Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets [5 ed.] 9814273562, 9789814273565

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PATH INTEGRALS in Quantum Mechanics, Statistics, Polynter Physics, and Financ ial Markets 5th Edition

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PATH INTEGRALS in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets 51h Edilion

Hagen KLEINERT Professor of Physics freie Universirot Berlin, Germany !CRANe! Pescara, /mfy, and Nice, France

,~ World Scientific NeW JERSEY · LONDO N· SI NGAPORE · BEIJI NG · SHANGHAI · HO NG KONG · TAIPEI · CHE NN AI

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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

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PATH INTEGRALS IN QUANTUM MECHANICS, STATISTICS, POLYMER PHYSICS, AND FINANCIAL MARKETS (5th Edition) Copyright © 2009 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-13 978-981-4273-55-8 ISBN-13 978-981-4273-56-5 (pbk)

Printed in Singapore by Mainland Press Pte Ltd.

Kim - Path Integrals (5th Edn).pmd

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3/31/2009, 4:21 PM

To Annemarie and Hagen 11

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Nature alone knows what she wants . GOETHE

Preface The paperback version of the fourth edition of this book was sold out in Fall 2008. This gave me a chance to revise it at many places. In particular, I improved considerably Chapter 20 on fi nancial markets and removed some technical sections of Chapter 5. Among the many people who spotted printing erron; and suggested changes of text passages are Dr. A. Pelster, Dr. A. Redondo, and especially Dr. Annemarie Kleinert .

H. Kleinert Berlin, January 2009

vii

Preface to Fourth Edition The third edition of t his book appeared in 2004 and was reprinted in the same year without improvements. The present fourt h edition contains several extensions. Chapter 4 includes now semiclassical expansions of higher order. Chapter 8 offers an additional path integral formulation of spinning particles whose action contains a vector field and a Wess-Zumino term. From this, the Landau-Lifshitz equation for spin precession is derived which governs the behavior of quantum spin liquids. The path integral demonstrates that fermions can be described by Bose fields- the basis of Skyrmion theories. A further new section introduces the Berry phase, a useful tool to explain many interesting physical phenomena. Chapter to gives more details on magnetic monopoles and multivalued fields . Another feature is new in t his edition: sections of a more technical nature are printed in smaller font size. They can well be omitted in a first reading of the book. Among the many people who spott ed printing errors and helped me improve various text passages arc Dr. A. Chervyakov, Dr. A. Pelster, Dr. F. Nogueira, Dr. M. Weyrauch , Dr. H. Ba ur, Dr. T . Iguchi, V. Bezerra, D. Jahn , S. Overesch, and especially Dr. Annemarie Kleinert.

H. Kleinert Berlin, June 2006

viii

Preface to Third Edition This t hird edition of the hook improves and extends considerably the second edition of 1995: • Chapter 2 now contains a path integral representation of t he scattering amplitude and new methods of calculating functional determinants for timedependent second-order d ifferential operators. Most importantly, it introduces the quantum field-theoretic definition of path integrals, based on pert ur bation expansions around t he trivial harmonic theory.

• Chapter 3 presents morc exactly solvable path integrals than in the previous editions. It also extends the Bcnder-Wu recursion relations for calculating perturbation expansions to morc general types of potentials. • Chapter 4 discusses now in detail the quasiclassical approximation to the scattering a mplitude and Thomas-Fermi approximation to atoms. • Chapter 5 proves the convergence of variational perturba tion theory. It also discusses atoms in strong magnetic fields and the polaron problem. • Chapter 6 shows how to obtain the spectrum of systems with infinitely high walls from perturbation expansions. • Chapter 7 offers a many-path t reatment of Boso-Einst ein condensation and degenerate Fermi gases. • Chapter 10 develops the quantum theory of a particle in curved space, treated before only in the time-sliced formalism, to perturbatively defined path integrals. Their reparametrization invariance imposes severe constraints upon integrals over products of distributions. We derive unique rules for evaluating these integrals, thus extending the linear space of distributions to a semigroup. • Chapter 15 offers a closed expression for the end-to-end distribution of stiff polymers valid for all persistence lengths . • Chapter 18 derives the operator Langevin equation a nd t he Fokker-Planck equation from the forward- backward path integral. The derivation in the literature was incomplet e, and the gap was elosed only roccntly by an elegant calculation of the J acobian functional determinant of a second-order d ifferential operator with dissipation. ix

x • Chapter 20 is completely new. It introduces t he reader into t he applications of path integrals to the fascinating new field of econophysics. For a few years, the third edition has been freely available on the internet , and several readers have sent useful comments, fo r instance E. Babaev, H. Baur, B. Budnyj , Chen Li-ming, A.A. Dragulescu, K. Claum, L Grigorenko, T .S. Hatamian, P. Hollist er, P. Jizba, B. Kastening, M. Kramer, W.-F. Lu, S. Mukhin, A. Pelster, C. Dpllr, M.B. Pinto, C. Schubert, S. Schmidt, R. Scalettar, C. Tangui, and M. van Vugt. Re ported errors are corrected in the internet edition. When writing the new part of Chapter 2 on the path integral representation of t he scattering amplitude I profited from discussions with R. Rosenfelder. In t he new parts of Chapter 5 on poiarons, many useful comments came from J .T . Devreese, F.M. Peeters, and F. Brosens. In the new Chapter 20, I profited from discussions with F. Nogueira, A.A . Dragulescu, E. Eberlein, J . Kallsen, M. Schweizer, P. Bank, M. Tenney, and E.C. Chang. As in all my books, many printing errors were detected by my secretary S. Endrias and many improvements are due to my wife Annemarie without whose permanent encouragement this book would never have been fi nished.

H. Kleinert Berlin, August 2003

Preface to Second Edition Since this book first a ppeared three years ago, a number of important developments have taken place calling for various extensions to the text. Chapter 4 now contains a discussion of the feat ures of the semiclassical quantization which are relevant for multidimensional chaotic systems. Chapter 3 derives perturbation expansions in terms of Feynman graphs, whose usc is customary in quantum field theory. Correspondence is established with Rayleigh-SchrOdinger perturbation theory. Graphical expansions are used in Chapter 5 to extend t he Fcynman-Klcinert variat ional approach into a systematic variational perturbation theory. Analytically inaccessible path integrals can now be evaluated with arbitrary accuracy. In contrast to ordinary perturbation expansions which always diverge, the new expansions are convergent for all coupling strengths, including the strong-coupling limit. Chapter 10 contains now a new action principle which is necessary to derive the correct classical equations of motion in spaces with curvature and a certain class of torsion (gradicnt torsion). Chapter 19 is new. It deals with relativistic path integrals, which were previously discussed only briefly in two sections at the end of Chapter 15. As an application, the path integral of the relativistic hydrogen a tom is solved.. Chapter 16 is extended by a theory of particles with fract ional statistics (anyons), from which I develop a theory of polymer entanglement. For t his r introduce nonabelian Chern-Simons fields and show their relationship with various knot polynomia ls (Jones, HOMFLY). The successful explanation of the fractio nal quantum Hall effect by anyon theory is discussed - also the failure to explain high-temperature superconductivity via a Chern-Simons interaction. Chapter 17 offers a novel variational approach to tunneling amplitudes. It extend;; the semiclassical range of validity from high to low barriers. As an application, I increase the range of validity of the currently used large-order perturbation theory far into the regime of low orders. This suggests a possibility of greatly improving existing resummation procedures for divergent perturbation series of quantum fie ld theories. The Index now also contains the names of authors cited in the text. This may help the reader searching for topics associated with t hese names. Due to their great number, it was impossible to cite all the authors who have made important contributions. I apologize to all those who vainly search for t heir names. xi

xii In writing t he new sections in Chapters 4 and 16, discussions with Dr. D. Wintgen and , in particular, Dr. A. Schakel have been extremely useful. I also thank Professors G. Gerlich, P. Hanggi, H. Grabert, M. Roncadelli, as well as Dr. A. Pelster, and Mr. R. Karrlein for many relevant comments. Printing errors were corrected by my secretary Ms. S. Endrias and by my editor Ms. Lim Feng Nee of World Scientific. Many improvements arc duc to my wife Anncmarie.

H. Kleinert Berlin, December 1994

Preface to First Ed ition These are extended lecture notes of a course on path integrals which I delivered at the Freie Universitat Berlin during winter 1989/1990. My interest in this subject dates back to 1972 when t he late R. P. Feynman drew my attention to the unsolved path integral of the hydrogen atom. I was then 5pending my sabbatical year at Caltech, where Feynman told me during a discussion how embarrassed he was, not being able t o solve t he path integral of this moot fundament al quantum system. In fact, t his had made him quit teaching t his subject in his course on quantum mechanics as he had initially done.! Feynman challenged me: "Kleinert, you figured out all that grouptheoretic stuff of the hydrogen atom, why don't you solve t he path integraW He was referring to my 1967 Ph.D. thesis2 where I had demonstrated that all dynamical questions on the hydrogen atom could be answered using only operations within a dynamical group 0(4 ,2). Indeed, in that work, t he four-dimensional oscillator played a crucial role and the missing steps to the solution of the path integral were latcr found to be very few. After returning to Berlin, I forgot about the problem since I was busy applying path integrals in another context, developing a field-theoretic passage from quark theories to a collective field t heory of hadrons. 3 Later, I carried these techniques over into condensed matter (superconductors, superfluid 3He) and nuclear physics. Path integrals have made it possible to build a unified field theory of collective phenomena in quite different physical systems. 4 The hydrogen problem came up again in 1978 as I was teaching a course on quantum mechanics. To explain the concept of quantum fluctuations, I gave an introduction to path integrals. At t he same time, a postdoc from Turkey, I. H. Duru, joined my group as a Humboldt fellow. Since he was fam iliar with quantum mechanics, I suggested that we should try solving t he path integral of the hydrogen atom. He quickly acquired the basic techniques, and soon we found the most important ingredient to the solution: The transformation of time in the path int egral to a new path-dependent pseudotime, combined with a transformation of the coordinates to i Quoting from t he preface of the textbook by n.p. Feynman and A.n. Hibb6, Quantum Mechanics and Path Integrals, 1IIcGraw-HiU, New York, 1965: "Over t he succeeding years, . Dr. Fcynman's approach to teaching thc subject of quantum mechanics evolved Il()mcwhat a.way from t he initial path integral approoch .~ 2H. Kleinert, Fortschr. Phys. 6, 1, (1968), and Group Dynamics of the Hydrogen Atom, Lectures presented at the 19671loulder Summer School, published in Lectures in Theoretical Physics, Vol. X B, Pi>. 427-482, ed. by A.O. Barut and W.E. Brittin, Gordon and Breach, New York, 1968. 3Scc my 1976 Erice lectures, Hadronization of Quark Theories, publ ished in Understanding the Fundamental Constituenl.i! of Matter, Plenu m press, New York, 1978, p. 289, ed. by A. Zichichi. 4H. Kleinert, Phys. Leu. B 69, 9 (1977); Fortschr. Phys. 26, 565 (1978); 30, 187, 351 (1982).

xiii

xiv "square root coordinates" (to be explained in Chapters 13 and 14).5 These transformat ions led to t he correct result, however, only due to good fort une. In fact, our procedure was immediately criticized for its sloppy treatment of the time slicing.6 A proper treatment could, in principle, have rendered unwanted extra terms which our treatment would have missed . Other authors went t hrough the detailed timeslicing procedure,1 but the correct result emerged only by transforming the measure of path integration inconsistently. When I calculated t he extra terms according to the standard rules I found them to be zero only in two spacc dimensions. s The same t reatment in three dimensions gave nonzero "corrections" which spoiled the beautiful result, leaving me puzzled. Only recently I happened to loca te t he place where the t hree-dimensional treatment went wrong. I had just finished a book on t he use of gauge fields in condensed matter physics. 9 The second volume deals with eru;embles of defects which are defi ned and classified by means of operational cutting and pasting procedures on an ideal crystal. Mathematically, t hese procedures correspond to nonholonomic mappings. Geometrically, they lead from a flat space to a space with curvature and torsion. While proofreading that book, I realized t hat t he transformation by which t he path integral of t he hydrogen atom is solved also produces a certain type of torsion (gradient torsion). Moreover, this happens only in three dimensions. In two dimensions, where the t ime-sliced path integral had been solved without problems, torsion is absent. Thus I realized that t he transformat ion of the time-sliced measure had a hitherto unknown sensit ivity to torsion. It was t herefore essential to find a. correct path integral for a. particle in a space with curvature and gradient torsion . This was a nontrivial task si nce the literature was ambiguous already for a purely curved space, offering several prescriptions to choose from. The corresponding equivalent SchrOdinger equations differ by multiples of the curvature scalar. 10 T he ambiguities are path integral analogs of t he so-called opemtor-omering problem in quantum mechanics. When trying to apply t he existing prescript ions to spaces with torsion, I always ran into a disaster, some even yielding noncovariant answers. So, something had to be wrong with all of t hem. Guided by t he idea that in spaces with constant curvature the path integral should produce t he same result as an operator quantum mechanics based on a quantization of angular momenta, I was eventually able to find a consistent quantum equivalence principle 5I. H. Duru and H. Kleinert, P hys. Lett. B 84 , 30 (1 979), Fortschr. Phys. 30 , 401 (1982). 6C.A. Ringwood and J.T. Devreese, J . Math. Phys. 21, 1390 (1980) . 1R. Ho and A. inomata, Phys. Rev. Lett. 48, 231 (1982); A. inomata, Phys. Lett. A 87, 387 (1981). 8H. Kleinert, Phys. Lett. B 189, 187 (1987); contains also a crit icism of Ref. 7. 9H. Kleinert, Gauge FieltJ.j in Condensed Matter, World Scientific, Singapore, 1989, Vol. i, pp. 1- 744, Super/fow and Vortex Lines , and Vol. II, pp. 745- 1456, Stre8ses and Defects. 10B.S. DeWitt, Rev. Mod. Phys. 1!9, 377 (1957); K.S. Cheng, J. Math. P hys. 13, 1723 (1 972), H. Kamo and T. Kawai, Prog. Thoor. Phys. 50 , 680, (1973); T . Kawai , Found. P hys. 5, 143 (1975), H. Dekker, Physica A 103, 586 (1980), C.M. Cavazzi, Nuovo Cimento 101 A, 241 (1 981); M.s. Marinov, Physia; Reports 60, I (1 980) .

xv for path integrals in spaces with curvature and gradient torsion,1l thus offeri ng also a unique solution to the operator-orderi ng problem. T his was the key to the leftover problem in the Coulomb path integral in three dimensions - the proof of the absence of t he extra time slicing contributions prcscnted in C hapter 13. Chapter 14 solves a variety of one-dimensional systems by the new techniques. Special emphasis is given in Chapter 8 t o instability (path collapse) problems in the Euclidean version of Feynman's time-sliced path integraL These arise for actions containing bottomless potentials. A general stabilization procedure is developed in Chapter 12. It must be applied whenever centrifugal barriers, angular barriers, or Coulomb potentials are prescntYl Another project suggested to me by Feynman, the improvement of a variational approach to path integrals explained in his book on statistical mecha nics l 3 , found a faster solution. We started work d uring my sabbatical stay at the University of California at Santa Barbara in 1982. After a few meetings and discussions, the problem was solved and the preprint drafted. Unfortunately, Feynman 's illness prevented him from reading the fin al proof of t he paper. He was able to do this only t hree years later when T came to the University of California at San Diego for another sabbatical leave. Only then could the paper be submitted. 14 Due to recent int erest in lattice theories, I have found it useful t o exhibit the solution of several path integrals for a finite number of time slices, without going immediately to the continuum limit . T his should hclp identify typical lattice effects seen in t he Monte Carlo simulation data of various systems. The path integral description of polymers is introduced in Chapter 15 where stiffness as well as the famous excluded-volume problem are discussed. P arallels are drawn to path integrals of relativistic particle orbits. This chapter is a preparation for ongoing research in t he theory of fluctuating tmrfaces with extrinsic curvature stiffness, and their application to world sheets of strings in particle physics. I:> I have a lso int roduced the field-t heoretic description of a polymer to account for its increasing relevance to the understanding of various phase transitions driven by fluctuating line-like excitations (vortex lines in superfluicls and superconductors, defect lines in crystals and liquid crystals) .16 Special attention has been devoted in C ha pter 16 to simple topological questions of polymers and particle orbits, the latter arising by the presence of magnetic flux tubes (Aharonov-Bohm effect) . Their relationship to Bose and Fermi statistics of particles is pointed out and the recently popular top ic of fractional statistics is introduced. A survey of entanglement phenomena of single orbits and pairs of t hem (ribbons) is given and their application to biophysics is indicated . I1 H. Klei nert, Mod. P hys. Lett. A 4, 2329 (1 989); Phys. Lett. B 296, 315 (1990). 12H. Kleinert, Phys. Lett. B 224,313 (1989). 13R.p. Feynman, Statistical Mechanics, Benja.min, Read ing, 1912, Section 3.5. t° R.P. Feynman and H. Kleinert, Phy:l. Rev. A 34,5080, (1986). IS A.M. Polyakov, Nuel. P hys. B 268, 406 (1986), H. Kleinert , Phy:l. Lett. B 174,335 (1986). 16Sec Ref. 9.

xvi

Finally, Chapter 18 contains a brief introduction to the path integral approach of nonequilibrium quantum-statistical mechanics, deriving from it the standard Langevin and Fokker-Planck equat ions. I want to thank several students in my class, my graduate students, and my postdocs for many useful discussions. In particular, T. Eris, F. Langhammer, B. Meller, I. Mustapic, T . Sauer, L. Semig, J. Zaun, and Drs. G. German, C. Holm, D. Johnston, and P. Kornilovitch have all cont ributed with constructive criticism. Dr. U. Eckern from Karlsruhe University clarified some points in the path integral derivation of the Fokker-Planck equation in Chapter 18. Useful comments are due to Dr. P.A . Horvathy, Dr. J . Whit enton, and to my colleague Prof. W. Theis. T heir careful reading uncovered many shortcomings in the first draft of t he manuscript. Special t hanks go to Dr. W. Janke with whom I had a fert ile collaboration over the years and many discussions on various aspect s of path integration. T hanks go also to my secretary S. Endrias for her help in preparing the manuscript in WTBX, thus making it readable at an early stage, and to U. Grimm for drawing the figures. Finally, a nd most importantly, I am grateful to my wife Dr. Annemarie Kleinert for her inexhaustible patience and constant encouragement.

H. Kleinert Berlin, January 1990

Contents

Preface

vii

Preface to Fourth Edition

viii

Preface to Third Edition

ix

Preface to Second Edition

xi

Preface to First Edition

xiii

1 Fundamentals 1. 1 Classical Mechanics 1.2 Relativistic Mechanics in Curved Spacetime l.3 Quantum Mechanics 1.3.1 Bragg Reflections and Interference 1.3.2 Matter Waves . SchrOdinger Equation 1.3.3 1.3.4 Particle Current Conservation . 1.4 Dirac's Bra- Ket Formalism. 1.4.1 Basis Transformations 1.4.2 Bracket Notation 1.4.3 Continuum Limit . . . 1.4.4 Generalized Functions 1.4.5 SchrOdinger Equation in Dirac Notation 1.4.6 Momentum States 1.4.7 Incompleteness and Poisson's Summation Formula 1. 5 Obscrvables 1.5.1 Uncertainty Relation. 1.5.2 Density Matrix and Wigner Function. 1.5.3 Generalization to Many Particles 1.6 Time Evolution Operator . l.7 Properties of the Time Evolution Operator 1.8 Heisenberg Picture of Quant um Mechanics l.9 Interaction Picture and Perturbation Expansion 1.10 Time Evolution Amplitude 1.11 Fixed-Energy Amplitude xvii

1 1 10 11

12 13 15 17 18 18

20 22 23 25 26 28 31 32 33 34 34 37 39

42 43 45

xviii 1.12 1.13 1.14 1.15 1.16

Free-Particle Amplitudes. . . . . . . . . . . . . . . . Quantum Mechanics of General Lagrangian Systems. Particle on the Surface of a Sphere Spinning Top . . . . . . Scattering 1. 16.1 Scattering Matrix 1.16.2 Cross Section . . . 1.16.3 Born Approximation 1. 16.4 Partial Wave Expansion and Eikonal Approximation 1. 16.5 Scattering Amplitude from Time Evolution Amplitude 1. 16.6 Lippmann-Schwinger Equation 1.17 Classical and Quantum Statistics 1. 17.1 Canonical Ensemble. . 1. 17.2 Grand-Canonical Ensemble 1.18 Density of States and Ttacelog . Appendix l A Simple Time Evolution Operator. Appendix I B Convergence of the Fresnel Integral Appendix l C The Asymmetric Top Notes and References. . . . . . . . . . . . . . . .

47 51 57 59 67 67 68 70 70 72 72 76 77 77 82 84 84 85 87

2 P ath Integrals - E le mentary Properties and Simple Solutions 2.1 Path Integral Representation of Time Evolution Amplitudcs 2.1.1 Sliced Time Evolution Amplitude. 2.1.2 Zero-Hamiltonian Path Integral. . 2.1.3 SchrOdinger Equation for Time Evolution Amplitude 2.1.4 Convergence of of the Time-Sliced Evolution Amplitude 2.1.5 Time Evolution Amplitude in Momentum Space. 2.1.6 Quantum-Mechanical Partition Function. . . 2.1.7 Feynman's Configurat ion Space Pat h Integral 2.2 Exact Solution for the Frec Particle 2.2. 1 Direct Solution 2.2.2 Fluctuations around the Classical Path . 2.2.3 Fluctuation Factor . . . . . . . . . . . . 2.2.4 Finite Slicing Properties of Free-Particle Amplitude. 2.3 Exact Solution for Harmonic Oscillator . 2.3.1 Fluctuations around the Classical Path. . . . 2.3.2 Fluctuation Factor . 2.3.3 The i7]-Prescription and Maslov-Morse Index 2.3.4 Continuum Limit. . . . . . . . . . . . . 2.3.5 Useful Fluctuation Formulas. 2.3.6 Oscillator Amplitude on Finite Time Lattice . 2.4 Gelfand-Yaglom Formula. . . 2.4.1 Recursive Calculation of Fluctuation Detcrminant. 2.4.2 Examples

89 89 89 91 92 93 94 96 97 101 101 102 104 111 112 112 114 115 116 117 119 120 121 121

xix

2.5

2.6

2.7 2.8 2.9 2.10

2.11 2.12 2.13 2.14 2.15

2.16 2.17 2.18

2.19 2.20 2.21 2.22

2.23

2.4.3 Calculation on Unsliced Time Axis 123 2.4.4 D'Alembert 's Construction 124 2.4 .5 Another Simple Formula. 125 2.4.6 Generalization t o D Dimensions 127 Harmonic Oscillator with Time-Dependent Frequency 127 2.5.1 Coordinate Space. 128 2.5.2 Momentum Space .. . . . . . . 130 Free-Particle and Oscillator Wave Functions 132 General Time-Dependent Harmonic Action 134 Path Integrals and Quantum Statistics 135 Density Matrix 138 Quantum Statistics of the Harmonic Oscillator 143 Time-Dependent Harmonic Potential 148 Functional Measure in Fourier Space .. 151 154 Classical Limit. . . . . . . . . . . . . . . Calculation Techniques on Sliced Time Axis via the Poisson Formula 155 Ficld·Theorctic Definition of Harmonic Path Integrals by Analytic Regularization . . . . .. . . . . . . . . . . 158 2. 15.1 Zero--Temperature Evaluation of the Frequency Sum. 159 2. 15.2 Finite-Temperature Evaluation of the Frequency Sum. 162 2. 15.3 Quantum.Mechanical Harmonic Oscillator . 164 2. 15.4 Tracelog of the First·Order Differential Operator 165 2. 15.5 Gradient Expansion of the One-Dimensional Tracclog . 167 2. 15.6 Duality Transformation and Low· Temperature Expansion 168 Finite-N Behavior of Thermodynamic Quantities 175 Time Evolution Amplitude of Freely Falling Particle. 177 Charged Particle in Magnetic Field 179 2. 18.1 Action. 179 2. 18.2 Gauge Properties. 182 2. 18.3 Time-Sliced Path Integration 182 2. 18.4 Classical Action 184 2. 18.5 Translational Invariancc 185 Charged Particle in Magnetic Field plus Harmonic Potentia] 186 Gauge Invariance and Alternative Path Integral Representation 188 Velocity Path Integral. . . . . . . . . . . .. . . . . . 189 Path Integral Representation of the Scattering Matrix . 190 2.22.1 General Development 190 2.22.2 Improved Formulation . 193 2.22.3 Eikonal Approximation to the Scattering Amplitude 194 Heisenberg Operator Approach to Time Evolution Amplitude. 194 2.23.1 Free Particle . . . . . . . . . . . . 195 2.23.2 Harmonic Oscillator .. 197 2.23.3 Charged Particle in Magnetic Field 197

xx Appendix 2A Baker-Campbell-Hausdorff Formula and Magnus Expansion . 201 Appendix 28 Direct Calculation of the Time-Sliced Oscillator Amplitudc204 Appendix 2C Derivation of Mehler Formula. 205 Notes and References 206

3 External Sources, Correlations, and Perturbation Theory 209 3.1 External Sources .. . . . . . . . . . 209 3.2 Green Function of Harmonic Oscillator 213 3.2.1 Wronski Construction 213 3.2.2 Spectral Representation . . . . . 217 3.3 Green Functions of First-Order Differential Equation 219 3.3.1 Time-Independent Frequency 219 3.3.2 Time-Dependent Frequency . . . . . . . . . . . 226 3.4 Summing Spectral Representation of Green Function 229 3.5 Wronski Construction for Periodic and Antiperiodic Green F\mctions 231 3.6 Time Evolution Amplitude in Presence of Source Term 232 3.7 T ime Evolution Amplitude at Fixed Path Average 236 3.8 External Source in Quantum-Statistical Path Integral 237 3.8.1 Continuation of Real-Time Result 238 3.8.2 Calculation at Imaginary Time 242 3.9 Lattice Green Function .. 249 3.10 Correlation Functions, Generating Functional, a nd Wick Expansion 249 3.10.1 Real-Time Correlation Functions. . . . . . . . . . . . 252 3.11 Correlation Functions of Charged Particle in Magnetic Field 254 3.12 Correlation Functions in Canonical Path Integral. 255 3.12.1 Harmonic Correlation Functions 256 3.12.2 Relations between Various Amplitudes 258 3.12.3 Harmonic Generating Functionals. 259 3.13 Particle in Heat Bath . . . . . 262 Heat Bath of Photons. 266 3.14 3.15 Harmonic Oscillator in Ohmic Heat Bath 268 3.16 Harmonic Oscillator in Photon Heat Bath 271 3.17 Perturbation Expansion of Anharmonic Systems 272 3.18 Rayleigh-SchrOdinger and Brillouin-Wigner Perturbation Expansion 276 Level-Shifts and Perturbed Wave Functions from SchrOdinger 3.19 Equation . 280 3.20 CrucuiatioIl of Perturbation Series via Fcynman Diagrams. 282 287 3.21 Perturbative Definition of Interacting Path Integrals. 3.22 Generating Functional of Connected Correlation Functions 288 3.22.1 Connectedness Structure of Correlation Functions. 289 3.22.2 Correlation Functions versus Connected Correlation Functions . 292

xxi 3.22.3 FUnctional Generation of Vacuum Diagrams. . . . . . .. 294 3.22.4 Correlation Functions from Vacuum Diagra.ms . 298 3.22.5 Generating Functional for Vertex Functions. Effective Action 300 3.22.6 Ginzburg-Landau Approximation to Generating Functional 305 3.22.7 Composite Fields. 306 3.23 Path Integral Calculation of Effective Action by Loop Expansion 307 3.23.1 General Formalism 307 3.23.2 Mean-Field Approximation 308 3.23.3 Corrections from Quadratic Fluctuations. 312 3.23.4 Effective Action to Second Order in Ii 315 3.23.5 Finite-Temperature Two-Loop Effective Action 319 3.23.6 Background Field Method for Effective Action 321 3.24 Nambu-Goldstone Theorem 324 3.25 Effective Classical Potential . . . 326 3.25.1 Effective Classical Boltzmann Factor 327 3.25.2 Effective Classical Hamiltonian . . . 330 3.25.3 High- and Low-Temperature Behavior 331 3.25.4 Alternative Candidate for Effective Classical Potential 332 3.25.5 Harmonic Correlation FUnction without Zero Mode 333 3.25.6 Perturbation Expansion 334 3.25.7 Effective Potential and Magnetization Curves 336 3.25.8 First-Order Perturbative Result. . . . . . 338 3.26 Perturbativc Approach to Scattering Amplitude 340 3.26.1 Generating Functional 340 3.26.2 Application to Scattering Amplitude . . . 341 3.26.3 First Correction to Eikonal Approximation 341 3.26.4 Rayleigh-SehrOdinger Expansion of Scattering Amplitude. 342 3.27 Functional Determinants from Green FUnctions 344 Appendix 3A Matrix Elements for General Potential. . 350 4 Appendix 3B Energy Shifts for gx /4-Interaction. 351 Appendix 3C Recursion Relations for Perturbation Coefficients. 353 3C.l One-Dimensional Interaction X4 • 353 3C.2 General One-Dimensional Interaction. 356 3C.3 Cumulative Treatment of Interactions X4 and x 3 . 356 Ground-State Energy with External Current . . . 3C.4 358 3C.5 Recursion Relation for Effective Potential 360 Interaction r 4 in D-Dimensional Radial Oscillator. 3C.6 363 3C.7 Interaction r2q in D Dimensions. 36' 3C.8 Polynomial Interaction in D Dimensions 364 Appendix 3D Feynman Integrals for T =f 0 36' Notes and References . 367

xxii 4

Semiclassical Time Evolution Amplitude 369 4. 1 Wentzel-Kramers-Brillouin (WKB) Approximation. 369 4.2 Saddle Point Approximation 376 4.2. 1 Ordinary Integrals . . . . . . . . 376 4.2.2 Path Integrals 379 4.3 Van Vlcek-Pauli-Morette Determinant. 385 404 Fundamental Composition Law for Semiclassical Time Evolution Amplitude 389 4.5 Semiclassical Fixed-Energy Amplitude . 391 4.6 Semiclassical Amplitude in Momentum Space. 393 4.7 Semiclassical Quantum-Mechanical Partition Function . 395 4.8 Multi-Dimensional Systems . . . . . 400 4.9 Quantum Corrections to Classical Density of States 405 4.9. 1 One-Dimensional Case . 406 4.9.2 Arbitrary Dimensions . . . . . . . . . . . . . 408 4.9.3 Bilocal Density of States. 409 4.904 Gradient Expansion of Tracelog of Hamiltonian Operator. 411 4.9.5 Local Density of States on Circle . . . . .. . . . . . . 415 4.9.6 Quantum Corrections to Bohr-Sommerfeld Approximation 416 4. 10 T homas-Fermi Model of Neutral Atoms. 419 4.10.1 Semiclassical Limit. 419 4.10.2 Self-Consistent Field Equation 421 4.10.3 Encrgy Functional of Thomas-Fermi Atom. 423 4.1004 Calculation of Energies 424 4.10.5 Virial Theorem 427 4.10.6 Exchange Energy. . 428 429 4.10.7 Quantum Correction Near Origin 4.10.8 Systematic Quant um Corrections to T homas-Fermi Energies 432 4. 11 Classical Action of Coulomb System. 436 4. 12 Semiclassical Scattering. . 444 4.12.1 General Formulation 444 4.12.2 Semiclassical Cross Section of Mott Scatt ering . 448 Appendix 4A Semiclassical Quantization for Pure Power Potentials . 449 Appendix 4B Derivation of Semiclassical T ime Evolution Amplitude 451 Notes and References. . . . . .. ................... 455

5 Variation al Perturbation T heory 458 5.1 Variational Approach to Effective Classical Partition Function 458 5.2 Local Harmonic Trial Partition Function 459 5.3 Optimal Upper Bound . . 464 504 Accuracy of Variational Approximation . 465 5.5 Weakly Bound Ground State Energy in Finite-Range Potential Well 468 5.6 Possible Direct Generalizations . 469 5.7 Effective Classical Potential for Anharmonic Oscillator 470

xxiii Particle Densities . . Ext ension to D Dimensions Application to Coulomb and Yukawa Potentials Hydrogen Atom in Strong Magnetic Field . 5. 11.1 Weak-Field Behavior .. 5.11.2 Effective Classical Hamiltonian .. 5.12 Variational Approach to Excitation Energies 5.13 SY5tematic Improvement of Feynman-Kleinert Approximation ... 5.14 Application5 of Variational Perturbation Expansion 5.14. 1 Anharmonic Oscillator at T = O. 5.14.2 Anharmonic Oscillator for T > 0 . . . . . . . 5.15 Convergence of Variational Perturbation Expansion 5.16 Variational Perturbation T heory for Strong-Coupling Expansion 5.17 General Strong-Coupling Expansions . 5.18 Variational Interpolation betwccn Weak and Strong-Coupling Expansions . . . . . . . . . . . . . . . . . . . . . . 5.19 Systematic Improvement of Excited Energies 5.20 Variational Treat ment of Double-Well Potential 5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions .. . . . . . . . 5.21.1 Evaluation of Path Integrals . 5.21. 2 Higher-Order Smearing Formula in D Dimensions. 5.21.3 Isotropic Second-Order Approximation to Coulomb Problem 5.21.4 Anisotropic Second-Order Approximation to Coulomb Problem . . . 5.21.5 Zero-Temperature Limit 5.22 Polarons 5.22.1 Partition Function 5.22.2 Harmonic Trial System 5.22.3 Effective Mass 5.22.4 Second-Order Correction. 5.22.5 Polaron in Magnetic Field, Bipoiarons, etc. 5.22.6 Variational Interpolation for Polaron Energy and Mass 5.23 Density Matrices . 5.23.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 5.23.2 Variational Perturbation Theory for Density Matrices. 5.23.3 Smearing Formula for Density Matrices .. 5.23.4 First-Order Variational Approximation . 5.23.5 Smearing Formula in Higher Spatial Dimensions. Appendix 5A Feynman Integrals for T =f 0 without Zero Frequency Appendix 5B Proof of Scaling Relation for the Extrema of W N Appendix 5C Second-Order Shift of Polaron Energy Notes and References . 5.8 5.9 5.10 5.11

475 479 48 1 484 488 488 492 496 498 499 501 505 512 515

518 520 521 523 524 525 527 528 529 533 535 537 542 543 544 545 548 548 550 552 554 558 560 562 564 565

xxiv 6 Path Integrals with Topological Constraints 6.1 Point Particle on Circle. 6.2 Infinite Wall 6.3 Point Particle in Box . . . 6.4 Strong-Coupling Theory for Particle in Box. 6.4. 1 Partition Function . 6.4.2 Perturbation Expansion 6.4.3 Variational Strong-Coupling Approximations 6.4.4 Special Properties of Expansion . 6.4.5 ExponentiaJly Fast Convergence . Notes and References .

571

7 Many Particle Orbits - Statistics and Second Quantization 7.1 Ensembles of Bose and Fermi Particle Orbits 7.2 Bose-Einstein Condensation . 7.2. 1 Free Bose Gas 7.2.2 Bose Gas in Finite Box 7.2.3 Effect of Interactions . 7.2.4 Bose-Einstein Condensation in Harmonic 'frap 7.2.5 Thermodynamic Functions . 7.2.6 Critical Temperature . 7.2.7 More General Anisotropic Trap 7.2.8 Rotating Bose-Einstein Gas 7.2.9 Finite-Size Corrections . . . . 7.2.10 Entropy and Specific Heat .. 7.2. 11 Interactions in Harmonic Trap Gas of Free Fermions 7.3 Statistics Interaction . . . 7.4 7.5 Fractional Statistics . Second-Quantized Bose Fields 7.6 7.7 Fluctuating Bose Fields. 7.8 Coherent St ates Second-Quantized Fermi Fields 7.9 Fluctuating Fermi Fields . 7.10 7.10.1 Grassmann Variables . 7.10.2 Fermionic Functional Determinant 7.10.3 Coherent States for Fermions 7.11 Hilbert SPac

(P., F1) (Q"Q')

0, ~

o.

(1.47)

8

1 Fundamentals

This follows from the fact that the 2N x 2N matrix formed from the Lagrange brackets

-(Q; ,Q,) ) (P; ,Qj) can be written as Poisson brackets

(£_1)_1

p

E? ) -1,

=(

(1.48)

while an analogous matrix formed from the

{P;,Qj } {Q;, Qj }

- {P;, Pj

}

)

- {Q;, Pj}

(1.49)

is equal to J(E- 1JE)T. Hence C = p - l, so that (1.43) and (1.47) arc equivalent to each other. Note that the Lagrange brackets (1.43) [and thus the Poisson brackets (1.47)1 ens ure p/J; - P/:tj to be a total differential of some function of Pj and Qj in the 2N-dimensional phase space: (1.50)

The Poisson brackets (1.47) for P;, Q. have the same form as those in Eqs. (1.25) for the original phase space variables Pi, q;. The other two equations (1.42) relate the new Hamiltonian to t he old one. They can always be used to construct H'(Pj , Qj, t) from H(p;, q;, t}. The Lagrange brackets ( 1.43) or Poisson brackets (1.47) a.rc therefore both nc (t)).

•

(LJ36)

This is, of course, just an a pproximation to the integral

Jd'xh· (x)" (x l'l>(t)) .

(1.137)

The completeness of the basis hn(x) may therefore be expressed via t he abstract relation (1.138)

•

The approximate sign turns into an equality sign in t he limit of zero mesh size, 1:-0.

22

1.4.3

1 Fundamentals

Continuum Limit

In ordinary calculus, fine r and fine r sums are eventually replaced by integrals. T he same thing is done here. We define new continuous scalar products (1.139)

where X n are t he lattice points closest to x. With (1.1 34), the right-hand side is equal to lI1(xn,t) . In the limit t. ---- 0, x and X n coincide a.nd we have

(xlw(,» = w(x,').

(1.140)

The completeness relation can be used to write

(al,,(t»

~

2)alx. )(x. I,,(t» •

~ L"(a l x)(x l w('» I "~""

(1.141)

•

which in t he limit

t. ____

0 becomes

Jd'x (alx)(x IW(t».

(a lw(,» ~

(1.142)

This may be viewed as t he result of inserting the formal completeness relation of the limiting local bra. and ket basis vectors (x l and Ix) ,

Jd'x Ix)(xl ~

1,

(1.143)

evaluated between the vect ors (al and 11l«t)). With the limiting local basis, the wave functions can be treated as components of t he st ate vectors 11lr(t)) wit h respect to t he local basis Ix) in the same way as a.ny other set of components in an arbitrary basis la). In fact , the expansion

Jd'x (alx)(x l"'('»

(al"'(t» ~

(1.144)

may be viewed as a re-expansion of a component of 11l«t)) in one basis, la), into those of another basis, Ix) , just as in (1.129). In order to express all t hese transformation properties in a most compact nota.tion, it has become customary to deal with an arbitrary physical state vector in a basis-independent way and denote it by a ket vect or IIlr(t) . T his vector may be specified in any convenient basis by multiplying it with the corresponding completeness relation Da)(al ~ 1, (1.145)

,

resulting in t he expansion

Iw(t»

~

Da)(alw(,». ,

(1.146)

23

1.4 Dirac's Bra-Kct Formalism

This can be multiplied with any bra vector, say (bl, from the left to obtain the expansion formula (1. 131):

(bl"' (')) ~ Dbla)(al"'('» ,

•

(1.1 47)

The continuum version of the completeness relation (1.1 38) reads

Ji'x Ix )(x l ~

1,

(1.1 48)

and leads to t he expansion

1"' ('))

J

~ i'x Ix )(x l"' (')) ,

(1.1 49)

in which the wave function 1lr(x, t) = (x lw(t)) plays t he role of an xth component of the state vector II11(t)} in the local basis Ix). This, in turn, is t he limit of the discrete basis vectors Ix n), 1 (1.150) Ix ) ~ 0< Ix.) , V"

with Xn being the lattice points closest to x. A vector can be described equally well in bra or in ket form. To apply t he above formalism consistently, we observe that t he scalar products

(a lb) ~ (b la)

Jd'x f"( x )"f'(x ), Jd'x l'(x), f"(x)

(Ll5!)

satisfy the identity

(bla) " (alb)",

(1.1 52)

Therefore, when expanding a ket vector as

L: la)(a l"' (')),

(1.1 53)

("' (' )1 ~ D"' (t) la)(al,

(1.1 54)

1"'(')) ~ or a bra vector as

•

a multiplication of the first equation with the bra (x l and of the second with the ket Ix ) produces equations which are complex-conjugate to each other.

1.4.4

G e n eralize d Functions

Dirac's bra-ket formalism is elegant and easy to handle. As far as t he vectors Ix) are concerned t here is, however, one inconsistency with some fundame ntal postulates of quantum mechanics: When introducing state vectors, the norm was required to be unity in order to permit a proper probability interpretation of single-particle states.

24

1 Fundamentals

The limiting states Ix ) introduced above do not satisfy this requirement. In fact , the scalar product between two different states (x l and Ix') is

where Xn and X n, are the lattice points closest to x a nd x'. For x of x' , the states are orthogonal. For x = X , on t he other hand, the limit t: --+ 0 is infinite, approached in such a way that (1.156)

Therefore, the limiting state Ix) is not a properly normalizablc vector in the Hilbert space. For the sake of elegance, it is useful to weaken the requirement of normaliza bility (1.96) by admitting the limiting states Ix) to the physical Hilbert space. In fact , one admits all states which can be obtained by a limiting sequence from properly normalized state vectors. The scalar product between states (xlx') is not a proper function. It is denoted by the symbol 6(3)(X - x) and called Dimc J-junction: (x ix') =: 6(3)(x - x').

(1.157)

The right-hand side vanishes everywhere, except in the infinitely small box of width f around x Rj x. Thus the J-function satisfies J(3)(x -

x) =

0

for

x =F

x.

(1.158)

At x = x', it is so large that its volume integral is unity: (1.159)

Obviously, t here exists no proper functio n that can satisfy both requirements, (1 .158) and (1.159). Only the finite-f approximation in (1.155) to the 6-function are proper functions. In this respect, t he scalar product (xix') behaves just like the states Ix ) themselves: Both are f --+ 0 -limit.s of properly defined mathematical objects. Notc that thc integral Eq. (1.159) implies the following propcrty of the J function: o(3)(a(x - x'))

~ I~t)(x -

x').

(1.160)

In one dimension, this leads to the more general relation 1

o(f(x))

where

~ ~ II'(x,)lo(x - x;},

x. are the simple zeros of f(x).

(1.161 )

25

1.4 Dirac's Bra-Kct Formalism

In mathematics, one calls the a-function a generalized function or a distribution. It defines a linear functional of arbitrary smooth test functions f(x) which yields its value at any desired place x :

bl!, xl "

J

Jd'x (plx)(xl w(t)) J

(plw(t))

(1.1 91)

d 3 x e- ;px/hW(x , t) .

=

T he amplitudes (p llJ1 (t)) are referred to as momentum space wave functions. By inserting the completeness relation

Jd'xlx)(xl ~

1

(1.192)

between t he moment um states on t he left-hand side of the orthogonality relation (1. 186), we obtain the Fourier representation of the a-function

Jd'x (pl x)(xlp') J

(pip') =

1.4.7

(1.193)

dJxe- ·( p- p'}x /h.

Incomple t e n ess a n d P oisson 's Summa tion Formula

For many physical a pplications it is important to find out what happens to t he completeness relation (1. 148) if one restrict the int egral so a subset of positions. Most relevant will be the one-dimensional integral,

Jdx Ix)(x l ~ I , restrict ed to a sum over equally spaced points

(1.194)

Xn = na:

N

l:

n=- N

Ix.)(x. l·

(1.195)

1.4 Dirac's Bra-Kct Formalism

29

Taking this sum between momentum eigenstates Ip}, we obtain N

I:

(P1".)(x.IP') ~

n :_ N

For N

----> 00

N

I:

(Plx.)(x.IP')

~

N

I:

,'~-")=I'

(1.196)

n :_ N

n =_ N

we can perform t he sum with t he help of Poisson's summation fonnula (1.1 97) n", -oo

m "' - OO

Identifying p, wit h (p - rI)aj211"fl, we find using Eq. (1.160):

.t(Plx.)(x.IP') ~mt'C ;:2a-m) ~mt p

2:"

o(.-P' -2,:m).

(1.1 98)

In order to prove the Poisson formula (1.197), we observe that the sum 8(p,) == Lm 8(p, - m) on the right-hand side is periodic in p, with a unit period and has the Fourier series 8(1-') = L:::": _oo8 ne2".il'n. T he Fourier coefficients are given by 2lfi Sn = f~{~2 dp. s(l-')e- l'n == 1. T hese are precisely the Fouricr coefficients on the left-hand side. For a finite N, the sum over n on the left-hand sidc of (1.197) yields

- 1+ ( =

1+

1 _ e2lfiJl (N+1 ) 1

e2".il'

+ C.c.

)

(1.1 99)

e21fi l' _ e21fi l'(N+l) sin 1I"Jt(2N + 1) + c.C. = 1 e 2.n1' sin 11"11

T his function is well known in wave optics (see Fig. 2.4). It tion pattern of light behind a grating with 2N + 1 slits. Jt = 0, ± 1, ± 2, ±3,.. . and N - 1 small maxima between ing peaks, at II = (1 + 4k)/2(2N + 1) for k = 1, ... , N v ~ (1+ 2k)/(2N + 1) fo' k ~ 1, ... , N - 1. Inserting p, = (p - rI)a/211"n into (1.1 99), we obtain

determines t he diffracIt has large peaks at each pair of neighbor1. There are zeros at

~ (PI )( I ') ~ 'in (P - p')a(2N + 1)/21i x. x. P . (P p ') a /2 It' . Sill

L"

(1.200)

n :-N

Let us see how the right-hand side of (1.1 99) t urns into the right-hand side of (1.197) in the limit N ----> 00. In this limit , the area under each large peak can be calculated by an integral over the central large peak plus a number n of small maxima next to it: n/2N d sin 1I"p.(2N + 1) = ;n/ 2N d sin 211"p.N cos 1I"p.+ cos211"/lN sin 1I"/l . ; - n/2N I-' sin 1I"p. - n/2N I-' sin 1I"p.

