given particle. In the case of photons the setting up of the equations is facilitated by the existence of a classical analog, viz., of the equations of [1]

2

QUANTUM ELECTRODYNAMICS

the classical electromagnetic field. It is natural to use Maxwell’s equations as the quantum mechanical equations of motion for the photon and it follows that the wave properties of the photon will coincide with the properties of the electromagnetic field. As will be shown later, this, together with the relation w = ha>, suffices for the formulation of a theory of photons and of their interaction with charged particles. Our first problem is the study of photons in the absence of electric charges. Although properties of particles become apparent only as a result of their interaction with other particles, such an investigation is necessary as a preparatory stage for the study of interactions. 1.2.

The Photon Wave Function in k Space The electromagnetic field is described by the electric field vector E and the magnetic field vector H, which in the absence of charges satisfy Maxwell’s equations for a vacuum: dH curl E = IT ’ di v H = 0, dE curl H = ~dT’ div E

( 1. 1)

0.

(Here and subsequently we make use of the system of units in which the velocity of light is equal to unity, c = 1.) In accordance with the foregoing we shall interpret the vectors E and H as quantities describing the quantum mechanical photon state. In order to give such a “corpuscular” interpretation to the system of equations (1.1) we compare this system with the Schrodinger equa­ tion in ordinary quantum mechanics. This may be done conveniently if we first subject equations (l.l) to a Fourier transformation with respect to the space coordinates r, i.e., if we go over into k space. By writing E and H in the form1 E= H=

f Ekeikrdk, ,

Hkeikr dk,

(1-2)

1 Following the authors’ usage, the scalar product of two three-dimensional vectors will be denoted by boldface letters standing next to each other without an

QUANTUM MECHANICS OF PHOTON

3

we can easily see that in virtue of (1.1) the Fourier components Ek and Hk satisfy the following system of equations: Hk = - i [ k , E k\, kH k = 0, Ek = i[ k ,H k], kEh - 0,

(1.3)

where dots denote differentation with respect to time. (For the sake of brevity we shall omit here and subsequently the argument t in the functions Ek = E {k ,t) and Hk = H (k, t).) To this system we must also add the condition that the fields be real: E -k = E*, H - k =

(1.4)

H *.

Instead of the two vectors Eh and Hk we can use the vectors Ek and Ek by eliminating Hk with the aid of (1.3): (1-5) It is convenient to eliminate the necessity of taking into aecount the requirement of reality. In order to do this we introduce a transfor­ mation which automatically guarantees that the relations (1.4) will be satisfied: Ek = N (k )(fk+ f* k), Ek = —ikN (k) (fk—f * k). where N(k) is a certain normalizing factor, which, as we shall see later, it is convenient to choose equal to

( 1 '7 )

The substitution (1.6) means that instead of the two functions Ek and Ek, which in virtue of (1.4) are actually defined in a half-space, we intro­ duce the single function f k, which is independently defined over the whole &-space. intervening dot, and either with or without parentheses: thus kr or (kr ), but not k r. Similarly, the vector product will be denoted by the square brackets [A, B] with or without a comma, but not by A x B .

QUANTUM ELECTRODYNAMICS

4

If we make use of (1.6), the expansion (1.2) will assume the following form: £ '= g + ( y * , H = < q (y=

f N{k)fkeikrdk,

$ =

^ ^

/* ]< * < * •

We can easily obtain the equation satisfied by f k. By eliminating Hk from (1.3) we obtain d2 \-k2\E k = Q , dt2 which may be rewritten in the following form:

This second-order equation may be transformed into a first-order equation for f k. Indeed, by utilizing the relation - ik^Ek = -2 iN (k ) k f k, which follows from (1.6), we obtain !~ d f= k J *■

° - 9)

The substitution of expressions (1.5) and (1.6) into (1.3) yields * fk = 0.

(1.10)

Equations (1.9) and (1.10) may be combined into one equation. On multiplying (1.10) by k jk and on adding it to (1.9), we obtain (1. 11)

where

QUANTUM MECHANICS OF PHOTON

5

It follows from (1.11) that - ( * / ; > = o. Therefore (1.10) will always hold if it holds at some initial time. Equation (1.11) together with the initial condition (1.10) are equiv­ alent to the system of Maxwell’s equations. Equation (1.11) has the form of a Schrodinger equation in which w is the Hamiltonian operator. (Here and subsequently we shall make use of the system of units in which ft = 1.) The eigenvalues of this opera­ tor are equal to k. This would amount to nothing other than the quantum relation between the energy and the frequency of the photon, if we could identify the Hamiltonian operator formally introduced in the foregoing with the physical operator for the photon energy. We shall show later that such an identification can be justified, and that the function f k can be interpreted as the photon wave function in A>space in the usual quantum mechanical sense of this word. We shall also be able to define in A>space operators for other physical quantities referring to the photon, for example, the operators for the momentum, the angular momentum, etc. 1.3.

Energy

We shall show that the operator w which we have just introduced can indeed be interpreted as the operator for the photon energy. In order to do this we introduce the expression for the energy w of the electromagnetic field corresponding to a given photon as

w = \ J (E2+ H 2)dr.

(1.12)

(We make use here of Heaviside units for E and H.) This expression represents a space integral of quantities quadratic in the field vectors. On the other hand, space integrals of expressions quadratic in the wave function are interpreted in quantum mechanics as expectation values of the corresponding physical quantities. Since we are considering the field corresponding to a single photon, it would appear that the natural generalization is the interpretation of expression (1.12) as the expectation value of the photon energy.

QUANTUM ELECTRODYNAMICS

6

We show that w may be represented in the form w = J f ^ w f kdk,

(U 3 )

where w is determined by formula (1.11). In order to do this we substitute expansion (1.2) into (1.12): w = i f {EkEv + H kHv} e"k+k)rdkdk'dr. After carrying out the integration over r with the aid of the re­ lation J el