(1.201)

30

1 Fundamentals

";nlfl'(2N +1)

"n If"

, Figure 1.2 Relevant function E;:,.,- Ne2ripn in Poisson's summation for mula. In the limit N --> 00 , Jl. is squeezed to the integer values.

Keeping a fixed ratio n/N «: 1, we may replace in the integrand sin 1r1l by 1f1l and COS1rll by 1. T hen t he integral becomes, for N --+ 00 at fixed njN, "J2N Sin1TJ.I(2N + 1) N_ oo ;n/2N sin27rJ1-N . --> dl' ; - n / 'lN dp. sm 7rJ.l. - n/2N 7rJ.l. N_oo ~

sinx -1 ;"".. dx11: -If" x

+ -1- ;

.."

21l'N _,....

dxcosx

+

N_oo -->

;n/'lN - n/2N

1

'

dJl,cos21rI-tN

(1.202)

where we have used the integral formula 00 ;

- 00

dxsinx x

=71".

(1.203)

In the limit N ____ 00, we find indeed (1.197) and thus (1.205), 3.'; well as the expression (2.464) for t he free energy. There exists another useful way of expressing Poisson's formula. Consider an arbit rary smooth fu nction f(p,) which possesses a convergent sum 00

2:

f(m) .

(1.204)

m=-oo

T hen Poisson's formula. (1.197) implies t hat the sum can be rewritten as an integral and an a uxiliary sum:

(1.205) T he auxiliary sum over n squeezes J1. to t he integer numbers.

1.5 Observablcs

1.5

31

Observables

Cha nges of basis vectors a rc an important tool in analyzing t he physically observable content of a wave vector. Let A = A{p, x) be an arbitrary time-independent real function of the phase space variables p and x. Important examples for such an A are p and x themselves, the Hamiltonian H( p , x) , and the angular momentum L = x x p . Quantum-mechanically, there will be an observable operator associated with each such quantity. It is obtained by simply replacing the variables p and x in A by the corresponding operators f> and x:

A = A(p, x).

(1.206)

T his replacement rule is the extension of the correspondence principle for the Hamiltonian operator (1.92) to more general functions in phase space, converting them into observable operators. It must be assumed that the replacement leads to a unique Hermitian operator, i.e. , that there is no ordering problem of the type discussed in context with the Hamiltonian (1. 101).8 If there are ambiguities, the naive correspondence principle is insufficient to determine the observable operator. Then the correct ordering must be decided by comparison with experiment, unless it can be specified by means of simple geometric principles. This will be done for the Hamiltonian operator in Chapter 8. Once an observable operator A is Hermitian, it has the useful property that t he set of all eigenvectors la) obtained by solving the equation

A la) ~ ala)

(1.207)

can be used as a basis to span the Hilbert spacc. Among the eigenvectors, there is always a choice of orthonormal vectors la) fulfilling the complet enes.,> relation

z= la)(al ~ 1.

(1.208)

T he vectors la} can be used to extract physical information concerning the observable A from arbitrary state vector IIIr(t)). For t his we expand this vector in the basis la}: (1.209) 1'"

,

XN)(X l, .. . , XNI = 1,

(1.229)

and define

(1.230) {XJ, ... , x Nlx"

=

x" {x l> ", , x NI·

T he SchrOdingcr equation for N particles (1.107) follows from (1.228) by multiplying it from the left with the bra vectors (Xl, ... , xN I. In t he same way, all other formulas given above can be generalized to N · body state vectors.

1.6

Time Evolution Operator

If the Hamiltonian operator posse&;es no explicit time dependence, the basisindependent SchrOdinger equation (1.163) can be integrated to find t he wave function IIV (t») at any time tb from the state at any other time ta

(1.231 ) The operator

O(tb,ta )

= e-i{to-t$)Hlh

(1.232)

is called the time evolution opemtor. It satisfies t he differential equation irI8t .U(tI>, ta) = H U(tb' tal.

(1.233)

Its inverse is obtained by interchanging t he order of tb and ta :

O-I(tb,ta) =: ei{t.-t~) iflh

=

O(ta,tl».

(1.234)

As an exponential of i times a Hermitian operator, 0 is a unitary operator satisfying

Of =

(;-1 .

(1.235)

lndood ,

(1.236)

35

1.6 Time Evolution Operator

If H(p,x, t) depends explicitly on t ime, t he integration of the SchrOdinger equation (1.163) is somewhat more involved. The solution may be found iteratively: For tb > t a , the time interval is sliced into a large number N + 1 of small pieces of thickness f. with f. == (tb - ta)/(N + 1), slicing once at each time t,. = ta + nf. for n = 0, ... , N + 1. We t hen use the SchrOdinger equation (1. 163) to relate the wave function in each slice approximately to t he previous one:

IW('o + ,))

~

(1- ;. t

+·

IIli (ta + 2£))

~

(l-*l:::'ltal also for the anticausal (or advanced) case where tb lies before tao To be consistent with the above composition law (1.254), we must have (1.256)

Indeed, when considering two states at successive times (1.257)

the order of succession is inverted by multiplying both sides by (;-I(t", tb): I ~(@ ~ O(to . t,) - 'I ~(to)) .

t, < to.

(1.258)

The operator on the right-hand side is defined to be the time evolution operator U(tb' ta) from the later time ta to the earlier time tbIf the Hamiltonian is independent of time, with the time evolution operator being (1.259)

t he unitarity of the operator U(tb' ta) is obvious: A

U

t

(t,. to) ~ U(t,. to) A

_ I

.

(1.260)

Let us verify this property for a general time-dependent Hamiltonian. There, a direct solution of the SchrOdinger equation (1.163) for the state vector shows that the operator U(Lb , La) for tb < La has a representation just like (1 .252), except for a reversed time order of its arguments. One writes this in the form [compare (1.252)] (1.261)

where T denotes the t ime-ant iordering operator, with an obvious definition analogous to (1.241), (1.242). This operator satisfies the relation (1.262)

1.8 Heisenberg Picture of Quantum Mechanics

39

with an obvious generalization to the product of n operators. We can therefore conclude right away that (1.263)

With O(ta, tb) =: O(tb, ta)- I, this proves the unitarity relation (1.260), in general.

c) Schrodinger equation for U(tb, tal Since the operator U(tb' ta) rules the relation between a rbitrary wave functions at different times, 1"' ( t a , on the other hand, the poles in the lower half-plane give, via Cauchy'S residue

47

1.12 Free-Particle Amplitudes

theorem, the spectral representation (1.3 19) of the propagator. An i1}-prescription will appear in another context in Section 2.3. If the eigenstates are nondegenerate, the residues at the poles of (1.321) render directly the products of eigenfunctions (barring degeneracies which must be discussed separately). For a system with a continuum of energy eigenvalues, there is a cut in the complex energy plane which may be thought of as a closely spaced seQuence of poles. In general, the wave functions are recovered from the discontinuity of the amplitudes (JIQ, I ~)E across the cut, using the formula disc

in

(E iftEn) == "E;--ii~C-o-'+-:i="

"E'---;::---:'= ~ 2nM(E - Eo). E"

tT/

(1.324)

Here we have employed the reiationiO, valid inside integrals over E: E

1

En±iT/

~

E

PE~ ~ i7ro(E ..

(1.325)

E,,},

where P indicates that the principal value of the integral has to be taken. The energy integral over the discontinuity of the fixed-energy amplitude (1.321) (JIQ,lxah: reproduces t he completeness relation (1.318) taken between the local states (",I ""d I",,), (00

2d~ di" (",I""),, ~ I: >l'o(",)>I';(x o) ~ (",I",,) ~ ,(DJ(x, -

~oo 7r1~

x o)'

(1.326)

"

The completeness relation reflects the fonowing property of the resolvent operator: (00 dE

•

J-OIO 2Jrh discR (E)

• = 1.

(1.327)

In general, the system possesses also a cont inuous spectrum, in which case the completeness relation contains a spectral integral and (1.318) has the form LI>I'o)(>I'ol o

+

Jdv 1>1'")(>1'"1 ~

1.

(1.328)

The continuum causes a branch cut along in the complex energy plane, and (1.326) includes an integral over the discontinuity along the cut. The cut will mostly be omitted, for brevity.

1.12

Free-Particle Amplitudes

For a free particle with a Hamiltonian operator if = p2 j 2M , the spectrum is continuous. The eigenfunctions are (1.189) with energies E(p ) = p 2j2M. Inserting IOThis is often referred to as Sochocki's JOffllula. It is the beginning of an expansion in + 1'/ ['lf6'(x) ± idzPlx] + 0(1'/2).

of 1'/ > 0: 1/(x ± il'/) = Pix 'fi 'lf6(x)

p0WCn!

48

1 Fundamentals

the completeness relation (1. 187) into Eq. (1.296) , we obtain for the time evolution amplit ude of a free particle t he Fourier representation (1.329)

The momentum integrals can easily be done. First we perform a quadratic completion in the exponent and rewrite it as

1

P(Xb - x..) - -

2M

,

1 (

p (tb - ta) = -

2M

1

Xb- x..)2 (tb -

p - ---

M tl> - La

M{Xb - x..f . 2 il> ta

tal - -

(1.330) Then we replace the integration variables by the shifted momenta p' = p (xt. - Xa) /( tb - t".)M, and the amplitude (1.329) becomes (1.331)

where F(tb -

tal

is the integral over the shifted momenta (1.332)

This can be performed using t he Fresnel integml jO'fmula

f

OO

- 00

(.a ') vfaI 1 { Vi, l / vt,

dp

.;r;:;cxP ~ 2P

=

a> 0, a < O.

(1.333)

Here the square root .jt denotes the phase factor ei1r / 4 : This follows from t he Gauss formula 1 Reo> 0, (1.334) _oo ,;21rcxp - "2 P =

f OOdP

(0,) va'

by continuing a analytically from positive values into the right complex half-plane. As long as Rea> 0, this is straightforward. On the boundaries, i.e. , on the positive and negative imaginary axes, one has to be careful. At a = ±ia + 1] with a ~ and infinitesimal 1] > 0, the integral i;; certainly convergent yielding ( 1. 333). But t he integral also converges for 1] = 0, as can easily be seen by substituting x 2 = z. See Appendix lB. Note t hat differentiation of Eq. (1.334) with respect to a yields the more general Gaussian integral formula

°

f

oo

- 00

dp .j21f

-- p

2n

(0 ') = 1- (2n - 1)1!

exp - - p

2..;a

an

Rea> 0,

(1.335)

whcre (2n - I)!! is defined as t he product (2n - 1) · (2n - 3) ···1. For odd powers n +1, the integral vanishes. In the Fresnel formula (1.333) , an extra integrand p2n produces a factor (ija)n .

rl

49

1.12 Free-Particle Amplitudes

Since t he Fresnel formula is a special analytically continued case of the Gauss formula, we shall in the sequel always speak of Gaussian integrations and use Fresnel's name only if t he imaginary nature of the quadratic exponent is to be emphasized. Applying this formula to (1. 332), we obtain

F(tb-ta)

1

=

(1.336)

D'

/2rri;,(', - 'o)IM so t hat the full time evolution amplitude of a free massive point particle is 1

(Xbtb lxata) =

D

/2rri;,(', - 'o)IM

exp

[iM(",-X.)'] h2 t t . '

(1.337)

0

In the limit tb -+ t a , the left-hand side becomes the scalar product - x..), implying the following limiting formula for the a-function

(",Ix.)

~

a(D) (X{.

a(D) ( X b -

x,,) =

1

lim

t. -t~_O ..j27rifi.(tb _ ta)IM

D

[i

exp -M - ( X b - x..f ] . Ii 2 tb ta

(1.338)

Inserting Eq. (1.331) into (1.313) , we have for the fixed-e nergy amplitude the integral representation

(",Ixo)'

~

f

d(t, - to)

J(2~~D exp { ~ [P('" - xo) + (t, - to) (E - :~)] } . (1.339)

Performing the time integration yields iii

dVp.

(Xblx..)s =

J(21Th)D exp [t p (Xb - x a) ] E _ p2/2M + iT!,

(1.340)

where we have inserted a damping facto r e - '1{tb - t..) into the integral to ensure convergence at large tb - tao For a more explicit result it is more convenient to calculate the Fourier transform (1.337): 00

(XbIXa)s = in1 d(tb - tal o

1

..j21Tih(tb _ ta)/M

D exp

{ifi [E(tb - tal + 2"M (X.t - x.)'] } b

l

t

a

.

(1.34 1)

For E < 0, we set

0,

(1.347)

to obtain (1.348)

This result can be continued analytically to D > 2, which will be needed later (for example in Subsection 4.9.4) . ForE>Oweset (1.349) ls and use the formula (1.350)

where HPl(z) is the Hankel function, to find (1.351)

T he relation 16 (1.352) 12ibid., Formula 8.486.16 13/1,1. Abramowitz and I. Stcgun, Handbook 0/ Mathematical FUnctions, Dover, New York, 1965, Formula 10.2.17. 14ibid., Formula 9.6.9. 15ibid., Formulas 3.471.11 and 8.421.7. 16ibid., Formula 8.407.1.

1.13 Quantum Mechwljcs of Gelleral Lagrangian Systems

51

connects the two formulas with each other when continuing the energy from negative to positive values, which replaces ,.. by C i " J2 k = - ik . For large distances, the asymptotic behavior 17 (1.353)

shows t hat the fixed-energy amplitude behaves for E < 0 like

(>Co IXa) E ~

. M D- ' (2lr)(D --;::"'; ;::,,,, ,=ci:'=",e- "RI' , -tr;"" 1)/ 2 (""R)(D 1)/2

(1.354)

and for E > 0 like M

Ii.

kD-2",..."~'=",,,,='iL=e;kRJ" (21ri)(D I)J2 (kR)(D 1)/2 .

(1.355)

For D = 1 and 3, these asymptotic expressions hold for all R.

1.13

Quantum Mechanics of General Lagrangian Syste ms

An extension of the quantum-mechanical formalis m to systems described by a set of completely general Lagrange coordinates ql, ... , qN is not straightforward. Only in the special case of qi (i = 1, ... , N) being merely a curvilinear reparametrization of a D-dimensional Euclidean space are t he above correspondence rules sufficient to quantize the system. Then N = D and a variable change from X i to Qj in the SchrOdinger equation leads to the correct quantum mechanics. It will be useful to label the curvilinear coordinates by Greck superscripts and write ql' instead of Qj . This will help when we write all ensuing equations in a form that is manifestly covariant under coordinate transformations. In the original definition of generalized coordinates in Eq. (1. 1), t his was unnecessary since transformation properties were ignored. For the Cartesian coordinates we shall use Latin indices a lternatively as sub- or superscripts. The coordinate transformation Xi = Xi(ql') implies the relation between t he derivatives 01' == ojfJql' and Oi == fJjfJx i : (1.356)

with the transformation matrix (1.357)

called basis D-ad (in 3 dimensions t riad , in 4 dimensions tetrad, etc.). Let e;l'(q) = fJql' jfJxi be the inverse matrix (assuming it exists) called t he reciprocal D-ad, satisfying with e;1' the orthogonality and completeness relations (1.358)

52

1 Fundamentals

Then, (1.356) is inverted to (1.359)

fJj = e;"'(q)fJ"

and yields the curvilinear transform of the Carteian quantum-mecha.nical momentum operators Pi = -ihfJ; = -ilie/,(q)8,.. (1.360) T he free-particle Hamiltonian operator

iI = T =_I_ P,2 =_ !!....V2

(1.36 1)

• h' Ho = -2M!:::..'

(1.362)

o

2M

2M

goes over into

where t1 is the Laplacian expressed in curvilinear coordinates:

o? =

!:::. =

ei l'fJl' c{ov

eil'e{fJpfJ"

+ (e i P.(}pet)8".

(1.363)

At this point one int roduces the metric tensor (1.364)

its inverse

gI'V(q) = eil'(q)e{(q) ,

(1.365)

defined by 91'''9.,>. = 6")' , a nd the so-called affine connection (1.366)

Then the Laplacian takes t he form (1.367)

with

r I-' >.v being defined as the contmction r !' >v -~ g"r.." ".

(1.368)

T he reason why (1.364) is called a metric tensor is obvious: An infinitesimal square distance between two points in the original Cartesian coordinates ds'! =: dx?

(1.369)

becomes in curvilinear coordinates (1.370)

1.13 Quantum MechwljCS of GeIleral Lagrangian Systems

53

The infinitesimal volume element dDx is given by (1.371)

where (1.372)

is the determinant of the metric tensor. Using this determinant, we form the quantity (1.373)

and sec t hat it is equal to the once-contracted connection (1.374)

With the inverse metric (1.365) we have furthe rmore (1.375)

ov

We now take advantage of the fact t hat the derivatives fJ", applied to the coordinate transformation Xi{q) commute causing r"v~ to be symmetric in /-tv , i.e. , r"v A = rv/ and hence r"v" = rv. Together with (1.373) we find the rotation (1.376)

which allows the Laplace operator t:. to be rewritten in the more compact form (1.377)

This expression is called the Laplace-Beltrumi operutOT".18 Thus we have shown that for a Hamiltonian in a Euclidean space 1

H(p, x) = 2M P'l + V(x) ,

(1.378)

the SchrOdinger equation in curvilinear coordinates becomes

.

["'

1

H,p(q , t) '" - 2Mt. + V(q) ,p(q, t) ~ ih8,,p(q, t) ,

(1.379)

where V (q) is short for V(x(q)). T he scalar product of two wave functions which determines the transition amplitudes of thc system, transforms into (1.380) dDq.Jij ,p;(q,t),p,(q,t).

f dDxtf;;(x, t) tf;l(X, t),

J

--;'··'~!~ o'~'~d~"~~~ · .~w ~;~ll~"" C-:g~ ;'~'n:C;: I.~"~'~;n:-;Eqs. (11.12)- (11.18).

54

1 Fundamentals

It is important to realize t hat t his SchrOdinger equation would not be obtained by a straightforward application of t he canonical formalism to the coordinatetransformed version of the Cartesian Lagrangian

L(x,x) =

~ x.'l - V(x).

(1.381)

With the velocities transforming as

x' = e\,(q)q''',

(1.382)

the Lagrangian becomes

L(q, q)

~

M

,9,"(q)q'q" - V(q).

(1.383)

Up to a factor M , t he metric is equal to the Hessian metric of t he system, which depends here only on q" [recall (1.1 2)1: (1.384)

The canonical momenta a re

aL

PI' == 8it' =

M

."

9t",q·

(1.385)

T he associated quantum-mechanical momentum operators PI' have to he Hermitian in the scalar product (1.380) and must satisfy the canonical commutation rules (1.268),

IP"

q"] [q", q"]

IP",,"]

~

- iM.. ", (1.386)

0, ~

O.

An obvious solution is (1.387)

The commutation rules are t rue for _ilig- Z 8I' gZ with any power z, but only z = 1/4 produces a Hermit ian momentum operator:

JJlq..;gw;(q , t)[_ilig- 1/48",gl/4W1(q, t)1 ! Jlq JJlq..;g [-ilig-I/481,gl /4WZ(q, t) ]* w](q, t), =

g l/4 w; (q, t)[-i1i8",gl/4WI(q , t)1

=

(1.388)

as is easily verified by partial integration. In terms of the quantity (l.373), this can also be rewritten as (1.389)

1.13 Quantum MechwljCS of GeIleral Lagrangian Systems

55

Consider now the classical Hamiltonian associated with the Lagrangian (1. 383) , which by (1.385) is simply

H

=

Pl"ql" - L

=

2~gl",,(q)pI'p" + V(q).

(1.390)

When trying t o turn this expression into a Hamiltonian operator, we encounter the operator-ordering problem discussed in connection with Eq. (1.101 ). The correspondence principle requires replacing the momenta PI" by the momentum operators PI" but it does not specify the position of these operators with respect to the coordinates ql" contained in the inverse metric gl""(q). An important constraint is provided by the required Hermiticity of the Hamiltonian operator, but this is not sufficient for a unique specification. We may, for instance, define the canonicaJ Hamiltonian operator as (1.391)

in which the momentum operators have been arranged symmetrically around the inverse metric to achieve Hermiticity. This operator, however, is not equal to the corrcct SchrOdinger operator in (1.379). The kinetic term contains what we may call the canonical Laplacian .6.a

n

=

(01"+ ! rl")gl""(q)(o"

+ ! r,,).

(1.392)

It differs from the Laplace-Beltrami operator (1.377) in (1.379) by .6. - .6.ean =

-~ 81"(gl"" r,,)

- igl""r "r,...

(1.393)

The correct Hamiltonia n operator could be obtained by suitably distributing pairs of dummy factors of gl/4 and g- I/4 symmetrically between the canonical operators [5]:

fl = 2~g-I/4pl"gl/4g1""(q)gl/4p"g-1/4 + V(q).

(1.394)

This operator has the same classical limit (1.390) as (1.391). Unfortunately, the correspondence principle docs not specify how the classical factors have to be ordered before being replaced by operators. The simplest system exhibiting the breakdown of the canonical quantization rules is a free particle in a plane described by radial coordinates ql = r, q2 = '(): (1.395)

Since the infinitesimal square distance is ds 2 = dr 2 + r 2 dr, the metric reads

~) It has a determinant

g=

r2

.

(1.396)

"" (1.397)

56 and an inverse

1 Fundamentals

g~V= (~ ~-2

rv.

(1.398)

The Laplace-Beltrami operator becomes 1

,

6. = - (}rrfJr

1

+ "2(}cp ,

,

.

(1.399)

The canonical La placian, on the other hand, reads

(1.400)

The discrepancy (1.393) is t herefore (1.401)

Note that this discrepancy arises even though t here is no apparent ordering problem in the naively quantized canonical expression {JI'9p,,(q) if' in (1.400). Only the need to introduce dummy 9 1/ 4_ and g- I/ 4_factors creates such problems, and a specification of the order is required to obtain the correct result. If the Lagrangian coordinates qi do not merely reparametrize a Euclidean space

but specify the points of a general geometry, we cannot proceed as above and derive the Laplace- Beltrami operator by a coordinate t ransformation of a Cartesian Laplacian. With the canonical quantization rules being unreliable in curvilinear coordinates there a re, at first sight , severe difficulties in quantizing such a system. This is why t he literature contains many proposals for handling this problem (6]. Fortunately, a large class of non-Cartesian systems allows for a unique quantummechanical description on completely different grounds. These syst ems have t he common property that their Hamiltonian can be expressed in terms of the generators of a group of motion in the general coordinate frame. For symmetry reasons, the correspondence principle must then be imposed not on the Poisson brackets of the canonical variables p and q, but on those of the group generators and the coordinates. The bracket s containing two group gcnerators specify the structure of the group, those containing a generator and a coordinate specify thc defining representation of the group in configuration space. The replacement of t hese brackets by commutation rules constitutes the proper generalization of the canonical quantizat ion from Cartesian to non-Cartesian coordinates. It is called group quantization. The replacement rule will be referred to as the group correspondence principle. T he canonical commutation rules in Euclidean space may be viewed as a special case of the commutation rules between group generators, i.e., of the Lie algebm of t he group. In a Cartesian coordinate frame, the group of motion is the Euclidean group containing translations a nd rotations. The generators of translations and rotations arc the momenta and thc angular momenta, res~ti vely. According to thc group

57

1.14 Particle on the Surface of a Sphere

correspondence principle, the Poisson brackets between t he generators and the coordinates are to be replaced by commutation rules. Thus, in a Euclidean space, t he commutation rules between group generators and coordinates lead to the canonical quantization rules, and this appears to be the deeper reason why the canonical rules are correct. In systems whose energy depends on generators of t he group of motion other than those of translations, for instance on t he angular momenta, the commutators between the generators have to be used for quantization rather than the canonical commutators between positions and momenta. The prime examples for such systems are a particle on t he surface of a sphere or a spinning top whose quantization will now be discussed.

1. 14

Particle on the Surface of a Sphere

For a particle moving on the surface of a sphere of radius r with coordinates rsin Ocos4', x 2

Xl =

=

rsin Osin 4', x 3

=

rcosO ,

(1.402)

the Lagrangian reads (1.403)

The canonical momenta are (1.404)

and t he classical Hamiltonian is given by

1 (p'e +--p' 1) . sin20

H ~--

2Mr2

(1.405)

'P

According to the canonical quantization rules, the momenta should become opera..

to"' •

_

Po -

a

·to 1-,· 1/ 2lJ - tw-:-i 2 - II Sill (1 , Sill 0

• P'P

=

·ton

- tl«J'P'

(1406) •

But as explained in the previous section , these momentum operators are not expected to give the correct Hamiltonian operator when inserted into the Hamiltonian (1.405) . Moreover, there exists no proper coordinate transformation from the surface of the sphere to Cartesian coordinates lll such that a particle on a sphere cannot be treated via the safe Cartesian quantization rules (1.268);

w;,

X' ] ]x\X' ]

W;,p, ]

~

-iM/ , 0,

(1.407)

o.

19There exist, however, certain infinitesimal nonholonomic coordinate transformations which arc multivaluoo and can be used to t ransform infinitesimal distances in a curvlXl space into those in a flat one. They are introduced and applied in Sections 10.2 and Appendix lOA, leading once more to the same quantum mechanics 88 the one described here.

58

1 Fundamentals

The only help comes from the group properties of the motion on the surface of the sphere. The angular momentum

L =xx p

(1.408)

can be quantized uniquely in Cartesian coordinates and becomes an operator (1.409)

whose components satisfy the commutation rules of the Lie algebra of the rotation group

(i,j, k cyclic).

(1.410)

Note that there is no factor-ordering problem since the x"s and the Pi'S appear with different iudices in each Lk • An important property of the angular momentum operator is its homogeneity in x. It has the consequence that when going fro m Cartesian to spherical coordinates Xl =

rsinOcos'P, x 2 = rsin ()sin rp, x 3 = r cos 0,

(1.4 11 )

the radial coordinate cancels making the angular momentum a differential operator involving only t he angles fJ,

'P, t he

63

1.15 Spimling Top

For j = 1/2, t hese form the spinor representation of t he rotations around the y-axis d1/'1 (/3) = ( COS{3/2 -sin {3/2 ) m'm sin{3/2 cos{3/2 .

(1.444)

The indices have t he order + 1/2, - 1/2. The full spinor representation function DI/'1(a,{3, "y) in (1.442) is most easily obtained by inserting into the general expression (1.429) the representation matrices of spin 1/2 for the generators £. with the commutation rules (1.410) , t he famous Pauli spin matrices:

u' -_

(0 o' )' ,(O-i) ,( ' 0) 1

(1

=

i

0

'

(1

0 - 1

=

.

(1.445)

Thus we can writ e (1.446)

The first and the third factor yield the pure phase factors in (1.442). The function 2 d!!.,'!,.({3) is obtained by a simple power series expansion of e- ·/kJ / 2 , using the fact that «(12)'ln = 1 and «(12frt+l = (12: (1.447)

which is equal to (1.444). For j = 1, t he representation functions (1.443) form the vector representation

d:"'m(j3) =

!(1 +cos /3) -7zsin {3 ! (1 -cos/3) ) ~sin {3 cos{3 -~sin {3 . ( !(1-cos{3) -32sin {3!(1 +cos{3)

(1.448)

where the indices have the order +1/2,-1/2. The vector representation goes over into the ordinary rotation matrices R;j({3) by mapping the states 11m) onto the spherical unit vectors 10(0) = i , E(±I) = =f(x ± iy )/2 using the matrix elements (illm) = f'(m) . Hence R({3)E(m) = E:"'=_ l E(m')d:"'m(j3)· The rcpresentation functions DI(a,{3,"Y) can also be obtained by inserting into the general exponential (1.429) the representation matrices of spin 1 for t he generators L; with the commutation rules (1.410). In Cartesian coordinates, these arc simply (£i)jk = -if;j k, where fijk is the completely antisymmetric tensor with fl'13 = 1. In t he spherical basis, these become (L;)mm' = (m li)(L.:)ij(j lm') = fi(m)(L');jEj(m') . The exponential (e- itJL2 )mm' is equal to (1.448). The functions p,(oJJ\z) are the Jacobi polynomials [8], which can be expressed in terms of hypergeometric functions as

(ofl,_(- 1)'r(I+P+l)

P,

~-l!-

.

.

r (p+ 1) F(-I ,I+ l+ a+p,l+p,(1 H)/2) ,

(1.449)

64

1 Fundamentals

where

ab c

F(a ,bjc;z)=l+ -z +

a(a+l)b(b+l)z2 ( ) ,+ cc + l 2.

...

(1.450)

The rotation functions df:,m,({3) satisfy the differential equation

T he scalar products of two wave functions have to be calculated with a. measure of integration that is invariant under rotations;

(tf12I1J!I) == 121< [ l~" dadfJsinf3d"'( 'I/J'2 (0:, {J, "Y)'1/7J(a,{3,"'().

(1.452)

The above eigenstates (1.442) satisfy t he orthogonality relation

(1.453)

Let us also contrast in this example the correct q uantization via the commutation rules between group generators with the canonical approach which would start out with the classical Lagrangian. In terms of Euler angles, the Lagrangian reads (1.454)

where w{, w'I'w( are the angular velocities measured along t he principal axes of t he top. To find t hese we note that the components in the rest system Wl>~,W3 are obta.ined from the rela.tion (1.455)

WI

W2 W3

-/3sincr + tsin{jcoscr, = /3coscr+tsin (jsincr, = tcos(j+a. =

(1.456)

After the rotation (1.436) into the body-fixed system, these become

w, w, w(

~

~

/3 sin-y - a sin (j 008'Y, /3 cos 'Y + a sin {j sin 'Y, acos(j + 1'.

(1.457)

Explicit ly, the Lagrangian is L =

4[/{(jj2 + a

2

sin 2(j) + It;{a cos (j + t)2J .

(1.458)

65

1.15 Spimling Top

Considering 0:',/3, "f as Lagrange coordinates ql' with IJ. = 1, 2,3, this can be written in the fo rm (1.383) with the Hessian metric [recall (1.1 2) and (1.384)1: I{ sin2 /3 + I, cos2 /3 0 ~v =

(

whose determinant is

0

~

I(cos /3

0

(1.459)

9 = Iile sin2 /3.

(1.460) Hence the measure f rflq..j9 in the scalar product (1.380) agrees with the rotationinvariant measure (1.452) up to a trivial constant factor. Incidentally, t his is also true for the asymmetric top with h =f II) =f 1( , where 9 = Iii, sin 2 /3, although the mctric gl'v is then much morc complicated (sec Appendix I C) . T he canonical momenta associated with t he Lagrangian (1.454) are, according to (1.383) , =

Po

8LI8o = i{osin2 /3+ I(cos /3(ocos/3+1'),

i!L/i!/3 ~ [, /3,

p,

P"/ = 8L I81' = Jc(ocos /3 + 1') .

(1.461)

After inverting the metric to

(1.462) we find the classical Hamiltonian

[2. ,+ (000' [ . P 2.)' + /

H - ~ - 2 IV,

[ {sm2a+ fJ'

{

V,

1·

P .1 2a Pa , _ [200, . 2a PaP"/ {sm fJ

{Sill fJ

(1.463)

This Hamiltonian has no apparent ordering problem. One is therefore tempted to replace the momenta simply by the corresponding Hermitian operators which are, according to (1.387),

Po

=

-ifi.8a ,

pp

1 - in(sinj3) - 1/28p(sin/3) 1/2 = -ih(8p + 2cot /3),

p"/ =

-ina"/.

(IA64)

Inserting these into (1.463) gives the canonical Hamiltonian operator

Han = H + Hdisc"

(1.465)

with

(1.466)

66 and

1 Fundamentals

_ 1 1 2 1 3 H,· =-(8" cotf.l)+ - cot f.I= _ _ __ • '''''' 2 " JJ 4 fJ 4sin 2 {3 4 A

(1.467)

T he first term if agrees with the correct quantum-mechanical operator derived above. Indeed, inserting the differential operators for the body-fixed angular momenta (1.437) into the Hamiltonian (1.423), we find f.J. The term Hdi1>cr is t he discrepancy between the canonical and the correct Hamiltonian operator. It exists even though t here is no apparent orderi ng problem, j ust as in t he radial coordinate expression (1.400). T he correct Hamiltonian could be obtained by replacing the classical pi term in H by the operator g- I/4 ppgl f 2fjpg- l/4, by analogy with the treatment of the radial coordinates in if of Eq. (1.394). As another similarity with t he two-dimensional system in radial coordinates and the particle on t he surface of the sphere, we observe that while the canonical quantization fails, the Hamiltonian operator of the symmetric spinning top is correctly given by t he Laplace-Beltrami operator (1.377) after inserting the metric (1.459) and the inverse (1.462). It is straightforward although tedious to verify that this is also true for the completely asymmetric top [which has quite a complicated metric given in Appendix l C, see Eqs. (lC.2), and (lC.4)]. This is an important nontrivial result, since for a spinning top, the Lagrangian cannot be obtained by reparametrizing a particle in a Euclidean space with curvilinear coordinates. The result suggests that a replacement (1.468)

produces the correct Hamiltonian operator in any non- Euclidean space.:!! W hat is t he characteristic non-Euclidean property of the a, (3, I space? As we shall see in detail in Chapter 10, t he relevant quantity is the curvature scalar R. T he exact definition will be found in Eq. (10.42) . For the asymmetric spinning top we find (see Appendix lC) (1.469)

Thus, just like a particle on the surface of a sphere, the spinning top corresponds to a particle moving in a space with constant curvature. In this space, the correct correspondence principle can also be deduced from symmetry arguments. The gcometry is most easily understood by observing t hat the a ,{3, "f space may be considered as the surface of a sphere in four dimensions, as we shall see in more detail in Chapter 8. An important non-Euclidean space of physical interest is encountered in t he context of general relativity. Originally, gravitating matter was assumed to move in a spacetime with an arbitrary local curvature. In newer developments of the t heory one also allows for the presence of a nonvanishing torsion. In such a general situation, 21 If the space has curvature and no torsion, t his is t he correct ansWtJr. If torsion is present, t he corroct answer will be given in Chapters 10 and 8.

1.16 Scattering

67

where the group quantization rule is ina pplicable, t he correspondence principle has always been a matter of controversy [see the references after (1.401)1to be resolved in this text . In Chapters 10 a nd 8 we shall present a new quantum equivalence principle which is based on an application of simple geometrical principles to path integrals and which will specify a natural and unique passage from classical to quantum mechanics in any coordina te frame. 22 The configuration space may carry curvature and a certain class of torsions (gradient torsion). Several arguments suggest t hat our principle is correct. For t he above systems with a Hamiltonian which can be expressed entirely in t erms of generators of a group of motion in t he underlying space, the new quantum equivalence principle will give t he same results as t he group quantization rule.

1.16

Scattering

Most observations of quantum phenomena arc obtained from scatt ering processes of fundamental particles.

1.16.1

Scattering Matrix

Consider a particle impinging with a moment um p" a nd energy E = E" = p!/2M upon a nonzero potent ial concentrated around t he origin. After a long t ime, it will be found far from the potential with some momentum Pb. The energy will be uncha nged: E = Eb = pV 2M. The probability amplitude for such a process is given by the t ime evolution amplitude in the moment um representation (1.470)

where the limit tb ----> 00 and t" ----> -00 has to be taken. Long before and after t he collision, t his amplitude oscillates with a frequency w = Elf! characteristic for frcc particles of energy E. In order to have a time-independent limit , we remove these oscillations, from (1.470), and define the scattering matrix (S-matrix) by t he limit (1.471 )

Most of the impinging particles will not scatter at all, so that this amplitude must contain a leading term, which is separated as follows: (1.472)

where (1.473)

shows the normalization of the states [recall (1.186)1. T his leading term is commonly subtracted from (1.471 ) to find the true scatt ering amplitude. Morcover, Z2 H.

Kleinert, Mod. Phys. Lett. A 4, 2329 (1989) (http: //vww .physik .fu- ber lin.de/ Phys. Lett. B 236, 3U'i (1990) (i~d. http/202) .

~kleinert/ 199) ;

68

1 Fundamentals

since potential scattering conserves energy, the remaining amplit ude cont ains a 6function ensuring energy conservation, and it is useful to divide this out, defining the so-called T-matrix by t he decomposition (1.474)

From t he definition (1.471) and the hermiticity of fl it follows that the scattering matrix is a unitary ma trix. This expresses the physical fact that t he total probability of an incident particle to fe-emerge at some t ime is unity (ill quantum field t heory the situation is morc complicated d ue to emission and absorption processes). In the basis states IpID) introduced in Eq. (1.180) which satisfy the completeness relation (1. 182) and are normalized to unity in a finite volume V, the unitarity is expressed as (1.475) m'

m'

Remembering the relation (1.185) between the discrete states IpIn) and their cont inuous limits Ip), we see t hat (1.476)

where £3 is t he spatial volume, and Pb and P:;' a re the discrete momenta closest to Pt> a nd p". In the continuous basis Ip), the unitarity relation reads (1.477)

1.16.2

Cross Section

The absolute square of (pt>ISl p,,) gives the probability PPb-P~ for t he scattering from the initial momentum state p" to t he final momentum state Pt>. Omitting the unscattercd particles, we have

p,.-"

1

~ L'

"

,

2.M(0)2.M(E, - Eo)l(p, ITI Po) 1 .

(1.478)

The factor 0(0) at zero energy is made finite by imagining the scattering process to take place with a n incident t ime--indcpendent plane wave over a. finite total time T . Then 21TM(0) = J dteil>'t/Ti le=o = T, and the probability is proportional to the time To (1.479)

By summing this over all discrete final momenta, or equivalently, by integrating this over the phase space of t he final momenta [reca ll (1.184 )], we find t he total probability per unit time for t he scattering to take place (1.480)

69

1,16 Scattering

The momentum integral can be split into an integral over the fi nal energy and the final solid angle, For non-relativistic particles, this goes as follows (1.481)

where dO = d¢>bd cos Bb is the element of solid angle into which the particle is scattered, The energy integral removes the a-function in (1.480), and makes Pb equal to Pa ' The differential scattering cross section du I dO is defined as the probability t hat a single impinging particle ends up in a solid angle dO per unit time and unit current density, From (1.480) we identify

du dF l i M p zl dO = dO] = £3(21r!i)321rrIITp~pJ ],

(1.482)

where we have set (1.483)

for brevity, In a volume £3 , t he current density of a single impinging particle is given by the velocity v = plM as

.

1 P

(1.484)

J = £3 M '

so that the differential cross section becomes (1.485)

If the scattered particle moves relativistically, we have to replace t he constant mass M in (1.481) by E = Ji? + M2 inside the moment um integral, where P = Ipl, so that =

J {JO (2:!i)3 J 10"'" dEEp, (2:!i)3

dO dO

dpp2

(1.486)

In the relativistic case, the initial current density is not proportional to plM but to the relativistic velocity v = pi E so that

. J =

1 P

£3£'

(1.487)

Hence the cross section becomes (1.488)

70

1.16.3

1 Fundamentals

Born Approximation

To lowest order in t he interaction strength, the operator

S'"

1-

S in

iV /n.

(1.471) is (1.489)

For a time-independent scattering potential , this implies (1.490)

where (1.491 )

is a function of the momentum transfer q =: Pb - p" only. Then (1.488) reduces to the so called Darn approximation (Dorn 1926) (1.492)

T he amplit ude whose square is equal to the differential cross section is usually denoted by fp~ P6' i.e. , one writes

dpe - il>' = e- i"'/4,J;,

(1B.6)

which goes into Fresnel's integral formula (1.333) by substituting p -- pM.

Appendix

Ie

The Asymmetric Top

The Lagrangian of the asymmetric top with three different moments of inertia reads

(IC.I)

86

1 Fundamentals

It has the Hessian metric [recall (1.12) and (1.384)] gil

!hI

U31

Ie sin2 fJ + I( cos2 f3 - (Ie - I.,l sin 2 thin2 'Y, - (Ie - 1'l)sin .Bsin,),cos')', I, oos{3, + (Ie - 1'1) 5in7 ,)"

977

1'1

932

0,

(IC.2) rather than (1.459). The determinant is

(IC.3) and the inverse metric has the components

~{I'l +(h - I'l)Sin7'Y}I< ,

9 11 9

21

g

1 - sin!3sin1C061'(1e - I,,) / {,

=

9

931

!{cosl3l-siU7 1'(Ie -

l2

!{Sin2 Plh

9 9

-

I1}l - I"llI"

sin 2 "'1(Ie - I")]}!,,

~{sin.6cosl3sin-YOOS 1'(I'I - I{)}I(,

y32

9 9

1

_ {mnl PIe!.,

33

9

+ coo 2 PI,,!, + cos7 {3sin2 i (Ie - I'I)I,}.

(ICA)

From t his we find the components of t he llicmann connection, the Christoffel symbol defined in Eq. (1.70):

Tl + [{fell/hI."

fnl

[cos.6cosl'siwy(l~ - I'll, -

f211

{cosfJIsin 2 -y(Il - I~ - (Ie - It)lId + I'I(Ie

f3J

1

[':n l ['32 1'33

{COS1'sin 1'[I~ -

II + (Ie -

+ [ '1 -

Idllj2sin .6 I e I '1 ' I,,)I(]} j2Ie I'I'

0,

1

[sin 2 "({ll - I~ - (Ie - J'I)Id - [qUe - 1'1 + IdIl2sin.BI~I",

1

0,

I' n 2 = (cosfJsin ,8[sin1 "'f(ll - I~ - Idle - I,,)) - Ie(le - I dl} / leI" , 1'21 2 1':11 2

{cosfJcos "'fsill"'f[Il - I~ - Idle - I'I)]}/21eI'I' {sinfJ[sin2 "'f (ll - I~ - I d l e - 1'1» - Idle - I'I - IdJ}/2IeI'I'

f':n 2 f'al

0,

2

0,

1'33

[cos"'fsin"'f(ll - I~ - I ((Ie - I 'I))]/2IeI'I'

f'n 3 = {cos "'fsin "'f[sin2 fJ( I eI 'I( I e - 1'1) - IdIl - I~) + IlUe - I~» + Ul - I~)I( - fl(Ie - I'I)]}/I eI'Ih, 1'21

3

{sil? fJ[sin 2 "'f(2I{I"(I,, - Ie) + IdIl - I~ ) - Il(Ie - I ,,)) +IeI,,(Ie - 1'1)

+ I'IId I "

- Id] - sin "f«(Il - I~)I( - Il(Ie - 1'1» 2

Notes 8Jld

87

RcfcrellCCS

1'31 3

-['lie (I! + 1'1 - l d}/2SiUp l!I'1 I" [C06PC06'YSin'y(I1- l~ - ld/{ - l'1» I/21{/'1'

006'Ysin'Y(I'1-/e)/lc, {cosp[sin 2 'Y (I~ - 11 + (l{ - 1'1)/d

+ l'1(1{

- 1'1 + Id]} /2sin p I'1 l{, (IC.5)

O.

The other components follow from the sy mmetry in the first two indices 1'1' and r 1''' \ respectively.

Notes and References For more details see some standard textbooks: I. Newton, Mathematische Prinzipien der Naturkhre, Wiss. Buchgcscllschaft, Darmstadt, 1963, J.L. Lagrange, Analytische Mechanik, Springer, Berlin, 1887; G. Hamel, Theoretische Mechanik , Springer, Berlin, 1949; A. Sommerfeld, Mechanik, Harri Deutsch, Frankfurt, 1977; W. Weizel, Lehrbuch der Theoretischen Physik, Springer, Berlin, 1963; H. Goldstein, Classical Mechanics , Addison-Wu'x(t) "

~Ix(t) , -

x(' -

,)1 ·

(2 .90)

Thcy are two differcnt discrete vcrsions of the time derivative Ot, to which both reduce in the continuum limit € --+ 0: (2.91)

if they act upon differentiable functions. Since the discretized timc axis with N + 1 steps constitutes a one-dimensional lattice, the difference operators 'V , 'V are also called lllttia derivatives.

2.2 Exact Solution for the.Free Particle

105

For the coordinates x .. = x(t .. ) at the discrete t imes t.. we write 1

, 1 -(x ,

-(X .. +l n -

N

x .. ),

~

n

~

0,

N+l~n~ 1.

x .. -t} ,

(2.92)

The time-sliced action (2.89) can then be expressed in terms of 'Vxn or 'Vx .. as (writing x .. instead of oXn)

(2.93)

°

In this notation, the limit f ...... is most obvious: The sum f 2::.. goes into the integral ft~dt, whereas both ('VX .. )2 and ('Vx .. )2 tend to i;2, so that

~

. . . 1~· dt~

(2.94)

i;2.

Thus, the time-sliced action becomes t he Lagrangian action. Lattice derivatives have properties Quite similar to ordinary derivatives. One only has to be careful in distinguishing 'V and 'V . For example, they a1low for the useful operation summation by parts which is analogous to integration by parts. Recall the rule for the integration by parts

f

d'g(,)j(,)

~ 9(')f(')I:: -

f

d'g(t)f(')·

(2.95)

On the lattice, this relation yields for functions f(t) ...... x .. and g(t) ...... P.. : N

N+l f

,,

L P..-'\i'x ..

= p.. x ..

" loN -H f L ('VPn)x ...

(2.96)

This follows directly by rewriting (2.35) . For functions vanishing at the endpoints, i.e., for XN+ l = Xo = 0, we can omit the surface terms and shift the range of the sum on t he right-hand side to obtain the simple formula [see also Eq. (2.47)] N+l

N

N+l

L P. '''". ~ - L("P.)x. ~ - L ("P.)x •.

(2.97)

The same thing holds if both pet) and x(t) are periodic in the interval tt> - t a , 50 that 1'1 = PN+l, Xo = XN+l. In this case, it is possible to shift the sum on the right-hand side by one unit arriving at the more symmetric-looking formula N+l

.-,

N+l

L P. "x. ~ - L ("P.)x • . n= 1

(2.98)

2 Path Integrals -

106

Elementary Properties and Simple Solutions

In the time-sliced action (2.89) t he qua ntum fluctuations xn (=ox,,) vanish at t he ends, so that (2.97) can be used to rewrite N+l

L

N

('\7xn)2 = -

n: l

L

(2.99)

xn'V'Vx".

n: l

In the 'Vx" -form of t he action (2.93), the same expression is obtained by applying formula (2.97) from the right- to the left-hand side and using the vanishing of Xo and IN+!: N+l

N

N

L ('lx. )' ~ - L x.'ii'lx. ~ - L x.'ii'lx•.

(2 .100)

n= ]

n= O

The right-hand sides in (2.99) and (2.100) can be written in mat rix form as N

- L xn'V\lx"

N

- -

n=1

L

x"(VV),,n,Xnl,

",n'=]

N

- L In \7\i'x"

N ~

,,=1

-

L X"{VV')"n'X,,I, n,,,' = 1

(2.1Ol)

with the same N x N -matrix

_ \7\7

_

==

- 2 1 0 1 - 2

1

,

0 0

0 0

0 0 (2. 102)

'\7\7 =: 2"

0 0

1 -2 1 1 -2 0

0 0 0 0

This is obviously the lattice version of the double time derivative fit, to which it reduces in the continuum limit l -+ O. It will therefore be called the lattice Laplacian. A furt her common property of lattice and ordinary derivatives is that thcy can both be diagonalized by going to Fourier components. When decomposing

(2. 103)

and applying the lattice derivative V, we find

I:

Vx(t n }

~

dw

~ (e- iw(t,,+ - TQ this gives directly (2.345) . Physically speaking, thc path has at high temperatures "no (imaginary) time" to fluctuate , and only one term in the product of integrals needs to be considered .

139

2.9 DellSity Matrix

If H (p, x) has the standard form

1" + V(x) ,

Hlp, x) ~ 2M

(2.348)

the momentum integral is Gaussian in p and can be done using the formula 00 / - 00

dp

e- ar /211

21rn

___ 1_

(2.349)

- J21rlia'

This leads to the pure x-integral for the classical partition function

(2.350) In the second expression we have introduced the length

(2.351) It is t he thermal (or Euclidean) analog of the characteristic length l(tb - t a ) intro-duced before in (2.126) . It is called the de Broglie wavelength associated with the tempemture T = l/k n f3 or, in short, t he thermal de Broglie wavelength. Omit ting t he x-integration in (2.350) renders t he hU'ge-T limit p(x), the classical particle distribution ~ Z- l cl

1

Ie (liP) e

- 9(.,)

(2.352)

For a free particle, the integral over x in (2.350) diverges. If we imagine the length of t he x-axis to be very large but finite , say equal to L, the partition function is equal to

(2.353) In D dimensions, this becomes

(2.354) where Vo is t he volume of t he D-dimensional system. For a harmonic oscillator with potential MW 2 X 2 /2, the integral over x in (2.350) is finite and yields, in the D-dimensional generalization ID (2.355) Z" ~ IDti,fi)' where

C ==

{lj; f3Mw 2

(2.356)

140

2 Path Integrals -

Elementary Properties and Simple Solutions

denotes t he classical length scale defined by the frequency of the harmonic oscillator. It is related to t he quantum-mechanical one -\.., of Eq. (2.301 ) by (2.357) Thus we obtain the mnemonic rule for going over from the partition function of a harmonic oscillator to that of a free particle: we must simply replace __ 0 l.,--L,

oc

(2.358)

"~!fiM

w

- 0

21f

L •

(2.359)

The real-time version of this is, of course, 1

(2.360)

-~

w

w_ o

Let us write down a path integral representation for p(x). Omitting in (2.337) the final trace integration over XI> == Xa and normalizing the expression by a factor Z - l , we obtain

(2.361) The thermal equilibrium expectation of an arbitrary Hermitian operator 6 is given by (2 .362)

In the local basis Ix), this becomes

" (2.363)

An arbitrary function of the position operator

x has the expectation

T he particle density p(x,,) determines the therma l averages of local obscrvablcs. If f depends also on t he momentum operator p, then the off-diagonal matrix element s (xb le- Plf lx,,) arc also needed. They arc contained in the density matrix introduced for pure quantum systems in &t. (1.221), and reads now in a thermal ensemble of temperature T: (2.365)

141

2.9 DellSity Matrix

whose diagonal values coincide with t he above particle density p(x,,). It is useful to keep the a nalogy between qua ntum mechanics and qua ntum statistics as close as possible and to introduce the time t ranslation operator along the imaginary time axis (2.366)

defining its local matrix elements as imaginary or Euclidean t ime evolution a mplitudes (2.367)

As in the real-time case, we shall only considcr t he causal time-ordering r~ > T". Otherwise the partition function and the density matrix do not exist in systems with energies up to infinity. Given the imaginary-time amplitudes, the partition function is found by integrating over the diagonal elements

z ~ [: dx(x hPlx 0),

(2.368)

and the density matrix (2.369)

For the sake of generality we may sometimes also consider the imaginary-time evolution operators for time-dependent Hamiltonians a nd the associated amplitudes. They are obtained by time-slicing the local matrix elements of the operator (2.370)

Here T.. is an ordering operator along the imaginary-time axis. It must be emphasized t hat the usefulness of the operator (2.370) in describing thermodynamic phenomena is restricted to the Hamiltonian operator H(t) depending very weakly on the physical time t . T he system has to remain close to equilibrium at all times. T his is t he range of validity of the so-called linear response theory (see Chapter 18 for more dctails). The imaginary-time evolution amplitude (2.367) has a path integral representation which is obtained by dropping t he final integration in (2.334) and relaxing the condition X~ = X,,: (2.371)

The time-sliced Euclidean action is

A;' ~

N+'

z= (- ip"(x" - x"_.J + 1°O

dx

Dx~ II

n=l

-CQ

•.

(2.377)

J2rrfll'.jM

It contains no extra 1/J27r1if./M factor , as in (2.374), due to t he trace integration over t he exterior x. The condition x{Ii(3) = x(O) is most easily enforced by expanding x{r) into a Fourier series X

()r = ;c.. ~

m= - CQ

1

I7ff'"'7"1 e

yN

+I

- >w""

x m,

(2 .378)

143

2.10 Quantum Statistics of the HarmoIlic Oscillator

with the Matsubara frequencies 2.m wm=27rmkBT/Ii= 1if3'

m=O,± I,±2, ....

(2.379)

When considered as functions on the entire T-axis, the paths are periodic in hf3 at any T , i.e., (2.380) Thus the path integral for the quantum-statistical partition function comprises all periodic paths with a period r~f3. In the time-sliced path integral (2.374), the coordinates X(T) arc needed only at the discrete times T" = nf. Correspondingly, the sum over m in (2.378) can be restricted to run from m = - N/2 to N/2 for even N and from - (N - 1)/2 to (N + 1)/2 for odd N (see Fig. 2.3). In order to have a real X(T,,), we must require that

Xm = x :m

(modulo N

+ I ).

(2.381)

Note t hat the Matsubara frequencies in t he expansion of the paths x( T) are now twicc as big as the frequencies 11m in t he quantum fluctuations (2. 110) (after analytic continuation of tb - ta to -ili/kBT). Still, t hey have about the same total number, since they run over positive llnd negative integers. An exccption is the zero freque ncy Wm = 0, which is included here, in contrast to the frequencies 11m in (2.110) which . This is necessary to describe paths wit h run only over positive m = 1, 2, 3, . arbitrary nonzero endpoints XI> = Xa = X (included in the trace) .

2.10

Quantum Statist ics of the Harmonic Oscillator

The harmonic oscillator is a good example for solving the quantum-statistical path integral. The T-axis is sliced at T" = m, with ( :::::: hj3/(N + I) (n = 0, ... , N + 1), ILIld the partition function is given by the N -- 00 -limit of the product of integrals (2.382) where ~ is the time-sliCtJd Euclidcan oscillator action (2.383) Integrating out t he x" 's, we find immediately ZN =

....

I VdetN+l (_(2 \1\1 + ( 2w 2 ) .

(2.384)

Let us evaluate the fluctuation determinant via the product of eigenvalues which diagonali1.e the matrix _€z \11i7' + E2 W Z in the s liced act ion (2.383). They are (2.385)

144

2 Path Integrals -

Sin fW

Elementary Properties and Simple Solutions

sin (W~

2

2

N"'ewll

N z odd

Nn l - cos€:lJ)

-,

- NO

N+L 2

-

-2

I-cos

(lJ)

-

N- ' 2

- 2

F ig ure 2.3 Illu~t ration of the eigenvalues (2.385) of t he fluct uation matrix in the action (2.383) for even and odd N. with the Matsubara frequencies

Wm .

For w = 0, the eigenvalues arc pictured in Fig. 2.3. The

action (2.383) becomes diagonal after going to the Fourier components X m . To do this we arrange the real and imaginary parts Rex", and Irnx", in a row vector (Re x\, 1m

XI;

Re x2, 1m %2; ... ; Re :1:", 1m x .. ; . .. j,

a nd see t hat it is related to the time-sliced positions x" = X(T.. ) by a traru;formation matrix with the rows

= (Tm)n x "

JN:

1

(~, ~

N:

12lf . 1,sin

N:

12,.. . 1,

c~~21f'2 sin~21r·2 . N+l

'N+l

m

'

m

)

... ,cos N+127f.n,siO N+ 12'1f.n, ... "x" .

(2.386)

For each row index m = 0, ... ,N, t he column index n runs from zero to N / 2 for even N, and to (N + 1)/2 for odd N. In the odd case, the last column sin ffi 211"· n with n = (N + 1)/2 vanishes

identically and must be dropped, so t hat the number of columus in T mn is in both CIlSCll N + 1 , as it should be. For odd N, t he second-last column of Tmn is an alternating sequence ± I. Thus, for a proper normalization, it ha:; to be multipli(.x1 by an extra normalization factor I/.[i, ju~t as the elements in the first column. An argument similar to (2.1 15), (2.116) shows that the resulting matrix is o rthogonal. Th us, we can diagonalize the sliced action in (2.383) as folloW1j for

N = even,

(2.387) for Thanks to t he orthogonality of T mn, t he measure

fl n f~ dx{Tn)

transforms simply into for

1- 1-

dxo

-

(N -'1/'1-

dxu'+I)n Q I

_ 00

1-

N = odd.

dRex m _oc dlmxm

N = even,

(2.388) for

N = odd.

145

2.10 Quantum Statistics of the HarmoIlic Oscillator By performing the Gaussian integrals '\\ll ohtain the partition function

(2.389) T hanks to the periodicity of t he eigenvalues under the replacement n __ n + N + I, the result hIlS become a unique product expression for both even and odd N . It is important to realize t ha.t contrary to the fluct uation factor (2.160) in the real-time ampli t ude, the partition function (2.389) contains t he square root of only positive eigcnmodes lIS a unique result of Gaussian integrations. There are no phase subtleties as in the F"rcsnel integral

(1.333). To calculate the product, wc observe that upon decomposing

sin~ W;f = (I + ~ W;f) (I - COS W;f) ,

(2.390)

the sequence of first factors l + cosW",f = l + cos~

(2.391)

N+l

2

runs for m = I , . . . N through the same values as the sequence of second factors

, - = -Wmf -= 2

Jrffi N+I- m 1 - cos - - := 1 + cos Jr "-",'ceo""

N+ l

N+I

(2.392)

'

except in an opposite order. ThIlS, separating Ollt the m = 0 -term, we rewrite (2.389) in the form

["

z~ = ~ II 2(1_~WmE) (W

,., _ 1

2

]-' U"II (1 + .. I

~~w~

4 Slll'2

)]-'" .

!o!m.!

2

(2.393)

Thc first factor on the right- hand side is the quantum·mechanical fluctua.tion determ inant of the free-part icle determinant detN( - f 2V'V ) = N + 1 [sec (2. 123)], so that we obtai n for both even and oddN (2.394) To evaluate the remaining product, we must distinguish again between even and odd For even N, where every eigenvalue occurs twice (see Fig. 2.3), we obtain

ca8e!i

of N .

(2.395) For odd N, the term with m = (N + 1)/2 occurs only once lI.Ild must be treated separately so t hat

II

•. T [( l + fW " ) '" (" - "1' ( 1+ fW ")] - ' Z~ = ~ 1iw 4 ",.1 4 sin~ Nfl

.

(2.396)

We now introduce t he parameter We, the Euclidean analog of (2.161), via the equations (2.391)

146

2 Path Integrals -

Elementary Properties and Simple Solutions

In the odd case, the product formu la l6

., 1

(N_.", [

2 slu[(N + l)xJ sin2x (N+ l )

sm x 1 - si n 2 ..."

II

7i"1m

... .. 1

(2.398)

[similar to (2. 163)] yields, with x = (;V/2, ZN = kaT [

'"

fiw

Sinh[(N+l)W.~/2Il - · .

1

Sinh(W. f/2)

N

+1

(2.399)

In the even case, the formula l ?

1 sin[(N + l)x] sinx (N + 1)

(2.400)

produces once more t he same ruml t !IS in Eq. (2.399 ). Im;crting Eq. (2.397) leads to the partition function on the sliced imaginary time axis:

1

N

(2.401)

Z'" = 2 sin h (liWeP/ 2) " T he partition function can be ex pandod into t he following series

Z:: =

e - row. nkBT

+ e - 3ft,;.,· n k

IJ

T

+ e - M i;;. ! 2k s T + .

By comparison with t he general spectral expansion (2.326), the system:

W{)

display the energy eigenvalues of

(n+DliWe.

En =

(2.402)

(2.403)

T hey show the ty pica l linearly rising O6Cillator sequence wi th _ 2 . h ""f w. = ,€8.I"SLll "'2

(2.404)

playing the role of t he frequency on t he sliced timc axis, and hWe /2 being the zero-point energy. In the continuum limit € ..... 0, t he t ime-sliccd partition function goes over into the usual OlSCil1ator partition fUIlction 1 (2.405) Z", = 2sinh(.81iw/2) .

Z:

In D dimensions this becomes, of course, [2 sin h(.BtlW/2)[- J), due to the additivity ofthe action in each component of x . Note that the continuum limit of the product in (2.394) can also be taken factor hy factor. Then Z", becomes (2.406)

According to formula (2.171 ), t he product a nd we find with :r: =

rr: ",1(1+ ",(, ) converges rapidly against sin hx/x

fu.J.B/2 Z", = kaT

ru..;/ 2k a T

hw sinh{Iiw/2knT ) """:-;;:::-:=-:c:-:;-;.,.,-;;:~ 161.S. Gradshtcyn and I. ~I. Ryzhik , op.

17ibid., formula 1.391.3.

2si nh{.BIiw/2)·

c it., Formula 1.391.1.

(2.407)

14 7

2.11 Time-Dependent Harmonic PotelltiaI

As discussed after &j. (2. 181 ), the continuum limit can be taken in each factor since the product in (2.394) contains only ratios of frequencies. Just lIS in the quantum-mechanical case, this procedure of obtaining the continuum limit can be summarized in the sequence of equations arriving at a ratio of differential operators 12 ZN [detN+I( _ f2VV + f2W2) r /

"

(_ f2VV)r 1/2 [det N+ L(_ e V'fJ + ~2W2)l - l/1 2

[det'

det'",+1( t2VV)

N+l

k.T

[det(-a~ det'(

h

2 +w )] - 1/2 = kBT

an

rr [w~ + oo

hw m=l

2 w ] -'

(2.408)

w~

In the W = 0 -determinants, the zero Matsubara frequency is excluded to obtain a finite expression. This is indicated by a prime. The differential operator - ~ a.ct~ on real functions which are periodic under the replacement T -> T+ hJ3. Remember that ca.ch eigenvalue w~ of - ~ occurs twice, except for the zero frequency Wo = 0, which appears only once. Let us finally mention that the resu lts of this section could also have hccn obtained directly from the quantu m-mochanical amplitude (2.l73) [or wit h the discrete t imes from (2.197)1 by an analytic continuation of t he time difference t~ - ta to imaginary values - i(Tb - Ta):

(X~TbI XaTa)

=

~VSinhW(:

x

exP{ - 2~sinhw~: T)(x~+X!)COSh W(Tb - Ta) - 2X~Xa] }.

ra) (2.409)

By setting x = X~ = Xa and integrating over x, we obtain [compare (2.331 )]

2sinh[w(Tb

ra)/2]'

(2.410)

Upon equating Tb - Ta = hJ3, we retrieve the partition function (2.405). A similar treatment of the discrete-time version (2. 197) would have led to (2.401). The main reason for presenting an independent direct evaluation in the space of real periodic functions was to display t he frequency structure of periodic paths and to see the difference with respect to the quantum-mechanical paths with fixed ends. \Ve also wanted to show how to handle the ensui ng product expressions. For applications in polymcr physics (see Chapter 15) one also needs the partition function of all path fluctuations with open ends

/

00 dxb / 00 dxa (xbTb IXara) =

_ oc

_ oc

V2,,-h

W(Tb

I

\/2,,-h{Tb

ra)/M

sin h[w(Tb

Ta) 21Th Ta)] Mw

1

Mw.,Isinh [w(Tb

(2.411)

Ta)['

The prefactor is .jfi times the length scale >..., of &j. (2.301).

2.11

T ime-Dep e nde nt Harmonic P ote ntia l

It is often necessary to calculate thermal fluctuation determinants for the case of s time-dependent

frequency

neT) wh ich is periodic under T __ r + hJ3. (XbTb IXaTa )

=

~

JV X

J'D

As in Section 2.3.6,

v.'C

consider t he anlplitude

P - r'dT[-ip~+p'nM + M{l'(T)r'nl/~

21Tn e

b a a xo'

(2.663)

After a quadratic completion in xo , the total exponent in (2.663) reads

iMwt. - 2" [- (Xb,

, tan IWt. (tl> + xa)

, ta)/2I + (XI> - xa).

5lfi WL

(1 tb

.MWL ( X b + xa Yb - Ya )' -'-h- tan [wI.{t b - ta)/2] Xo - - 2 - - 02t;-:a-=nf[w~,-';(t~,"-ot'}"/2"1 o +i Mwt, [(Xb + xa)2 tan[wI.{tb _ t a)/2] + (Yb - Ya)2 21i 2 2tan[wL(tb - t a)/2] .MWL (2.664) +t21i(Xb + Xa)(Yb - Ya).

1

The integration MWL f~oo dxo/21r1i removes the second term and results in a factor

MWt. ,I 1r1i 21r1i ViMwL tan[wL(tb

(2.665)

ta)/2]·

By rearranging the remaining terms, we arrive at t he a mplitude 1 M 3 Wdtb - t a)/2 a (X!>tb lx..t ) = V21rili(tb _ tal sin[wd tb _ ta)/2] exp h(AcI + .A..rl ,

[i

1

(2.666) with a n action

M { (z, - zo}' Acl = -2 tb

ta

+ WL -2 cot[WL(tb +

ta)/2] [ (Xb - xa) '

W...(XaYb - XbYa)}

(2.667)

I'

(2 .668)

and the surface term

A.f = -MWL 2- (XbYb - Xa Ya)

+ (Yb - Ya) 'J

e

= 2cBxy a·

184

2.18.4

2 Path Integrals -

Elementary Properties and Simple Solutions

C lassical Action

Since t he action is harmonic, the amplitude is again a product of a phase ei..A", and a fluct uation factor. A comparison with (2.639) and (2 .656) shows that t he surface term would be absent if the amplitude (Xbtb lx..t"h were calculat ed with the vect or potential in the axially symmetric gauge (2.638). Thus Acl must be equal to t he classical act ion in this gauge. Indeed, t he orthogonal part can be rewritten as

AJ = i~~ dt { ~ ~(XX + YiJ) + ~ [x(- x + Wl,y) + y(- y -

Wl,X)] }.

(2.669)

T he equations of motion are (2.670)

redueing the action of a classical orbit to M(. "la="2 M ([. . J+ [ . - Y,,Y,, . [) · "'",' M. Sw.uki, Physica A 191 , 501 (1992).

The path integral representation of the scattering amplltudc is developed in W.B. Campbell, P. Finkler, C.E. Jones, and M.N. Mishcloff, Phys. Rev. D 12, 12, 2363 (1975). See also: H.D.I. Abarband and C. Itzykson, Phys. Rev , k tt. 23, 53 (1 969); R. Ro3cnfelder, sec Footnote 37. The alternative path integral representation in Section 2.18 is dlle to M. Roncadelli, Europhys. Lett. 16,609 (1991 ); J. Phys. A 25, L997 (1992); A. Defendi and M. RoncadelIi, Europhys. Lett. 21 , 127 (\993).

AI>iV1/'Tfi I>Wtv;.

You stir what should not be stirred. H EROOOTUS

3 External Sources, Correlations, and Perturbation Theory Important informatio n on every quantum-mechanical system is carried by thc correlation functions of the path x(t). They are defined as the expectation values of products of path positions at different times, x(td . .. x(t n ), to be calculated as functional averages. Quantities of this type are observable in simple scattering experiments. The most efficient extraction of correlation functions from a path integral proceeds by adding to the Lagrangian an external time-dependent mechanical force term disturbing the system linearly, and by studying the response to t he disturbance. A similar linear term is used extensively in quantum field theory, for instance in quantum electrodynamics where it is no longer a mechanical force, but a source of fields , i.e., a charge or a current dcnsity. For t his reason wc shall call this term generically source or current term . In this chapter, the procedure is developed for the harmonic action, where a linear source term does not destroy the solvability of the path integral. The resulting amplitude is a simple functional of t he current. Its functional derivatives will supply all correlation fu nct ions of t he system, and for t his reason it is called the genemting junctional of t he theory. It serves to derive the celebrated Wick rule for calculat ing the correlation functions of an arbitrary number of x(t). This forms the basis for pcrt urbat ion expansions of anharmonic t heories.

3 .1

External Sources

Consider a harmonic oscillator with an action

(3.1) Let it be disturbed by an external source or current j(t) coupled linearly to t he particle coordinate x(t) . The source action is A;

~

', 1'.

dt x(t)j(t).

209

(3.2)

210

3 External Sources, Correlations, and Perturbation Theory

The total action

A = A.+Aj

(3.3)

is still harmonic in x and X, which makes it is easy to solve the path integral in t he presence of a source t erm. In particular, the source t erm does not destroy the factorization property (2.151) of the time evolution amplitude into a classical amplitude eiA ;, _ t) t~

+ Xbsi n w(t - ta) ]i(t).

211

3.1 ExtCrIlW SOllI"Cffi

Consider now t he fluctuating part of t he action, A ft = ..4..,,8 + A i,H. Since Xcl(t) extremizes the action without the source, A ft contains a term linear in ox(t) . After a partial integration [making use of the vanishing of ox(t) at the ends] it can be written as

(3. 12) where D",2(t,t') is t he differential operator

Dw,(t, t!) ~ (- if, - w'),(t - t!) ~ ,(t - t')( - a?,

-

w') ,

t, t'

E

(to, to) .

(3. 13)

It may be considered as a functional matrix in the space of the t-dependent functions va nishing at ta, tb. The equality of the two expressions is seen as follows. By partial integrations one has

f,_I" dtf(t)if,g(t ) ~ f,_I" dtif,f(t)g(t) ,

(3. 14)

for any f(t) and g(t) vanishing at t he boundaries (or for periodic functions in t he interval) . The left-hand side can directly be rewritten as It~ dtdt! f(t)o(t - t')8;9{t!), the right-hand side as If: dtdt' 8i f(t)o(t - t')9(t') , and after further partial integratiom;, os J dtdt' f(t)if,'(t - ng(t) . The inverse D:!(t, t') of t he functional matrix. (3.13) is formally defined by the relation

i'-"dt' D",,{t",

t')D~!(t' , t) = o(t" - t) ,

(3.15)

which shows that it is the standard classical Green function of the harmonic oscillator of frequency w;

This definition is not unique since it leaves room for an additional arbitrary solution H (t , t') of t he homogeneous equation II~ dt' Dw2(t", t') H (t' , t) = O. This freedom will be removed below by imposing appropriate boundary condit ions. In the fluctuation action (3. 12), we now perform a quadmtic completion by a shift of ox(t) to

,x(t)

1 (',

= 'x(t) + M f,

,-

(3. 17)

dt'Gw,(t , t')j(t').

Then the action becomes quadrat ic in both

ox and j:

[M

1

I" dt f,_I" de "2,x(t )Dw,(t,t'),x(t') - 2M 1 j (t)Gw,(t, t')j(t') . A . ~ f,_

(3.18)

The Green function obeys the same boundary cond it ion as the fluct uations ox(t): G",.(t , t')=O

for

f

t=

tb ,

1 t arbitrary,

t' arbit rary, t' = ta .

(3.19)

212

3 External Sources, Correlations, and Perturbation Theory

Thus, the shifted fluctuations ox(t) of (3. 17) also vanish at the ends and run through the same functional space as the original ox(t). T he measure of path integration f Vox(l) is obviously unchanged by the simple shift (3.1 7) . Hence t he path integral f V ox over eiAft / h with the action (3.18) gives, via the first term in A ft, the harmonic fluctuat ion factor F",(t b - tal calculated in (2. 169):

w

1

(3.20)

The source part in (3. 18) contributes only a trivial exponential factor PUt = cxp

{*A,f1 } ,

(3.21)

j

whose exponent is quadratic in j(t) : A j,H = - 2M 1

1" 1" Ie

dt t~ dt'j(t)Gw~(t , t')j(t').

(3.22)

The total time evolution amplitude in the presence of a source term can therefore be written as t he product (3.23)

where (Xbtb lxat)", is t he source-free t ime evolution amplitude

(3.24)

and Pj,cl is an amplitude containing the classical action (3. 11 ): Pj,cI =

e(i/f "

(3.43)

the second line in (3.41) can be rewritten as

(3.44) By choosing the initial conditions

6.(t, t)

=

0,

Li(t, t') I,':=, = -1 ,

(3.45)

we satisfy the inhomogeneous differential equation (3.27) provided 6.(t , t') obeys the homogeneous differential equation

[-if - n'(t)I£>(t, t')

~ 0,

fo, t > t .

(3.46)

This equation is solved by a linear combination

£>(t, t')

~

o(f)«t) + ptt')q(t)

of any two independent solutions 1J(t) and

[-if - n'(t)I((t)

~ 0,

~(t)

(3.47)

of the homogeneous equation

[-if - n'(t)Mt) ~ O.

(3.48)

T heir Wronski determinant W = ~(t)1j( t ) - ~(t)1J( t) is nonzero and, of course, timeindependent, so that we can dctermine t he coefficients in t he linear combination (3.47) from (3.45) and find

£>(t,

n ~ ~ [«tMt') - «t')"(t )l·

(3.49)

The right-hand side contains the so-called Jacobi commutator of the two funct ions ~(t) and 1J(t). Here we list a few useful algebraic properties of 6.(t, t'):

(3.50) (3.51)

216

3 External Sources, Correlations, and Perturbation Theory

Il(', 'o)8"Il(t" to) - Il('" ,) ~ Il(t" 'o)8,1l(', (0).

(3.52)

The retarded Green function (3.40) is so far not the unique solution of t he differential equation (3.27), since one may always add a general solution of t he homogeneous differential equation (3.48):

Gn,(',

n ~ a(, - nll(t, n + a(nW) H(")"(' ),

(3.53)

with arbitrary coefficients a(t') and b(t'). This ambiguity is removed by the Dirichlet boundary conditions 0,

tb

=I- t, (3.54)

Imposing these upon (3.53) leads to a simple algebraic pair of equations

a(')Wo) a(')W,)

+ b(,)"(,o) + b(')"(t,)

~

0, Il(t ,t,).

(3.55) (3.56)

Denoting the 2 x 2 -coefficient matrix by

(3.57) we observe that under the condition (3 .58) the system (3.56) has a unique solution for the coefficients a(t} and b(t) in the Green function (3.53). Inserting this into (3.54) and using the identity (3.50), we obtain from t his Wronski's general formula corresponding to (3.36)

At t his point it is useful to realize that t he funct ions in the numerator coincide with the two specific linearly independent solutions Da(t) and Db(t) of the homogenous differential equations (3.48) which were introduced in Eqs. (2.226) and (2.227). Comparing the initial conditions of Da(t ) and Db(t) with t hat of the function .6..(t, t') in Eq. (3.45) , we readily identify (3.60) and formula (3.59) can be rewritten as (3.61)

3. 2 Green FUnction of Harmonic Oscillator

217

It should be pointed out t hat this equation renders a unique and well-defined Green function if the differential equat ion [-0; - ffl(t) [y(t) = 0 has no solutions with Dirichlet boundary conditions y(t a) = y(tb) = 0, generally called zero-modes . A zero mode would cause problems since it would certainly be one of t he independent solutions of (3.49), say 1/(t). Due to the property 1/(ta) = 1/(tb) = 0, however, the determinant of A would vanish, thus destroying t he condition (3.58) which was necessary to find (3.59). Indeed, t he funct ion A(t,t') in (3.49) would remain undetermined since t he boundary condit ion 17(ta ) = 0 together with (3.55) impHes that also ~(ta) = 0, making W = ~(t)Jj(t) - {(t)17(t) vanish at t he initial time t a, and t hus for all times.

3 .2 .2

Spectral Representat ion

A second way of specifying the Green function explicitly is via its spectral representation. Cons tant Freque n cy For constant frequency O(t) == equat ion and vanish at the ends t = orthonormal functions:

W,

t he fl uctuations ox(t) which satisfy the differential

(- if - w') ;x(t) ~ 0, (3.62) ta and t = tb , are expanded into a complete set of

(3.63)

with t he frequencies [compare (2. 110)] (3.64)

These functions satisfy the orthonormality relations (3.65)

Since the operator -8i - w 2 is diagonal on x,,(t) , this is also true for the Green function CoAt, t') = (- 8i - w2) - 16(t - t'). Let C n be it s eigenvalues defined by (3.66)

T hen we expand Cwo(t, t') as follows:

..

00

Gw,(t, t') ~

L

"

G"x"(t)x"(t') .

(3.67)

218

3 External Sources, Correlations, and Perturbation Theory

By definition, the eigenvalues of Cwl(t , t') are the inverse eigenvalues of the differential operator (- 8; - w'l), which are v~ - w 2 • Thus (3.68)

and we arrive at the spectral representation of Cw,(t , t'l:

') ~ _ 2_ G.,2 (t, t tb -

~ sinvn(t - t,,)8inv,,(t' -

t"

w

2

,,= 1

2

t,,)

(3.69)

.

W

tin

We may use t he trigonometric relation sin V,,(tb - t) = - sin v,,[(t - t,,) - (lb - t,,)] = -(-1)" sin tI,,(t - t",) to reWTite (3.69) as

G",2 (t,l

') ~ _2_ ~(_ ),,+1 sin Vn(tb - t) sinvn(t'W 1 2 2 tb - ta "=l If" - W

tal

.

(3.70)

T hese expressions make sense only if tb - ta is not equal to an integer multiple of 1fjw, where one of the denominators in the sums vanishes. This is the same range of tb - ta as in t he Wronski expression (3.36).

Time-D e p e nde nt Freque ncy The spectral representation can also be written down for t he more general Green fu nction with a time-dependent frequency defined by t he differential equation (3.27). If y,,(t) are the eigenfunctions solving the d ifferential equation with eigenvalue ..\"

K(t)y.(t)

~

'.y.(t),

(3.71 )

and if these eigenfunctions satisfy the orthogonality and completeness relations to

l'.

dty"(t)y,,,(t) =

I: y.(t)y.(t')

(i"""

(3.72)

6(t - t') ,

(3.73)

and if, moreover, there exists no zero-mode for which ..\" = 0, then spectral representation

Go,(t, t') ~

I: y.(t~y.(t') . •

•

Gn~(t ,

t') has the (3.74)

This is easily verified by multiplication with K(t) using (3.71) and (3.73). It is instructive to prove t he equality between thc Wronskian construction and the spectral representations (3.36) alld (3.70). It will be useful to do this in several steps. In t he present context, some of these may appcar redundant. They will, however, yield intermediate results which will be needed in Chapters 7 and 18 when discussing path integrals occurring in quantum field theories.

219

3.3 Oreell Functions of First-Order Differential Equation

3.3

Green FUnc tions of First-Order Differentia l Equation

An important quantity of statistical mechanics arc the Green functions G:'J(t, t') which solve the first-order differential equation

[Wt

-

O(t) ]Go(t, t')

=

t - t' E [0, tb - t,,) ,

iJ(t - t') ,

(3.75)

or it s Euclidean version Gl;,,(r, r") which solves the differential equation, obtained ' from (3.75) for t = -iT: [0, - !l(T)[ Gn .• (T, T') ~ 8(T - T'),

r E [0, "fi).

T-

(3.76)

These can be calcula ted for an arbit rary funct ion O(t).

3.3.1

Time-Independent Frequency

Consider first t he simplest case of a Green function which solves the first-order differential equation (iiJ, - w)G~(t , n ~ ;o(t - f ),

~(t,

t') with fixed frequency W

t - f E [0, t. - to).

(3.77)

The equation determines ~(t, t') only up to a solution H (t , t') of t he homogeneous differential equation (Wt - w) H (t, t') = O. The ambiguity is removed by imposing t he periodic boundary condition G~(t , t')

" G~(t - t') ~ G~(t - f

+ t. -

to),

(3.78)

indicated by t he superscript p. With this boundary condition, the Green function G~(t,t') is translationally invariant in time. It depends only on the difference between t and t' and is periodic in it. T he spectral representation of ~(t, t') can immediately be written down, assuming t hat tb - ta does not coincide with an even multiple of 1r/w:

G~(t - t') = _ 1_

f

tb - ta m= - oo

e - ,:..,,,,(t- t, ) _ _i _ Wm - W

The frequencies Wm are twice as large as the previous Wm

2.m

== - -, t b - ta

rn =

11m'S

.

(3.79)

in (3.64):

0, ± 1, ± 2, ± 3, ....

(3.80)

As for the periodic orbits in Section 2.9, there arc "about as many" Wm as 11m, since there is an Wm for each positive and negative integer m , whereas the 11m arc all positive (see the last paragraph in that section). The frequencies (3 .80) are the real-time analogs of t he Matsubara frequencies (2.379) of quantum statistics wit h the usual correspondence tb - ta = - ih/k8T of Eq. (2.328).

220

3 External Sources, Correlations, and Perturbation Theory

To calculate the spectral sum, we use t he Poisson summation formula in the form (1.197) : (3.81)

Accordingly, we rewrite t he sum over wm as an integral over Wi, followed by an auxiliary sum over n which squeezes the variable Wi onto the proper discrete values Wm = 21rm/(t1> - tal : (3.82)

At this point it is useful to introduce another Green function Gw(t - t') associated with the first40rder differential equation (3. 77) on a n infinite time interval:

Gw(t)

f"'"

=

i_co

dw' e- iw't_i_. 271" w' - w

(3.83)

In terms of this function, t he periodic Green function (3.82) can be written as a sum which exhibits in a most obvious way t he periodicity under t -+ t + (tb - tal: 00

G~(t) ~

I:

Gw(t - (t. - to)n).

(3.84)

n : - oo

The advantage of using Gw(t - t') is that the integral over Wi in (3.83) can easily be done. We merely have to prescribe how to treat the singularity at w' = w. This also removes the freedom of adding a homogeneous solution H(t , t'). To make the integral unique, we replace w by w - iT} where T} is a very small positive number, i.e., by the iT}-prescription introduced after Eq. (2.166). This moves the pole in the integrand of (3.83) into the lower half of the complex wi-plane, making the integral over W i in G.,(t) fundamentally different for t < 0 and for t > o. For t < 0, the contour of integration can be closed in the complex wi-plane by a semicircle in t he upper half-plane at no extra cost, since e- iw't is exponentially small t here (see Fig. 3.1) . With t he integrand being analytic in the upper half-plane we can contract t he contour to zero and find that the integral vanishes. For t > 0, on the ot her hand, the contour is closed in the lower half-plane cont aining a pole at Wi = W - iT} . When contracting the contour to zero, the integral picks up the residue at this pole and yields a factor -27ri. At the point t = 0, fin ally, we can close the contour either way. The integral over the semicircles is now nOllzero, :r=1/ 2, which has to be subtracted from t he residues 0 and 1, respectively, yielding 1/ 2. Hence we find

-211" ,-wt w' (]W'

[.

- 00

=

. { I e- lWt x !

o

for for for

i W

+ iT} t > 0, t = 0, t < O.

(3.85)

3.3 Oreell Functions of First-Order Differential Equation

221

1m",'

F igure 3.1 Pole in Fourier transform of Green functions O~" (t), and infinite semicircles in the upper (lower) half-plane which extend the integrals to a closed contour for t < 0 (t > 0).

T he vanishing of thc Green function for t < 0 is the causality property of G..,(t) discussed in (1.306) and (1.307). It is a general property of functions whose Fouricr transforms are analytic in the upper half-plane. T he three cases in (3.85) can be collected into a single formula using the Heaviside function 6(t) of Eq. (1.309) :

Gw(t) ~ , - ""e(t).

(3.86)

The periodic Green function (3.84) can then be written as 00

L

~(t) =

e- iw ll - (lb - I~)nI9(t -

(tb - t,,)n).

(3.87)

n: - ""

Being periodic in tb - t", its explicit evaluation can be restricted to the basic interval

t E [O , tb - t"J

(3.88)

Inside the interval (0, tb - tOll , t he sum can be performed as follows:

0::,(') ~ e- iwlt - (tb - t~)/2l

- i 2sin [w( tb

ta )12l'

t E (0, tb - t,,).

(3.89)

At t he point t = 0, the initial term wit h 9(0) contributes only 1/2 SO that

G~(O) ~ G~(O+) - ~.

(3.90)

Outside the basic interval (3.88), the Green function is determined by its periodicity. For instance,

(3.91)

222

3 External Sources, Correlations, and Perturbation Theory

Note that as t crosses the upper end of the interval [0, tb - ta), the sum in (3.87) picks up an additional term (the term with n = 1). This causes a jump in C!:,(t) which enforces the periodicity. At t he upper point t = tb - t a , there is again a reduction by 1/2 so that C::,(tb - tal lies in t he middle of the jump, just as the value 1/2 lies in the middle of the jump of the Heaviside function 6(t). The periodic Green function is of great importance in the quantum statistics of Bose particles (see Chapter 7). After a continuation of the time to imaginary values, t ----> -iT, tt. - t" ----> -ifijkBT, it takes the form

Ct,e(T) = 1

e 1""'lkHT C - """,

T E (O,ht3),

(3.92)

where the subscript c records the Euclidean character of the time. The prefactor is related to the average boson occupation number of a particle state of energy nw, given by the Bose-Einstein distribution [unction nb = w

1

e",",,/knT

(3.93)

1·

In terms of it, G~,.(T} ~ (l+n~Vw',

T E (O,hm .

(3.94)

The T-behavior of the subtracted periodic Green function Gt:',e( T) == G!,e(T) - l jht3w is shown in Fig. 3.2.

Figure 3.2 Subtracted periodic Grecn function Ce: e =:;; Ce,e(T)- l /h{3w and antiperiodic GreeD functioll G~,cCT) for frequencies w = (0, 5, 1O)/h{3 (with illcreasing dash length). The points show the values at the j umps of the three functions (with increasing point size) corresponding to the relation (3.90).

As a next step, we consider a Green funct ion order differential operator -vi - w2 ,

G~~(t)

associated with the second(3.95)

which satisfies the periodic boundary condition: (3.96)

223

3.3 Creell Functions of First-Order Differential Equation

1m",'

F igure 3.3 Two poles in Fourier t ransform of Green function

G~,a (t) .

J ust like Ct{t, t'l , this periodic Green function depends only t - t! . It obviously has the spectral representation

0 11

the time difference

(3.97) which makes sense as long as tb - ta is not equal to an even multiple of rr/w. At infinite tb - ta , the sum becomes a n integral over Wm with singularities at ±w which must be avoided by an i1]--prescr iption, which adds a negative imaginary part to the frequency w [compare the discussion after Eq. (2.166)]. This fixes also the continuation from small tb - ta beyond the multiple values of rr/w. By decomposing 1

1 (

w2 + if] = 2iw

,

-w'-c--w'--;+-,'C'"

-w~'~+-w'---'CC'1]) ,

(3.98)

t he calculation of the Green function (3.97) can be reduced to t he previous case. The posit ions of the two poles of (3.98) in the complex wi-plane are illustrated in Fig. 3.3. In this way we find, using (3.89) , 1 2wi [GC(t) - t > f > ta: (3 .122)

This equation is solved by a t'-independent C(t'): 1 C -- n IIP = - -",.-::::0:;:;;;--. r'b rk" 0(1") e )'4 - 1

(3 .123)

227

3.3 Oreell Functions of First-Order Differential Equation

Hence we obtain the periodic Green funct ion (3 .1 24)

For antiperiodic boundary conditions we obtain the same equation with n~ replaced by - n~ where • _ 1 ( )

no =

i f'bdHl(t )

e],6

3. 125

+1

Note that a sign change in t he time derivative of the firs t-order differential equation (3.75) to (3. 126) [-iiJ, - 0(')1 Gn ('" ,) ~ "(t - ") has the effect of interchanging in the time variable t and t' of the Green function Eq. (3.120). If the frequency O(t) is a matrix, all exponentials have to be replaced by timeordered exponentials [recall (1.252)[ (3. 127)

As remarked in Subsection 2.15.4, this integra.l cannot, in general, be calculated explicitly. A simple formula is obtained only if the matrix r2(t) varies only little around a fixed matrix 0 0 , For imaginary times 7 = it we generalize t he results (3.92) and (3. 110) for the periodic and antiperiodic imaginary-time Green functions of the first-order differential equation (3.76) to time-dependent periodic frequencies n(7) . Here the Green function (3. 120) becomes (3. 128)

and t he periodic Green function (3.124): (3. 129)

where

1

n b = ~~--''-----

-

(3. 130)

efo~fJ 01.,."0(.,." ) _ 1

is the generalization of the Bose distribution function in Eq. (3.93). For antiperiodic boundary conditions we obt ain the same equation, except that the generalized Bose distribution fu nction is replaced by the negative of t he generalized Fermi distribution function in Eq. (3. lIl ):

:,-,,---_ - --.,.,-,-,,1 r~" +1

nf =

e10

(3. 131)

d.,."!l(.,." )

For the opposite sign of the time derivative in (3. 128), the arguments interchanged.

7

and r' are

228

3 External Sources, Correlations, and Perturbation Theory

From the Green functions (3. 124) or (3.128) we may find directly the t race of the logarithm of the operators [- iOt + fl(t) ] or [8.- + 0(7)]. At imaginary time, we multiply 0(7") wit h a strength parameter g, and use the formula Tr log [a.. + 90(7)] =

fo9 dg' Gg'o(r, r) .

(3 .132)

Inserting on the right-hand side of Eq. (3.129), we find for 9 = 1: Trlog [8.. + fl(r) ]

=

!Og{2Sinh

[tf'P

dT'O(r")]}

~ fo"fJ dr" 0(7"") + log [1 -

e- 1:.8 d,..,'O(T" )], (3. 133)

which reduces at low temperature to

~ "2 j I" ,

dT" !l(T").

(3. 134)

The result is the same for the opposite sign of the time derivative and the t race of the logarithm is sensitive only to 6(r - r) at T = 7', where it is equal to 1/2. As a n exercise for dealing with distributions it is instructive to rederive t his result in the following perturbative way. For a positive 0(7) , we introduce an infinitesimal positive quantity 1J and decompose

Tdog [±a, + !l(T)] ~ '[i-iog [±a, + "[ + '[i-iog = Tr log [±8T

[1+ (±&, + ")- ' !l(T)[

(3.135)

+ 1J] + Tr log [1+ (±8 + 11)-10(7)] . T

The first term Tr log [±8 + 111 = Tr log [±8 + 111 = f~OQ dw log w vanishes since f~OQ dw logw = 0 in dimensional regularization by Veltman 's rule [see (2.506)]. Using the Green functions T

T

(3.136) the second t erm can be expanded in a Taylor series

For the lower sign of ±8T , the Heaviside functions have reversed arguments 72 72, ... , 71 - 7 n . The integrals over a cyclic product of Heaviside funct ions in (3 .1 37) are zero since the a rguments 71> ... ,Tn are time-ordered which makes t he argument of the last factor 9(7n - 71) [or 9(71 - 7n ) ] negative and t hus 9(7n - 71) = 0 [or 9 (7] - 7 n ) ]. Only t he first term survives yielding 71, 73 -

(3.138)

229

3.4 Summing Spectral Representation of Oreell Function

such that we re-obtain the result (3.134). This expansion (3. 133) can easily be generalized to an arbitrary matrix O(r) or a time-dependent operator, H(r). Since H(r) a nd H(r') do not necessarily commute, the generalization is Tr log[MT + H(r)] =

~ Tr

2h

[

r Jo

hfJ

dr H(r) ]-

f: .!.Tr [fe-nf: n

iJ

dr' li(r' )!"] , (3. 139)

n: 1

where f is the time ordering operator (1.241). Each term in the sum contains a power of the t ime evolution operator (1.255).

3.4

Summing Spectral Representation of Green Function

After these preparations we are ready to perform the spectral sum (3.70) for the Green function of the differential equation of second order with Dirichlet boundary conditions. Setting t2 =: tb - t, tl =: t' - t" , we rewrite (3.70) as

(3. 140)

We now separate even and odd frequencies lin and write these as bosonic and ferm ionic Matsubara frequencies Wm = 112m and wfn = I12m+! , respectively, recalling the definitions (3.80) and (3.104) . In this way we obtain

Cw,(t, t') = (3. 141)

Inserting on the right-hand side the periodic and antiperiodic Green functions (3.99) and (3. 108), we obtain the decomposition

aw,(t, t')

=

~ [a!:,(tz + td -

a:'(tz + t l )

-

a!:,(t2 - td

+ a:'(t2 - tdl. (3. 142)

Using (3.99) and (3. 11 3) we find that

'( ) _ G'( _ ) _ sin w[tz - (tb - t,,)/21sinwt l Gw t2 + tl w tz t] wsin[w(tb t,,)/2] ,

)-G'( _ G'( w t2 + t ] w tz

) _ _ cosw[t, - (tb - t.)/2[,in wt, wcOS[W(tb t,,)/2] ,

t\ -

(3. 143) (3. 144)

2 30

3 External Sources, Correlations, and Perturbation Theory

such t hat (3. 142) becomes GW~(t , t') =

.

I

(

WSlllW tb -

tIl

)

Sinwt2sinwt\ ,

(3. 145)

in agreement with the earlier result (3.36) . An important limiting case is

t"

---> -00 ,

(3. 146)

tb ----> 00.

T hen t he boundary conditions become irrelevant and the Green function reduces to G i(I,) 'f .i(t.)1 ~ - a,LI.(t" I) .

(3.162)

232

3 External Sources, Correlations, and Perturbation Theory

Defining now the constant 2 x 2 -matrices

iV ,O(t t) ~ ( {(to) 'f {(t.) .. 0 «to) 'f «t.)

(3. 163)

the condition analogous to (3.58), det AP,J,(t" , tb)

=

W dP"(t", tb) =f 0,

(3. 164)

with (3. 165)

enables us to obtain the unique solution to Eqs. (3. 162). After some algebra using the identities (3.51) and (3.52), the expression (3.159) for Green functions with periodic and antiperiodic boundary conditions can be cast into the form

') _ G 2 (

G" O(

n~ t, t -

0

')

t, t =F

[" (t, t.) ± ,,(to, t))[" (O.) ± ,,(to, t')[ .6,p,&(t t )"(t t) , ii ,

b

'"

(3. 166)

b

where C 02(t, t') is the Green function (3.59) with Dirichlet boundary conditions. As in (3.59) we may replace the fUllctions on the right-hand side by the solutions DB(t) and Db(t) defined in Eqs. (2.226) and (2.227) wit h t he help of (3.60). The right-hand side of (3.166) is well-defined unless the operator K(t) = - o'f (f.!(t) has a zero-mode, say 1](t) , with periodic or antiperiodic boundary conditions 7}(tb) = ±1](t,,), 1)(tb) = ±1)(t,,), which would make the determinant of the 2 x 2 -matrix ii,P,& vanish.

3.6

Time Evolution Amplitude in Prese nce of Source Term

Given thc Green function Cw,(t ,t'), we can write down an explicit expression for the timc evolution amplitude. Thc quadratic source contribut ion to the fluctuation factor (3 .21) is given explicitly by A j ,'

~

-

2

M i l" dtl" dfG,.(t,t')j(t)j(t') f~

t~

1 1 M wsin w(tb

to

)

(3. 167)

r·dtl1 dt' sin w(tb _ t)sinw(t' -t,,)j(t)j(t').

itA

t"

Altogether, the path integral in the presence of an external source j(t) reads

with a total classical action 1 ' ~ -2

Mw ' , ( )[ (Xb+Xa)COSW(tb-ta)-2xbXa tb to

Sill W

+

I

t

I ) • dt [xa sin W(tb _ t) sin w(tb t" j~

1

+ xbsinw(t - ta)b(t),

(3. 169)

233

3.6 Time Evolution Amplitude in Presence of Source Term

and the fluctuation factor composed of (2. 169) and a contribution from t he current term eiAj,R/":

Fw,jb,a-_ (t t) _ F. .(t" t.)eiAj,R/" _ xexp

{

TiM

1 J27ritt/ M

=

(W sin w tb

ta

)

}

'.'

. t. ( ) [ dt [ dt'sinw(tb - t)sinw(t' - ta)j(t)j(t') . (3. 170) wsm w tb - ta it~ it~

T his expression is easily generalized to arbitrary time-dependent frequencies . Using the two independent solutions Da(t) and Db(t) of the homogenous differential equations (3.48) , which were introduced in Eqs. (2.226) and (2.227) , we fi nd for the action (3.169) the general expression, composed of the harmonic action (2.266) and the current term fl~ dtxcl (t)j(t) with t he classical solution (2.246):

The fluctuation factor is composed of the expression (2.261) for the current-free action , and t he generalization of (3. 167) with the Green function (3.61): _

(

)iAj,R/" _

Fw,j(tb,ta) - Fw tb ,ta e X

-

r" dt iI't" dt' j(t) [Sit -

it~

1

1 {_ i ~ exp fiMD ( )

J21tih/M V Do.(tb)

t')D,(t) Do(t')

+ S(t! -

2~

a

tb

t)Do(t)D,(t')l j(t')}. (3.172)

For applications to statistical meclIanics which becomes possible after an analytic continuation to imaginary times, it is useful to write (3.169) and (3.170) in another form. We introduce the Fourier t ransforms of the current

A(w)

(3.173)

B(w)

(3 .1 74)

and see that t he classical source term in the exponent of (3.168) can be written as A-cl = -i . Mw }, smw(tb

tal

{ [xb(e.:w(t.-t~)A - B)l + Xa(e.:w(t·-t~)B - A)l.

(3. 175)

T he source contribution to the quadratic fluctuations in Eq. (3. 167) , on the other hand, can be rearranged to yield

234

3 External Sources, Correlations, and Perturbation Theory

This is seen as follows: We write the Creen function between j(t),j(t') in (3.168) as - [8inw(lb - t)sin w(t' - t,,)9(t - t') =

~

[ ( eiw (lb- t4 )e- iw (t-t')

+ c.c')

+ sinw(tb -

t')sin w(t - t,,)e(t' - t)l

- (e"",(t b+t4)e-iw(Hf')

+ c.c.) ]8(t -

t')

+{t_t'). Using

eCt -

t')

~ {_

+ e(t' - t)

=

1, this becomes

(eu...(lbHA)e- iw(t'H)

+eiw(t.-tAl

(3. 177)

+ c.c')

(e- iw(t-t'18(t _ t') + e-

+e- iw(to- I"l [eiw(/ - I')(l

-

SCt' -

iw {l'- t1

t»

6(t' -

t»)

+ eu..(t'- t){1_ e(t -

(3. 178)

t'» ]}.

A multi plication by j(t),j(t') and an integration over the times t, I! yield

H-

eiw(t.-t"14M 2w2(B2

+ (eiw(t. -t.. 1 _

e-iw(t.-t A

+ A2)

»)

(3. 179)

r· dt I dt'e-iw1t-t'lj(t)j(t') + 4M w 2AB], itA I. t

2 2

•

thus leading to (3.176). If the source jet) is time-independent, the integrals in the current terms of the exponential of (3.169) and (3.170) can be done, yielding the j-dependent exponent

(3. 180)

Substituting (I - coso:) by sin a tan(a/2), this yields the total source action becomes

This result could aL'lO have been obtained more directly by taking the potential plus a constant-current term in the action

- 1~· dt (~ W2X2 -

Xj) ,

(3. 182)

and by completing it quadra.tically to t hc form (3. 183)

235

3.6 Time Evolution Amplitude in Presence of Source Term

This is a harmonic potential shifted in x by - jj Mw 2 . The time evolution amplitude can thus immediately be written down as

(3. 184)

i) }+ ~it.-t".,) i )(Xa - Mw2 2Mw 2 J .

(

-2 Xl> - Mw 2 In the free-particle limit w

->

0, the result becomes particularly simple:

As a cross check, we verify that the total exponent is equal to ijli t imes the classical action (3. 186)

calculated for the classical orbit Xj,d(t) connecting x" and XI> in the presence of t he constant current j. This satisfies the Euler-Lagrange equation Xj,d

=

jjM,

(3. 187)

which is solved by

Xj,cl () t

=

Xa

i ( ib - ta ),]ttb-_ t"ta + 2M i ( t - ta )' . + [Xb - Xa - 2M

(3. 188)

Inserting this into the action yields (3. 189)

just as in the exponent of (3 .185) . Let us remark that the calculation of the oscillator amplitude (xatblxa t )L in (3 .168) could have proceeded alternatively by using t he orbital separation

x(t) ~ x",,(t) + 'x(t) ,

(3. 190)

where Xj,cl(t) satisfies the Euler-Lagrange equations wit h t he time-dependent source term (3. 191)

236

3 External Sources, Correlations, and Perturbation Theory

rather t han the orbital separation of Eq. (3.7),

x(t)

~

x,,(t)

+ 6x('),

where Xd(t) satisfied the Euler-Lagrange equation with no source. For this inhomogeneous differential equa.tion we would have found the following solution passing through x" at t = ta and Xb at t = tb:

sin w(tb - t) xi.,,(t) ~ x. . (t t) smw b "

sinw(t - ta)

I1t~

+ x,.SlllW (t b t) +M a

la

I

I

.

dt Go,(t,' )J(t').

(3. 192)

T he Green fundion Gw 2(t, t') appears now at the classical level. T he separation (3.190) in the total action would have had the advantage over (3.7) that the source causes no linear term in !Sx(t). Thus, there would be no need for a quadratic completion; the classical action would be found from a pure surface term plus one half of the source part of the action

Inserting Xj,cl from (3 .1 92) and Cw,(t, t') from (3.36) leads once more to the exponent in (3.168). T he fluctuating action quadratic in ox(t) would have given the same fluctuation factor as in t he j = 0 -case, i.e., the prefactor in (3.168) with no further F (due to the absence of a quadratic completion).

3.7

Time Evolution Amplitude at Fixed P ath Average

Another interesting quantity to be needed in Cha pter 15 is the Fourier transform of the amplitude (3. 184):

(Xbtblxata)~

=

(tb - ta)

L: 2;'"

e-;j(t6-t~)"'oJ"(Xbtblxata)~.

(3. 194)

T his is the amplitude for a particle to run from Xa to Xb along restricted paths whose temporal average x == (tb - ta)-l Jt~ dt x(t) is held fixed at Xu :

(Xbtblxata):o =

JV xo(xo - x)

cxp

{ ~1:· dt ~ (±2 -

2

(3. 195)

w x2) } .

This property of t he paths follows directly from the fact t hat the integral over the time-independent source j (3.194) produces a o-function 8((tb - ta )xo dtx(t)) . Restrict ed amplitudes of this type will turn out to have important applications later in Subsection 3.25.1 and in Chapters 5, 10, and 15.

J::

237

3.8 External Source in Quantum-Statistical Path Integral

The integral over j in (3. 194) is done after a quadratic completion in A j - j(t/) ta)xo with A j of (3.181) :

A j - J.( t/) - ta )Xo

=

1 [W( tb - ta ) - 2 tan w(t, 2- t.)] (J. - Jo . )' + A:to, 2Mw 3

(3. 196)

with

.

Jo =

Mw' [ w(t, - t.) ] ~ W(tb - ta)xo - tan (x/) + x a) , w(tb-ta)- 2 tan W Z ~ 2

and

With t he completed quadratic exponent (3.196), the Gaussian integral over j in (3.194 ) can immediately be done, yielding "'0

(Xbtblx"ta)..,

=

(Xbtb lxata)

(i

iMw 3 /27rh % 0) . ~ exp;;-A w(tb-t,,)-2tan '" z ~ n

(3. 197)

If we set Xb = x" and integrate over Xb = Xa, we find the quantum-mechanical version of the partition function at fixed Xo: • Z,} =

) ,,] ( ) J27rh(tb 1- tall Mi .w(t,-t.)j2 [W(tb ta )j2[ exp [i( - 2"' tb - ta Mw xo' 3.198 I~ Sill

As a check we integrate this over Xo and recover the correct Zw of Eq. (2.410). We may also integrate over both ends independently to obtain the partition function

-

~n,"' and T< in the first line (3.206) denote the larger and the smaller of t he Euclidean times T and r, respectively.

3.8 External Source in Quantum-Statistical Path Integral

239

The source terms (3.204) and (3.205) can be rewritten as follows: (3.209) and

We have introduced the Euclidcan vcrsions of the functions A(w) and B(w) in Eqs. (3. 173) and (3. 174) as (3.211)

(3.212) From (3.201) we now calculate thc quantum-statistical partition function . Set;. ting Xb = Xu = x, t he firs t term in the action (3.202) becomes . ,(wfl/3 /2)x. ' A.,= . Mw hal< 2smb sm f.' llW

(3.213)

If we ignore t he second and t hird action terms in (3.202) and integrate (3.201) over x , we obtain, of course, the free partition function 1

Zw

~ 2,inh(p nw/2r

(3.21 4)

In the presence of j, we perform a quadratic completion in x and obtain a sourcedependent part of the action (3.202) : Aj =

~

where t he additional term reads

Ai

-"fI,e

+

Aj

(3.215)

"""",e'

At,c is the remainder left by a quadratic completion.

j Mw R1u.>( A re, = - 2 sm . h W f.'ae'" Ae

+ Be )' .

It

(3.216)

Combining this with A~ ,e of (3.210) gives i

A:J,c

j

_

+ ~,e -

fj 1 fh P fh I -,1 "-.-'1 . . r' Mw 8 1u.>/2 - 4Mw Jo dT Jo dT e )(T);() - sinh(/31iw/2) e'" AeBc·

(3.217)

240

3 External Sources, Correlations, and Perturbation Theory

This can be rearranged to the total source term

~ = __

1_

r"fJ dT rhP dr'cosh w( lr - 7'1 Jo

4Mw Jo

hfJ/2) j(r)j(T') .

sinh{/HIW/2)

(3.218)

This is proved by rewriting the latter integrand as 1

2sinh(phw/2)

{ [ewC"-- "")e - fJfow/2

+ (w ---> - w)] a(T _ T')

+ [e{r'- 1")e- Pllw/2 + (w -+ - w)j 8(7' -

T) } j(T)j(T').

In the second and fourth terms we replace e {Jhw/ 2 by e - (J1u.J/2 + 2sinh(j31lwj2) and integrate over T, r, with the m,ult (3.217). The expression between the currents in (3.218) is recognized as the Euclidean version of t he periodic Green function G~~ (T) in (3.99): G~2.e(T)

-

iG~2( - iT) l tb _j~= _;h.8

1 cosh w(r - h/3/2) 2w ,inh(phw/2)

T

E

10,lipl .

(3.219)

In terms of (3.218) , the partition function of an oscillator in the presence of the source term is

(3.220) For completeness, let us also calculate the partition funct ion of all paths with open ends in the presence of the 50urce j(t), t hus generalizing the result (2.4 11 ). Integrating (3.201) over initial and final positions x" and Xb we obtain

(3.221) whcre

(3.222) with 1 . 3 {3 {cosh wh{3[sinh w(1i{3- r) sinh w(1i{3- r') + sinh wr sinh wr'l 2wsmh wh +sinhw(1i{3-r) sinh wr ' + sinh w(fl{3- r') sinhwr} . (3.223)

By some trigonometric idcntities, this can be simplified to (3.224)

241

3.8 External Source in Quantum-Statistical Path Integral

The first step is to rewrite the curly brackets in (3.223) as sinh wT [ cosh wli(3 sinh WT ' + sinh w(Ii(3 - r')]

+ sinh w(f!(3 -T') [ coshwf~(3 sinh W(Ii(3 -T) + sinh w(Ii(3- «f~(3 - T»].

(3.225)

The first bracket is equal to sinh (3/kv cosh WT, t he second to sinh (3flW cosh w(li(3 -T'), so that we arrive at sinhwli(3[ sinh wT coshwr'

+ sinh w(Ii(3- T) oo;hw(fi.(3 - T') ].

(3.226)

The bracket is now rewritten as

~ [sinh w{T + T') + sinh w(T -

T')

+ sinh w(21i(3 - T - T') + sinh w(r' - T) ], (3 .227)

which is equal to

~ [ sinh w(li(3 + T + r' -

1i(3)

+ sinh w(li(3 + 1i(3 - T - T') ],

(3.228)

and thus to

~ [2sinh wli(3cosh w(li(3 - T - T')],

(3 .229)

such t hat we arrive indeed at (3.224). The source action in the exponent in (3.221) is t herefore:

(A~. e+ . .A~ e) =

- Ml fh{J dT dT' j(T)c:~.n(T, T')j(r'), 10)0 .

r

(3.230)

with (3. 205)

Cf''::: (T, T')

~

coshw(li(3 - IT - T'I) + cosh w{fl(3 - T - T') 2w sinh wli(3 coshw(li(3 - T» cosh wT< w sinh wfi.(3

(3.231)

This Green function coincides precisely with t he Euclidean version of Green function G~,(t, t') in Eq. (3.151 ) using t hc relation (3 .208) . T his coincidence should have been expected after having seen in Section 2.12 that t he partition function of all paths with open ends can be calculated, up to a trivial fact or lo(Ii(3 ) of Eq. (2 .351 ), as a sum over al1 paths satisfying Neumann boundary conditions (2.449), which is calculated using the measure (2.452) for t he Fourier components. In the limit of small-w, the Green function (3.231 ) reduces to _"(TT ') ::::: - 1 + P - -' 1T - T'I - -21 (T + T') w .e ' w'",o (3w2 3 2

L>' . ~,

which is the imaginary-time version of (3.157).

1 (T' +T") + -2(3

,

(3 .232)

2 42

3 .8 .2

3 External Sources, Correlations, and Perturbation Theory

Calculation a t Imaginar y Time

Let us now see how the partition function with a source term is calculated directly in the imaginary-time formulation, where the periodic boundary condition is used from t he outset. Thus we consider

Z",[j j =

JV x(r)e- A.lill",

(3.233)

with the Euclidean action (3.234)

Since x(r) satisfies the periodic boundary condition, we can perform a partial integration of the kinetic term without picking up a boundary term xii:: . T he action becomes (3.235)

Let De(r, r') be the functional matrix D"" ,~Jr, r')

== (- eJ: + w2 )o(r - r) , r - r' E [0, hpj .

(3.236)

Its functional inverse is t he Euclidean Green function,

with the periodic boundary condition. Next we perform a quadratic completion by shifting the path: X""" X

,

1

=x - M

cw2P ,cJ·.

(3.238)

This brings the Euclidean action to the form

Aelil =

(M dr M x'( 2

Jo

-eJ: + w2 )x' _ _2MJo 1_ ( "fj dr (M dr j(r)G (r _ r)j(r'). Jo w ,e P,

(3.239)

The fluctuations over the periodic paths x'(r) can now be integrated out and yield for j(r) == 0 Zw = Oct D:,i,~2 . (3.240) As in Subsection 2.1 5.2, we find the functional determinant by rewriting t he product of eigenvalues as Oct Dw2,c =

mD (w; + w2) = exp [m~.., log(w; + W2)] ,

(3.24 1)

3.8 External Source in Quantum-Statistical Path Integral

243

and evaluating the sum in the exponent according to t he rules of analytic regularization. This leads directly to the partition function of the harmonic oscillator as in Eq. (2.40n Z = 1 (3.242) w 2 einh(pfow/2) The generating functional for j(r)

f- 0 is therefore (3.243)

with the source term: (3.244)

T he Green function of imaginary time is calculated as follows. The eigenfunctions of the differential operator - 8; are e - iw",r with eigenvalues w~, and the periodic boundary condition forces Wm to be equal to the thermal Matsubara frequencies Wm = 27rm/h/3 with m = 0, ±1, ±2, . Hence we have the Fourier expansion (3.245)

In the zero-temperature limit, the Matsubara sum becomes an integral, yielding (3.246)

The frequency sum in (3.245) may be written as such an integral over W m , provided the integrand contains an additional Poisson sum (3.81): 00

00

L

o(m - m)

=

m=-oo

00

L

e,2"."m

=

"=-00

L

e'nwmhP.

(3.247)

"=-00

T his implies that the finite-temperature Green function (3.245) is obtained from (3.246) by a periodic repetition:

1 coshw(r - h/3/2) einh (pliw/2)

2w

, E [0, hpj.

(3.248)

A comparison with (3.97), (3.99) shows t hat G~2 ,e(r) coincides with G~2(t) at imaginary times, as it should. Note that for small w, the Green function has the expansion 1

p

Gw1,~(r)

=

li/3w2

r2

+ 21i/3

rh/3

-

'2 + 12 + ..

(3.249)

244

3 External Sources, Correlations, and Perturbation Theory

The first term diverges in the limit w --+ O. Comparison with the spectral representation (3.245) shows t hat it stems from the zero Matsubara frequency contribution to the sum. If this term is omitted, the subtracted Green function

G~~,e(T) has a well-defined w

-+

G~2,e(T) -

-

(3.250)

h;w2

0 limit

G"O,e (T ) =

1

'"'

2

1

h/3 _ ~

m_± 1,±2,..

-iwm'r _

w2 e

m

-

7

2ft{3

T

Ti{3

-"2 + 12'

(3.251)

the right-hand side being correct only for T E [0, h.8J. Outside t his interval it must be continued periodically. The subtracted Green function G~~,, (T) is plotted for different frequencies w in Fig. 3.4. '

Figure 3.4 Subtracted periodic Grccn function GP~ (7);::;: GP, (T) _ 1/1ij3w 2 and an'" ,e '" ,e tipcriodic Green fun ction G~2.,,(T) for freque ncies w = (0, 5, lO )/h{3 (with increasing dash

length). Compare Fig. 3.2. The limiting expression (3.251 ) can, incidentally, be derived using the methods developed in Subsection 2.15.6. We rewrite the sum as 1

hfi _L

(3.252)

m_±l,±2 ....

and expand (_1)m-' - -2 (nfi)' -'- L -1l [ -i-2rr (r-r.fij2) ]" L '--',b 2 00

lij3

2Jr

,,=0.2.4, .. n

lij3

m=l

m

(3.253)

n

The sum over m on the right-hand side is Riemann's eta function l

=L 00

"(z)

m=l

(

l)m-'

L-~_

m'

1M. Abramowitz and I. StCglm, op. cit., Formula. 23.2.19.

(3.254)

245

3.8 External Source in Quantum-Statistical Path Integral

which is related to the I'.eta function (2.519) by

"I') ~ (1 - 2' - (T"T)] (3280) T,7 T I\p,a ( ) , • '--' 7a,7b) u 'Ta(, 7b

6Sce H. Kleinert and A. Chervyakov, P hys. Lett. A 2~5, 345 (1998) (quant-ph/9803016); ~O , 6044 (1999) (physics/9712048).

J. Math. P hys. B

248

3 External Sources, Correlations, and Perturbation Theory

where C",~,e(T, 7') is the imaginary-time Green function with Dirichlet boundary conditions corresponding to (3.206): G

r') ~ 8(T-T')6.(T"T)6.(T', To) + 8(T-r')6.(T"r')6.(T,To)

( w',e 7,

"(

)

u Ta, Tb

,

(3.281 )

with (3.282)

and (3.283)

Let also write down the imaginary-time versions of the periodic or antiperiodic Green functions for time-dependent frequencies. Reca ll the expressions for constant frequency ~(t) and G!(t) of Eqs. (3.94) and (3.112) for 7 E (O,Iij3): G~,,,(T)

~

-1

1i{J

L ,-""",. m

- 1

iwm

-

= e- w(" - "fJ/2) W

1 2 einh(fiIiw12)

(l +n:)e- "'''' ,

(3.284)

and

C::',,,(T)

~

-1 h{J

~

L ,- it.l.,. . m

- 1

iw!,.

=

e- w(7 - r./Jf2j

1

2 cooh (filiw 12)

w

( l -n:)e- W "

(3.285)

the first sum ext ending over the even Matsubara frequencies, the second over t he odd ones. The Bose and Fermi distribution functions were defined in Eqs. (3.93) and (3. 111 ). For 7 < 0, periodicity or antiperiodicity determine

nY

G""(T) w,e

~

±G""(T + hfi). w,e

(3.286)

The generalil.ation of these expressions to t ime-dependent periodic and antiperiodic frequencies 11(7) satisfying t he differential equations

[- 8.,. has for

11(7)l G~: (7, 7') =

6P '''(7 - 7')

(3 .287)

/3 --- 00 the form G~,: (7, 7') = 9(7 _ r)e- fo' d""I1(,.'j .

Its periodic superposition yields for finite 00 e- Jo C'" ' ' d,.'O( ...'j P '" ( GO,e ') = '("" T,7 L ,. ",0

/3 a sum analogous

{I} (- I)"

,

(3.288)

to (3.277):

h/3>7>r>O,

which reduces to (3.284), (3.285) for a constant frequency 11(7) =: w.

(3.289)

249

3.9 Lattice Green FUn ction

3.9

Lattice Gree n FUnction

As in Chapter 2, it is easy to calculate thc above results also on a sliced time a.xis. This is useful when it comes to comparing ana.lytic results with Monte Carlo lattice simulations. We consider here only the Euclidean versions; the quantum-mechanica.l ones can be obtained by ana.lytic conti nuation to real times. The Green function Gw,(T, T') on an imaginary_time lattice with infinitely many lattice points of spacing ~ reads [instead of the Euclidean version of (3. 147)]: 1 1 _.;..1 ... _ ... '1 G ' (T,T') = 2 . fh - e - ';'. 1.. - .,.'1 = 2w h( - 12)' , sm €We coo €We where We is given,

118

in (2.404), hy

_ 2 1 'w "'e = -, arsin )-2 .

(3.290)

(3.291)

T his is derived from the spectral representation (3.292) by rewriting it

118

(3.293) with n ;;; (T' - T)/(, performing the ",'-integral which produces a Bessel function I("' _"") I« 28/(~) , and subsequently t he integra.l over s with the help of formula (2.473). The Green function (3.290) is defined only at discrete 7"" = nhp/(N + 1). If it is summed over a.ll periodic repetitions n ...... n + k(N + 1) with k = 0, ±1, ±2, ... , one obtains the lattice analog of the periodic Green function (3.248):

-

,~ """_00 I: 2(1 1

1

2t..> COSh(fW/2)

3.10

cosbw(T - hP/2) sinh(,",,",p/2)

TE [O,htJ] .

(3.294)

Correlation FUnctions, Generating FUnctional, and Wick Expa nsion

Equipped with the path integra.l of the harmonic oscillator in the presence of an external source it is easy to calculate the correlation functions of any number of position variables X(T). We consider here only a system in thermal equilibrium and study the behavior a t imaginary times. The real-time correlation functions can be discussed similarly. The precise relation between them will be worked out in Chapter 18. In general, i.e. , also for nonharmonic actions, the t herma l correlation functions of flo-varia bles X(T} a re defined as thc functional avcrages

G~~)(Tl, ... , Tn) _ _

(x{Tdxh)···X{Tn)) Z - lJVXX(Tdx(T2) ... X(Tn)exp( - ~A,).

(3.295)

250

3 External Sources, Correlations, and Perturbation Theory

They are also referred to as n-point functions. In operator quantum mechanics, the same quantities are obtained fro m the t hermal expectation values of t ime-ordered products of Heisenberg position operators iH(T):

G~~)(T" ... , Tn)

=

Z - ITr

{t" [XH(TdxH(T2)"

. XIl(Tn)e- il/knT]} ,

(3.296)

where Z is the partition function Z = e- F/k"T = Tr(e- lilk8T)

(3 .297)

and tT is t he time-ordering operator. Indeed, by slicing the imaginary-time evolution operator cJ'iTI" at discrete times in such a way t hat the t imes T; of the n position operators x( Ti) arc among them, we find t hat G~".{ (7\, .. . ,Tn) has precisely the path integral representation (3.295) . By definition, the path integral with the product of X(Ti} in the integrand is calculated as follows. First we sort the times 1'; according to their time order, denoting the reordered times by Tt(;) . We also set Tb == 1'1(..+1 ) and Ta == Tt(O) ' Assuming that the times Tt(;) are different from one another, we slice the time axis l' E [Ta , Tb ) into the intervals [7b , Tt(fI» ) , [Tt(fI),Tt(fI _ l»), !rt( .. - 2),Tt( ..- 3»)," " [TI(4),Tt(3» ) , [TI(2) , Tt(I) ], [Tt(l), Ta ]. For each of these intervals we calculate the time evolution amplitude (Xt(HI)Tt(i+1) IXt(i)Tt(i») as usual. Finally, we recombine the amplitudes by performing the intermediate x(Tt(;j)-integrations, with a n extra factor X(T;) at each Ti,

i.e.,

'(Xt(Hl)Tt(H l)IXt(i)Tt(i»)' X(Tt(i »)' (Xt(i)Tt(i) IXt(i_l)Tt(i_l»)'

X(Tt(i _ l))

(3. 298 )

•.... (Xt(2)Tt(2)IXt(I)Tt(I»)' X(Tt( I »)' (Xt(I)Tt(1) IXt(O)Ta).

We have set Xt (n+1) == Xb = Xa == xt(O), in accordance with the periodic boundary condition. If two or more of the times Ti are equal, the intermediate intcgrals arc accompanioo by t he corresponding power of X(Ti). Fortunately, t his rather complicated-looking expression can be rcplaced by a much simpler one involving functional derivatives of the t hermal part ition function Zfj ] in the presence of an external current j . From t he definition of Zfj ] in (3.233) it is easy to see that all correlation functions of the system are obtained by the functional formula

G~':'(,,, ...,'.) ~

[zur't. ;j~,.) ... t, ;jf,./U1Lo .

(3.299)

T his is why Zfj ] is called the gene1llting functional of the theory. In the present case of a harmonic action , Zfj ] has t he simple form (3.243), (3.244), and we can write

G~~)(Ti> ... ,T.. )

=

[t.-;j(,,) ;_... /l. -;j(,.) ;-

x

exp

{2;M 10

M dT

lo"tJ drj(T)G~1,e(T -

(3.300) r)j(T') }

L~

3.10 Correlation Functions, Generating FuIICtiOlllll, wid Wick Expansion

251

where G~2,e(T - T') is the Euclidean Green function (3.248) . Expanding the exponential into a Taylor series, the differentiations are easy to perform. Obviously, any odd number of derivatives vanishes. Differentia ting (3.243) twice yields the two-point function [recall (3.248)[

G~Z;(T,T') =

(x(T)x(r» =

~G~2,e(T -

T') .

(3.301)

Thus, up to the constant prefactor, t he two-point function coincides with t he Euclidean Green function (3.248). Inserting (3.301 ) into (3.300), all n-point functions are expressed in terms of t he two-point function G~/(T, r): Expanding the exponential into a power series, the expansion term of order nj2 earries the numeric prefactors Ij(nj2)! . Ij2 n/ Z and consists of a product of nj2 factors fohP dT'j(T)G~;(T, r)j(T')j1i? The n-point function is obtained by functionally differentiating this term n times. The result is a sum over products of nj2 factors G~Z;(T, T') with n ! permutations of t he n time arguments. Most of these products coincide, for symmetry reasons. First, G~2;(T,T') is symmetric in its arguments. Hence 2n / Z of t he permutations correspond to identical terms, their number canceling one of t he prefactors. Second, t he nj2 Green functions G~;(T, r) in t he produet are identical. Of the n ! permutations, subsets of (nj2)! permuta.tions produce identical terms, their number canceling the other prefactor. Only n!j[(nj2)!2n/Z] = (n - 1)· (n - 3)···1 = (n - I )! ! terms are different. T hey all carry a unit prefactor and their sum is given by the so-called Wick rule or Wick expanswn:

G~~)(Ti> . .. , Tn) =

L

G~/(Tp(l) ' Tp(2»··· G~2;(Tp(n_l}' Tp(n}) .

(3.302)

"';'

Each term is characterized by a different pair configurations of the time arguments in the Green funct ions. These pair configura tions are found most simply by t he following rule: Write down all time arguments in the n-point function T1TZT3T4 ... Tn . Indicate a pair by a common symbol, say 7"p(;)7"P(Hl)' and call it a paiT contraction to symbolize a Green function G~;(Tp(;), TP(H l » . T he desired (n - 1)!! pair configurations in t he Wick expansion (3.302) are t hen found iteratively by forming n - 1 singlc contractions (3.303)

and by t reating t he remaining n - 2 uncontracted variables in each of these terms likewise, using a different contraction symbol. The procedure is continued until all variables are contracted. In the literature, one sometimes another shorter formula. under the name of Wick's rule, stating that a single harmonically fluctua ting variable satisfies t he equality of expectations: (3.304)

252

3 External Sources, Correlations, and Perturbation Theory

This follows from the observation that the generating functional (3.233) may also be viewed as Z", times the expectation value of t he source exponential (3.305)

Thus we can express the result (3.243) also as (3.306)

Since

(h/M)G~~,e(T,r')

G~z;(T, r')

=

in t he exponent is equal to the correlation function

(x(r)x( r'» by Eq. (3.301 ), we may also write (ef dTj(r)z(r)/")

=

ef dr J dr'j{r)(z{r}z(r'J);{T'1/2h'.

(3.307)

Considering now a discrete time axis sliced at t = tn. and inserting the special source current j(Tn} = K6n,o, for instance, we find directly (3.304). The Wick theorem in this form has an important physical application. The intensity of the sharp diffraction peaks observed in Emgg scattering of X -rays on crystal planes is reduced by therma1 fluctuations of the atoms in the periodic lattice. T he roouction factor is usually written as e-:2W and called the Debye- Waller factor. In the Gaussian approximation it is given by (3.308)

where u (x) is the atomic displacement field. If the fluctuations take place around (x(r)} into

i

0, then (3.304) goes obviously over

(3.309)

3 .10.1

R eal-Time Corr elation Functions

T he translation of these results to real times is simple. Consider, for example, the harmonic fluctuations ox(t) with Dirichlet boundary conditions, which vanish at tb and to.. Their correlation funct ions can be found by using t he amplitude (3.23) as a generating funct ional, if we replace x(t) -+ ox(t) and Xb = Xa -+ O. Differentiating twice with respect to the external currents j(t) we obtain

c~i(t, n ~ (x(t)x(t')) ~ i !Cw,(t - r),

(3.310)

with the Green funct ion Gw 2(t - t') of Eq. (3.36), which vanishes if t = tb or t = to.' T he correlation function of i:(t) is

.().(')) (x txt

.hcosw(t.-t»cosw(t,D,ZI>,

(X(T)p,,(T'» = iMa~,G~2J,B,z:r,(T,T') - MWBG~J,B,%~(T,T'),

(3.3M)

2 ) d "",H,~1>y ( T ,'T) 2J d...',B,.", (T,T ')

(X(T)p~(r'» = iMl}T,G~J,B,,,,~(T, T')

(3.355)

C ..'"',B,p.,,_ (T,'T)

+ MWBG~lJ,B,n{T,r'),

(3.356) (Z(T)P.(T'» = iMl}T, G~l,B, .. (T,r'), 2 2 (Pz (T )p:z( T'» = - M l}T8T•G~lJ,B,u (T, :r') - 2iM w B8TG~1,B ,:z~ (T, r')

+ M2w~ G~l,B,u(T, T') + hM6(T - T'l, C...'"',B,p. ". (T,T ')

(3.357)

(P.,(T)p~(T')}= -M2 8TaT,G~1.B,:z~(T,T') + iM28TG~1.B ,n(T, T') 2 ) + M 2 w2B d"",B.z~ (n') , ,

(3.358)

G~l,B.p,p, (T,T') =: (P.(T)P.(r')}= - M2aTaT,G~1.R, .. (T,r') + hM6(T - T').

(3.359)

Only diagonal correlations between momenta contain the extra 6-function on the right-hand side according to t he ru le (3.352). Note that 8TIJT'G~1'BloI>(T' T') = -1J:G~2J,B,"b{T, T'). Each correlation function is, of course, invariant under time trans ations, depending on ly on the time difference T - T'.

The correlation functions (X(T)X(T')} and (X( T)Y(T'» are the same as before in Eqs. (3.324) a nd (3.325).

3.12.2

R elations between Various Amplitudes

A slight generahation of the genera ting functional (3.330) contains paths with fixed end points rather than all periodic paths. If the endpoints are held fixed in configura tion space, one defines (x~h.Blx .. O) [j,kl =

1""'-"

{l

Dp~ exp - "i"A,,[j,kj Dx-,

"'(O)=z.

1f«

«

}.

(3.360)

If t he e ndpoints arc held fixed in momentum space, one defincs

(3.361) The two are related by a Fourier transformation

(3.362) We now observe that in the canonical pa.th integral, the am plitudes (3.360) and (3.361) with fixed endpoints can be reduced to those with vanishing endpoints with modified sourecs. T he modification consists in shifting the current k(T) in t he action by the source term ix~6(1"b - T) -

259

3.12 Correlation Functions in C8.ll0Ilical Path Integral

iX"J(T - '1,,) and observe that this produces in (3.361) an overall phase factor in the limit 1), 1 hf3 and Ta J 0: lim lim (Pb hf3IPa O)li(T), k(r)

... 1~~r. IO

= exp

+ iXbJ(1), - '1) - 1x"J(r - Ta) 1

{* (PbXb - Pax,,) } (Pb hPiPa O)li(r), k(T)].

(3.363)

By inserting (3.363) into the inverse of the Fourier transformation (3.362) , (3.364) "''il

obtain

(Xb hPlx a O)li, k]= lim lim (OTif3100)li(T), k(T) ... lh/Jr. IO

+ ixbJ(1), -r ) - ix"J(T - ra )] .

(3.365)

In th is way, the fixed-endpoint path integral (3.360) can be rWuced to a path integral with vanish ing endpoints but addi t ional J-terms in the current k(T) coupled to the momentum P{T). T hen: is also a simple n:lation betwccn path integrals with fixed equal endpoints and periodic path intcgrals. T hc measuI"CS of integration an: related by "(11/1)=,,

1

"(0) _,,

V V ~=

f

21ft,

V Vp

- "- J(x(O) - x) . 21fh

(3.366)

Using the Fourier decomposition of the delta function, we rewrite (3.366) as (3.367) Inserting nOW (3.367) into (3.365) leads to t he announced dcsirW relation

(Xb li/JIX" O)[k,j]

=

lim lim lim

roo

... 111/J .... IO ...!IO} _oo

xZ li(T) - ip"J(T -

,dP: 1f"

T~), k(T)

+ iXbJ(1), -

'0) - ix"J( '0 - '0,,)] ,

(3.368)

where Zli, kJ is the thennodynamic part ition function (3.330) summi ng all periodic paths. When using (3.368) we must oc careful in evaluati ng the thf(lC limits. T he limit T~ J 0 has to oc evaluated prior to the other limits 1), 1 hf3 and '1" 1 o.

3.12.3

Harmonic Generating Functionals

Here we write down explicitly the harmonic generating funetionals with the above shifted source terms:

f(T) = k(T) + ixbJ(1), - '0) - iX"J(T - Ta),

J(T) = jeT) - ipJ(T - ~),

(3.369)

leadi ng to the factorized generating functional

Z", [k,Ji = z~O)[O,OJ Z~I) [k , jiZ![k,jJ. T he respective terms on the right-hand side of (3.370) read in detail

Z~O)[O,OJ

=

Z",exp

C~2 { _p2G~,,(~,~) -

2p

[x"G~p(~,T,,) + xbG~,,(~, 1),)]

(3.370)

260

3 External Sources, Correlations, and Perturbation Theory -z! G~(1"c, 'Te) - x~G~(1)" 1),) + 2xQXbG~h, Til) }) , exp

(3.371)

(:1 [ '/3 dr {j(T)[-ipG~%(T, T~) + iX~G~p(T,1"b) - iX"G~p(T,T,,)J

+k(T) [-ipG~k,r,;,) + ixbCl:,,cr, 1),) - ix"G~(T,T,,)J} ) ,

~p { ~ ["tJ dT) r~tJ dT2[Uhl,k(T2)) 2h Jo Jo

ZE lk,;]

x ( G~r(T" rJ) Gi;,,{Tl> 1"2) G~(Tl,Tl)

G~(Tl,Tl)

(3.372)

1( l]} j(7"2) k(1',)

(3.373)

,

where Z.., is given by (3.342) and G~p(TJ,T2) etc. are the periodic E uclidean Green functions defined in Eqs. (3.346)- (3.349) in an abbrev iated notation. Inserting (3.370) into

all} beTh 7"2)

(3~3;)

and performing the Gaussian momentum integration, over the exponentials in and Z~l) [ k,jl , the result is

(Xb 1i.Blxa O)[k, j] = (Xb 1i.Blxa 0)[0, 01 x cxp { xexp

{ 1""r dT)

--;.

21i

0

dT:!lU(TJ),k(T2»

)0

Z~O) [0, 0]

~ ['tJ dT [Xel(T)j(T) + PCI(T) k(T) ]}

(

101 101 ) ( .J(Tl) G(~)(Tt,Tl) G(:;t1'1'2)

Gp~ (T!,1'2)

G"" (Tl ,T2)

k (1'2)

l]}

,

(3.374) where t he Green funct ions G~?)(Tl' T:l) have now Dirichlet boundary conditions. In particular, the Green function G~?\T!,Tl) is equal to (3,36) continued to imaginary timc. Thc Green funct ions G~~)(Tl ' 1'2) and G~)(Tl' 1'2) arc Dirichlet versions of Eqs, (3,346)- (3,349) which arise from t he above Gaussian momentum integrals. After performing t he integrals, the first factor without eurrents is

Performing the limits using h lim lim G~p«, T,,) = -i - ,

~.IO~~IO

2

(3.376)

where t he order of t he respect ive li mits turns out to be important, we obtain t he amplitude (2.409):

Mw 21fhsinh h{3w

x cx p {

2h::h\3w[(X~+X~) COSh li.8w -2xax~1 } .

The first exponential in (3.374) contains a complicated representation of the classical path

(3.377)

261

3.12 Correlation Functions in C8.ll0Ilical Path Integral

(3.378)

and of the classical momentum

(3.379)

Indeed, lI\lierting the explicit periodic Green functions (3.346)-(3.349) and going to the limits we obtain X B sinh..,(h/3

- T) + sinhh{3w

xb~inh"'T

(3.380)

(3.381)

the first being the imaginary-time version of the elassical path (3.6), the second being related to it by the classical relation PdCr) = iMd:rd(T)/dT. The second exponential in (3.374) quadratic in the currents contains the Green functions with Dirichlet boundary condit ions (3.382) (3.383)

(3.384) (3.3SS) After applying some trigonomtric identities, thc:;c take the form h

G~~)(T!, Ta) = 2M.., sinh h/L [cosh..,(IiP-IT) - T21) - cosh..,(h/3 - T) - T2) ],

G~~)(T!, Ta)

2

si:

h{3w (O(T) - Tal sinh ..,(h/3- IT) -Tall

-B(T2-Tilsinh..,(h/3 - ITZ - T)i)+sinh..,(h/3 - T) - T2)},

G~)(TJ,72)

(3.386)

(3.387)

.h

2sinhh/L {O{T) - 72)smh..,{h/3 - IT) - T21) -D{1'J - TJ) sinh..,(h/3 - 172 - T)il - sinh ..,{h/3 - T) - 1'J)},

At"'"h{3w [cosh ..,{h/3 - ITI -Tal) + cosh..,(h/3-TI - Ta)] . G~)(TJ, Ta) = 2 sinh

(3.388) (3.389)

T he first correlation function is, of course, the imagi nary-time version of the Groen function (3.206). Observe t he symmetry properties under interchange of the time argumenl.$:

G~~){TL'Tz) = G~~){T2,Tll,

G~~)h,T2) = - G~~)(Tz, TI) ,

(3.390)

G~~){TL' Ta) = -G~){T2,Tll,

G~)(ThT2) = G~){Tz, TI),

(3.391)

262

3 External Sources, Correlations, and Perturbation Theory

and t he identity (3.392)

In addition, there are the following derivative relations between the Groon functions with Dirichlet boundary conditions: (3 ,393) (3 ,394) (3 ,395) Note that Eq. (3.382) is It. nonlinear alternative to the additive decomposition (3. 142) of It. Green function with Dirichlet boundary condit ions: into Green functions with periodic boundary

conditions.

3.13

Particle in Heat Bath

The results of Section 3.8 are the key to understanding t he behavior of a quantummechanical particle moving through a dissipative medium a t a. fixed temperature T . We imagine the coordinate x(t) a particle of mass M to be coupled linearly to a heat bath consisting of a great number of ha rmonic oscillators X i(T) (i = 1, 2,3, . .. ) with various masses M; and frequencies 0 ;. The imaginary-t ime path integral in t his heat bath is given by

(x,"Plx. O) ~ x

I>" "'" f.I f VX;(r) 1>(0)=>. V x(r)

exp{-~fo"i3 dr ~ [~i(X/+ O~Xl)]}

x exp {

(3. 396 )

-~ fo"fJ dr [~ :i;2 + V(x(r)) - ~ C;X;(r)x(r) ]} x n;lz;'

where we have allowed for an arbitrary potential V (x ). The partition functions of t he individual bat h oscillators

Z; == ~

f VX;(r)exp { - ~ fo"P dr [~;(X/ + O~X;)]} 1 2 sinh("j3il;(2)

(3.397)

have been divided out, since their t hermal behavior is trivial and will be of no interest in the sequcl. The path intcgrals ovcr Xi(r) can be performed as in Section 3.1 leading for each oscillator label i to a source expression like (3.243), in which c;x(r) plays t hc role of a current j(r). T he result can be written as

(Xb h.6lx"O)

=

r(~)=Zb V x(r)cxp { _ _"I r"fJ dr [M:i;2 + V (x(r » ]- _,1, .Abat.b [XI} ,

Jz(O )_zG

Jo

2

(3.398)

263

3.13 Partide in Heat Bath

where Abath [x] is a nonlocal action for the particle motion generated by the bath

d7 Io'P d7'X(7)O:(7 - r)x(r). 'Io'P 2

Abatb[X] =

- -

0

0

(3.399)

The function a(7 - 7') is t he weighted periodic correlation function (3.248) :

a(7 - 7')

=

~~ ~i G:;~.e (7 -

~

L: --=L w;h fl,( IT -

7')

T'I - oN 2) , mh(fl,op /2)

,2M,fl,

Its Fourier expansion has the Matsubara frequenc ies Wm

=

(3.400)

21rk n T/h

(3.401) with the coefficients

"

am = L. •

d

1

~.• W2m + w~• ·

(3.402)

Alternatively, we can write the bath action in the form corresponding to (3.276)

Abath[X] =

00 d7 1 d7'X(7)ao(7 11," 2

--

0

- 00

7')x(r),

(3.403)

with the weighted nonperiodic correlation function [recall (3.277)]

(3.404) T he bath properties are conveniently summarized by the spectml density of the bath

Po(w') " 2.

L: --=Lo(w' 2M;n; i

fl,) .

(3.405)

T he frequencies n; are by definition positive numbers. The spect.ral densit.y allows us to express 0:0(7 - r) as t he spectral integral

(3.406) and similarly

(

') 1,00 -Pow ,,",' (,)coshw'( IT -

a7-7 -

0

211"

,-'1- oN 2) sinh(w'MJ/2)·

(3.407)

264

3 External Sources, Correlations, and Perturbation Theory

For t he Fourier coefficients (3.402), t he spectral integral reads (3.408)

It is useful to subtract from these coefficients t he first term 0'0 , and to invert the sign of t he remainder making it positive definite. Thus we split Ct m

=2

dd ",,(w') ( W!) -- 1wI2 1,o00 21f w' w",+ 2

=0'0-9",·

(3.409)

T hen the Fourier expansion (3.401) separates as a(T - T') = O'oQP(r - r) - 9(r - T'),

(3.410)

where JP(r - r) is t he periodic o-function (3.279) :

b'(r - r) ~ -

1

ti/3

0000

L

e-"'-('-" ) ~

"' = - 00

L

b(r - r' -

""/3),

(3.4 11)

1>= - 00

the right-hand sum following from Poisson's summation formula (1.197). The subtracted correlation function

has t he coefficients

(3.4 13)

The corresponding decomposition of the bath action (3.399) is (3.4 14)

where

11," dr 1," dT'X(T)g(T - r)x(T'), 2 , ,

A~th [xl = -

(3.4 15)

and (3.4 16)

is a local action which can be added to t he origina l action in Eq. (3.398) , changing merely t he curvature of t he potential V (x). Because of t his effect, it is useful to introduce a frequency shift ~'l via t he equation M!:::.w

, =-Qo= _ 1,00 ",,'-",,(w') Ci -2 - = - 2: --, . o

21f

w'

i

Mini

(3.4 17)

265

3.13 Partide in Heat Bath

Then the local action (3.41 6) becomes A loe

M fM 2 = "2AW2 Jo dTX (T) .

(3.418)

This can be absorbed into the potential of the path integral (3.398) , yielding a renormalized potential (3.4 19)

With the decomposition (3.414), the path integral (3.398) acquires the for m

(x/,1i.8lx aO) =

_ VX(T)exp (-~1 1" dT [M _ ±2 + Vn;,n(X(T))11'('''~'' 2 0: (0)-0:4

/I

0

-h1 Ab..th[X]} . (3.420)

The subtracted correlation function (3.4 12) has the property

r"

Jo

(3.421)

dTg(T - T') = 0.

Thus, if we rewrite in (3.415) (3.422)

the first two terms do not contribute, and we remain with

'1" dT 1, " drg(T - r)[X(T) - X(T'W .

Ab..th [x] = - -

4 ,

(3.423)

If the oscillator frequencies fl; are densely distributed, the function Pb(w') is continuous. As will be shown later in Eqs. (18.208) and (18.317) , an oscillator bath introduces in general a friction force into classical equations of motion. If t his is to have t he usual form - M -yx(t) , the spectral density of the bath must have the approximation Pb(W')::::,: 2M,,{w' (3.424)

[see Eqs. (18.208), (18.31 7)]. T his approximation is characteristic for Ohmic dissipation. In general, a typical friction force increases with W only for small frequencies ; for larger w , it decreases again. An often applicable phenomenological approximation is the so-called Drude form

Pb(w')::::,: 2M,,{w'

2

w'0

Wo + w'

2'

(3.425)

where l / wf) == TO is Drude 's relaxation time . For times much shorter than the Drude time TO , there is no dissipation. In the limit of large Wv , the Drude form describes again Ohmic dissipation.

266

3 External Sources, Correlations, and Perturbation Theory

Inserting (3 .425) into (3.41 3) , we obtain the Fourier coefficients for Drude d issipation

, 10

9m = 2M"(wl)

00

o

dw -2

2w~

1 2

2

2

IrWo + w wm +w

2

= MIWm l'Yl

WD Wm

I

+wO

.

(3.426)

It is customary, to factorize 9m == M lwml'Ym .

(3.427)

so that Drude dissipation corresponds to (3.428)

and Ohmic dissipation to 1m =: 'Y. The Drude form of the spectral density gives rise to a frequency shift (3.4 17) (3.429) which goes to infinity in the Ohmic limit Wo -- 00.

3.14

Heat Bath of Photons

The heat bath in the last section was a convenient phenomenological tool to reproduce the Ohmic friction observed in many physical systems. In nature, t here can be various different sources of dissipation. The most elementary of these is the deexcitation of atoms by radiation, which at zero temperature gives rise t o t he natural line width of atoms. The photons may form a t hermally equilibrated gas, the most famous example being the cosmic bl ack~body radiation which is a gas of the photons of 3 K left over from the big bang 15 billion years ago (and which create a sizable fraction of the blips on our television screens) . The theoretical description is quite simple. We decompose the vector potential A (x , t) of electromagnetism into Fourier components of wave vector k

A(x, t) ~

L: ",,(x)X, (t),

,

(3.430)

The Fourier components Xk (t) can be considered as a sum of harmonic oscillators of frequency l1k = elk l, where e is the light velocity. A photon of wave vector k is a quantum of Xk(t ). A certain number N of photons with the same wave vector can be described as the Nth excited state of the oscillator Xk (t). The statistical sum of thesc harmonic oscillators led Planck to his famous formula for the energy of black-body radiation for photons in an otherwise empty cavity whose walls have a temperature T. These will form the bath, and we shall now study its effect on the quantum mechanics of a charged point particle. Its coupling to the vector potential is given by the interaction (2.632) . Comparison with the coupling to t he

267

3.14 Heat Bath of Photons

heat bath in Eq. (3.396) shows t hat we simply have t o replace - E,c; X ,(r)x(r) by - EkCJ. Xk (r):ic(r). T he bath action (3.399) takes then t he form Aba~h[xl

= --1 2

i" dr i"~ dr'x'(T)a"( . ..x (T),T; X(T') , r').i'(T') . , 0

(3.431)

0

where a,j(x,T;X',r') is a 3 x 3 matrix generalization of the correlation func tion (3.400),

,

Q;j(X,T; x' ,r) ~ ;c'I:C-k(x ),.(x')(X:k(T)XI(r)).

(3.432)

k

We now have to account for the fact that there arc two polarization states for each photon , which are transverse to the momentum d irection. We therefore introduce a transverse Kronecker symbol (3 .433)

and write the correlation function of a single oscillator X~ k (T) as

G;j_ k'k (T - T')

=

(X';- k' () T X'jk' (')) T

=

hTOj , G'",,2 ,., k (T - T') , 0kOkk'

(3.434)

with (3.435)

Thus we fi nd ii(

. '

') _ "

(t x,r, x ,T - dl

J(271")3 fi3k

1b":i

e,k (x - x')

k

20 k

cosh Ok(IT - T'I - hfJj2) sinh(l1k ;tfJj2) .

(3.436)

At zero temperature, and expressing 11k = elkl, this simplifies to

.. (t"(x, r;x,r')

e fi3k c1! (271")3 i5'~ 2

=

T . . eik (x - x') - cl k ll.,. - .,.'I

21kl

.

(3.437)

Forgetting for a moment the transverse Kronecker symbol and the prefactor e2 jc?, the integral yields ) 1 GcR ( X,T; X" ,T = 47T2dl(r

1

T')2 + (X

x ,)2/dl'

(3.438)

which is t he imaginary-time version of t he well-known retarded Green function used in electromagnetism. If the system is small compared to the average wavelengths in the bath we can neglect the retardation and omit the term (x - x')2/r:? In t he finite-temperature expression (3.437) this amounts to neglecting the x-dependencc.

268

3 External Sources, Correlations, and Perturbation Theory

The t ransverse Kronecker symbol can then be averaged over all directions of the wave vector and yields simply 26;j /3, and we obtain the a pproximate function . x'

ii( 0'

x , T,

2

') _ 2e {/i _ 1_ ,T 3& 2Jrc2

J21T w coshw(lr - 7'1 - ftf3/ 2) sinh(wh.8/2) . dw

(3.439)

This has the generic form (3.407) with the spectral function of the photon bath (3.440)

T his has precisely the Ohmic form (3.424), but there is now an important difference: the bath action (3.431) contains now the time derivatives of the paths x(r). This gives rise to an extra factor wI'.! in (3.424), so that we may define a spectral density for the photon bath: (3.44 1)

In contrast to t he usual friction constant J in the previous section, this has the dimension I Jfrequcncy.

3.15

Harmonic Oscillator in Ohmic Heat Bath

For a harmonic oscillator in an Ohmic heat bath, the partition function can be calculated as follows. Setting

M , , V""(x) ~ ,w x ,

(3.442)

the Fourier decomposition of the action (3.420) reads

A.,

=

Mh klJT

{W', ~ ["+ "'2xo + ;;:1 Wm

W

+ Wm 'Ym1IXm

I'}

.

(3.443)

The harmonic potential is the full renormalized potential (3.4 19). Performing the Gaussian integrals using the measure (2.445), we obtain the partition function for the damped harmonic oscillator of frequency W [compare (2.406») (3.444)

For the Drude dissipation (3.426) , t his can be written as ZdlUIlp

'" Let

WI, W'l, W3

fi

= koT w~(wm + WD) 1iw m= Iw;. + W;'WO + wm(W'l + 'YWD) + WOW'l·

(3.445)

be the roots of t he cubic equation w 3 _ W'lWD

+ w(w'l + 'YWD) -

W'lWD

= O.

(3.446)

269

3.15 Harmonic Oscillator in Ohmic Heat Bath

Then we can rewrite (3.445) as z""mp

= knT

liw

w

fi

Wm

Wm

Wm

Wm

m = IWm + W\Wm + W2Wm + W3

+ WD

(3.447)

Wm

Using the product representation of t he Gamma functionS

nZ r(Z) ~ lim -

II -m+-z "

(3.448)

.. _ 00 Z m = 1 m

and t he fact that (3.449)

t he partition function (3.447) becomes zdam p =

'"

~ W f(w';wd f ("Il'2/wdr (W3/wd 2Jr WI r(wD/wd '

where WI = 2Jrk n T/h is the first Matsubara frequency, such that In the Ohmic limit WD --lo 00 , the roots WI, W2 , W3 reduce to

WI = ,,(/2 - i6,

WI = 1/2 + i6,

W3

(3.450) Wi/WI

= w;(3/2Jr.

(3.451)

= WD - 1 ,

with

J " JW'-1' /4 ,

(3.452)

and (3.450) simplifies further to

z~amp =

1 -2 ":::" r (wdwdf(wdwd. rrw,

For vanishing fr iction, the roots WI and the formula9

W2

become simply WI = iw,

(3.453) W2

= - iw, and

rr sm Jrz

r( 1 - z)r(z) ~ -.-

(3.454)

can be used to calculate (3.455)

showing that (3.450) goes properly over into the partition function (3.214) of the undamped harmonic oscillator. The free energy of the system is

F(T) =

- knT [log(w/2Jrwd - IOgr(WD/wd + log f (wdwd + log f(W2/WI) + log f(W3/wd l .

----;:;;;-;;:::-=:-:-=~:::;:;;:-

sI.S. Gradshtcyn and I.M. Ryzhik , op. cit. , Formula 8.322. "ibid., Formula 8.334.3.

(3.456)

270

3 External Sources, Correlations, and Perturbation Theory

Using t he large-z behavior of log r(Z)lO

1) 1 1 1 log r (z}= ( z-2" logz-z + 2"log21r+ 12z - 360z3 - O (l/z't

(3.457)

we find the free energy at low temperature

1 F(T) - E,- ( WI

+ -1 + -1 W2

WI

W2)1I" 2 '111" 2 - (kRT) ~ Eo- - (kRT) , W\W2W3

6w 2h

6h

(3.458)

where

Eo

= - -

Ii

2.

[WJlog(wdwD)

+ 1V21og(w~dwD) + W3!Og(W3/WV)]

(3.459)

is t he ground state energy. For small fr iction, this reduces to

'Y 21T

flW

Wv

"r'~

W

16w

Eo~ - + - Iog - - -

2

( 1+ -4w

)

7rWo

_> + Ob)·

(3.460)

The P-behavior of F(T) in Eq. (3.458) is typical for Ohmic dissipation. At zero temperature, t he Matsubara frequencies Wm = 21fmknTjh move arbitrarily close together, so t hat Matsubara sums become integrals according to t he rule (3.461) Applying t his limiting procedure to t he logarit hm of the producl. formula (3.445), t he ground state energy can also be written as an integral

E _ h roo dJ..J I [W~ + W!.WD+wm(wZ+l'WD)+WDWZl 0 - 21!"Jo m og w;,(wm+WO) ,

(3.462)

which 5hows that thc encrgy Eo incrcases with thc friction coefficient 'l'It is instructive to calculate the density of states defined in (1.580). Inverting the Laplace transform (l. 579) , we have to evaluate

p(.;:)

1 1"+;00 d{3 eit{i z~"mp({3), 21!"t ,,- ;00

= -.

(3.463)

whcre TJ is an infinitesimally sma.lI posit ive number. In thc a.bsence of friction , t he integral over Z.,({3) = L~=o e - {ihw(n+ l /2j y ields 00

p(,) ~

l: b(, -

(n + 1/2)1iw) .

(3.464)

n=O

In the presence of friction, we expect t he sharp 6-function spikes to be broadened. The calculation is done as follows: The vertical line of integration in the complex {3-planc in (3.463) is moved all the way to the left, thereby picking up the poles

271

3.16 Harmonic Oscillator in Photon Heat B ath

c·

c

~

R.,

Figure 3.5 Poles ill complex p-pJane of Fourier integral (3.463) coming from the Gamma functions of (3.450) of the Gamma funct ions which lie at negative integer values of w;(J/27r. From the representation of the Gamma function l l

f (z)

=

1.

00

dt e - 1e- t

1

1)"

+ L: -+,---,:,---, n=O n !(z + n) 00

(

(3.465)

we see the size of the residues. Thus we obtain the sum (3.466)

R,. _ w (_ I)n- 1 f (- nwdwdr(- nw3/ wd ,I - w~ (n I)! r ( 'T!WO/WI)

,

(3.467)

with analogous expressions for R..,2 and R..,3' T he sum can be done numerically and yields the curves shown in Fig. 3.6 for typical underdamped and overdamped situations. T here is an isolated a-function at the ground stat e energy Eo of (3.459) which is not widened by t he friction. Right above Eo , the curve continues from a finite value p( Eo + 0) = '""(1r /&i determined by the first expansion term in (3.458).

3.16

Harmonic Oscillator in Photon Heat Bath

It is straightforward to extend this result to a photon bath where the spectral density is given by (3.441) and (3.468) becomes

Z;~, ~ lOibid., Formula 8.327. 11 ibid., Formula 8.3l4.

kBT Iiw

{IT [w~ +w:wm+w::.0j }-' m= l

(3,468)

272

3 External Sources, Correlations, and Perturbation Theory

,

3'

,

pet ) 2.5

3.'

,

p(e) 2_5

,

,.,

",

I - ----

0.'

0.'

0,LJec-o,e-'3--., -"C-"C-',--;s tjw O,L, _,\--;,e-., -',c-', ---;''--'8 € /w

Figure 3.6 Density of states for weak and strong damping in natural units. On the left, t he parameters are ,,(/w = 0.2, WD / W = 10, on the right 'Y/w = 5, wo /w = 10. For more details see Hanke and Zwerger in Notes and References.

with"( = e 2 j621rM . The power of Wm accompanying the friction constant is increased by two units. Adding a Drude correction for the high-frequency behavior we replace I by wm/(Wm + wo) and obtain instead of (3.445)

zdamp = kBT

'"

Iiw

fi: ma l

W~(wm+ wD)(l+'YwD) ~(1

.

+ ')'WD) + W~WD + wm w 2 + WDW 2

(3.469)

The resulting partition function has again t he form (3.450), except that W123 are t he solutions of the cubic equation

(3.470) Since the electromagnetic coupling is small, we can solve this equation to lowest order in -y. If we also assume Wv to be large compared to w, we find t he roots

where we have introduced an effective friction constant of the photon bath ~ff 1'pb

=

e2 '). 6c27rM w ,

(3.472)

which has the dimension of a frequency, just as the usual friction constant l' in the previous heat bath equations (3.451) .

3.17

Perturbation Expansion of Anharmonic Systems

If a harmonic system is disturbed by an additional anharmonic potential V(x ), to be called interaction, the path integral can be solved exactly only in exceptional cases. These will be treated in Chapters 8, 13, and 14. For sufficiently smoot h and small potent ials V(x) , it is always possible to expa nd the full partition in powers of t he interaction strength. The result is the so-called perturbation series. Unfortunately, it only renders reliable numerical results for very small V (x) since, as we shall prove

3.17 Perturbation Expansion of Anharmonic Systems

273

in Chapter 17, the expansion coefficients grow for large orders k like k!, making t he series strongly divergent. The can only be used for extremely small perturbations. Such expansions arc called asymptotic (more in Subsection 17.10.1). For this reason we arc forced to develop a more powerful technique of studying anharmonic systems in Chapter 5. It combines the perturbation series with a variational approach and will yield very accurate energy levels up to arbitrarily large interaction strengths. It is t herefore worthwhile to find the formal expansion in spite of its divergence. Consider the quantum-mechanical amplitude

(Xbtblxata)

=/.%(!bl "'%b Vxexp { ~Ii. ltbt~ dt [M2 :t2 _ Mw'l2 x %(t~l "'%~

2-

V(X)] }

(3.473)

(3.475)

A similar expansion can be given for the Euclidean path integral of a partition function (3.476)

(3.477)

The individual terms arc obviously expectation values of powers of the Euclidean interaction A nte ,

,

== 10" drV(x(r)),

(3.478)

274

3 External Sources, Correlations, and Perturbation Theory

calculated within t he harmonic-oscillator partition function Z",. The expectation values are defined by (3.479)

With t hese, the perturbation series can be written in the form

Z

(1-

=

~(Ain~,c)", + 2!~2{A~,t.c)w - 3!~3{A~nt'Dh" + ... )Z",.

(3.480)

As we shall see immediately, it is preferable to resum the prefactor into an exponential of a series 1=

~(A;n~,")w + 2!~2 (A;~~,c)w - 3!~J (Al

nl ,,,),,,

exp { - *(AiDt,e}w

+ ...

+ 2!~2 (Afnt,,,}w,e - 3!~3 (A~nt.e}w,c + .. .} .

(3.481)

T he expectation values {}",,c are called cumulanls. They are related to the original expectation vaJues by the cumulant expansionY

(A~nt,e}w,c -

(A~nt,e}."'c =

{Afnl, .. )", - (Aint,,,): {[A nt,., - {A nt,e)wfl ) ... , (A~Dt,e)w - 3{A~nt,e)w(Aint,e)w {[A im,., - {Aint,e)...

(3.482)

+ 2{Aint,e)!

(3.483)

f')...,

T he eumulants contribute directly to the free energy F = - (l/,8)logZ. From (3.481) and (3.480) we conclude that the anha rmonic potential V(x) shifts t he free energy of t he harmonic oscillator Fw = (1/,8) log[2sinh (h,8w/2)] by fJ.F =

~ (~(Aillt,C)'" - 2!~2(A~nt,e ) ...,c + 3!~3(A~nt,e)w,c + ... ).

(3.484)

Whereas the original expectation values ( A [;.t,e) ... grow for large ,8 with the nth power of ,8, d ue to contributions of n disconnected diagrams of first order in 9 which are integrated independently over 1" from 0 to A,8, t he cumulants (A~t"e) ... are proportional to ,8, thus ensuring that the free energy F has a fin ite limit, t he ground state energy Eo. In comparison with the ground state energy of the unperturbed harmonic system, the energy Eo is shifted by

!leo = J!.?'. ;, P1 (1h(A;nt,e)w -

1

2

1

3

2!1i2 (Ai"t,e)w,c + 3!IiJ (Ai"t,e)w,c

) + ....

(3.485)

12Note that the subtracted expI'CS8iollll in the second lines of these equations are particularly simple only for the lowest two cumulants given here.

3.17 Perturbation Expansion of Anharmonic Systems

275

There exists a simple functional formula for the perturbation expansion of the partition function in terms of the generating functional Z... [j] of the unperturbed harmonic system. Adding a source term into the action of the path integral (3.476) , we define the generating functional of the interacting theory:

Z[j]

=

f Vx exp { -~ fo" Pdr [~ (X2 +

w

2X2) + V(x) - jX] }.

(3.486)

T he interaction can be brought outside the path integral in the for m

Z[j]

=

e- i

f:

1l

dr V(J/ Jj(r»

Z... [j ]

(3.487)

The interacting partition fu nction is obviously Z~Z[OI ·

(3.488)

Indeed, after inserting on the right-hand side the explicit path integral expression for Z[j ] from (3.233): (3.489)

and expanding the exponential in t he prefactor • Jo r Ail drV(6J6J(r» . e- K

+;;;;:; 1 2.n

i'" i"

=

0

dT,V(ojJj(T,))

0

- ;;;;:;. 1 3.llO

Ii" drV(o!oj(r» i" i"

1- Ii

(3.490)

dT, V(OjOj(T,))

0

dT3V(OjOj(T3))

dT,V(ojo(T,))

0

i"

dT, V(OjO(T.»)

0

+.

the functional derivatives of Z[j] with respect to the source j(r) generate inside the path integral precisely the expansion (3.480) , whose cumulants lead to formula (3.484) for the shift in t he free energy. Before continuing, let us mention that the partition function (3.4 76) can, of course, be viewed as a generating functional for the calculation of the expectation values of the action and its powers. We simply have to form the derivatives with respect to /i -I : (3.491)

For a. harmonic oscillat or where Z is given by (3.242) , this yields (3.492)

The same result is, incidenta.lly, obtained by calculating the expectation value of the action with analytic regularization:

2 (x2(r»).., + w (x2(r»)..,=

I

dw

l

wt2 271"w12+w2+

l,w w' Id"" 2;w12+w2=

The integral vanishes by Veltman's rule (2.506).

271" =0.(3.493)

276

3 External Sources, Correlations, and Perturbation Theory

3_18

Rayle igh-Schrodinger and Brillouin-Wigne r P e rturbation E x p a n sion

T he expectation values in formula (3.484) can be evaluated by means of the socalled Rayleigh-Schriidinger perturbation expansion, also referred to as old-fashioned perturbation expansion. This expansion is particularly useful if the potential V(x) is not a polynomial in x. Examples are V( x) = 6(x) and V(x) = l/x. In these two cases the perturbation expansions can be summed to all orders, as will be shown for the first example in Section 9.5. For the second example the reader is referred to t he literature. 13 We shall explicitly demonstrat e t he procedure for the ground state and the excited energies of an anharmonic oscillator. Later we shall also give expansions for scattering amplitudes. To calculate the free-energy shift ~F in Eq. (3.484) to first order in V(x), we need t he expectation (3.494)

The time evolution amplitude on the right describes the temporal development of the harmonic oscillator locat ed initially at the point x , fro m the imaginary time 0 up to 7]. At the time 7], the state is subject to the interaction depending on its position XI = x(7d with the amplitude V(xd. After that, the state is carried to the final state at the point x by the other time evolution amplitude. To second order we have to calculate the expectation in V(x): 1( ,

"2 A int ,.,

)

_

'"

=

(3.495)

T he integrat ion over 71 is taken only up to 72 since the contribution with 7] > 72 would merely render a factor 2. The explicit evaluation of t he integrals is facilitated by t he spectral expa nsion (2.298). T he t ime evolution amplitude at imaginary times is given in terms of the eigenstatcs l/Jn(X) of the harmonic oscilla tor with the energy En = Tiw(n + 1/ 2): 00

(Xb7b lx"7,,),,,

=

L

1j.on (xb) 1j.>~(x,,) e- £" ("" -T~)/h .

(3.496)

n=O

The same type of expansion exists also for the real-time evolution amplitude. T his leads to the Rayleigh-SchrOdinger perturbation expansion for the energy shifts of all excited states, as we now show. The amplitude can be projected onto the eigenstates of the harmonic oscillator. For this, the two sides are multiplied by the harmonic wave functions 1j.o~(Xb) and llM.J. Goo~rts and J .T. Devr~, J. Math. Phys. 13, 1070 (1 972).

3.18

Rayleigll~SchrOdinger

277

and Brillouin-Wigner Perturbation Expansioll

tP.,(x,,) of quantum number n and integrated over in the expansion

Xb

and x"' respectively, resulting

(3.497)

with the interaction

A;••

1'."dt V(x(t)).

=-

(3.498)

The expectation values arc defined by

where

Z

=

QM,w ,n

e- iw (n+1/Z}{t. - I

Q )

(3.500)

is the projection of the quantum-mechanical partition function of the harmonic oscillator

[see (2.40)] onto the nth excited state. The expectation values are calculated as in (3.494), (3.495) . To first order in V(x), one has (nIA nt ln)w =: - ZQ:"',w ,n

1:"dt Jdxbdx"dxltP~(Xb)(Xbtblxltdw l

x V{Xl)(Xlt 1Ix"ta)wtPn(Xa).

(3.501)

The time evolution amplitude on the right-hand side describes the temporal development of the initial state tPn{x,,) from t he time t" to the time t l , where t he interaction takes place with an amplitude - V(xd. After that, the time evolution amplitude on the left~hand side carries the state to tP~(Xb). To second order in V(x), the expectation value is given by the double integral

(3.502)

As in (3.495) , the integral over t, ends at t z. By analogy with (3.481), we resum the corrections in (3.497) to bring t hem into the exponent: i 1 z i 3 1 + 'h(nIAint ln)w - 2!1i2(n l ~ntln)w - 3!h3 (nIAntln)w + . =

exp {*(nIAint[n)W -

(3.503)

2!~2 (nIA~,t ln)w,e - 3!~3 (nIA~>tln)w,c + ... } .

278

3 External Sources, Correlations, and Perturbation Theory

The cumula nts in the exponent are

(nIAi.. ln). - (nIA'odn)~

(nI Ai.. ln)."

~ (nIlA o, - (nIA,"dn).I'ln).,

(nI AI", ln)."

(3.504)

_ (nIAi.. ln). - 3(nIAio, ln).(nIA,"dn). + 2(n I A"dn)~ (nl[A int - (nIA indn)",]3In)"" (3.505)

=

From (3.503) , we obtain the energy shift of the nth oscillator energy dE.. =

{i

. ih 1 2 j.J:~co tb _ ta h(nIA;nd n )", - 2!hz {nIA ;nt ln)""c -

3!~3(nIA~nt ln)""c + .. .},

(3.506)

which is a generalization of formula (3.485) which was valid only for the ground state energy. At n = 0, the new formula goes over into (3.485), after the usual analytic continuation of the time variable. The cumulants can be evaluated furthe r with the help of the rcal-time version of the spectral expansion (3.496): 00

(Xt,lb lx"ta)", =

L ¢n(Xb)t,b;(xa)e- E"(t6- I,,l /h. i

(3.507)

.. =0

Th first order in V (x), it leads to (3.508)

To second order in V(x) , it yields (3.509)

The right-hand side ean also be written as (3.510)

and bocomes, after the time integrations,

(3 .511)

3.18

Rayleigll~SchrOdinger

and Brillouin-Wigner Perturbation Expansioll

279

As it stands, the sum makes sense only for the Elc #- En -terms. In these, the second term in the curly bracket s can be neglect ed in the limit of large t ime differences tb - ta o T he term with Ek = En must be treated separately by doing the integral directly in (3.51O). This yields (3.512)

so t hat

I ) =- '" 'I( 2 n lA'intn"' L

",,0: ..

m

•. (- ) EV.mV Elhtb ta + VnnVnn m

(t. -2 'oJ' .

(3.513)

n

The same result could have been obtained without the special treatment of the Ek = En -term by introducing artificially an infin itesimal energy difference Ek - En = (; in (3.511), and by expanding the curly brackets in powers of tt> - tao When going over to the cumulants !(nIA?ntln)..,.c according to (3.504) , t he k = nterm is eliminated a nd we obtain (3.514)

For the energy shifts up to second order in V(x), we thus arrive at the simple formula (3.515)

The higher expansion coeffi cients become rapidly complicated. T he correction of third order in V(x) , for example, is (3.516)

For comparison, we recall the well-known formula of Brillouin- Wigner equation 14

ll. En = R...n(En + ll.En),

(3.517)

~here R..n(E) are the diagonal matrix elements (n IR(E) ln) of the level shift operator R(E) which solves the integral equation

R(E)

~ II + II 1- p.

E - H",

R(E)

(3.518)

The operator Pn =: In)(nl is the projection operator onto the state In}. The factors 1 - P ensure that t he sums over the intermediate states exclude the quantum R

14L. Brillouin and E.P. Wigner, J . Ph~. Radium 4 , 1 (1933); M.L. Goldberger and K.M. Watson, Collision Theory , John Wiley & Sons, New York, 1964, pp. 425-430.

280

3 External Sources, Correlations, and Perturbation Theory

number n of the state under consideration. The integral equation is solved by t he series expansion in powers of V:

R(E)

=

V + V 1 - ~" V + if E - H",

Up to the third order in

V, Eq.

I - ~" V 1 - ~" V + .. E - H", E - H",

(3 .519)

(3.5 17) leads to the Brillouin- Wigner perturbation

expansion

Vnk Vkl Vi.. E,)(E E,)

+ ... ,

() 3.520

which is an implicit equation for t:::.E" = E - En. The Brillouin-Wigner equation (3.517) may be converted into a n explicit equa tion for the level shift t::.E,,:

C> E" ~ R."(E") + R."(E")R;."(E")+ [R."(E")R;."(E")' + 1il!"(E")R;;"(E") [ + [R." ( E") R;." (E")' + I il!" (E") R;." (E") R;;" (E") + ,R!" (E" )R;7" ( E") [+ .. . .(3 .521) Inserting (3.520) on the right-hand side, we recover the standard RayleighSchriidinger perturbation expansion of Quantum mechanics, which coincides precisely with the above perturbation expansion of the path integral whose fi rst three terms were given in (3.515) a nd (3.516). Note that starting from t he t hird order, t hc explicit solution (3.52 1) for t he level shift introduces more and morc extra disconnccted terms with respect to t he simple systematics in t he Brillouin-Wigncr expansion (3.520). For arbitrary potentials, the calculation of the matrix elements VnJ; can become quite tedious. A simple technique to find them is presented in Appendix 3A. The calculation of t he energy shifts for the particular interaction V ex) = gx 4 /4 is described in Appendix 3B. Up to order if, t he result is

n;

!:lEn

(2n

+ 1) + ~3(2n2 + 2n + l)a4

(~r 2(34n 3 + 51n2 + 59n + 21)a8~ +

(3 .522)

(~r 4· 3(125n4 + 250n3 + 472n2 + 347n + 111 )a 12 n2~2'

The perturbation series for this as well as arbitrary polynomial potentials can be carried out to high orders via recursion relations for the expansion coefficients. This is done in Appendix 3C.

3 .19

Leve l-Shifts and P ertur b ed W ave Funct io n s from Schrodinger E quation

It is illStructive to rederive the perturbation eXp8l1Sion from ordinary operator SchrOdinger theory. T his derivation provides us a1so with the perturbed eigenstates to any desired order.

3.19 LeveJ-Sllifts and Perturbed Wave FunctioIlS from Schriidinger Equ8tioIl

fl

T he Hamilton ian operator

is split into a free and an interacting part

fl = flo+v. Let

In)

be the eigelllltates of

281

(3.523)

those of fl :

flo and I,,(n»

(3.524) We shall assume that the two sets of states In) and 1,,("» a re orthogonal sets, the first with unit norm, t he latter normalized by scalar products a~») ::::::

Due to t he completeness of t he states

In),

(nl,,(»» = I.

(3.525)

the states 11b(n» can be expanded as

L

11b(n» = In) +

a~)lm),

(3.526)

m#

where

are the components of the interacting states in the free basis. P rojecting the right-hand Schrooinger equation in (3.524) onto (ml a nd using (3.527), we obtain

(3.528) Inserting here (3.526), th is becomes

+ (mW ln) + L

E~m)(l~) and for m =

fl.,

a~"){mWlk) = E(n)a;;;),

(3.529)

""

due to the special normahation (3.525),

E~n)

+ (nIVln) + L a~"){n I V l k) =

E (nJ.

(3.530)

"" ;\Iulti plying th is equation with a;;;) and subtract ing it from (3.529), we eliminate the unknown exact energy E(n) , and obtain a set of coupled algebraic equations for a;;;):

a;;;) =

E(") 1 0 -

e(m)

[(m - a;;;JnIVln) + L ainJ(m -

0

k,ton

a~)nIVlk)l '

(3.531)

where we have introduced the notation {m - a;;;)nl for the combina tion of states (ml- a;;;)(nl , for brevity. This equation can now easily be solved pcrturbatively order by order in powers of t he interaction strength. To count these, we replace V by gV a nd expand a;;;) as well as the energies E(n) in powers of 9 as: ~

a;;;)(g) =

L a~~/( - .q)'

(m#n),

(3.532)

1=[

and ~

E(n) = E~n) - L (- g)'Ei n ).

,-,

(3.533)

282

3 External Sources, Correlations, and Perturbation Theory

Inserting these expansions into (3.530), and equating the coefficients of 9, we immediately find t he perturbation ex pansion of the energy of the nth level

(3.534)

(nIVln),

" ,.)

-

L,.. a k,I _ 1 (n lV lk)

I

".

> l.

(3.535)

T he expansion coefficients a~~ are now determined by inserting t he ansatz (3.532) into (3.531). T his yields '

(mIV ln)

o(n) _

I.I.lldforl> 1: (n) a ... ,I=

(m)

Eo

1

(nl

Eo

[

(n)

-

(3.536)

E'.) E(n)' o - 0

m,1 -

- Om,l_l(nIVln)+

L

(n) (1.1:,1 _ 1 (mlV lk)

10'1"

-

,-,L

I' ~ l

(n) 0 ",,1'

L (lk,l _ l_I,(n lV lk)1. (n)

,

k,. ..

(3.53 7)

Using (3.534) and (3.535), this can be si mplified to

(3.538) Together with (3.534), (3.535), and (3.536), this is a set of recursion relations for the coefficients a (n) ... ,1

and

E(n) . I

T he recursion rela tions allow us to recover the perturbation expansions (3.515) and (3.51 6) for the energy s hift. T he seoond-order result (3.515), for example, follows directly from (3.537) and (3.538), the latter giving

E(n) = ' " a(n)(nIV lk) = ' " (kIVln){nIV lk) . 2

L

k;in

k,t

L

~....

E(k) 0

E (n) 0

(3.539)

If the potential V = V(X) ilia polynomial in X, its matrix elements (n IVlk) are nonzero only for n in a finite neighborhood of k, and the recursion relations consist of finite sums which ean be solved exactly.

3.20

Calculation of Perturbation Series via Feynman Diagrams

The expectation values in formula (3.484) can be evaluated also in anot her way which can be applied to all potent ials which are simple polynomials pf x. T hen t he partition function can be expanded into a sum of integrals associated with certain Feynman diagrams . The procedure is rooted in t he Wick expansion of correlation functions in Sect ion 3.10. To be specific, we assume t he anharmonic potential to have the form (3.540)

The graphical expansion terms to be found will be typical for all so-called cp4. theories of quantum field theory.

283

3.20 Calculation of PerturbatiolJ Series via Feynman Diagrams

To calculate the free energy shift (3.484) to first order in 9, we have to evaluate t he harmonic expectation of A int ,,,. This is written as (A;o~,c}.., =

'4 ior"fJ dT(X 4 (Tn..,·

9

(3.54 1)

T he integrand contains the correlation function

(X(TJ)X(TZ)xh)x(T4))..,

=

G~4J(Th TZ, 1'3 , 1'4)

at identical time arguments. According to the Wick rule (3.302), this can be expanded into the sum of t hree pair terms

G~l(TJ, Tz)G~j(T3' 1'4)

+ G~j(TJ, T3)G~~(T2' 1'4) + G~2j(T!, T4)G~l(T2' 1'3),

where G~l(T, 1") are the periodic Euclidean Green functions of the harmonic oscillator [see (3.301) and (3.248)]. T he expectation (3.541) is therefore equal to the integral

,',( 1', l' )' . '4 ioI"~ dTG..,~

(Ant,c ).., -- 3 9

(3.542)

The right-hand side is pictured by the Feynman diagram

CD 3

Because of its shape this is called a two-loop diagram. In general, a Feynman diagram consists of lines meeting at Pzjints called vertices . A line connecting two points represent-s the Green function G..,21(T}, TZ). A vertex indicates a factor 9/4n and a variable l' to be integrated over the interval (0, h{3) . The present simple diagram has ollly olle point, and the T-arguments of the Green functions coincide. The number underneath counts how often the int egral occurs. It is called the multiplicity of t he diagram. To second order in V(x), the harmonic expcct ation to be evaluated is (3.543)

The integral now contains the correlation fu nction G~lh , . .. ,1'8) with eight time arguments. According to the Wick rule, it decomposes into a sum of 7!! = 105 products of four Green functions G~21(T, 1") . Due to t he coincidence of the time arguments, there are only three different types of contributions to the integral (3.543):

,',( TZ,T2 )G..,. ,',(1'2,1'1 )'G.."TJ,TI ",( ) ( ') (9)' '4 ioI"~ dTZ io['" dTI [72G",2 A int,e'" =

+24G~21(Tz, Td 4 + 9G~1(T2' T2)2G~21h, Td 2].

(3.544)

The integrals are pict ured by the following Feynman diagrams composed of three loops:

284

3 External Sources, Correlations, and Perturbation Theory

coco 24

72

9

They contain two vertices indicating two integration variables 1"1> 1"2. The first two diagrams with the shape of t hree bubbles in a chain and of a wat ermelon, respectively, are connected diagrams, the third is disconnected. When going over to the cumuiant {Afnt,e)w,,;, the d isconnected d iagram is eliminated. To higher orders, the counting becomes increasingly tedious a nd it is worth developing computer-algebraic tochniqucs for this purpose. Figure 3.7 shows t he diagrams for the free-energy shift up to four loops. The cumulants eliminate precisely all disconnected diagrams. This diagram-rearranging property of the logarithm is very general and happens to every order in g, as can be shown with the help of functional differential equations.

~-~(o;p + g) ,. +M

Cf:rD + V' + § 2592

1728

""

+

© )+ m,

Figure 3.7 Perturbation expansion of free energy up to order

rI

(four loops).

The lowest-order term {3Fw containing the free energy of t he harmonic oscillator [,ecrul Eq,. (3. 242) and (2.524)[

Fw

=

1 log ( 2 sinh -Pt~) P 2-

(3.545)

is often represented by the one-loop diagram (3.546)

With it, the graphical expansion in Fig. 3.7 starts more systematically with one loop rather than two. The systematics is, however, not perfect since t he line in t he one-loop diagram does not show that integrand contains a logarithm. In addition, the line is not connected to any vertex. All r-variables in t he diagrams are integrated out. The diagrams have no open lines a nd arc called vacuum diagrams . The calculation of t he diagrams in Fig. 3.7 is simplified with the help of a factorizat ion property: If a diagram consists of two subdiagrams touching each other

285

3.20 Calculation of Perturbatioll Series via Feynman Diagrams

at a single vertex, its Feynman integral factori:t.es into those of the subdiagrams. Thanks t o this property, we only have to evaluate the following integrals (omitting the factors g/41i for each vertex)

0 0

0 g g

~

~

fo"/3dTG~}(T' T)

= h{3a 2,

l"'oloMdT1dT2 G",,(TI,T2) =: 1i{3-1 a:;., l"'olo"'ln"PdTldT2dT3 G""(Tl,T2)G",,(T2,T3)G,,,,(T3,Td (2)

2

0

~

0

4

W

(2)

0

(2)

(2)

- h{3 (~f a~, ~

l"'o ln"PdTldT2 G ,(TI,T2) (2) '"

0

~

4

=: h{3-1 a82, w

1"10"10"/ 3dTldT2dT.~ G",1(TI,T2)G""(T2,T3)G,,,, (T3,Td o 0

0

(2)

(2)

(2)

3

- h{3 (~f a~O,

@

~

1h{11"1oh{1 o 0 0 dT1dT2dT3 0 Z"I} > = Z - lli]OjtTJ) { [Wj(TJ)j(TOJ) = Z - Ili] {Wj {1'l.)j{TOJ)jh)

+ Wj{TOJ) Wj(TJ)] Zli]}

(3.580)

+ (Wj('l)Wj{TOJ)jh) + Wj (TOJ) Wj {1'l.)j{TJ)

+ Wj {n)Wj (1l) Wi (TJ) } Zb] + [Cii)(Td Ci2)(T2, T3) + 2 perm] + C?)(T])C~I)(T2)Cii)(T3)' + Wj {TJ) Wj ('l)j{1l»)

Gi3 ) (TJ, T2, T3)

=

and for the four-point function C(4) (Til " . , 74) =

ci4) (7J, ... , T4) + [C~3) (TI' T2 , TJ) Gi]) (T4) + [c i 2) (TJ, T2) c i 2) (T3, T4) + 2 perm] + [ci2) (TJ, T2) ci1)(T3)Cil )(T4)

+

+ 3 perm]

+ 5 perm]

Gil )(TJ) ... cil)(T4)'

(3.581)

In the pure x 4 -theory there are no odd correlation functions , because of the symmetry of the potential. For t he general correlation function G{n} , t he total number of terms is most easily retrieved by dropping all indices and differentiating with respect to j (the arguments TI, ... , Tn of the currents arc again suppressed) : =

= =

G(4) =

e-W(eW)j = Wi = Gil) e- W (e W ) = W .. + W ·2 = G(2) + C{l}2 e- (e jij = W· ·· + 3W··W· + W·3 = C(3) + 3G(2)C{l) + G{l)3 1JJJ cc e- W (e W ) J]]J .... = Wjjjj + 4WjjjWj + 3Wj / + 6W W/ + W/ jj

W

W

)

JJJCC

11J

C

= G~4)

+ 4Gi3)C~I) + 3Gi +

C

jj

2 )2

6Gi2 )Gil )2

+ G?}4.

(3.582)

All equations follow from the recursion relation G(n) = G~n- I)

+ G(n- i)G~I),

n 2:

2,

(3.583)

if one uses G~n - l ) = Gin) and t he initial relation G( l ) = G~I). By comparing the first four relations with the explicit expressions (3.579)- (3.581) we see that the numerical factors on the right-hand side of (3.582) refer to the permutations of the arguments TI , T2, T3, ... of otherwise equal expressions. Since there is no problem in

294

3 External Sources, Correlations, and Perturbation Theory

reconstructing the explicit permutations we shall henceforth write all composition laws in the short-hand notation (3.582) . The formula (3.582) and its generalization is often referred to as cluster decomposition, or also as the cumulant expansion, of t he correlation functions. We can now prove that the connected correlation functions collect precisely all connected diagrams in t he n-point functions. For this we observe that the decomposition rules can be inverted by repeatedly differentiating both sides of t he equation WIi] = log ZIJ] fWlctionally with respect to the current j; c(1)

=

a(2) _ C(l)C(i) e(3) _ 3C(2)C(1)

+ 2C(I)3

C(4j _ 4G(3)G(i)

+ 12C(2)C(I)2 _

3C(::!)::! _ 6C(I}4.

(3.584)

Each equation follows from t he previous one by one more derivative with respect to j, and by replacing the derivatives on the right-hand side according to the rule

(3.585)

Again the numerical factors imply different permutations of thc arguments and the subscript j denotes functional differentiations with respect to j. Note that Eqs. (3.584) for t he connected correlation functions arc valid for symmetric as well as asymmetric potentials V ex). For symmetric potentials, the equations simplify, since all terms involving G{l) = X = (x) vanish. It is obvious that any connected diagram contained in G(n) must also be contained in Gin), since all the terms added or subtracted in (3.584) are products of Gjn)s, and thus necessarily disconnected. Together with the proof in Section 3.22.1 that the correlation functions Gin) contain only t he connected parts of G("), we can now be sure that Gi") contains precisely the connected parts of G(n).

3.22.3

Functional Generation of Vacuum Diagrams

The functional differential equation (3.570) for Wfj ] contains all information on the connected correlation functions of t he system. However, it does not t ell us anything about the vacuum diagrams of the theory. These are contained in WIO] , which remains an undetermined constant of functional integration of these equations. In order to gain information on the vacuum diagrams, we consider a modificat ion of t he generating functional (3.558), in which we set the external source j equal to zero, but gencralize the source j(r) in (3.558) coupled linearly to x(r) to a bilocal form K(r,r') coupled linearly to x(r) x (r'): ZIK] =

J

Vx(r)e - A.,[z,Kl,

(3.586)

where Ae [x, K ] is the Euclidean action

A.[x,K]

=A.[x]+ A;"· [x]+ ~J dr J

dr'x(r)K(r,r')x(r).

(3.587)

295

3.22 Generating Functiollal of COllllectoo Correlation Fullctions

When forming t he functional derivative with respect to K(7, 7') we obtain the correlation function in the presence of K(7, 7'): ;Z G"'( 7,7') = -2Z - ' [K ] OK(7,T')'

(3.588)

At the end we shaH set K(7,7') = 0, just as previously the source j. When differentiating Z[K] twice, we obtain the four-point function (3.589) As before, we introduce the functional W[K] == 10gZ[K]. Inserting this into (3.588) and (3.589), we find

(3.590) (3.591)

With the same short notat ion as before, we shall use again a subscript K to denote functional differentiation with respect to K, and write (3.592)

From Eq. (3.582) we know that in the absence of a source j and for a symmetric potential, G(4) has the connectedness structure (3.593)

This shows t hat in contrast to W jjjj , the derivative W KK does not directly yield a connected four-point function, but two disconnected parts: (3.594)

the two-point functions being automatically connected for a symmetric potential. More explicitly, (3.594) reads 40'W

OK(7J> 72)oK(73 , 74) = G~4)h, 72, 73, 74)

+ G~2)(7J, 73)G~2)(72' 74) + G~2)h , 74)CF)(72, 73) ' (3.595)

Let us derive functional differential equations for Z [K ) and W[K). By analogy with (3.560) we start out with t he trivial functional differential equation

I

VX x(T)_'- c A.I. 1 (3.600) Go ZK + 3ZKK = 2Z. Inserting Z[K] = e W1KI, this becomes (3.601) It is useful to reconsider the functional W(KJ as a functional W[Go]. JG%K = Gl, a nd t he derivatives of W[KJ become

Then

(3.602)

and (3.601) takes the form A( , , , ) 1 GoWGo + 3" GoWGoGo + 2GoWGo + GoWGo WGo = 2'

(3.603)

This equation is represented diagra mmatically in Fig. 3.9. The zeroth-order solution

Figure 3.9 Diagrammatic reprcscntation of funct ional differential equation (3.603). For t he purpose of finding the multiplicities of the diagrams, it is convenient to reprcscnt here by a vertcx the coupling strength - )"/4! rather than g/4 in Section 3.20.

297

3.22 Generating Functiollal of COllllectoo Correlation Fullctions

to this equation is obtained by setting ,\ = 0:

W (O' (Go] ~

1 ,TIlog(Go).

(3.604)

Explicitly, the right-hand side is equal to the one-loop contribution to the free energy in Eq. (3.546) , apart from a factor -{J. T he corrections are found by iteration. For systematic treatment, we write W [Gol as a sum of a free and an interacting part,

(3.605) insert this into Eq. (3.603) , and find the differential equation fo r the interacting part:

(3.606) This equation is solved it eratively. Setting wint [Gol = 0 in all terms proportional to'\ , we obtain the first-order contribution to wint[Gol:

wint (G ] =3 - ).Gz 4!

o

(3.607)

o'

T his is precisely the contribut ion (3.542) of the two-loop Feynman diagram (apart from the different normalization of g). In order to sec how t he iteration of Eq. (3.606) may be solved systematically, let us ignore for the moment the functional nature of Eq. (3.606) , and treat Go as an ordinary real variable rather than a functional matrix. We expand W [Gol in a Taylor series:

wint [Gol =

f AW (~; )P (GO )2p,

(3.608)

p

p=lP,

.

and find for the expansion coefficients the recursion relation Wp+l = 4 { [2P (2 P - 1)

+ 3(2p)1Wp + ~ ( : )

Solving this with the initial number WI connected vacuum diagrams of pth order:

=

2q Wq x 2(P - q)Wp _ q } . (3.609)

3, we obtain the multiplicities of the

3, 96,9504, 1880064, 616108032, 301093355520, 205062331760640, 185587468924354560, 215430701800551874560, 312052349085504377978880.(3.610) To check these numbers, we go over to ZIG] = eWIGol, and find the expansion:

Z [Gol = exp

[~Tr log Go + ~ ~Wp (~t

D,,'I' (Go]

r

{GO)2P]

[1+ f: ~z, (-;)' (Go)"]. p=!

p.

4.

(3.611)

298

3 External Sources, Correlations, and Perturbation Theory

The expansion coeffi cients zp count the total number of vacuum diagrams of order p. T he exponentiation (3.611) yields zp = (4p - I) !!, which is the correct number of Wick contractions of p interactions X4. In fact , by comparing coefficients in t he two expansions in (3.611 ), we may derive another recursion relation for Wp:

which is fulfilled by the solutions of (3.609). In order to find the associated Fcynman diagrams, we must perform the differentiations in Eq. (3.606) functionally. The numbers Wp become then a sum of diagraJDl;, for which the recursion relation (3.609) reads

WI>+1= 4 [G~d~Z W p+ 3·Cg

ddn W

p+ ~(:) (d~Wq)G~ ' ~CdnWp-q)l ' (3.6 13)

where the differentiation dldn removes one line connecting two vertices in all possible ways. This equation is solved diagrammatically, as shown in Fig. 3.10.

Wp+1= 4[ G~d~~WP

+3

~ /nWp + ~(:)(/n Wq)G~.G~ (ddn Wp-q)]

Figure 3.10 Diagrammatic representation of recursion relation (3.609). A vertex repre-

sents the coupling strength - A. Starting t he iteration with WI = 3 CD , we have dWp/dn = 6 0 and iPw p/ d n 1 = 6 X. Proceeding to order five loops and going back to t he usual vertex notation -.x, we find the vacuum diagrams with their weight factors as shown in Fig. 3.11 . For more t han five loops, the reader is referred to t he paper quoted in Notes and References, and to t he internet address from which Mathematica programs can be downloaded which solve the recursion relations and plot all diagrams of W[O] and the resulting two- and four-point functions.

3.22.4

Correlation Functions from Vacuum Diagrams

The vacuum diagrams contain information on all correlation functions of the thcory. One may rightly say that t he vacuum is the world. The two- and four-point funct ions are given by t he functional derivatives (3.592) of the vacuum functional W[K]. Diagrammatically, a derivative with respect to K corresponds to cutting one line of a vacuum diagram in all possible way:;. T hus, all diagrams of the two.-point function

299

3.22 Generating Functiollal of COllllectoo Correlation Fullctions

II

diagrams and mnlt iplicities

g'

300

g'

~( 1128 ©

i'

g'

~ (62208 0

+ 66296

165888~ +

~(24 §

+

3456@

+

+

qp +

124416

248832

o::xx:o

@

12

ceo

2592

I

)

CXXXJ

1728~

)

g+ ~+ cxxg + ~

+ 491664

+ 248832

+

165888

62208

248832

~

)

Vacuum diagrams up to five loops and their multiplicities. The total numbers to orders gn are 3, 96,9504, 1880064, respectively. In contrast to Fig. 3.10, and to the previous diagrammatic notation in Fig. 3.7, a vertex stands here for -Aj4! for brevity. For more than five loops see the tables on the internet

Figure 3.11

(http ://www .physik.fu- berlin/Nkleinert/b3/programs).

can be derived from such cuts, multiplied by a factor 2. As an example, consider the first-order vacuum diagram of W[K ] in Fig. 3.11. Cutting one line, which is possible in two ways, and recalling that in Fig. 3.11 a vertex stands for - ,\/41 rather than - A, as in the other diagrams, we find

C (Z)

(3.614)

The second equation in (3.592) tells us that all connected contributions to the four-point function C (4) may be obtained by cutting two lines in all combinations, and multiplying the result by a factor 4. As an example, take the second-order vacuum diagrams of W[O] with the proper translation of vertices by a factor 4! , which are (3.6 15)

Cutting two lines in all possible ways yields the following contributions to the connected diagrams of the two-point function: C(4} =

4 x (2.1 . ~ 16

+ 4 . 3 .~) ;()< 48

(3.616)

3 00

3 External Sources, Correlations, and Perturbation Theory

It is also possible to find all diagrams of the four-point function from the vacuum diagrams by forming a derivative of W[O] with respect to t he coupling constant - >., and multiplying the result by a factor 4!. This follows directly from the fact that this differentiation applied to Z[O] yields the correlation function f dr{x 4 }. As an example, take the first diagram of order rT in Table 3.11 (with t he same vertex convention as in Fig. 3.11 ):

W, [O[ ~

4~

©

(3.6 17)

Removing one vertex in the t hree possible ways and multiplying by a factor 4! yields C(4) =

3.22.5

1 4! x 48 3

U .

(3.618)

G e nerating Functional for Vertex FUnctions. Effective Action

Apart from the connectedness structure, the most important step in economizing the calculation of Feynman diagrams consists in the decomposition of higher connected correlation functions into one-particle irreducible vertex functions and one-particle irreducible two-particle correlation functions, from which the full amplitudes can easily be reconstructed. A diagram is called one-particle irreducible if it cannot be decomposed into two disconnected pieces by cutting a single line. There is, in fact , a simple algorithm which supplies us in general with such a decomposition. For t his purpose let us introduce a new generating functional fiX ], to be called thc effective action of t he theory. It is defined via a Lcgendre transform ation of WU ]:

- fiX [ = Wb[ - Wjj.

(3.619)

Here and in the following , we use a short-hand notation for the functional multiplication, Wjj = fd7Wj (7)j(7), which considers fields as vectors with a continuous index 7 . The new variable X is the functional derivative of WU ] with respect to j(T) [cecali (3.569)[, (3.620)

and thus gives t he ground state expectation of the field operator in t he presence of the current j. When rcwriting (3.619) as

- fiX [ = Wb[ - X j,

(3.62 1)

and fUllctionally differentiating this with respect to X, we obtain the equation

fx [X[

~ j.

(3.622)

3.22 Generating Functiollal of COllllectoo Correlation Fullctions

301

This equation shows that t he physical path expectat ion X(7) = (X(7)), where t he ext ernal current is zero, extremizes the effective action:

f x iXi

~ O.

(3.623)

We shall study here only physical systems for which the path expectation value is a constant X(7) =: Xo. Thus we shall not consider systems which possess a timedependent XO(7), although such systems can also be described by x 4-theories by admitting more general types of gradient terms, for instance x({P - k~)2X. The ensuing 7-dependence of XO(7) may be oscillatory. 11 Thus we shall assume a constant

(3.624) which may be zero or non-zero, depending on the phase of the system. Let us now demonstrate that the effective action contains all the information on the proper vertex functions of the theory. These can be found directly from t he functional derivatives:

(3.625) We shall sec that the proper vertex functions are obtained from these functions by a Fouricr transform and a simple removal of an overall factor (211,)DJ (2:::=1Wi) to ensure moment um conservation. The functions r{n) (71, .. . , 7n ) will therefore be called vertex junctions , without t he adjective proper which indicates t he absence of the a-function. In particular, the Fourier transforms of t he vertex functions r (2) (71, 72) and r (1) (71, 72,73 , 74) are related to their proper versions by r (2)(Wl, W;!) = r (4}(Wl,W;!, WJ,W4) =

21rJ (WI 21rJ

+ W;!) r (2}(wd ,

(~ W;) r (4}(Wl,W;!, WJ,W4).

(3.626) (3 .627)

For the functional derivatives (3.625) we shall use t he same short-hand notation as for the functional derivatives (3.569) of WliJ, setting

(3 .628) The argumcnt s 71, .. . ,7n will usually be suppressed. In order to derive relations between the derivatives of the effective action and the connected correlation functions, we first observe t hat the connected one-point function Gil) at a nonzero source j is simply the path expectation X [recall (3.578)]:

(3.629) 17In higher dimensions t here can be crystal- or qua.sicrystal-likc modulations. Sec, for example, H. Kleinert and K. Maki, Fortscl!r. P hys. 29, I (1981) (bttp;f/""".pbyai k.fu-berlin.der kleinert/75). T hill paper was the first to investigate in detail icosahedral quasicrystal line structures discovered later in aluminum.

3 02

3 External Sources, Correlations, and Perturbation Theory

Second , we see that the connected two-point function at a nonzero source j is given by 0(2) = c

C(l) = ]

W . , = oX =

oj

]1

(OJ )-1= r-I oX xx-

(3.630)

T he inverse symbols OIl the right-hand side arc to be understood in the functional sense, i.e., rxi denotes the functional matrix:

8'r

r X(~)X(y) == [JX(T)JX (r')

]-'

(3.631)

which satisfies

(3.632)

Relation (3.630) states that the second derivative of the effective action determines directly the connected correlation function G~:l)(w) of the interacting theory in the presence of t he external source j. Since j is an auxiliary quantity, which eventually be set equal to zero t hus making X equal to X o, the actual physica1 propagator is given by

a("C 1 ~ r xx -'IX = X o . j=O

(3.633)

By Fourier-transforming th b relation and removing a a-function for the overall momentum conservation, the full propagator Gw 2(W) is related to the vertex function r (2)(w), defined in (3.626) by _ _ ("(

Gw 2(W ) = G

_ 1 k) - f(2)(W)'

(3.634)

T he t hird derivative of the generating functional Wli] is obtained by functionally differentiating Wjj in Eq. (3.630) once more with respect to j , and applying t he chain rule: W jjj =

-rx~rxxx ~~

=

-rx~rxxx = _G~2)3 rxxx .

(3.635)

This equation has a simple physical meaning. The third derivative of Wfj] on the left-hand side is the full t hree-point function at a nonzero source j, so that (3.636)

This equation states that the full three-point function arises from a t hird derivative of r[X] by attaching to each derivation a full propagator, apart from a minus sign. We shall express Eq. (3.636) diagrammatically as follows:

3 03

3.22 Generating Functiollal of COllllectoo Correlation Fullctions

where II

,, -1

~

I

·: - C{")

:

'~

2

denotes the connected n-point fu nction, and

n~

1~2

... - - [ XI ... X"

the negative n-point vertex function. For the general analysis of the diagrammatic content of the effective action, we observe that according to Eq. (3.635), the functional derivative of the correlation function G with respect to t he current j satisfies (3.637)

This is pictured diagrammatically as follows: (3.638)

T his equation may be differentiated furt her with respect to j in a diagrammatic way. From the defin ition (3.557) we deduce the trivial re 2. By applying 6j6j repeatedly to the left-hand side of Eq. (3.637), we generate all higher connected correlation functions. On t he right-hand side of (3.637), the chain rule leads to a derivative of all correlation functions G = G12) with respect to j, thereby changing a line into a line with an extra t hree-point vertex as indicated in the diagrammatic equation (3.638). On the ot her hand, t he vertex function rxxx must be differentiated with respect to j. Using the chain rule, we obtain for any n-point vertex function: (3.640)

3 04

3 External Sources, Correlations, and Perturbation Theory

which may be represented diagrammatically as

n

n- l

t 1~2 "rh, .

n+~~ l

1~2

With these diagrammatic rules, we can differentiate (3.635) any number of times, and derive the diagrammatic structure of the connected correlation functions with an arbitrary number of external legs. The result up to n = 5 is shown in Fig. 3.12.

$YMDOLS

COMP OSITION

=i;

:0: -C~.,~

I 14 p'nn )

+ (~19P"m) + ~ Fig ure 3 .12 Diagrammatic differentiations for deriving t ree decomposition of connected correlation functions. The last term in each decomposition yields, after amputation and removal of an overall .,-function of momentum conservation, precisely all one-particle irreducible diagrams.

The diagrams generated in t his way have a t ree-like structure, and for t his reason they are called tree diagrams. The tree decomposition reduces all diagrams to their one-particle irreducible contents.

3.22 Generating Functiollal of COllllectoo Correlation Fullctions

305

The effective action r[X ] can be used to prove an important composition theorem: T he full propagator G can be expressed as a geometric series involving the so-called self-eneryy. Let us decompose the vertex function as f(2)

--

a-' 0 +

rint

xx,

(3.641)

such that the full propagator (3.633) can be rewritten as

G = (1 + Gor~\-

r

1

Go.

(3.642)

Expanding t he denominator, this can also be expressed in the form of an integral equation:

G = Go - Gof'~\:Go + Go f'~\-Gof'~\:Go - ...

(3 .643)

The quantity -f~~ is called the self-energy, commonly denoted by E: (3 .644)

i.e., the self-energy is given by the interacting part of t he second functional derivative of t he effective action, except for t he opposite sign. According to Eq. (3.643) , all diagranls in G can be obtained from a repetit ion of self-energy diagrams connected by a single line. In terms of E, the full propagator reads, according to Eq. (3.642): (3 .645)

This equation can, incidentally, be rewritten in the form of an integral equation for the correlation function G:

G = Go + GoEG. 3.22.6

(3.646)

Ginzburg-Landau Approximation to Generating Functional

Since the vertex functions are the functional derivatives of the effective action [sec (3.625)], we can expand the effective action into a functional Taylor series

r[X]

~ f: ~ n=on.

Jdr, ... dr. r "'(r" ... , r.)X(r,) ... X(r.).

(3.647)

The expansion in the number of loops of the generating functional r[X ] collects systematically the contributions of fluctuations. To zeroth order, all fluctuations are neglected, and the cffective action reduces to the initial action, which is t he mean-field approximation to the effective action. In fact, in the absence of loop diagrams, t he vertex fu nctions contain only the lowest-order terms in r {2) and r (4) :

r&2)(T"T2) r &4)(T"T2 , T3 ,T4) =

M(-a~ +W2)J(Tl-T2),

(3.648)

AJ(Tl - T2)J(TI - T3)J(Tl - T4) .

(3.649)

306

3 External Sources, Correlations, and Perturbation Theory

Inserted into (3.647), this yields the zero-loop approximation to r[X ]: (3.650)

This is precisely the original action functional (3.559). By generalizing X(T) to be a magnetization vector field, X(r) --+ M (x ), which depends on the three-dimensional space variables x rather t han the Euclidean t ime, t he functional (3.650) coincides with the phenomenological energy functional set up by Ginzburg a nd La ndau to describe t he behavior of magnetic materials near the Curie point, which they wrote

os"

(3.65 1)

The use of this functional is also referred to as mean-field theory or mean-field approximation to the full theory.

3 .22.7

Composite Fields

Sometimes it is of interest to study also correlation functions in which two fields

coincide at one point , for instance (3.652)

If multiplied by a facto r Mw 2, the composite operator Mw 2x 2(r)/2 is precisely the frequency term in the action energy functional (3.559) . For this reason one speaks of a frequency insertion, or, since in the Ginzburg- Landau action (3.651) the frequency w is denoted by t he mass symbol m , one speaks of a mass insertion into the correlation function c(n)(rl>' .. , rn). Actually, we shall never make use of the full correlation function (3.652), but only of thc integral over r in (3.652) . This can be obtained directly from the generating functional Zlil of all correlation fu nctions by differentiation with respect to t he square mass in addition to the sourcc tcrms

Z - l () J J zr ~ll jdTG(I,n)(T,T" .. . , T) n - M8w2 Jj(rd ... Jj(rn) lJ j = o ' By going ovcr to the generating functional connected parts:

Wli J,

- __ a___, _ jdTGc(I ,n)(T"T, .. . , T) n M8w2 Jj(rd ISL.O. Landau, J. E.T.P. 7, 627 (1937).

(3 .653)

we obtain in a similar way the

-'-W "ll

... Jj(rn}

lJ j=o'

(3.654)

3.23 Path Integr81 C81culatioll of Effective Action by Loop ExpllJlsion

307

The right-hand side can be rewritten as

J

dT cil ,n) (T, TI, . .. ,Tn) = -

M~2Cin)(TJ,'"

, Tn) .

(3.655)

The connected correlation funct ions Gil ,n)(T, TI, . . . ,Tn ) can be decomposed into tree diagrams consisting of lines and one-particle irreducible vertex functions r (1,n)(T, TI," " Tn). If integrated over T, these are defined from Legendre transform (3.6 19) by a further differentiation with respect to Mw 2: (3.656)

implying the relation

J

dT

3.23

r (I,n)(T, TI,

. . . ,Tn) = - M:w 2 r

Cn) (Tl, .. .

, Tn) .

(3.657)

Path Integral Calculation of Effective Action by Loop Expansion

Path integral;; givc the most direct access to the effective action of a thcory avoiding the cumbersome Legendre transforms. The derivation will proceed diagrammaticruly loop by loop, which will turn out to be organized by the powers of the P lanck constant h. This will now be kept explicit in all formulas. For later applications to quantum mechanics we shall work with real time.

3.23.1

General Formalism

Consider the generating functional of all Green functions ZUJ ~ "w Olf ' ,

(3.658)

where Wlil is the generating functional of all C()nnected Green functions. The vac· uum expectation of the field , the average X (') " (x(t)),

(3.659)

is given by the first functional dcrivative

XI')

~ ;W Ul/;j(t).

(3.660)

This can be inverted to yield j(t) as a functional of X(t): j(t)

~

j [X](t),

(3.661)

which leads to the Legendre transform of Wlil:

r[XJ " W UJ -

Jd'j(t)X('),

(3.662)

308

3 External Sources, Correlations, and Perturbation Theory

where the right-hand side is replaced by (3.661 ). T his is the effective action of the theory. The effective action for time independent X (t) == X defines the effective potential

V "" (X)

=-

-1- fiX ]. (3.663) tb - t" The first functional derivative of the effective action gives back the current 'fiX] . H(.) ~ - J(t).

(3.664)

The generating functional of all connected Green functions can be recovered from the effective action by the inverse Legendre transform

WU] ~ fi X ] +

Jdtj(t)X(t).

(3.665)

We now calculate these quantities from the path integral formula (3.558) for the generating functional Zlil:

Zlil =

J

V x(t)e(iJh){A[a:1 + jd!j(t)z(t) }.

(3.666)

With (3.658) , t his amounts to the pat h integral formula for r[X):

J

V x(t) e(i/h){ A(zl+ f dtj(tlz(tl }.

eHqxl+f dt;(t)X(t)} =

(3 .667)

The action quantum fl is a measure for the size of quantum fluctuat ions. Under many physical circumstances, quantum fluctuations arc small, which makes it desirable to develop a method of evaluating (3.667) as an expansion in powers of Ii.

3.23.2

Mean-Field Approximation

For Ii ----t 0, the path integral over the path x(t) in (3.666) is dominated by the classical solution Xcl(t) which cxt remizes t he exponent

M ix] I ox(t)

_ -j(t)

%=", 0, we perform the fluctuation expansion around the minimum of the potential (3.663) at X = 0, where the two Green functions coincide, both having t he same frequency w: (3.690)

For w2 < 0, however, where the minimum lies at the vector Xn of length (3.680), they are different: (3.691)

Since the curvature of the potential at the minimum in radial direction of X is positive at the minimum, the longitudinal part has now the positive frequency _ 2w 2 . The movement along the valley of the minimum, Oil the other hand, does not increase the energy. For this reason, the transverse part has zero frequency. This feature, observed here in lowest order of the fluct uation expansion, is a very general one, and can be fou nd in the effective action to a ny loop order. In quantum field t heory, there exists a theorem asserting this called Nambu-Coldstone theorem. It states t hat if a quantum field t heory without long-range interactions has a continuous symmetry which is broken by a nonzero expectation value of t he field corresponding to the present X [recall (3 .659)]' then the fl uctuations transverse to it have a zero

3 12

3 External Sources, Correlations, and Perturbation Theory

mass. They are called Nambu-GoldBtone modes or, because of their bosonic nature, Nambu-Goldstone bosoM. T he exclusion of long-range interactions is necessary, since these can mix with the zero-mass modes and make it massive. This happens, for example, in a superconductor where they make the magnetic field massive, giving it a finite penetration depth, the famous Meissner effect. One expresses this pictorially by saying t hat the long-range mode can eat up the Nambu-Goldstone modes and become massive. The same mechanism is used in elementary particle physics to explain t he mass of the W ± and ZO vector bowns as a consequence of having eaten up a would be Nambu-Goldstone boson of an auxiliary Higgs-field theory. In quantum-mechanical systems, however, a nonzero expedation value with t he associated zero frequency mode in the transverse direction is found only as an artifact of perturbation theory. If all fluctuation correction;; are summed, the minimum of the effective potential lies always at the origin. For example, it is well known, that the ground state wave functions of a particle in a double-well potential is symmetric, implying a zero expectation value of the particle position. This symmetry is caused by quantum-mechanical tunneling, a phenomenon which will be discussed in detail in Chapter 17. This phenomenon is of a nonperturbative nature which cannot be described by an effective potential calculated order by order in the fluct uation expansion. Such a potential does, in general, posses a nonzero minimum at some Xo somewhere near the zero-order minimum (3.680). Due to this shortcoming, it is possible to derive the Nambu-Goldstone theorem from the quantum-mechanical effective action in the loop expansion, even t hough the nonzero expectation value Xo assumed in the derivation of the zero-frequency mode does not really exist in quantum mechanics. The derivation will be given in Section 3.24 . The use of the initial action to approximate the effective action neglecting corrections caused by the fl uctuations is referred to as mean-field approximation.

3.23.3

Corrections from Quadratic Fluctuations

In order to find the first /i-correction to the mean-field approximation we expand the action in powers of the fluct uations of the paths around t he classical solution (3.692)

6x(t) " x(t) - Xcl(t),

and perform a perturbation expansion. The quadratic term in 8x(t) is taken to be the free-part icle action, the higher powers in 8x(t) are the interactions. Up to second order in the fl uctuation;; 8x(t), the action is expanded as follows:

Jdtj(t) [Xcl(t) +6x(t)[ A[xcl] + Jdtj(t)Xcl(t) + Jdt { j(t) + 6!1,) I . ~J 6x(t) + Udtde 6x(t) 6X(~:'(t') I..... 6x(t') + ((6x)') .

A [xcl +6x[ +

0

(3.693)

3.23 Path Integr81 C81culatioll of Effective Action by Loop ExpllJlsion

313

The curly bracket mult iplying t he linear terms in the variation ox(t) vanish due to the extremality property of the classical pat h X ci expressed by the equation of motion (3.668). Inserting this expansion into (3.667) , we obtain the approximate expression

ZIi]

~ e(i/lIl {Aj"" I+ j dtj(tl%,,(ll } jVoxexp { ~ jdtdti ox(t) Ii

I

oZA OX(t') } . 8x(,)8x(") " -%01 _

(3.694) We now observe that the fluctuations Ox(t ) will be of average size.,fh due to the /i-denominator in the Fresnel exponent. Thus the fluctuations (Jx)n are of average size .,fhn. The approximate path integral (3.694) is of the Fresnel type and my be integrated to yield e(i/hl {A j%ol l+ j dtj(t)"

t",

) log{2"'in [w(t, - toll /Mw }.

(3.783)

If the boundary conditions are periodic, so that the analytic continuation of the result can be used for quantum statistical calculations, the result is

V'ff(X) ~ V(X) -

(tt>

' ) log{2"in [w(t, - to)/2[}. t"

(3.784)

It is important to keep in mind that a line in the above diagrams contains an infinite series of fu ndamental Feynman diagrams of the original perturbation expansion, as can be seen by expanding the denominators in the propagator go.l> in Eqs. (3 .686)- (3.688) in powers of X 2 . This expansion produces a sum of d iagrams which can be obtained from the loop diagrams in t he expansion of the trace of the logarithm in (3.707) by cutting the loop. If t he potential is a polynomial in X , the effective potentia l at zero temperature can be solved most efficiently t o high loop orders with t he help of recursion relations. T his is shown in detail in Appendix 3C.5 .

3 .24

Nambu- Goldstone Theorem

T he appearance of a zero-frequency mode as a consequence of a nonzero expecta· t ion value X can easily be proved for any continuous symmetry and to all orders in perturbation theory by using the full effective action. To be more specific we consider as before the case of O(N)-symmetry, and perform infinit esimal symmetry transformations on the currents j in the generating functiona l WU]: (3.785)

where Lcd a re the N(N - 1)/2 generators of O(N)·rotations with t he matrix elements (3.786)

and £1> are the infinitesimal angles of the rotations. Under these, the generating functional is asSllmed to be invariant: (3.787)

Expressing the integrand in terms of Legend re-transformed quantities via Eqs. (3.620) a nd (3.622), we obtain (3.788)

3 25

3.24 Nambu-GoldstollC Tlworcm

This expresses t he infinitesimal invariance of the effective action fiX] under infinitesimal rotations

The invariance property (3.788) is called the Wam-Takakashi identity for the functional r[X ]. It can be used to find an infinite set of equally named identities for all vertex functions by forming all r[X ] functional derivatives of r[X ] and setting X equal to the expectation vaJue at the minimum of r[X]. The fi rst derivative of r[X ] gives directly from (3.788) (dropping t he infinitesimal parameter fed)

(3.789)

Denoting the expectation value at the minimum of the effective potential by X , this yields

I ,- ,

J'r[X]

dt Xo,(t) (L",)o" OX (t'),X (t) b

a

I

_ ~ O.

(3.790)

X(/) = X

Now the second derivative is simply the vertex function r(2) (t' , t) which is the functional inverse of the correlation function C(2)(t' , t). The integral over t selects the zero-frequency component of the Fourier transform (3.791)

If we define the Fourier components of r (2)(t' , t) accordingly, we can write (3.790) in Fourier spacc as (3.792)

Inserting the matrix elcments (3.786) of the generators of the rotations, this equation shows that for X =F 0, the fully interacting transverse propagator has to possess a singularity at w' = O. In quantum field theory, this implies the existence of N - 1 massless particles, t he Nambu-Goldstone boson. The conclusion may be drawn only if there are no massless particles in the theory from the outset, which may be "eaten up" by the Nambu-Goldstone boson, as explained earlier in the context of Eq. (3.688) . As mentioned before at the end of Subsection 3.23.1 , the Nambu-Goldstone theorem does not have any consequences for quantum mechanics since fluctuations are too violent to allow for the existence of a nonzero expectation value X . The effective action calculated to a ny finite order in perturbation theory, however, is incapable of reproducing this physical property and does have a nonzero extremum and ensuing transverse zero-frequency modes.

3 26

3.25

3 External Sources, Correlations, and Perturbation Theory

Effective Classica l Potential

T he loop expansion of the effective action f[X ] in (3.768), consisting of the trace of the logarithm (3.702) and the one-particle irreducible diagrams (3.741), (3.781) and the associated effective potential V(X) in Eq. (3.663), can be continued in a straightforward way to imaginary times setting tb - ta -+ - ihi3 to form the Euclidean effective potential r e[X]. For the harmonic oscillator, where the expansion stops after the trace of t he logarithm and the effective pot ential reduces to t he simple expression (3.782), we find the imaginary-time version (3.793)

Since the effective action contains the effect of all fl uctuations, the minimum of the effective potential V(X) should yield directly the full quantum statistical partition functio n of a system: (3.794) z ~ exp(- pV(X) 1 ). moo Inserting the harmonic oscillator expression (3.793) we find indeed the correct result (2.405). For anharmonic systems, we expect t he loop expansion to be able to approximate V(X) rather well to yield a good approximation for the partition function via Eq. (3.794). It is easy to realize that this cannot be true. We have shown in Section 2.9 that for high temperatures, the partition function is given by the integral (,ecall (2.351)) (3.795)

T his integral can in principle be t reated by the same background field method as the path integral, albeit in a much simpler way. We may write x = X + Jx and find a loop expansion for an effective potential. T his expansion evaluated at the extremum will yield a good approximation to the integral (3.795) only if the potential is very close to a harmonic one. For any more complicated shape, the integral at small i3 will cover the entire range of x and can therefore only be evaluated numerically. T hus we can never expect a good result for the partition function of anharmonic systems at high temperatures, if it is calculated from Eq. (3.794). It is easy to find the culprit for this problem. In a one-dimensional system, the correlation functions of the fluctuations around X are given by the correlation function [compare (3.301), (3.248), and (3.687)] (JX(T)Jx(r)) = cgl{x)(T, T') = !Cb2{X),,,(T - T') h 1 coshll(X)((T - ,.' ( - hP/2) M 21l(X) 'inh(Il(X)hpj2)

(T -

,.'1E [0, hPJ,

(3.796)

3.25 Effective Classical Pot.ential

3 27

with the X -dependent frequency given by (3.797)

At equal times

7 = 7' ,

this specifies the square width of the fluctuations OX(7):

( [OX(T)]

')" 1 O(X)"P ~ M20(X) coth 2 .

(3.798)

The point is now that for large temperatures T, t his width grows linearly in T ')

(

[OX(T) ]

kBT

T _ oo

~ MO'.

(3.799)

T he linear behavior follows t he historic Dulong-Petit law for the classical fl uctuation width of a harmonic oscillator [compare with the Dulong-Petit law (2.601) for t he thermodynamic quantities]. It is a dircct consequence of the equiparlition theorem for purely thermal fluctuations, according to which the potential energy has an average keTj2: (3.800)

If we consider the spectral representation (3.245) of the correlation function ,

~ 1 e- ;"'",("- 1"') G'O',c (7 _ 7 ') -_ ~ liP L 2 , fJ m ,,",_oow + !v

(3.801 )

m

we see that the linear growth is entirely due to t erm with zero Matsubara frequency. The important observation is now that if we remove this zero frequency term from the correlation function and form t he subtmcted correlation function [recall (3.250)] (3.802)

we see that the subtracted square width (3.803) decrease for large T . This is shown in Fig. 3.14. Due to this decrease, there exists a method to substantially improve perturbation expansions with the help of the so-called effective classical potential.

3.25.1

Effective Class ical Boltzmann Factor

T he above considerations lead us to the conclusion that a useful approximation for partition function can. be obtained only by expanding the path integral in powers of

328

3 External Sources, Correlations, and Perturbation Theory

2

unrestricted ,/ " fluctuation width :::V'"

1. 5

,/

\

'"

".- '/' ' and yields {,OO

1

2sinh(,81i.w/2)

dA

L;oo 21Ti exp

(2Mw2{3 A')

1

=

1

2sinh(.8hwj2) lc(hfJ) wn/3.

(3.819)

Inserting t his into (3.816) we obtain the local Boltzmann factor

T he final int egral oyer Xo in (3.805) reproduces the correct partition function (2.407) of the harmonic oscillator.

3.25.2

Effective Class ical Hamiltonian

It is easy to generalize t he expression (3.813) to phase space, where we define t he effective classical Hamiltonian Heffcl(po,xo) a nd the associated Boltzmann factor B(Pu, xo) by t he path int egral

B(po, xo) == exp [_/3H effd (po,XO)] ==

f vx f ~:n J(xo - x)27rM(po - p)

e-A.[p,z)/h,

(3.821) where x = fo"PdTX(T)/ft/3 and p = fo"PdTp(r)/ft/3 are the temporal averages of position and momentum, and A[P, xl is the Euclidean action in phase space

A,fp,x] ~

10r" dT ]-;p(T);'(T) + Hfp(T) ,x(T))].

(3.822)

The full quantum-mechanical partition function is obtained from t he classicallooking expression [recall (2.344)] (3.823)

3.25 Effective Classical Pot.ential

331

The definition is such that in the classical limit, Hclfcl(Pu, xo)) becomes the ordinary Hamiltonian H(Pu,xo) . For a harmonic oscillator, the effective classical Ha miltonian ca n be directly deduced from Eq. (3.820) by "undoing" the Pu-integration :

B ,- x ) == e-H~" e'l.Po,z")lkHT w\Pl»

0

=

I (hP) f3tiw/ 2 e-P~/2M+Mw~zl) e sinh(f3tiw/2) .

(3.824)

Indeed , insert ing this into (3.823), we recover t he harmonic partition function (2.407) . Consider a particle in t hree dimensions moving in a constant magnetic fie ld B along t he z-axis. For the sake of generality, we allow for an additional harmonic oscillator centcred at t he origin with frequencies wil in z-direction and W.L in t he xy-plane (as in Section 2.19). It is t hen easy to calculate the effective classical Boltzmann factor for the Ha miltonian [recall (2 .687)] (3.825)

where lA p, x ) is t he z-componcnt of the angular momentum defined in Eq. (2.645). We have shifted the center of momentum integration to Po, for later convenience (see Subsection 5.11.2) . Thc vector x.L = (x, y) denotes t he orthogonal part of x. As in t he generalized magnetic field action (2.687), we have chosen different frequenc ies in front of t he harmonic oscillator potential and of the term proportional to lz , for generality. The effective classical Boltzmann factor follows immediately from (2.701 ) by "undoing" the momentum integrations in PZ ,PII' a.nd using (3.824) for t he motion in the z-direction:

B(Po Xo )=e-PH."e'(po,X(I )=l3(hf3) hf3w+/ 2 hf3w_/ 2 hf3wlI/2 e - PIl( pO,X(I ) , e sinh hf3w+/ 2 sinh hf3w_/ 2 sinh hf3wn/2 ' (3.826) where w± == W8 ± W.L, as in (2.697) . As in Eq. (3.820), the restrictions of the path integrals over x a nd p to the fixed averages Xo = x and Po = P give rise to the extra numerators in comparison to (2.701 ).

3.25_3

High- and Low-Temperature B e h avior

We have remarked before in Eq. (3.809) that in the limit T -+ 00, the effective classical potential Veffcl(XO) converges by construction against the initial potential V (xo) . There exists, in fact , a well-defined power series in tiw/kBT which describes t his approach. Let us study this limit explicitly for the effective classical potential of the harmonic oscillator calculated in (3.820), after rewriting it as (3.827)

332

3 External Sources, Correlations, and Perturbation Theory

Due to the subtracted logarithm of w in t he brackets, the effective classical potential has a power series M

effel

Vw

(xo)

=

TW

2 '2

[

1 fIw

1

( fiw )' + . .. 1.

Xo + fUJJ 24 kBT - 2880 k8 T

(3.828)

This pleasant high-temperature behavior is in contrast to that of the effective tential which reads for the harmonic oscillator

p0-

(3.829)

as we can sec from {3.793} . The logarithm of w prevents this from having a power series expansion in hw/k8T, reflecting the increasing width of the unsubtractcd fluct uations. Consider now the opposite limit T ..... 0, where the fi nal integral over the Boltzmann factor 8(xo) can be calculated exactly by the saddle-point method. In this limit, the effective classical potential Vcffcl(XO} coincides with the Euclidean version of the effective potential: VefFcl{XO)T-=O Veff{xo) =:

r c[Xl/,6lx=zo'

(3.830)

whose real-time definition was given in &t. (3.663). Let us study this limit again explicitly for the harmonic oscillator, where it becomes off" ) T_O 1i4J M 2 2 flW (3.83 1) VW (XO - + - w Xo - k8 T log k T' 2 2 B i.e., the additional constant tends to 1i4J/2. This is just the quantum-mechanical zero-point energy which guarantees the correct low-temperature limit

(3.832)

The limiting partition function is equal to t he Boltzmann factor with the zero-point energy tlW/2.

3.25.4

A lternative Candidate for Effective C lassica l Potential

It is instructive to compare this potential with a related expression which can be defined in terms of the partition function density defined in &t. (2.330): (3.833)

3.25 Effective Classical Pot.ential

333

This quantity shares with V:ffd(XO) the property that it also yields the partition function by forming the integral [compare (2.329)1: (3.834)

It may therefore be considered as an alternative candida te for an effective classical potential. For the harmonic oscillator, we find from Eq. (2.331) the explicit form

2"'" "'"

[ (

1

. Mw "'" x 2 .(3.835) V:ffd(x)= _ kaT __ log __ + _ + kBT log l _ e- 2"'" I kilT ) + -tanh-2 kBT 2 Ii kBT

This shares with the effective potential vcft"(X ) in Eq. (3.829) t he unpleasant property of possessing no power series representation in the high-temperature limit. The low-temperature limit of V:ffcl(x) looks at first sight quite similar to (3.83 1): -efFd( x, ) _ T-o-ruu kBT 2r"" + k aT -MW:2 x -- Iog V

n.

2

2

kBT'

(3.836)

and the integration leads to t he same result (3.832) in only a slightly different way:

,

- ""'/ 2k n T

J-kBT 21iw dx -Mw"~ - [ . ---, - 00

I"

lo (Ii~)

= e- ""'/ 2k nT .

(3.837)

There is, however, an important difference of (3.836) with respect to (3.831). The width of a local Boltzmann factor formed from the partition function density (2.330): (3 .838)

is much wider than that of the effective classical Boltzmann factor 8(xo) = Whereas 8(xo) has a finite width for T -+ 0, t he Boltzmann factor 8 (x) has a width growing to infinity in t his limit. Thus t he integral over x in (3.837) converges much more slowly t han t hat over Xo in (3.832) . This is t he principal reason for introducing Vclfd(xo) as an effective classical potential rather than Veffd(XO) .

e - v ",," . , the linear

term in >. yields, a fter a quadratic

(3.843) Combined with the second exponential in (3.84 1) t his leads to a generating functional for t he subtracted correlation functions (3.839):

~,",,/2 - pM_' .!/, exp 1_1 _ r"d r'P ,·( )G" ( _ ').( 'i ). Zw""ILJ - sin ({3hwj2) e 12MIi 10 T Jo dT J T w2,e T T J T (3.844)

For j(T)

3.25.6

== 0, this reduces to the local Boltl'.mann factor (3.820). P e rturbation E x pans ion

We can now apply the perturbation expansion (3.480) to the path integra1 over 1](r) in Eq. (3.813) for the effective classical Boltzmann factor 8(xo) . We take the action (3.845)

and rewrite it as (3.846)

3.25 Effective Classical Pot.ential

3 35

with an unperturbed action

O'(x,) " V"(x,)/M, and an interaction

Ant,,, [xO;1]]

=

I"

Jo

.

(3.847)

(3.848)

drvmt(xo ;1](r»,

containing the subtracted potential .

V'"'(x", "(r))

~

1

,

V(x, + "(r)) - V(x,) - V'(x,)ry(r) - 2V"(x,)" (r).

(3.849)

This has a. Taylor expansion starting with the cubic term

vint(xo; 1]) =

~ VIlt(xo)1]J + ~V(4) (XO)1]4 + .

(3.850)

Since 1](r) has a zero temporal average, the linear term ft fJ dr V'(xo)1](r) is absent in (3.847) . The effect ive classical Boltzmann fact or 8(xo) in (3.813) has then the perturbation expansion [compare (3.480)] '. B(xo) = ( 1 - h1 (Ant,")n

' (Ant, ' ).." + 2!fi? n -

' (A ';nt),e " 311i3 n

+ ... ) Bn(xo)·

(3.851)

The harmonic expect ation values are defined with respect to t he harmonic path integral (3.852)

For an arbitrary functional F [x] one has to calculate (3.853)

Some calculations of local expectation values are conveniently done with the explicit Fourier components of the path integral. Recalling (3.806) and expanding the action (3.814) in its Fourier components using (3.804), they are given by the product of integrals (3.854)

T his implies t he correlation functions for the Fourier components (3.855)

336

3 External Sources, Correlations, and Perturbation Theory

From these we can calculate once more the correlation fu nctions of the fluctuat ions 1](T) as follows:

("(T)"(r»;: ~ Performing the sum gives once more the subtracted correlation function Eq. (3.839), whose generating functiona.l was calculated in (3.844). The calculation of the harmonic averages in (3.851) leads to a similar loop expansion as for the effective potential in Subsection 3.23.6 using the background field method. The path average Xo takes over the role of the background X and the nonzero Matsubara frequency part of the paths 1](7) corresponds to the fluctuatio ns. The only difference with respect to the earlier calculations is that the correlation functions of 1](7) contain no zero-frequency contribution. T hus they are obtained from the subtracted Green functions GPo;,"'0 ),c(T) defined in Eq. (3.802). All Feynman diagrams in the loop expansion are one-particle irreducible, just as in t he loop expansion of the effective potential. T he reducible diagrams are absent since there is no linear term in the interaction (3.850). This trivial absence is an advantage with respect to the somewhat involved proof required for the effective action in Subsection 3.23.6. The diagrams in the two expansions are therefore precisely the same and can be read off from Eqs. (3.741) and (3.781). The only difference lies in the replacement X -+ Xo in the analytic expressions for the lines and vertices. In addition , there is the final integral over Xo to obtain the partition function Z in Eq. (3.808). This is in contrast to the partition function expressed in terms of the effective potential Veff(X ), where only t he extremum has to be taken.

3.25.7

Effective Pote ntia l a nd Magne tization Curves

The effective classical potential Veffd(XO) in the Boltzmann factor (3.807) allows US to estimate the effective potential defined in Eq. (3.663) . It can be derived from the generating functional Zlij restricted to time-independent external source j(T) =: j, in which case Zlij reduces to a mere function of j: (3.857)

where i is the path average of X(T). The function Z(j) is obtained from the effective classical potential by a simple integral over Xo:

Z( .)

(00

dxo

J = 1-00 ..j27r{3e

- ,8[v'''"

(3.894)

Th evaluate this we shall always change, as in (2.750), the t ime variables t l ,2 to length variables .1'1,2 =' Pl,2t/M along the direction of p. For spherically symmetric potentials V Cr) with r ;;; Ixl = ~, we may express the deriva tives parallel a.nd orthogonal to the incoming particle momentum p as follows:

V yV =zV'/r,

(3.895)

V l.V = bV' /r.

(3.896) The part of the integrand before the bracket is obviously symmetric under z ..... - z and under t he exchange ZI ..... Z2. For this reason we can rewrite (3.897) Now we use the relations (3.895) in the opposite direction as

zV'/r = 8.V, and performing a partial integration in

Zl

bV'jr = Ov..ers i;4(zdx4(z:d' .. X4(Zn) of the operator x(z) = (atz + az - I ) between the excited oscillator states (nl a.nd In). Here a and at are the usual creation and annihilation opera tors of t he harmonic oscillator, and In) = (at)n IO) /v'ni . To evaluate these expectations, we make repeated use of the commutation rules [a, all = 1 and of the ground sta.te property alO) = O. For n = 0 this gives (x 4 (z»", = 3,

(X 4(ZI)X 4(Z2»'" = 72z ;-2z1 + 24z;-4Z~ + 9, (X 4(zJ)X 4(Z2)X 4(Z3» ... = 27· 8 Z;-2z~ + 63· 32z;-2z22z~ + 351 . 8Z;-2Z~ + 9 . 8Z ;-44 + 63 . 32z;-4z~z~ + 369 . 8Z ;-4 z1 +27· 8Z;2Z~ + 9· 8z24Z~ + 27.

(3B.l)

T he cumulants a.re

(X 4(ZI)X 4(Z2)}""c = 72z ;-2z~ + 24z 14z~, (X 4(ZJ)X 4(Z2)X 4(Z3» ....C= 288(7z;-2Z;2Z1 + 9Z ;-2Z~ + 7Z;-4 4z~ + IOz;-4 z~).

(3B.2)

T he powers of z show by how many steps t he intermediate states have been excited. They determine the energy denominators in the formulas (3.515) and (3.516). Apart from a factor (g/4)" a nd a

352

3 External Sources, Correlations, and Perturbation Theory

factor 1 /(2w)~" which carries the correct length scaJe of xC:) , the energy shifts 6.E = A 1Eo t::. 2 Eo + 6. 3 £0 are thus found to be given by

6. 1 £0 = 3, 1l2Eo = -

(72. ~ + 24·

6.3 £0=288(7.

n'

+

(3B.3)

~ . ~ + 9 .~ . ~ + 7' ~. ~+ 10· ~.~) =333·4.

Between excited stales, the calculation is somewhat more tedious and yields {x4(Z)}", = fu,2 + On + 3, 4 4 (x (zdx (.zz»""c = (16'1 4 + 96,,3 + 212n2 + 204n + 72)z12z~ + (n 4 + 1On3 +35n2 + 5Ctn+24)z [ 44 + (n 4 _ 6n 3 +11112 _ 6n)::z24 + (16,,4 _ 32,,3 + 2(}n2 _ 4 n)z~Z;2,

[(16n6 + 24On~

(x4(zdx4(Z2lx4(za)) ...o

(3B.4)

(3B.5)

+ 1444n4 + 4440113 + 7324n 2 + 612011 + 2016)

+

4 X ( zl 2 zi z z~ +%1 z~z~) 4 2 (384,,5 + 288On + 8544,,3 + 12528n + 9072n + 2592):j'"2:j

+ +

(48n 5 (16n 6

+ GOOn4 + 288On3 + 66OOn 2 + 7152n + 2880)z I 4z~

144n5 + 484n4 - 744 n 3 + 508n2 - l2On)ztZ;2z;2 6 + (_ 48n + 360n4 _ 960n3 + 1080n2 _ 432n):tz;-4 + (16n6 + 48n 5 + 4n 4 _ 72n 3 - 2On 2 + 24n)z?z24z~ + ( _ 384n5 + 96On' - 864n 3 + 336n~ - 48n)zlz; 2 + (16n6 _ 144n5 + 484n' _ 744n 3 + 508n~ - l2On)zf~z;4 + (16n6 + 48n 5 + 4n' - 72n 3 - 2On 2 + 24n)z j'" 2z1z; 2]. -

(3B.6)

From these we obtain the reduced energy shifts:

d 1 E o = 6n 2 + 6n + 3, d 2Eo = _ (16n 4 + 96n 3 + 212n 2 + 204n + 72)· _ (n 4 + 1071 3 + 35n 2 + 50n + 24)· ! _ (n4 _ 6nl

(3B.7)

t

+ ll n 2 _

6n). ~l 3 2 - (I6n' - 32n + 2071 - 4n)· :t = 2· (34n 3 + 51n2 + 59n + 21), d 3EO = [(16n!> + 24071 5 + 1444n' + 444071 3 + 7324n~ + 6120n + 2016)· + (384n 5 + 288071 4 + 8544n 3 + 12528n 2 + 9072n + 2592)· t· + (48n 5 + 600n 4 + 288On 3 + 6600n2 + 7152n + 2880) ·1·1 +(16n6 - 144n5 + 484n4 - 744n 3 + 508n2 - 12071) . ! . ~ + ( _ 48n5 + 36On' - 960n 3 + 108On2 - 432n)· ! ·1 + (16n6 + 48n 5 + 4n4 - 72n3 - 2071 2 + 24n)· t·1 +( -384n 5 + 960n4 _ 864n 3 + 336n 2 - 48n) . ~ . ~ + (16n6 _ 144n5 + 484n4 - 744n3 + 508n 2 - 12071) . ! + (16n6 + 48n 5 + 4n4 - 72n 3 - 2071 2 + 24n)· t· ;!] = 4 ·3· (125n4 + 25On 3 + 472n 2 + 347n + 111).

n ·1+ 1· !)

(3B.8)

t

·1 (3B.9)

Appelldix 3C Recursion RelatiollS for Perturbation Coefficiellts

Appendix 3C

353

Recursion Relations for Perturbation Coefficients of Anharmonic Oscillator

Bender and WnZ4 were the first to solve to high orders recursion relations for the pert urbation coefficients of t he ground state energy of an anharmonic oscillator with a. potential xl/2 + 9x4/4. T heir relations are si milar to Eqs. (3.534), (3.535), and (3.536), but not t he same. Extending t heir method , we derive here a recursion relation for the perturbation coefficients of all energy levels of the anharmonic oscillator in a.ny uumber of dimensions D, where the radial poteutial is 1(1 + D - 2)/2r2 + r2/2 + (g/2)(a4r4 + !l6r6 + ... + a2 qx 2q ), where t he first term is the centrifugal barrier of angular momentu m I in D dimensions. We shall do t his in several steps.

3e.l

One-Dime nsional Interaction

X4

In natural p hysical units with h = l ,w = 1, M = 1, t he SchrOdinger equation to be solved reads (3C.l) At 9 = 0 , this is solved by the harmonic oscillator wave functions (3C.2) wit h p roper normalizat ion constant N", where H,,(x) are t he Hermite polynomial of nth degree

L" h!x".

H.. (x) =

(3C.3)

~o

Generalizing this to the anharmonic case, we solve the SchrOdingcr equation (3C.l) with the power series ansatz

t,l>(n)(x)

e- ,,"/2

f

(_ 9)k $~n)(x),

(3C A )

k=Q 00

E(n)

=

L lEi"J.

(3C.5)

k =O

To make room for derivative sy mbols, the superscript of $ ~n)(x) is now d ropped. Inserting (3C A ) and (3C.5) into (3C.l) a nd equating t he coefficients of equal powers of g, we obtai n the equa t ions

x$~(x) -

n $ k(x) =

~$~(x) -

L• (_ I )k' E!;) $ k_dx) ,

X4 $ k_ I(X) +

(3C.6)

k' s l

where we have inserted the u nperturbed energy

E~nJ = n

+ 1/2,

(3C.7)

and defined $ k(X) :::::: 0 for k < O. The funct ions $ k(X) are anharmon ic versions of the Hermite polynomials. T hey turn ont to be polynomials of (4k + n)th degree: 41:r-,

(3C.1O)

p~O

we see that t he f(!(;ursion starts with (3C.11 )

For levels with an even principal quantum number n, the functions ' = {h'f!"/h~ o 0

for O::; p' ::; n', otherwise.

(3C.23)

The energy expansion coefficients are given in either case by

Bin) = _(_ l)kC~.

(3C.24)

T he solution of the recursion relations proceeds in three steps as follows. Suppose ....'C have p' calculated for some value of k all coefficients Cr~l for an upper index in t he range 1

:s

:s

2(k-1)+n'.

:s :s

In a first step, we find Cr' for 1 pi 2k + n' by solving Eq. (3C.2Q) or (3C.22), starting with p' = 2k + n ' and lowering p' down to pi = n' + I . Note that the knowledge of the coefficients C~ (which determine the yet unknown energies and are contained in t he last tcrm of the recursion rola.tions) is not roquired for p' > n', sinoo t hey arc accompanied by factors which vanisb dne to (3C.23). Next we usc the recursion relation with rf = n' to find equa.tions for the coefficients Cl containl.'([ in the lalSt tcrm. The result is, for even k,

ct

(3C.25) For odd k , the factor (2n' + 1) is replaced hy (2n' + 3). These equations contain once more the coefficients C;' . Finally, we take the recursion relations for II < n', a nd relate the coefficients CI:' - \, ... , C~ to Cr'. Combining t he results ....'C determine from Eq. (3C.24) all expansion coefficients n ) . The relations can easily be extended to interactions which are a.n arbitrary linear combination

Ei

~

V (x) = L a~

..

fnX2n.

(3C.26)

.. =2

A short Mathematica program solving the relations can be downloaded from the internct.z~ The expansion coefficients have the roma.rkahle property of growing, for large order k, like

Ek(.. ) - - + - -1~12(. -1"-3)r(k + n + l/2). 11"

1I"n.

(3C.27)

T his will be shown in Eq. (17.323). Such a factorial growth implies the perturbation expansion to have a zero radius of convergence. The reason for this will be explai ned in Section 17.10. At the expansion point 9 = 0, the energies possess an CS'lCntial singularity. In order to extract meaningful numbers from a Tay lor series expansion around such a singularity, it will be nOCOSlSary to find a oonvergent resummation method. T his will be provided by the variational pert urbation t hoory to be developed in Section 5.14. 2~Sce http://www.physik.fu-berlin/-kleinert/b3/progra.m8 .

356

3 External Sources, Correlations, and Perturbation Theory

3C.2

G e ner a l One-Dimensional Interaction

Consid(lr now an arbirary inwraction which is (lxpandable in a power series

., ~

1)(x) = :L: 9k1)k+2Xk+2.

(3C.28)

Note t hat th(l coupling constant corresponds now to th(l square root of th(l previous one, the IOw3X3 + y21)4:t 4 + . The POwtlJ1; of 9 count the number of loops of the associated Feynman diagrams. Then Eqs. (3C.6) and (3C.I5) become

(p - n)A: =

~(p + 2)(P + I)At 2 -

•

2:)- 1)k'1)k'+2A:: r2 k' .1

•

+L

(- l)k' Ei~) A:_ k.. (3C.30)

k' . d

Th(l expansion oocffici(lnts of the energies are, as before, given by (3C.17) and (3C.IS) for even and odd n, respectively, but the recursion relation (3C.30) has to be solved now in fulL

3C.3

Cumulative Treatment of Interactions

x4

and

x3

T here exists a slightly different recursi ve treatment which we shall illustrate for the simplest mixed interaction potential

(3C.31) Instead of the ansatz (3C.4) we shall now factorize the wave fnnction of the ground state as follows:

(3C.32) i.e., we allow for powers series expansion in the exponent: ~

4>(n l (x) = :L:yk 4>~n>C:t ).

(3C.33)

k=1 \Ve shall find that t his expansion contains fewer terms than in th(l &nder·Wu (lxpansion of t he correction factor in Eq. (3C.4). For completeness, we keep here physical dimensions with explicit constants h,w, M. Inserting (3C.32) into t he SchrOdinger equat ion

[-

2~f :::2 + (~ "/X2 + 91>3X3 + l1)~x')

- E(n)] t/J(n)(x) =

0,

(3C.34)

wtl obtain, aft(lr dropping (lvcrywhere the superscript (n) , th(l differential equation for 4>(n)(x):

(3C.35) where ( denotes th(l correction to the harmonic energy

(3C.36)

Appelldix 3C Recursion RelatiollS for Perturbation Coefficiellts We shall calculate

£

357

as a power series in 9:

(3C.37) From now on we shall consuder only the ground state with n = O. Inserting expans ion (3C.33) into (3C.35), and comparing coefficients, we obtain the infinite set of differential equations for ¢.I:(x): "1

"1 ,\: _1

2~ ¢Z(x) + hwxoP~(x) - 2~ L oP~_I(X)oP;(x) + 05 k ,IV:iX 3 + 05t.2VtX4 =

-

,=,

fA: .

(3C.38)

Assuming that oPA:(x) is a polynomial, "''() can show by induction that it:! degree cannot be greater than k + 2, i.e., 00

oPt (x) =

L

withc!:: );;;; O fOrm > k + 2 ,

c!::) x"',

(3C.39)

",=1

T he lowest terms 4 .1:) have been omitted since they will be determined at the end the normalization of the wave function t{I(x) . Inserting (3C.39) into (3C.38) for k = I , v.'() find ~1) = 0,

£1

(3C.40)

= 0.

For k = 2, we obtain (1) c\

(2

3V4

7vj

(2)

= 0, Co2

r4"l) =

= SM2",4 - 4M",2'

0,

(3C.41)

llv~h"1 3V4h"1 8M3W4 + 4APw1 .

-

(3C.42)

For the higher-order terms we must solve the recursion relations

(m + 2)(m + l)h (t) _ ,_,_ ~ ~I ( 2mM", c"'+1 + 2mM", L L n m

(t ) _

em -

+

2 _

) (I) (.1:_ 1)

n

en

C",H_ .. '

(3C.43)

1=1 .. .. 1

"1

_ _

fA:-

~

"1 .1:- 1

(k) _

MCi

~" (I) (.1: _ 1) 2M L

CICI

(3C.44)

.

/= \

Evaluating t his for k = 3 yields 5v~h

(3)

c,

= -

M 4",7

+

6V:lv 4 h A{3",S'

~3) = 0 , (3)

c4

(3) _

= 0, c5

-

v~

- lOM 2 hw 5

V:lV4

+ 5Mfu.J3 '

f3

= 0,

(3C.45)

and for k = 4: (4)

305v~fl

Co2 = 32M s",9 -

123vjv4f' 8M4w1

21 v~h

+ 8M3",5' (3C.46) (3C.47)

358

3 External Sources, Correlations, and Perturbation Theory

The general form of the coefficients is, now in natural units with h = 1, At = 1,:

(3C.48) (3C.49) T his leads to the recursion relations (t) _ { m

em,)' -

k - lm + 1

{} + ' }m+ l k + 2,

~ (.1:) + 2X (k) + ....!!:..... ~ ( (k _ l) (I) + (.1:_ 1) (I») + -.!... ~'(X) Mw q q M"" L q c1 c1 q rn,., k , I_I

(30 .76) (30.77)

Appelldix 3C Recursion RelatiollS for Perturbation Coefficiellts

1

363

10 _1

_!!:.....'" (I) (10 - 1) . 2M~c I c i

(3C.78)

,.,

The results are Listed in Table 3.3,

3C.6

Inte r actio n r 4 in D -Dime n siona l Radial O scilla t or

It is easy to generalize these relations further to find the perturbation expansions for the eigenvalues of the radial SchrOdinger equation of an anharmonic oscillator in D dimensions

_! D [_~~ 2dr 2 2 r The case 9 =

°

l .!!.. + /(1 + D - 2) dr 2r2

+ ~r2 + !!.r4] 2

4

R.,(r) = E (")R.,(r) .

(3C.79)

will be solved in Soction 9,2, wi t h the energy eigenvalues

E(") = 2n'+I+ D/ 2 = n+D/2,

n = 0, 1,2,3, . .

,1 = 0,1, 2,3,.

(3C.80)

For a fixed princi pal quantum number n = 2n,. + I, t he angular momentnm runs through 1 = 0,2, ... , n for even, and I = 1, 3, . . . , n for odd n . There are (n + 1)(n + 2)/2 degenerate levels. Removing a factor r' from R.,( r ), and defining R.,(r) = r'wn(r), the SchrOdinger equation becomes +D- 1dI294) (.) dr2 - 2 r dr + 2r + 4"r w .. (r) = E w .. (r). ( -21~121

(3C.Sl)

The second term modifies the differential equation (3C.G) to ,

1 2

r k(r) - 2n'k(r ) = - ~ (r)+

(21 + D-1 ) 2r

1 0 , (n ;'(r)+r4k_1 (r)+ '" (_ 1)10 E k, )k_dr) . (3C.82) ~ 10'_1

The extra terms change t he recursion relation (3C.15) into

(p- 2n')A: =

~ [(p+ 2)(P +

1) + (p+ 2)(21 + D _ 1) ]A:+2 + A::~ +

L• (_ 1)10' Ek:')A:_ ,.(3C.83) k

10 ' ", 1

Fbr even n = 2n'

+ 1 with I = 0, 2, 4,. ,. ,n, we

C1 =

normalize the wave functions by setting

(21 + D)oSok ,

(3C.84)

rather than (3C.12), and obtain

2(p' -

n')Cr' = [(2p' + 1)(P' + 1) +(p' + 1)(I+D/2 -

1/2)]Cr'+1 + Cf~-12 -

L• c:' Cr'_k"

(3C.85)

instead of (3C.20). For odd n = 2n' + 1 with I = 1,3,5"." n, the equations analogous to (3C.13) and (3C.22) are

e! = 3(21 + D)6ok

(3C.56)

ru,d

2(P' -

n')er' = [(2p' + 3)(P' + 1)+(p' +3/2)(1+ D/ 2- 1/2)]er'H + er~-;2 -

L• el, er~ k"

(3C.87)

In either case, the expansion coefficients of the energy arc given by E(n ) _ _ k

-

(_1) 10 21+ D + lei

2

21

+D

k'

(3C.88)

364

3C.7

3 External Sources, Correlations, and Perturbation Theory

Interaction

r2q

in D Dimensions

A further extension of the recursion relation applies to interact ions gx 2q /4. Then Eqs. (3C. 20) and (3C.22) arc changtxj in the s(''COnd terms on the right-hand ~ide which bocome Cr' - q. In a first step, these equations are now solved for Cr' for I ::; P ::; qk + Il, starting with pi = qk + Il' and lowering pi down to pi = Il' + L As before, the knowlodge of the coefficients C~ (which determine the yet unknown energies and are contained in the last term of the recursion relations) is not required for pi > Il'. T he second and third steps are completely analogous to the case q = 2. T he same generalization applies to t he D-dimensional case.

3C.8

Polynomial Interaction in D Dimensions

If t he SchrOdinger equation has the general form

~ _~D[ _!2dr2 2 r

l~ dr

+ 1(I +

D - 2)

+ ~ r2

2r2

2

+ ~(a4r4 + a6r6 + ... + a2qx2q)j R,,(r) = E(n)R,,(r),

(3C.89)

we simply have to ruplaoo in t he recursion relations (3C.85) and (3C.87) the second term on t he

right-hand side as follows

2 C,,'- 3 C.,,'- 12 .... a4 C,,'~ _ ) + Clf, ~ _ ) + . . . +ll2q

c,,'-q k- l '

(3C.90)

a nd perform otherwise the same steps as for the potential gr 2q /4 alone.

Appendix 3D

Feynman Integrals for T-/ 0

T he calculation of the Feynman integrals (3.547) can be done straightforwardly with t he help of the symbolic p rogram Mathematica. T he first integral in Eqs. (3.547) is triv ial. The second and forth integrals arc simple, since one overall integration over, say, 1"3 yields merely a factor h{J , d ne 10 translational invariance of the integrand along the 1"-sxis. T he triple integrals can t hen 00 split ~

L"fjlo',fjlo',fjd1")d'l)d73 1(11"1 = h{J

Ll!.flo"8d1")d1"2 /(11") -

= h{J

(10

M d'l)

1"2 1, 1'1) - 731, 173 - 1") 1)

(3D.I)

'1)1, 11"2 1, l1"d )

L~ d1"t/('I) -1"1, '1), 1"1) + 10"8d'l)

1:8

d1"t/(1"1 - '1),'1), 1"J)) ,

10 ensure that the arguments of t he Green fWiction have the same sign in each term. T he lines

represent t he thennal correla tion function

C(2)(r r) = h ooshw ll1" - 1"'1 - h{J/21 , 2M,.., s in h{,..,h{J/2)

(30.2)

W ith the dimension less variable x :::::: ,..,h(j, the result for t he quantities O~L definod in (3.547) in the Feynman d iagrams with L lines and V vertices is

21 coth "2' 1

1

-, -.- , - (x + sinh x), smh ~

(3D.3) (3D.4)

Appelldix 3D Feynman Integrals for T

=I 0

365

1 - ( - 3 cosh -X +2x 2 cosh -X + 3 cosh -3x + 6xsmh . -x) -1 - 64sinh3~ 2 2 2 2 '

(30.5)

_1___\ _

(30 .6)

256 sinh

(6x+8 sinhx+sinh2x),

~

II (

x

x

3x

- - - - - - 40 cosh- + 24 x 2 OO8b -+ 35OO8h 4096 sinh~ 1 2 2 2 + 5 cosh 5; +

72xsinh~ +

1 1 - - - - , - ( - 48 + 32x2 - 3 cosh x 16384 sinh ~

12xsinh ~) ,

(30 .7)

+ 8X2 cosh x

+48 cosh2x + 3 cosh3x + 108xsinhx), 1 x), 24 ~(5+24cosh smh }

(30.8)

x x 3 xcosh"2+9sinh"2 +si nhT

(30.10)

1 1

1

72sinh3~ 1

(

3x)

1

'

. .

(30.11)

2304 -.- . - (lOx + 1048mhx + 5smh2x). smh

(30 .9)

~

For completeness, we have also listed the integrals ot o~, and o~9, corresponding to the three diagram.s

e ,

e ,

(30.12)

respectively, which occur in perturbation expansions with a cubic interaction potential x 3 . T hese will appear in a modified version in Chapter 5. In the low-temperature limit where x = wh{3 __ 00, the x-dependent factors o~L in Eqs. (30.3)(30 .11) converge towards the constants 1/2,

1/4,

3/ 16,

1/32,

5/(8.25 ),

3/(8.26),

1/12,

1/18,5/(9.25 ),

(30. 13)

respectively. From these numbers we deduce t he relations (3.550) and, in addition, ,,~

-.

__ _2 a6 3

'

0 ,,1 - J'

.....

~.'" 9 .

(30.1 4)

In the high- temperature limi t :r: ..... 0, t he Feynman integrals h{3(l/w)V - la~L witb L li ne; and V vertices diverge like {3V( l /{3)L. The first V factors are due to the V-integrals over T, the second are the consequence of t he product of n/2 factors a 2 . T hus, a~L behaves for x ..... 0 like (30.15) Indeed, the x-dependent factors o}L in (30.3) - (30 .11) grow like

a' al a! a! a'",

~ ~ ~ ~ ~

l/x + x/12 + 1/x + x 3/72O, 1/x+x 5/30240 + . .. , 1/x3 + x/ 12O - x 3/378O + x 5 /80640 + ... , 1/x3 + x/240 _ x 3/15120 + x 7 /6652800 +.

366

3 External Sources, Correlations, and Perturbation Theory a~~

~

.! ·1

~ ~

a1?

~

1/x4 + 1/240 + x~ /15120 - x6/4989600 + 701 x 8/34871316480 + ... , 1/x2 + x 2 j240 - x 4 /6048 + ... , l/r? + :r? /720 - x 6/518400 + ... , 1/x3 + x/31lO - x 5/1209600 + 629x9 /261534873600 +.. (30.16)

For the temperature behavior of t hClSC Fcynman integrals soo Fig. 3.16. We have plottod the reduced Fcynman integrals iitL(x) in which the low-temperature behaviors (3.550) a nd (3D.14) have been divided out of at"L.

1k-- --====-==--__ _

1.2

a'

0.8 0.6

0.4 0.2 -1---"'0-0 .1- -0 ""."2- - '0".30---= 0-' .4- -0 "" .5 L(x

Figure 3.16 Plot ofrcduccd Fcynman integrals o.~L(x) as a function of Llx = LknTjhI..J. The integrals (30Al- (3D,I I) arc indicated by decreasing dash-lengths. T he integrals (30.4) and (3D.5) for a~ and by the operation

n" (

"!Mn

8)"

- 8w2

ag can be obtained from the integral (30.3) for a2 n" (

= n!Mn

18)"

- 2w 8w

(30.17)

with n = 1 and n = 2, resp e .... (xcl)/h

1

00

-00

dJx

_ _ _ e ia"{Xd )(Jx]2/2h

=

eia(xc,)/h

.jall(xcl)

V211"itl

.

(4.45)

The right-hand side is t he saddle point approximation to the integral (4.41). The saddle point approximation may be viewed as the coru;cquenee of the classica! limit of the exponential funct ion: e

ia(x)/h

-

h_O

~ J( x -

r::;;;:::-;:. Va"(xcI)

Xci

)

.

(4.46)

Corrections can be calculated perturbatively by expanding t he integral in powers of Ii, lcading to what is called t he saddle point expansion . For this we expand the remaining exponent in powers of ox:

(4.47) and perform t he resulting integrals of the type

dJ __ 1 00

X_._eia"(Xcl)(6x)2/2h(Ox)fI =

-00 J21nli

{

(n - I)!! (. )"1' _ [a"(xcl)](l+fI)/2 tli , n - even, 0, n = odd.

(4.48)

Each factor ox in (4.47) introduces a power .jli/all(xcl). This is the average relative size of t he quantum fluctuations. The increasing powers of h ensure the decreasing

4 Semiclassical Time Evolution Amplitude

3 78

importance of the higher terms for small h. For instance, the fourth-orde r term a(41(xcl)(ox)4j4 ! is accompanied by fi , and the lowest correction amounts to a factor (4) 311 h 1 - ta (Xcl) 4! [a"(xcl)]2

a(31(xcJ)(ox)3j3!

The cubic term

1+

(4.49)

yields a factor

*

(3)

25

1

1

h

(4.50)

(Xcl)] 72 [a"(xcl) j3

Thus we obtain the saddle point expansion to the integral is dx

1

00

- -e

- (>0

>,(.)/"

v21rin

=

ei IPata)""

=

~[-fJ:l;6f)",~A{Xb' x,,; tb - t .. )l I/2ei[A(%~,z~;fb-ta)-PbZ.+p.,O:AlI",(4.163)

where H is the matrix

aZbaZAA(Xblx,,~tb ~AA(XblXa,tb

-

tal ) . tal

The exponent must be evaluated at the extremum with respect to lies at

XI>

(4.164)

and

Xa ,

which

(4. 165)

The exponent contains then the Legendre transform of the action A(Xb, X and p,,: (4. 166)

The inverse Legendre transformation to (4 .165) is (4. 167)

The important observation which greatly simplifies the result is that for a 2 x 2 matrix Hal> with (a, b = 1, 2) , the matrix element -H I 2/detH is equal to H 12 . By writing the matrix H and its inverse as

(4. 168)

we sec that, just as in the Eqs. (2.276) and (2.277):

H

"-

1 _

fJx a _ a'lA(PI»Pa ;t b - tal 8Pb8pa·

- -8Pb -

(4. 169)

As a result, the semiclassical time evolution amplitude in momentum space (4. 163) takes the simple form (4. 170)

In D dimensions, thb kcomes (4.171 )

4.7 $emic1l1SSical Quantum-Mechallical Partition Functioll

395

oc

(4.172)

these results being completely analogous to the x-space expression (4 .125) and (4.127), respectively. As before, the subscripts a and b can be interchanged in the determinant. If we apply these formulas to t he harmonic oscillator wit h a time-dependent frequency, we obtain precisely the amplit ude (2.283) . Thus in t his case, the semiclassical time evolution amplitude (Pbtb lp"t")",, happens to coincide with t he exact one. For a free particle wit h t he action A(Xb,X,,; tb - t,,) = M(Xb - x,,? j 2(tb - t,,) , t he formula (4. 163) cannot be applied since determinant of H vanishes, SO t hat the saddle point approximation is ina pplicable. The formal infinity one obtains when trying to a pply Eq. (4 .163) is a reflection of the 6-function in t he exact expression (2.138), which has no semiclassical approximation. T he Legendre transform of t he action can, however, be calculated correctly and yields via the derivatives Pa = Pb =: P = A(Xb, Xa; tb - t,,) = M(Xb - xa)j2(tb - tal the expression

A(Pb , p,,;tb - t,,)

=

r

-2(tb - t,,) ,

(4.173)

which agrees with the exponent of (2. 138).

4.7

Semiclassical Quantum-Mechanical Partition Function

From the result (4 .103) we can easily derive the qua ntum-mecha nical partition function (1.537) in semiclassical approximation:

ZQM(tb - tal =

JdXa(Xatb lxata)"" JdXaF(xa, Xa; tb - ta)e;A(z~,z~;tb-t,,)//i;. =

(4.174)

Within the semiclassical approximation t he path integral, as the final trace integral may be performed using t he saddle point approximation. At the saddle point one h", ['" in (4. 133)[

i.e" only classical orbits contribute whose momenta are equal at the coinciding endpoints. This restricts the orbits to periodic solutions of the equations of motion. The semiclassical limit selects, among aJl paths wit h x" = Xb, t he paths solving t he equation of motion, ensuring t he continuity of the internal momenta along these paths. The integration in (4.174) enforces the equality of the initial aud final momenta on these paths and permits a continuation of the equations of motion beyond

4 Semiclassical Time Evolution Amplitude

3 96

the final time tb in a periodic fashion , leading to periodic orbits. Along each of these orbits, t he energy E(x" , xo. , tb - tal and t he action A(xa, X a, tb - tal do not depend on t he choice of x" o The phase factor c,AI" in the integral (4.174) is t herefore a constant. T he integral must be performed over a full period between the turning points of each orbit in the fonvard and backward direction. It contains a nontrivial xa-depcndence only in t he fluctuation factor. Thus, (4.174) can be written as

(4.176) For the integration over the fl uctuation factor we use the expression (4. 123) and the equation

(4.177) fo llowing from (4.119) and (4.93), and have

F(Xb,xa;tb Inserting

Xa

=

Xb

tal =

1

[

82 Aj'/'

1

J2nin X(tb)X(t,,) 8t~

(4. 178)

leads to 1

1

[8'Aj'/'

F(xa ,x,,; tb - tal =..j . -:- i!t' 2Jrln Xa /)

(4. 179)

T he action of a periodic path does not depend on Xa,SO t hat the xo-integration in (4. 174) requires only integrating 1/3;" forward and back, which produces the total period: tb- t a=2

{x+dxa~= 2 {x+dx

1._

Xo

1._

M

J2M IE V( x)J

(4.180)

Hence we obtain from (4. 174) :

Z~ QM

(t _

b

t ) = to - ta I8'A Il/2eiA(t6- t..l/h- i". . a

..j2Jrin

&tl

(4. 181)

There is a phase factor e- i ". associated with a Maslov-Morse index /.I = 2, first introduced in the fluct uation factor (2.269). In the present context, this phase factor arises from the fact that when doing the integral (4.176), the periodic orbit passes t hrough the turning points x _ and x+ where the integrand of (4.180) becomes singular, even though the integral remains finite. Near the turning points, the semiclassical approximation breaks down, as discussed in Section 4.1 in the context of the WKB approximation to the SchrOdinger equation. This breakdown required special attention in t he derivation of t he connection formulas relating the wave functions on

4.7 $emic1l1SSical Quantum-Mechallical Partition Functioll

397

one side of the turning points to those on the other side. There, t he breakdown was circumvented by escaping into t he complex x-plane. When going around the singularity in the clockwise sense, the prefactor l /p(x) = 1/ J2M(E _ V(x» ]/2 acquired a phase factor e- i1f / 2 • For a periodic orbit, both turning points had to be encircled producing twice this phase factor, which is precisely the phase e- i 1f given in (4.181). The result (4.181) takes an especially simple form after a Fourier transform action:

(4. 182)

In the semiclassical approximation, the main contribution to the integral at a given energy E comes from the time where tb - ta is equal to the period of the particle orbit with this energy. It is determined as in (4.139) by the extremum of

A(t. - to)

+ (t. - tole.

(4.183)

Thus it satisfies (4.184)

As in (4. 140), the extremum determines the period tb - ta of the orbit with an energy E. It will be denoted by t(E). The second derivative of the exponent is (i/h)8 2A(tb - ta)/&t~. For t his reason, the quadratic correction in the saddle point approximation to the integral over tb cancels the corresponding prefactor in (4. 182) and leads t o t he simple expression (4. 185)

The exponent contains again the eikonal S(E) = A(t) + t(E)E, t he Legendre transform of the action A(t) defined by

S(E)

~ A(t) _ tV~~t),

(4.186)

where the variable t has to be replaced by E(t) = -8A(t)/8t. Via t he inverse Legendre transformation, the derivative 8S(E)/8E = t leads back to

A(t) ~ S(E) _ vS(E) E vE .

(4. 187)

Explicitly, S(E) is given by the integral (4.68):

S(E) ~ 2

1_ 0

10

•

dxp(x) ~ 2

1_ 0

10

•

dx "j2M [E - V(x)[ .

(4. 188)

4 Semiclassical Time Evolution Amplitude

398

Finally, we have to take into account t hat the periodic orbit is repeatedly traversed for an arbitrary number of times. Each period yields a phase factor eiS (E)l h- ;1r. The sum is Z~ (E) QM

is(E)/h

00

~ "t(E),;·,S(E)I' · V(x): p(E;x) =

{I- 1~~ VI/(x ) d~2

-

2:t~ [V'(xW d~3 + ... } Pd(E;x).

(4.254)

Inserting (4.253) and performing the differentiations with respect to V we obtain

E)

P( ;x

1

IM{ [E-V(x) 1 Ii' V"( )3 1 j1/2- 12M x 4 [E_V(X) ]5/2

=rrnV"2

r.' [ '

[2 15 - 24M V (x ) alE

1

V(x)J'/'

} ( + .... 4.255)

Note that t he proceeds in powers of higher gradients of the potential; it is a gradient expansion. The integral over (4 .255) yields a gradient expansion for p(E) [2J. The second term can be integrated by parts which, under the assumption that V(x) vanishes at the boundaries, simply changes the sign of the third term, so that we find

1

1M!

{

p() E ~

dt eil[E- V(Xl l/" [ dt' - ;e" / ~ dE _oo21rh(27riht/M)OnJo 7 e .

(4.282)

Integra.ting over the energy yields

[ .[ _ J :tJoroo Tr logH d _ _

d

D

dt

1

T (2trihtjM)Dl le

- iW(xl! "

.

(4.283)

Deforming the contour of integration by the substi tution t = - iT , "'() arrive at the inu'gTal roprt).. scntation of the Gamma. function (2.496) which rcproduCU'! immediately the result (4.281). T he quantwn corrections are obtained by multiplying this with the prdactor in curly brackets in the expa.nsion (4.258): Trlog if

(4.284) The !!COOnd term can 00 integrated by parUi, which replaces \7 2 V(x) ..... -['i7V(xWdjdV , so that we obtain the gradient expansion

TrlogH = -

M)DI' r{ - D/2) JdDx {h' l + - - [V V(xW - d', +. .} [V(x)]Ol l. (~ 2'1fh 24M dV

(4.285)

The curly brackets can obviously be replaced by

1'12 r(3 - D/ 2) [V V(x)J2 { 1 - 24M f( D/2) [V(x)J3

}

+. .. .

(4.286)

In one dimension and wi th M = 1/2, this amounts to the fonnula

(4.287) It is a useful exercise to rcderivc this with the help of the Gelfand-Yaglom method in Section 2.4. There exists another method for deriving the gradient expansion (4.284). We split V(x) into a coru;tant term V and a small x-depcndent term JV(xl, and rewrite Trlog [-

2~ V 2 + V(X)]

= Trlog [- 2":1 V 2 + V + OV(x)] = Trlog ( -

where 6. v denotcs the functional matrix

h' ) -' 6. v (x,x') = ( - 2M V 2 + V =

2~ V 2 + V) +Trlog (1 + 6. v oV) ,

J

dDp

e;p(X - x'j,A

(211"Iil1) p l/2M + V=6. v (x - x').

(4.288)

(4.289)

T his coincides with the fixed-energy am plitude (i/Ii)(x lx'l£ at E = - V [rocall Eq. (1.344) ]. T he first term in (4.288) is equal to (4.281) if we replace V (x) in that expression by t he constant V , so that ,,",'C may write Tr log if = [Tr log Hlell

V(xj_V

+ Tr log (1 +.6. v 5V) .

(4.290)

413

4.9 Quantum Corrections to ClllSSicai Density of States We now expand the remainder Tr log (1

+ .6. v 6V) = Trfl. v W - ~Tr(fl.V OV)2 +.

(4.291)

and evaluate the expansion terms. The first term is simply (4.292) where dDp 1 (21rh)D p2/2M

J

fl.V(O) =

"

(4.293)

+ V = llv [Trlog H lct!v(x)_ v'

The result of the integra.tion was given in Eq. (1.348). The second term in the rema.inder (4.291) reads explici t ly

- ~Tr We now make

(AVOV)2 =

U8C

-~

J J

dDx' Av(x,x')6V(x')Av{x', x)oV(x).

dDx

(4.294)

of t he operator relation

[/[A), B[ ~ f'(AIiA,B[ - ~f"(AIiA, [A, B[[ + .

(4.295)

to expand

WAy

=

AvW +At [t,ovj +A~ [t ,[t,WI] + .

(4.296)

where t is the operator of the kinetic energy p 2/2M. It commutes with any function I(x) as follows:

[T,I[

[1',[1',111

~

h'

,

- 'M il" 1[ + 2(" 11, " 1,

4~;2 {[(V 2)2 IJ + 4[VV2II· V + 4[ViVdl'ii'iV

(4.297) j ) ,

(4.298)

Inserting this into (4.294), we obtain a first contribution (4.299) The spatial integrals arc performed by going to momentum space, where we derive the general formula

(4.300)

This simpllfies (4.299) to (4.301) We may now combine the non-gradient terms of oV(x) consisting of the first term in (4.290), of (4.292), and of (4.301), and replace in the latter Av(O) according to (4.293), to obtain the first three expansion terms of [Trlog HId with the full x"dependent Vex) in Eq. (4.281).

414

4 Semiclassical Time Evolution Amplitude

The Ilext contribution to (4.294) coming from (4.297) is 4":f

J J J dDx

dDxl

dDx2L\.V(X, XJ)AV(Xl, X2 )Av(X2, X)

X ([V 2W(xl] W(x)

+ 2 [V6V(x)f + 2[V 5V(xl]6V(x)V } ,

(4.302)

where the last V acts on the first x in 6. V (X,X l), due to t he trace. It does not contribu te to the integral since it is odd in x - Xl' We now perform the integrals over Xl and Xz using formula (4.300) and find

(4.303) The first term can be integrated by parts, after which it removC8 half of the second term. A t hird contribution to (4.294) whi.:h contains only the lov.'CSt gradients of 6V(x) comes from

the third term in (4.298);

where the last 'iI;'Vj acts again on we mcn the first x in A v( x , xIl, as a consequence of the t race. in momentum space, we encounter the integral

(4.305) so that the third contribution to (4.294) reads, after an integration by parts,

h'J

- At

I .1)

d D x [V oV(x) ]2~(1"2 D 2"OV+i3Vuy

(4.306)

a v(O).

Combining all gradient terms in [V V(xW and replacing av(O) according to (4.293), we recover the previous result (4.285) with the curly brackets (4.286). For the one-dimensional tracelog, this leads to the formula

Trlog[- /i

a: + V (x)]

2

=

h1

J

c/:J: JV(x)

{ h Z [V'(xW } 1 + 32 V3(X) + ... .

(4.307)

It is a useful exen-:ise to rederive th is with t he help of the Gelfand-Yaglom method in Section 2.4. This expansion ca.n actually be deduced, and carried to much higher order, with the help of the gradient expansion of the t race of the logarithm of the operator - h 2 lP.. + W 2(T) derived in Subsection 2. 15.4. If we replace T by x/Ii and V(T) by V (x), we ohtain from (2.548)- (2.550): 1

h

J

{

V'

, (5V, 2

dx JV(x) 1 -Ii 4V3/2 -Ii

_

V" ) , ( 15V,3 9V'V" 32V3 - 8V2 - Ii 64V9j2 - 32V7/2

,(l105V,4 _ 221V,2V " Ii

2048V~

256V~

19V,,2

+ 128Vt +

V (3) )

+ 16V~j2

7V'V(3) _ V (4) ) } 32Vt 32V3 .

(4.308)

The h-term in the curly brackets vanishes if V (x) is the same at the boundaries, and the h2 term goes over into the til_term in (4.307).

415

4.9 Quantum Corrections to ClllSSicai Density of States

4.9.5

Local D e ns ity of States on Circle

Fbr future usc, let us also calculate t his d

pcl(b, E;x) = --; ~2L - (E) 11"1< 211"hlM .. = 0 p ;x

K 1/2{np(E;x)blh).

(4.31 3)

Inserting K 1/ 2 (z) = J'lr/2ze - ' [recall (2.559)], this becomes M -I PI(b E·x) = c

T he sum

"

1I"hp(E; x)

L::".. , a" is equal to al(1 -

( 1+2 L~

e- np(E,o:)b/h )

n_ 1

.

(4.31 4)

a) , so that we obtain

==th,-.j,;2~M~tEEi; V¥,(x:\,)Jr-b/"2,,h ( b E.x) = M coth[p(E;x)bI2h] = M CO Pel " 'lrh p{E;x) 'lrh J2M[E V ex)]

(4.315)

For b ..... 00, this reduces to t he previous density (4.253). If we include t he h igher powers of Tin (4.252), we ohtain the generalization of expression (4.254): pCb, E;x) = { 1 -

1~~ V" (x) d~2 - 2~l~ [V'(x) ]2 d~3 + ... } Pcl(b, E;x).

(4.31 6)

416

4 Semiclassical Time Evolution Amplitude

T he traoolog is obtained by integrat ing this over dElog E from Vex) to infinity. The integral d iverges, and we must employ analytic regularization. We proceed as in (4. 282), by usi ng t he real-time version of (4.311) a nd rewriting log E as a n integral - fo""(dt' /t')e - · Et '''', so that t he leading term in (4.316) is given by

, --1'

[TrlogH]Ol -

I)

1~ dx

1O..! !. . L

dE

_ 00

_ 00

2 Ii 11"

e - ;n'Mb'/2~'+j'(E - V(")I/~ (2

n~_ ""

'~

11"1 It

/Af)I/2

1"" 0

dt' - i E.' I"

t

J

e

. (4.317)

T he integral over E leads now to

(4.318) Deforming agai n the contour of integration by the substitution t = - iT, creating the denomi nator by a n integration over V(x) /h, we see that

[Tr logHJcl =

=

t dx ( to dVpc\(b,E;x)1 t ("" dV 1 coth 4.,jVJE;

-11"

)0

11" ) 0

lv(:_V(r)_V

dx iv(%) E6411"b

h/V/Eb

'

(4.319)

where E6 :::::: h/2Mb 2 is the energy associated with the length b. The integration over V produces a factor h/ T in the integrand. T hus we obtain

1>logH~ 1,'

,,+ +2:~[V'(X)['

:"

+p }~IO+iOh [~tg) ]}

(4.320)

For M = tI2 / 2, this give us the finite-b correction to formula (4.307). Replacing:r by the Euclidean time T, b by h/3, and V(x) by t he time-dependent square frequency W2(T), we obtain the gradient exparu;ion (4.321)

4.9 .6

Quantum Corrections to Bohr- Sommerfeld Approximation

T hc cxpansion (4.308) ca.n be used to obtain a higher-order expansion of the dcnsity of states peE), thereby extending Eq. (4.256). For this we rco-:a1l Eq. (1.592) according to which we can calculate the exact density of states from the formula 1 peE) = --8E 1m Trlog {-a!

,

+ [V ex) -

E I} .

(4.322)

Integrat ing thi~ over the energy yields, according to Eq. (1.593), t he number of Btatcs t imes a nd t hw; t he simple exact quantization condition for a nondegenerate ono-dimensional system: - h nTrlog {-a! By comparison with Eq. (4.209) eikonal:

"'1)

+ [v{x) -

EI} = 7t(n + 1/ 2).

7r,

(4.323)

may define a fullly quantum corrected version of the classical

Sqc(E) = - 2h lm Trlog {- a!

+ [v(x) - EJ}.

(4.324)

417

4.9 Quantum Corrections to ClllSSicai Density of States

The semiclassical expansion of t his can be obtained from our earlier result (4.308) by replacing V(x) ..... Vex) - E, so that JV(x) ..... -i JE Vex) , yielding Sqc(E) = 2 .[

-h

f

fix JE

V 2 V(X) { 1 + 1i [32(:,2 )J + 8(E "V)2] V

I I05V,4 22IV,2 V" 19V,,2 7V'V(J) V (4) ]) 2048(£ V )6 + 256(£ V)5 + 128(£ V )4 + 32(£ V )4 + 32(£ V )3 +... . (4.325)

The first term in the expansion corresponds to the Bohr-Sommerfeld approximation (4.27), the remain ing ones yield t he quantum corrections. The integrand agrees, of course, with the WKB expansion of the eikonal (4.14) after inserting the expansion terms So, S" . .. , obtained by integrating the relations (4.20) over the funct ions %(X) , qL(X) , . . . of Eq. (4.18). Using Eq. (4.322) we obtain from Sqc(E) the density of states

pee) = + h3

2 I I J I 3V"] -,.-Ii Sqe(E) = -, .-, dx -" -E-_V I\1- h [25V" 32 (E _ V )3 + ~8~(E:;.'--~V~)"

12155V,4 1989 V,2 V" 133V,,2 49V'V(3) 5V (~)] ) . (4.326) [ 2048(E V)6 + 256(E V) ~ + 128(E V )4 + 32(E V)4 + 32(E V)J

Let us calculate the quantum corrections to the semielassical energies for a purely quartic potential Vex) = gx4 /4, where the integral over the first term in (4.325) between the turning points ±Xf; = ±(4£/g)1 /4 gave the Bohr-Sommerfeld approximation (4.33). The integrals of the higher terms in (4.325) are divergent, but can be calculated in analyt ically regularized form using oncc more the integral formula (4.34). This extends t he Bohr·Sommerfeld oquation vee) = n+ 1/2 to the exact equation N(E):= Sqc/27rh = n + 1/2 [recall (1.593)1. If we express NCE) in terms of veE) defined in Eq. (4.33) rather than E, we obtain the ex pansion

N(v) _ v _ _1_ -

12lfv +

117r2

4

I0368ra(~)v3

46977r _ 3900657r + 1866240fSWv6 501645312f I6 (!)v 7

533528937r3 7739670528f 16 )v 9 + . . . = II + 1/2.

(!

(4.327)

The function is plotted in Fig. 4.327 for incre8lling orders in y. Given a solut ion equation, we obtain the energy E(") from Eq. (4.35) wit h

11(")

of t his

For large II, where 11(") ..... II, we recover the Bohr-Sommerfeld result (4.36). We can invert the series (4.327) and obtain v

' )[ 0.026525823 (.. ) _ ( - 11+ 2 1 + (n+~j2

0.002762954

(n+

D4

0.001299177

(n+ 1)6

+ o.oo(314009). 1 + 0.(007594)4,.97 + ... 11+ 1 11+ 1

J.

(4.329)

The results are compared with the exact ones ill Table 4. 1, which approach rapidly the BohrSommerfeld limit 0.688 253 702 ... The approach is illustrated in the right-hand part of Fig. 4.1 where log[II(")/(n + 1)4/ 3 - 1] = log[E(")/Ek1 - 11 is plotted oncc for the exact values and oncc for the semiclllSlSical expansion in Fig. 4. L The second cxeitL', it will be convenient to turn this into a local functionaL This is done as follows. We introduce an auxiliary local field · Let us introduce an auxiliary triplet of spherical coordinates =: (rb, Ob,rPb)' Then we factorize the determinant in (4.M3) as

PbPb, then Sb

We fu rt her calculate

Os.

det 8x., =

fh. 8x. 80. 8x. 8¢. 8x.

8,. 8y. 89. 8y. 8¢. 8y.

8,. 8,. 80. 8,. 8¢. 8,.

(4.544)

Long after the collision, for tb --+ 00, a small change of the starting point along t he trajectory dzo. will not affect the scattering angle. Thus we may approximate t he matrix clements in the third column by tho..o:::: &¢/8zo. ~ O. After the same amount of time the particle will only wi nd up at a slightly more distant 1'b, where drb ~ dzb. Thus we may replace the matrix element in the right upper corner by 1, so that the determinant (4.544) becomes in the limit

(4.545)

where dO. = sin ()bdOd¢b is the element of the solid angle of the emerging trajectories, and dn the area element in the x - y -plane, for which the trajectories arrive in an element of the final solid angle dO. Thus we obtain

&x.]-' ~ r~dO' 1 do [det &X"

(4.546)

The ratio dn / dO is precisely the classical differential cross section of the scattering process. Combining (4.543) and (4.545), we see that the contribution of a n individual trajectory to t he semiclassical amplitude is of the expected form [5]

Jdncl '"" f P.Pa=dOx~

.,is(p •• Pa;Ej/h -iVlf/2 .

...

(4.547)

cI ..... tr&j.

Note that this equation is also valid for some potentials which are not restricted to a finite regime around the origin , such as the Coulomb potentials. In the operator theory of quantum-mechanical scattering processes, such potentials always cause considerable problems since the outgoing wave funct ions remain d istorted even at large distances from the scattering center.

447

4.12 Semiclassical Scattering

Usually, there are only a few trajectories contribut ing t o a process with a given scatt ering angle. If the actions of t hese t rajectories differ by less than Ii, t he semiclassical approxima tion fails since the fluctuation integrals overlap. Examples arc the light scattering causing t he ordinary rainbow in nature, and glory effects seen at night around the moonlight. We now turn to a derivation of the amplitude (4.547) from t he more reliable formula (1.528) for t he interacting wave function

by isolating the factor of eip~rh Irb for large rb, as discussed a t the end of Section 1.16. On the right-hand side we now insert the x-space for m (4. 127) of the semiclassical amplitude, and use (4.87) to write

x ei rs(Xb,x..;E~)+iPGx. - i ""lzl/hl

. x~ =p~t~/M

(4.548)

Now we observe that

o)'/' [det3 (ilpo - th4 )]'/' -_[ det3 (ftxo th4 )]'/' '

-t (M

(4.549)

In Eq, (4 ,546) we have found that this determinant is equal to JdCYldfl.lrb, bringing Eq, (4 ,548) to t he form

(Xb IU[(O,t,,) lp,,)

=

lim .!. Vddcleirs(xp,,,,EG)+iP,,"" - iVlrIZlIhl

t~ __ oo

rb

dfl.

' (4.550)

x~= p.t~/M

For large Xb in the direction of the final momentum Pb, we can rewrite the exponent as [recalling (4, 240)1 (4.551)

so that (4.550) consist s of an outgoing spherical wave function eip"r.lh Iro multiplied by the scattering a mplitude (4.552)

t he same as in (4 .547).

448

4.12.2

4 Semiclassical Time Evolution Amplitude

Semiclassical Cross Section of Mott Scattering

If the scattering particle is distinguishable from the target particles, the extra phase in the semiclassical formula (4.552) docs not change the classical result (4.522). A quantum-mechanical effect becomes visible only if we consider electron-electron scattering, also referred to as Mott scattering . The potential is repulsive, and t he above Coulomb potential holds for the relative motion of the two identical particles in their center-of-mass frame. Moreover, the identity of particles requires us to add the amplitudes for the trajectories going to Pb and to - Pb [see Fig. 4.6]' so that t he differential cross section is (4.553)

The minus sign accounts for the Fermi statistics of the two electrons. For two identical bosons, we have to use a plus sign instead. Now the eikonal function S(p, p , E) enters into the result. According to Eq. (4.525), this is given by

Figure 4.6 Classical trajectories in Coulomb potential plotted in the center-of-mass frame. For identical particie:;, trajectories which merge with a scattering angle () and 7r - () arc indistinguishable. Their amplitudes must be subtracted from each other, yielding the differential cross section (4.553).

Mnoo ~+ 1 S( Pb, Pa; E) = ---p:- log ';1 + 6. _ l ' where Peo

=

(4.554)

v2M E is the impinging momentum at infinite distance, and 6. = (p~ - reo)(Pa'l - p~) r eolpb Palz .

(4.555)

The eikonai function is needed only for momenta Pb, Pa in the asymptotic regime where Pb ,Pa::::: Peo , so that d is small a nd

Mn",

S( Pb , Pa; E) -;;::: - - log 6., Poo

(4.556)

Appelldix 4A Semiclassical Quantization for Pure Power Potentials

449

which may be rewritten as Mfu:a

.

S(p , PaE) ~ 2(10 - - - log(sm 2 8/ 2) , Poo

with

Mt!CD: I [(P~ 0"0 = - 2- - og Poo

- P~)(Pa2 - p~)l ..-.4

'

(4.557)

(4.558)

1'00

and the scattering angle determined by cos8 = [Pb' P,,/PbPaj. The logarithmically diverging constant 0"0 for Pa = PbPoo does, fortunately, not depend on the scattering angle, and is t herefore the semiclassical approximation for t he phase shift at angular momentum l = O. It therefore drops, fortunately, out of the difference of the amplitudes in Eq. (4 .553) . Inserting (4.557) with (4.558) into (4 .552) and (4.553), we obtain the differential cross section for Mott scattering (sec Fig. 4.7 for a plot) dr' (1/;(r)'1j(r'» = , f"> dr , f"> dr'G;j(r, r') = 12hMAr(A r1 Jr. Jr. Jr.

J~.

1:

dr ('1i(r)'1j(r» = l : drGik, r) =

~

[t::.t

- r:] 6;j+

r;)o;j ,

~; t::.x; A Xj.

(48.14) (48.15)

T hus we ootain the se mi classical imaginary-time amplitude (4B.16)

T his agrees precisely with t he real- time amplitude (4. 265). For the partition function at inverse temperature f3 = (7b - r,,)/h, this implies the semiclassical approximation

(4Il.11) A partial integration simplifies this to

(48.18) Actually, it ill easy to calculate all terms in (4Il.16) proport ional to Vex) and its derivatives. Instead of the expansion (48.6), we evalnate

(48.19)

454

4 Semiclassical Time Evolution Amplitude

Dy rewriting V(X + 11 ) as a Fourier integral Vex

+ ,..) =

dDk (211")D V(k) cxp [i k (x + TtlJ,

J

(48.20)

we obtain

(48.2 1)

The expectation value can be ca.lculatcd using Wick's t heorem (3.307) as

f"

~

r'" dTe - k;.!('1;( ~)'1!(~»/2 =

dT(eikrj (~») =

~

f '" dTe - k;k;(A>(~,r)6;J + D'("',"')6"',6"'Jln .

~

(48.22)

where A2(r, T) , B2(T,r) a.re from (48. 12):

A2(r,r) =

M~1Jt::"T - T)T'

(r -

2

t/

B{T,T)=~ .

(48.23)

Inserting the inverse of the Fourier decomposition (48.20), (48.24)

where T] is now a time-independent variable of integration, we find

(48.25) After a quadratic completion of t he exponent, t he momentum integral can be performed and y ields

Using the transverse and longitudinal projection matrices p,L 'J

=

f),x i Llxj

(48.27)

(.6.x?'

satisfying p T2 = pT, pL2 = p T. , we can decompose Gij(r, T') as Gil' ;;; A~(T,r') Pi~

+ [A~(T, T'l + 8~(T, T')(.o.x)~l

Pi~·

(48.28)

It is t hen easy to find the determinant (48.29)

and t he inverse matrix

C- '( )

)

il' T = A2(T,T')

[,s

.o.Xi eu , ] i, -

(.6.xj2

1 .o.x,.o.Xj + A2(T, TI) + D2(T,T')(.6.xj2 (.o.xj2·

(48.30)

Inserting (4B.26) back into (48.21), and taking the correction into the exponent, we arrive at

(X1"(. Ix"Ta )

=

' /2 1"(. I(L.>X

m (X .... l ' / 2 Ta ) e - (I!hl!:·MV. '. ,

L.>X

(48.31)

Notes 8Jld

455

RcfcrellCCS

where v"m('i, '1) is the harmonically smeared potential

V"",(x, '1) == [det G('1, '1)r l /2

J

dD1) V(x

+ 'f) e-(l/2)'1,G;jL(~)'11.

(4B.32)

By expanding V (x + 'f) to second order in 'f) , the exponent in (4B.31 ) becomes (4B.33) According to Eq. (4B.15), we have (4B. 34)

so that we roobtain the first two correction terms in the curly brackets of (48.16) T he calculation of t he higher-order corrections bcoomcs quite tooious. One rewrites the cxpansion (4B.2) as V(X + 'f) (T» ) = e'l'(~)O· V (x) and (4B.19) as (4B.35) Now we apply Wick's rule (3.307) for harmonically fluct uating variables, to re-express (e'l(~)8,) = e (m{"-)'1J(~))/2 = eGjj(r,..-)8(OJ/2, (e'l.(~)O'e'l,(-r')B.) = e[('1'{"I")'1.J(r»B.0J + {'1'{"-')'1.J{..-'»O,8J+2(rn("I")'1J{..-'l)B.8J]1 2

= e [G'J ( ~,~)8,8J + G,.J (~', ~')B.O.J +2G'J ( ~. ~')8,O.J ]/2

(4B.36) Expanding the exponentials and performing the T-integrals in (4B.35) yields all desired higherorder corrections to (4B.16). For Ax = 0, the expansion has been driven to high orders in Ref. [6] (including a minimal interaction with a vector potential).

Notes and References For the eikonal expansion, see the original works by G. Wentzel, Z. Physik 38, 518 (1926); H.A. Kramers, Z. Physik 39, 828 (1926); L. Brillouin, C. R. Acad. Sci. Paris 183, 24 (1926); V.P. Mas!ov and M.V. Fcdoriuk, SemidMsirol A pproximation in Quantum Mechanics, Reidcl , Dordn,'Cht, 1982; J.B. Delos, Semida&ical Calculation of Quantum Mechanical Wave FUnctions, Adv. Chern. P hys. 65, 161 ( 1986); M.V. Berry and K.E. Mount, SemidMsirol Wave Mechanics, Rep. Prog. Phys. 35,315 (1 972); and the references quotoo in the footnotes. For the semiclassical exp8.l18ion of path integrals see R. Dashen, B. Hasslacher and A. Neveu, P hys. Rev. D 10, 4114, 4130 (1974) , R. Rajarama.n, Phys. Rep. 21C, 227 (1975); S. Coleman, Phys. Rev. D 15, 2929 (l977); and in The Whys of Su/mudear Physics , Erice Lect ures 1977, Plenum Press, 1979, 00. by A. Zichichi.

4 Semiclassical Time Evolution Amplitude

456

Recent semiclassical treatments of atomic systems are given in R.S. Manning and G.S. Ezra, Phys. Rev. 50,954 (1994). Chaos 2, 19 (1992). Scmiclassical scattering is tr 0,

(5.58)

we find an approximate ground state energy EiO) =

-~l. 27r

(5.59)

The exact value is (5.60)

469

5.6 Possible Direct Generalizations

The failure of the variational approximation is due to the fact that outside the range of the potential, the wave function is a simple exponential e- k1xl with k = ...; - 2E(0), and not a Gaussian. In fact, if we consider the expectation value of the Hamiltonian operator 1 d2 H = - - - - gt5(x) (5.61) 2dx 2 for a normalized trial wave function (5.62) we obtain a variational energy (5.63) whose minimum gives the exact ground state energy (5.60). Thus, problems of the present type call for the development of a variational perturbation theory for which Eq. (5.54) and (5.55) read (5.64) where (5.65) For an arbitrary attractive potential whose range is much shorter than a, this leads to the correct energy for a weakly bound ground state El(0) ~

5.6

-"21 [1 -

00

-00

dxoI V ( xo )] 2

(5.66)

Possible Direct Generalizations

Let us remark that there is a possible immediate generalization of the above variational procedure. One may treat higher components Xm with m > 0 accurately, say up to m = in - 1, where in is some integer> 1, using the ansatz

(5.67)

470

5 Variational Perturbation Theory

with the trial function Lm:

and a smearing square width of the potential a~ m

(5.69)

For the partition function alone the additional work turns out to be not very rewarding since it renders only small improvements. It turns out that in the low-temperature limit T -> 0, the free energy is still equal to the optimal expectation of the Hamiltonian operator in the Gaussian wave packet (5.49). Note that the ansatz (5.7) [as well as (5.67)] cannot be improved by allowing the trial frequency n(xo) to be a matrix nmm,(XO) in the space of Fourier components Xm [i.e., by using Lm,m' n mm, (XO)x;;'xm' instead of n(xo) Lm IxmI2 ]. This would also lead to an exactly integrable trial partition function. However, after going through the minimization procedure one would fall back to the diagonal solution nmm,(XO) = .=0

Inserting this into the right-hand side of (5.98), we find

VaL,aT (rO)

2

= [ v (rO

+ 0>. ) . /

1

2

1

VI - 2aLA /1 - 2afA V

D 1 e2r5>.2ai/(1-2ai>.l]

(5.101) >.=0

which has the expansion

+ v'(r5)[a"i + (D -l)a~l ~v"(r5)[3ai + 2(D - l)a"ia~ + (D2 -

VaL,aT(rO) = v(r5) +

l)af + 4r5a"il

+ ....

(5.102)

The prime abbreviates the derivative with respect to r5. In general it is useful to insert into (5.98) the Fourier representation for the potential

- J(27r)De dDk

V(x) -

ikx

(5.103)

V(k),

which makes the x-integration Gaussian, so that (5.99) becomes

Vai,aj,(ro)

=

dDk J(27r)D V(k)

(

a'i

2

a'f

2

.

)

exp -2"kL - 2"kT - zrOkL .

Exploiting the rotational symmetry of the potential by writing V(k) decompose the measure of integration as

(5.104)

== v(k 2 ),

we

(5.105)

481

5.10 Application to Coulomb and Yukawa Potentials

where

27rD/2 SD = f(D/2)

(5.106)

is the surface of a sphere in D dimensions, and further with kL = k cos ¢>, kT = k sin¢>: dDk SD-l D- l (5.107) (27r )D = (27r )D -1 d cos ¢> Jo dk k .

11

J

roo

This brings (5.104) to the form

(SD-)~ 27r

Va2 a2 (ro) = L' T

X

11

- 1

du1°O dkk D- 1V(k 2) - 1

exp {-~ [( aiu 2 + a~(1- u 2)) k 2 - irouJ} .

(5.108)

The final effective classical potential is found by minimizing W 1 (ro) at each ro in aL, aT, ~h , nT . To gain a rough idea about the solution, it is usually of advantage to study first the isotropic approximation obtained by assuming a~(ro) = ai(ro), and to proceed later to the anisotropic approximation.

5.10

Application to Coulomb and Yukawa Potentials

The effective classical potential can be useful also for singular potentials as long as the smearing procedure makes sense. An example is the Yukawa potential

V(r)

=

_(e 2/r)e- mr ,

which reduces to the Coulomb potential for m

(5.109)

== O. Using the Fourier representation (5.110)

we easily calculate the isotropic ally smeared potential

Va2(ro)

=

J.J27ra2

- e2

d3 x

3

[1

exp --(Xo - x) 2] V(r) 2a2

_e 2 27rem2a2 /21~ da'2 1 3 exp ( -r~/2a'2 - m 2a'2 /2) a .J27ra,2 2 2 _e2 2em a /2 _1 (0 / ..;20,2 a dte-(t2+m2r5/4t2).

Vir

ro Jo

(5.111)

In the Coulomb limit m ---.. 0, the smeared potential becomes equal to the Coulomb potential multiplied by an error function , (5.112)

482

5 Variational Perturbation Theory

where the error function is defined by erf(z)

=V7r2 iorz dxe-

X

2

(5.113)

•

The smeared potential is no longer singular at the origin, (5.114) The singularity has been removed by quantum fluctuations. In this way the effective classical potential explains the stability of matter in quantum physics, i.e. , the fact that atomic electrons do not fall into the origin. The effective classical potential of the Coulomb system is then by the isotropic version of (5.97) !l

3

WI (Xo) = 73 ln

sinh[n,60(xo)/2]

n,60(Xo)/2

e2

[ixol

1

3

2

2

- ~ erf J2a 2(xo) - "2 MO (xo)a (xo). (5.115)

Minimizing WI(ro) with respect to a2(ro) gives an equation analogous to Eq. (5.37) determining the frequency 02(ro) to be 2

_

o (ro) -

2~_1___1_ -r5/2a2

e 3 J21r (a2)3/2e

.

(5.116)

We have gone to atomic units in which e = M= n= kB= 1, so that energies and temperatures are measured in units of Eo = Me 4 /Mn 2 ~ 4.36 x lO-llerg ~ 27.21eV and To = Eo/kB ~ 31575K , respectively. Solving (5.116) together with (5.24) , we find a2 (ro) and the approximate effective classical potential (5.32). The result is shown in Fig. 5.9 as a dashed curve. The above approximation may now be improved by treating the fluctuations anisotropic ally, as described in the previous section, with different 02(XO) for radial and tangential fluctuations OXL and OXT, and the effective potential following Eqs. (5.97) and (5.98). For the anisotropically smeared Coulomb potential we calculate from (5.108):

Va L4(ro) =

-IfII

du J ai u2 +

e:f(l _u2)

exp

{-" 2-;-[a-; i'- u-: -2_+- ,r5'" 'a}".2.(""1-_-U-::-2~)]}' (5.117)

Introducing the variable ,\ = Jifu/ Jai,\2 + af(l - ,\2), which runs through the same interval [-1 ,1] as u, we rewrite this as (5.118)

483

5.10 Application to Coulomb and Yukawa Potentials

T=lxlO' K

~"-::;::;;.o"'~

4

Figure 5.9 Approximate effective classical potential W1(ro) of Coulomb system at various temperatures (in multiples of 104 K). It is calculated once in the isotropic (dashed curves) and once in the anisotropic approximation. The improvement is visible in the insert which shows WdV. The inverse temperature values of the different curves are j3 = 31.58, 15.78, 7.89, 3.945, 1.9725, 0.9863, 0 atomic units, respectively. Extremization of W1 (ro) in Eq. (5.97) yields the equations for the trial frequencies

~V2 84 aT,a

nT2 (ro)

2 L

= e2

raIl

y~

1 d)' ).2 exp [-(r5l 2a1J).2]

[aiC1 - ).2) + a}).2J2'

-1

82 + -31[n2L + 2nT2] = -31 [ uaL !l

(5 .119)

82 ] Va 2 a 2 uaT T' L

2!l

22 1 1 2 2) e -3 ~ r:2:4 exp( -rO/2a L . v27r yaia}

(5.120)

These equations have to be solved together with 2

(

) _

1

aL,T ro - /3ni,T(rO)

[/3n L,T(r o) 2

co

th /3n L,T(rO) _ 2

1] .

(5.121)

Upon inserting the solutions into (5.97) , we find the approximate effective classical potential plotted in Fig. 5.9 as a solid curve. Let us calculate the approximate particle distribution functions using a three-dimensional anisotropic version of Eq. (5.94) . With the potential W 1 (ro), we arrive at the integral [6]

(5.122)

484

5 Variational Perturbation Theory

2

4

8

6

10

12

14

r

Figure 5.10 Particle distribution g(r) == v27r(i p(r) in Coulomb potential at different temperatures T (the same as in Fig. 5.9), calculated once in the isotropic and once in the anisotropic approximation. The dotted curves show the classical distribution. For low and intermediate temperatures the exact distributions of R.G. Storer, J. Math. Phys. 9, 964 (1968) are well represented by the two lowest energy levels for which p(r)=7r- 1c 2T e,6/2+(1/87r)(1 - r + r2/2)e- Te,6/8+(1/387r)(243 - 324r + 216r2-48r3+4r4)e- 2T/3e,6/18.

The resulting curves to different temperatures are plotted in Fig. 5.10 and compared with the exact distribution as given by Storer. The distribution obtained from the earlier isotropic approximation (5.116) to the trial frequency [l2(ro) is also shown.

5.11

Hydrogen Atom in Strong Magnetic Field

The recent discovery of magnetars [7] has renewed interest in the behavior of charged particle systems in the presence of extremely strong external magnetic fields. In this new type of neutron stars, electrons and protons from decaying neutrons produce magnetic fields B reaching up to 1015 G, much larger than those in neutron stars and white dwarfs, where B is of order 1010 - 1012 G and 106 - 108 G, respectively. Analytic treatments of the strong-field properties of an atomic system are difficult, even in the zero-temperature limit. The reason is a logarithmic asymptotic behavior of the ground state energy to be derived in Eq. (5.132). In the weak-field limit , on the other hand, perturbative approaches yield well-known series expansions

485

5.11 Hydrogen Atom in Strong Magnetic Field

in powers of B2 up to B 60 [8]. These are useful, however, only for B « B o, where Bo is the atomic magnetic field strength Bo = e3 M2 In 3 ::::; 2.35 x 105 T = 2.35 X 109 G. So far, the most reliable values for strong uniform fields were obtained by numerical calculations [9]. The variational approach can be used to derive a single analytic expression for the effective classical potential applicable to all field strengths and temperatures [10]. The Hamiltonian of the electron in a hydrogen atom in a uniform external magnetic field pointing along the positive z-axis is the obvious extension of the expression in Eq. (2.644) by a Coulomb potential: H (p , x )

12M 2 2 = 2M P + 2WBX -

wBlz

e2

()

p , x -~ '

(5.123)

where WB denotes the B-dependent magnetic frequency wL/2 eB 12Mc of Eq. (2.649) , i.e., half the Landau or cyclotron frequency. The magnetic vector potential has been chosen in the symmetric gauge (2.638). Recall that lz is the z-com ponent of the or bi tal angular momentum l z (p, x) = (x x p) z [see (2.645)] . At first , we restrict ourselves here to zero temperature. From the imaginary-time version of the classical action (2.638) we see that the particle distribution function in the orthogonal direction of the magnetic field is, for Xb = 0, Yb = 0, proportional to (5.124) This is the same distribution as for a transverse harmonic oscillator with frequency Being at zero temperature, the first-order variational energy requires knowing the smeared potential at the origin. Allowing for a different smearing width an and al along an orthogonal to the magnetic field, we may use Eq. (5.118) to write

WB.

(5.125) Performing the integral yields 1

2

all

2) 2 arccosh-

27r (a ll - a.l

a.l

(5.126)

Since the ground state energies of the parallel and orthogonal oscillators are and 2 x 0.l/2, we obtain immediately the first-order variational energy

°11/2

(5.127)

°

with wil = and an,.l = 1/20Il,.l· In this expression we have ignored the second term in the Hamiltonian (5.123), since the angular momentum lz of the ground state must have a zero expectation value.

486

5 Variational Perturbation Theory

For very strong magnetic fields, the transverse variational frequency become equal to WB, such that in this limit

nT

will

(5.128) Extremizing this in

nil

yields n 4e4 HII :=::;-

7r

I

2 27rW B

og - , e4

(5.129)

and thus an approximate ground state energy E 1(O)

:=::; WB -

e4

-

7r

I

og

2 27rWB --4-'

e

(5.130)

The approach to very strong fields can be found by extremizing the energy (5.127) also in n1-. Going over to atomic units with e = 1 and m = 1, where energies are measured in units of fO = Me 4/!i2 == 2Ryd :=::; 27.21eV, temperatures in fo/kB :=::; 3.16 X 105 K, distances in Bohr radii aB = !i2/ M e2 :=::; 0.53 x 10- 8 cm, and magnetic field strengths in Bo = e3 M2 /!i 3 :=::; 2.35 X 105 T = 2.35 X 109 G, the extremization yields

~ y7r 2 ( 2a a2 ) yn1-= InB-2InlnB+lnB+ln2B+b

+ O(ln- 3 B) ,

(5.131 ) with abbreviations a = 2 - In2 :=::; 1.307 and b = In(7r/2) - 2 :=::; -1.548. The associated optimized ground state energy is, up to terms of order In- 2B, E(O)(B) = B _ .! {ln2 B - 4ln B lnln B + 4ln 2ln B - 4blnln B + 2(b + 2) In B + b2 2 7r -ln1B [8In2InB-8blnlnB+ 2b2] } +O(ln- 2B). (5.132)

The prefactor 1/7r of the leading In 2B-term using a variational ansatz of the type (5.64), (5.65) for the transverse degree of freedom is in contrast to the value 1/2 calculated in the textbook by Landau and Lifshitz [11] . The calculation of higher orders in variational perturbation theory would drive our value towards 1/2. The convergence of the expansion (5.132) is quite slow. At a magnetic field strength B = 105 B o, which corresponds to 2.35 x 1010 T = 2.35 X 1014 G, the contribution from the first six terms is 22.87 [2 Ryd]. The next three terms suppressed by a factor In- 1 B contribute -2.29 [2 Ryd], while an estimate for the In- 2B-terms yields nearly -0.3 [2 Ryd]. Thus we find c;(1)(10 5 ) = 20.58 ± 0.3 [2 Ryd]. Table 5.2 lists the values of the first six terms of Eq. (5.132). This shows in particular the significance of the second term in (5.132), which is of the same order of the leading first term, but with an opposite sign.

487

5.11 Hydrogen Atom in Strong Magnetic Field

Table 5.2 Example for competing leading six terms in large-B expansion (5.132) at B = 105 Bo ~ 2.35 X 1014 G. (l/rr)ln B

- (4/rr)lnBlnlnB

(4/rr) In InB

- (4b/rr)lnlnB

42.1912

-35.8181

7.6019

4.8173

[2(b+2)/rr]lnB 3.3098

b2 /7r 0.7632

30 ,-----------~------------~

Landau:-log 2 B2/2 ------

---

f(B)/2Ryd

10

20

10

,,(1) (B)/2Ryd

°O~----------~l~~=---------­

500

BIBo

1000 BIBo

1500

2000

Figure 5.11 First-order variational result for binding energy (5.133) as a function of the strength of the magnetic field. The dots indicate the values derived in the reference given in Ref. [12] . The long-dashed curve on the left-hand side shows the simple estimate 0.51n2 B of the textbook by Landau and Lifshitz [11]. The right-hand side shows the successive approximations from the strong-field expansion (5.134) for N = 0,1,2,3,4, with decreasing dash length. Fat curve is our variational approximation.

The field dependence of the binding energy

E(B)

=B

_

E(O)

(5.133)

2

is plotted in Fig. 5.11, where it is compared with the results of other authors who used completely different methods, with satisfactory agreement [12] . On the strongcoupling side we have plotted successive orders of a strong-field expansion [24]. The curves result from an iterative solution of the sequence of implicit equations for the quantity weB) = VE(4B)/2 for N = 1,2,3,4: (5.134) where

a4=

Pi y7rD VB 2 ' (5.135)

488

5 Variational Perturbation Theory

JY2+I

and D denotes the integral D == l' - 21000 dy (y / -1) log y ~ -0.03648, where l' ~ 0.5773 is the Euler-Mascheroni constant (2.467). Our results are of similar accuracy as those of other first-order calculations based on an operator optimization method [25] . The advantage of our variational approach is that it yields good results for all magnetic field strengths and temperatures, and that it can be improved systematically by methods to be developed in Section 5.13, with rapid convergence. The figure shows also the energy of Landau and Lifshitz which grossly overestimates the binding energies even at very large magnetic fields, such as 2000Bo ex: 10 12 G. Obviously, the nonleading terms in Eq. (5.132) give important contributions to the asymptotic behavior even at such large magnetic fields. As an peculiar property of the asymptotic behavior, the absolute value of the difference between the Landau-Lifshitz result and our approximation (5.132) diverges with increasing magnetic field strengths B. Only the relative difference decreases.

5.11.1

Weak-Field Behavior

Let us also calculate the weak-field behavior of the variational energy (5.127) . Setting 1] == 0 11 / 0 1- , we rewrite W1 (O) as W 1 (0 ) -- -01- ( 1 + -1])

2

2

B2 + V 1]011 - 1n 1 - J1=17 . + ---8017r J1=17 1 + J1=17

(5.136)

This is minimized in 1] and 01- by expanding 1](B) and O(B) in powers of B2 with unknown coefficients, and inserting these expansions into extremality equations. The expansion coefficients are then determined order by order. The optimal expansions are inserted into (5.136), yielding the optimized binding energy c(1)(B) as a power series 00

W 1 (O)

=

L

cnB2n.

(5.137)

n=O

The coefficients Cn are listed in Table 5.3 and compared with the exact ones. Of course, the higher-order coefficients of this first-order variational approximation become rapidly inaccurate, but the results can be improved, if desired, by going to higher orders in variational perturbation theory of Section 5.13.

5.11.2

Effective Classical Hamiltonian

The quantum statistical properties of the system at an arbitrary temperature are contained in the effective classical potential Heif cl (Po , xo) defined by the threedimensional version of Eq. (3.821):

B(po, xo) ==e-,l3Heffc1 {po ,x o) ==

f f (~~3 V 3x

J(3)(XO - X) (27rnlJ(po - p) e-A[p,xlln, (5.138)

489

5.11 Hydrogen Atom in Strong Magnetic Field

Table 5.3 Perturbation coefficients up to order B6 in weak-field expansions of variational parameters, and binding energy in comparison to exact ones (from J.E. Avron et al. and B.G. Adams et al. quoted in Notes and References). n

0 1.0

'In

fin en €~x

1 40571"2 - - - "" -0.5576 7168

32 - "" 1.1318 971" 4 - - "" - 0.4244 371"

9971" "" 1.3885 224 971" 128 "" 0.2209

- 0.5

2

_ 129397571"3 "" - 2.03982 19668992 801971"3 - - - "" - 0.1355 1835008

-~ "" 192

0.25

3 388699933207571"6 "" -4.2260 884272562962432

1682896571"4 "" 1.3023 1258815488

0.2760

52443166718771"5 "" 5.8077 27633517592576 25644980771"5 "" 0.2435 322256764928 5581 "" 1.2112 4608

where Ae[p , x] is the Euclidean action Ae[p,x]

(",(3 = Jo dT [-ip(T)X(T) + H(p(T),X(T))],

(5.139)

and x = fl,(3 dTX(T)/h(J and p = fOIi(3 dTp(T)lh(J are the temporal averages of position and momentum. Note that the deviations ofp(T) from the average Po share with x( T) - Xo the property that the averages of the squares go to zero with increasing temperatures like liT, and remains finite for T -+ O. while the expectation of p2 grows linearly with T (Dulong-Petit law). For T -+ 0, the averages of the squares of p( T) remain finite. This property is the basis for a reliable accuracy of the variational treatment. Thus we separate the action (5.139) (omitting the subscript e) as Ae[p, x]

= (JH(po, xo) + Aft XO [p, x] + A int [p, x],

(5.140)

where Aho,xO [p, x] is the most general harmonic trial action containing the magnetic field. It has the form (3.825), except that we use capital frequencies to emphasize that they are now variational parameters:

r (3

1

li

AhO,XO[P,x]

= Jo dT { - i[p(T) - Po] ' X(T) + 2M [p(T) - PO]2

+DBlAp(T)-Po,X(T)-Xo)

+ ~ Di [X.l(T)-xo.d 2+ ~ Drr[Z(T) - ZO]2}. (5.141)

The vector x.l = (x, y) is the projection of x orthogonal to B . The trial frequencies n = (DB, D.l, DII) are arbitrary functions of Po, XQ, and B. Inserting the decomposition (5 .140) into (5.138) , we expand the exponential of the interaction, exp {-AntlP, x] /h}, and obtain a series of expectation values of powers of the interaction (Annt[P, x] )hO ,XO, defined in general by the path integral ( O[p, x]

) .,~O,XO

=

zrt1

XO

f'1"\3

1)3p O[

v x(27rh)3

x exp { -~Aho,xO [p, x] } ,

] ,(3) (

p ,X u

-) (2 t 0 like (_l)N-l g N x (1 / 0)3N-I. Since the trial energy goes for large 0 to positive infinity, only the odd approximations are guaranteed to possess a minimum. The second-order approximation W 2 can nevertheless be used to find an improved energy value. As shown by Fig. 5.14, the frequency of least dependence O2 is well defined. It is the frequency where the O-dependence of W 2 has its minimal absolute value. Thus we optimize 0 with the condition

a2 wf

a0 2 = O.

(5 .193)

This leads to the energy values E~O)(g) listed in Table 5.6. They are more accurate than the values EiO) (g) by an order of magnitude.

5.14 Applications of Variational Perturbation Expansion

501

0.8 0.75

\

N=l

\

0.7

\

N=2 /

\

0.65

\

W 0.6 0.55 0.5 0.45 0.4

1

1.5

2

2.5

3

Figure 5.14 Typical O-dependence of approximations W1 ,2,3 at T = O. The coupling constant has the value 9 = 004. The second-order approximation wf has no extremum. Here the minimal O-dependence lies at the turning point, and the condition 8 2 Wf / 80 2 renders the best approximation to the energy (short dashes).

5.14.2

Anharmonic Oscillator for T

>0

Consider now the anharmonic oscillator at a finite temperature, where the expansion (5.183) consists of the sum of one-particle irreducible vacuum diagrams

Wr=-f3

1{

I?!

?!

--O+CD 2 3

?! --1 ( CJ:J:) +

2!

72

(5.194) 24

-

?!

2592

1728

+~( ciJ:o +V

?!

+

@ 3456

1728

648

648

The vertices represent the couplings 93(XO)/3!n, 94(xO)/4!n, whereas the lines stand for the correlation function G'~o (7, 7'). The numbers under each diagram are their multiplicities acting as factors . Only the five integrals associated with the diagrams

o

502

5 Variational Perturbation Theory

n

need to be evaluated explicitly; all others arise by the expansion of or by factorization. The explicit form of three of these integrals can be found in the first, forth, and sixth of Eqs. (3.547). The results of the integrations are listed in Appendix 5A. Only the second and fourth diagrams are new since they involve vertices with three legs. They can be found in Eqs. (3D.7) and (3D.8). In quantum field theory one usually calculates Feynman integrals in momentum space. At finite temperatures, this requires the evaluation of multiple sums over Matsubara frequencies. The present quantum-mechanical example corresponds to a D = 1 -dimensional quantum field theory. Here it is more convenient to evaluate the integrals in T-space. The diagrams

for example, are found by performing the integrals

Ti(3a 2 Ti(3

(~) 2 a~2

The factor Ti(3 on the left-hand side is due to an overall T-integral and reflects the temporal translation symmetry of the system; the factors l/w arise from the remaining T-integrations whose range is limited by the correlation time l/w. In general, the (3- and xo-dependent parameters a~L have the dimension of a length to the nth power and the associated diagrams consist of m vertices and n/2 lines (defining == 2 ). We now use the rule (5.188) to replace w by and expand everything in powers of 92 up to the third order. The expansion can be performed diagrammatically in each Feynman diagram. Letting a dot on a line indicate the coupling 92/2Ti, the one-loop diagram is expanded as follows:

ai a

n

fl

0=0-20+0

1

3

o·

The other diagrams are expanded likewise: fl

CD

1 1 CD +-6 CD +-6 CD,

1

2 E3

1 - 6

9

1 -6

CD 1

fl

2 E3

-

-

fl

1

-

2

9

1

-

2

E3 ,

g,

(5.195)

503

5.14 Applications of Variational Perturbation Expansion fl

lo::D

2

In this way, we obtain from (5.194) the complete graphical expansion for W:P(xo) including all vertices associated with the coupling g2(XO) :

jJW:P(xo)

=

-~ 0 +

-~[~ 0

+~[

0

+3

G0

+~

+~

CD

+3G

CD )

+~

o::D + 2~

9

+~

E7 ]

CD +~ CD)

(~4 (XX) + ~4 c:x:::D + ~6 \::::f ~ + ~2 E3 )

+ (136

CXXD

+~ cg +~ @ +~@

+~O +~g)l

(5.196)

In the latter diagrams, the vertices represent directly the couplings gn/n. The denominators n! of the previous vertices gn/n!n have been combined with the multiplicities of the diagrams yielding the indicated prefactors. The corresponding analytic expression for W:P(xo) is

(5.197)

The quantities a~L are ordered in the same way as the associated diagrams in (5.196). As before, we have omitted the variable Xo in all but the first three terms, for brevity. The optimal trial frequency O(xo) is found numerically by searching, at each value of Xo , for the real roots of the first derivate of W:P(xo) with respect to O(xo). Just as for zero temperature, the solution happens to be unique.

504

5 Variational Perturbation Theory

By calculating the integral (5 .198)

we obtain the approximate free energy (5 .199)

The results are listed in Table 5.7 for various coupling constants 9 and temperatures. They are compared with the exact free energy (5.200) whose energies En were obtained by numerically solving the Schrodinger equation.

9 0.002 0.4

2.0

4.0 20

200 2000

80000

f3 2.0 1.0 5.0 1.0 5.0 10.0 1.0 5.0 1.0 5.0 10.0 5.0 0.1 1.0 10.0 0.1 3.0

Fl 0.427937 0.226084 0.559155 0.492685 0.699431 0.700934 0.657396 0.809835 1.18102 1.24158 1.24353 2.54587 2.6997 5.40827 5.4525 18.1517 18.50l

F3 0.427937 0.226075 0.558678 0.492578 0.696180 0.696285 0.6571051 0.803911 1.17864 1.22516 1.22515 2.50117 2.69834 5.32319 5.3225 18.0470 18.146

Fex 0.427741 0.226074 0.558675 0.492579 0.696118 0.696176 0.6571049 0.803758 1.17863 1.22459 1.22459 2.49971 2.69834 5.31989 5.3199 18.0451 18.137

Table 5.1 Free energy of anharmonic oscillator with potential V(x) various coupling strengths 9 and f3 = l/kT.

= x 2 /2 + gx 4 /4

for

We see that to third order, the new approximation yields energies which are better than those of FI by a factor of 30 to 50. The remaining difference with respect to the exact energies lies in the fourth digit . In the high-temperature limit, all approximations WN(xo) tend to the classical result V(xo), as they should. Thus, for small (3, the approximations W3 (xo) and WI (xo) are practically indistinguishable.

505

5.15 Convergence of Variational Perturbation Expansion

The accuracy is worst at zero temperature. Using the T --+ 0 -limits of the Feynman integrals a~L given in (3.550) and (3D.14), the approximation wf takes the simple form

W:f(xo) = V(xo)

+ liO;xo) + ~a2 + ~a4 [g~ a4 + g2g4 a6 + g~ ~a6 + gl ~a8]

_ _ 1_ 211,0

2

2

6 3

24 2

(5.201)

1 [ 33 6 24 8 2 8 2 13 10 235 10 337 12] +611,202 g22 a +g2g33a +g2 g4 3a +g3g43 a +g2g416a +g464 a , where a2 = li/2MO. As in (5.197), we have omitted the arguments Xo in 0 and gi , for brevity. At zero temperature, the remaining integral over Xo in the partition function (5.198) receives its only contribution from the point Xo = 0, where W3(0) is minimal. There it reduces to the energy W3 of Eq. (5.192) .

5.15

Convergence of Variational Perturbation Expansion

For a single interaction x P , the approximation W N at zero temperature can easily be carried to high orders [13, 14]. The perturbation coefficients are available exactly from recursion relations, which were derived for the anharmonic oscillator with p = 4 in Appendix 3C . The starting point is the ordinary perturbation expansion for the energy levels of the anharmonic oscillator

E(n) = w ~ E(n) ~ k=O

k

(L) k 4w

(5.202)

3

It was remarked in (3C.27) and will be proved in Section 17.10 [see Eq. (17.323)] that the coefficients ELn) grow for large k like

(n) Ek

1 f;12n

---+ - 7r

- - I (-3) 7r

n.

k r(k + n + 1/2) .

(5.203)

Using Stirling's formula3 (5.204) this amounts to

E(n) k

---+ _

2vf3 (_4)n 7r

n!

[-3(k + e

n)] n+k

Ek

(5.205)

Thus, n ) grows faster than any power in k. Such a strong growth implies that the expansion has a zero radius of convergence. It is a manifestation of the fact that the energy possesses an essential singularity in the complex g-plane at the expansion point g = O. The series is a so-called asymptotic series. The precise form of the 3M. Abramowitz and 1. Stegun, op. cit., Formulas 6.1.37 and 6.1.38.

506

5 Variational Perturbation Theory

I

I

\ I

N=l

:

\

\ I

0.7

N=3

\ I

\

W

\

0.65

: :

N=5

\ I \ II

\1I

0.6

N=ll

\

3.5

4

0.59 r-~-~-----ry---~'--~-r-~---'

N=2

0.58 0.57

W 0.56 . 0.55

1

2.5

2

1.5

3

n

3.5

4

Figure 5.15 Typical !1-dependence of Nth approximations WN at T = 0 for increasing orders N . The coupling constant has the value g/4 = 0.1. The dashed horizontal line indicates the exact energy.

singularity will be calculated in Section 17.10 with the help of the semiclassical approximation. If we want to extract meaningful numbers from a divergent perturbation series such as (5.202) , it is necessary to find a convergent resummation procedure. Such a procedure is supplied by the variational perturbation expansion, as we now demonstrate for the ground state energy of the anharmonic oscillator.

Truncating the infinite sum (5.202) after the Nth term, the replacement (5.188) followed by a re-expansion in powers of g up to order N leads to the approximation WN at zero temperature:

n_

~

(0) (

wN-n~Cl 1=0

g

4fP

)1 ,

(5.206)

507

5.15 Convergence of Variational Perturbation Expansion

w

N=21

0.55914625

1.4

N=l1 1.6

2.2

2

n

Figure 5.16 New plateaus in W N developing for higher orders N 2: 15 in addition to the minimum which now gives worse results. For N = 11 the new plateau is not yet extremal, but it is the proper region of least !1-dependence yielding the best approximation to the exact energy indicated by the dashed horizontal line. The minimum has fallen far below this value and is no longer useful. The figure looks similar for all couplings (in the plot, 9 = 0.4). The reason is the scaling property (5.215) proved in Appendix 5B.

with the re-expansion coefficients Era) =

t

EjO) ( (1

~~j!/2

) (-40-)/-j.

(5.207)

J

J=O

Here 0- denotes the dimensionless function of D

0- == _~ Dr2 = D(D2 - 1) . 2

9

(5.208)

In Fig. 5.15 we have plotted the D-dependence of W N for increasing N at the coupling constant g/4 = 0.1. For odd and even N, an increasingly flat plateau develops at the optimal energy. At larger orders N :2: 15, the initially flat plateau is deformed into a minimum with a larger curvature and is no longer a good approximation. However, a new plateau has developed yielding the best energy. This is seen on the high-resolution plot in Fig. 5.16. At N = 11, the new plateau is not yet extremal but close to the correct energy. The worsening extrema in Fig. 5.16 correspond here to points leaving the optimal dashed into the upward direction. The newly forming plateaus lie always on the dashed curve. The set of all extremal DN-values for odd N up to N = 91 is shown in Fig. 5.17. The optimal frequencies with smallest curvature are marked by a fat dot. In Subsection 17.10.5. we shall derive that

(5.209)

508

5 Variational Perturbation Theory

10

8

6 ON

4

/.~

.;:-e/ ---

2

00

20

40

60

80

100

N

Figure 5.17 Trial frequencies ON extremizing the variational approximation WN at T = 0 for odd N ~ 91. The coupling is g/4 = 1. The dashed curve corresponds to the approximation (5.211) [related to ON via (5.208)]. The frequencies on this curve produce the fastest convergence. The worsening extrema in Fig. 5.16 correspond here to points leaving the optimal dashed into the upward direction. The newly forming plateaus lie always on the dashed curve.

grows for large N like IJN :;::,

eN,

e = 0.186047 .. . ,

(5.210)

so that ON grows like ON :;::, (eN g)1/3. For smaller N, the best ON-values in Fig. 5.17 can be fitted with the help of the corrected formula (5.210): IJN :;::,

6.85) eN ( 1 + N2/3 .

(5.211)

The associated ON-curve is shown as a dashed line. It is the lower envelope of the extremal frequencies. The set of extremal and turning point frequencies ON is shown in Fig. 5.18 for even and odd N up to N = 30. The optimal extrema with smallest curvature are again marked by a fat dot. The theoretical curve for an optimal convergence calculated from (5.211) and (5.209) is again plotted as a dashed line. In Table 5.8, we illustrate the precision reached for large orders N at various coupling constants 9 by a comparison with accurate energies derived from numerical solutions of the Schrodinger equation. The approach to the exact energy values is illustrated in Fig. 5.19 which shows that a good convergence is achieved by using the lowest of all extremal frequencies, which lie roughly on the dashed theoretical curve in Fig. 5.17 and specify the position of the plateaus. The frequencies ON on the higher branches leaving the dashed curve in that figure, on the other hand, do not yield converging energy values. The

509

5.15 Convergence of Variational Perturbation Expansion 6

,-------------~--------------~------------_,

4

2

o

10

20

N

30

Figure 5.18 Extremal and turning point frequencies ON in variational approximation W N at T = 0 for even and odd N :S 30. The coupling is 9/4 = 1. The dashed curve corresponds to the approximation (5.211) [related to ON via (5.208)].

-\ 10. 5

9/4 = 1

~:::::::::::::::::::::::::::::::::: ......-~

~ . .

~ BBBBBBBBBBBBBD

----.......'.-.-.-.... .. -

10.20

o

20

40

N

60

80

100

Figure 5.19 Difference between approximate ground state energies E = WN and exact energies Eex for odd N corresponding to the ON-values shown in Fig. 5.17. The coupling is 9/4 = 1. The lower curve follows roughly the error estimate to be derived in Eq. (17.409). The extrema in Fig. (5.17) which move away from the dashed curve lie here on horizontal curves whose accuracy does not increase.

sharper minima in Fig. 5.16 correspond precisely to those branches which no longer determine the region of weakest O-dependence. The anharmonic oscillator has the remarkable property that a plot of the ONvalues in the N, (IN-plane is universal in the coupling strengths g; the plots do not depend on g. To see the reason for this, we reinsert explicitly the frequency w (which

510

5 Variational Perturbation Theory N 1 2 3 4 5 10 15 20 25

exact

N 1 2 3 4 5 10 15 20 25

exact

g/4 =0.1 0.5603073711 0.5591521393 0.5591542188 0.5591457408 0.5591461596 0.5591463266 0.5591463272 0.5591463272 0.5591463272 0.5591463272 g/4 =50 2.5475803996 2.5031213253 2.5006996279 2.4995980125 2.4996213227 2.4997071960 2.4997089403 2.4997089079 2.4997087731 2.4997087726

g/4 = 0.3 0.6416298621 0.6380887347 0.6380357598 0.6379878713 0.6379899084 0.6379917677 0.6379917838 0.6379917836 0.6379917832 0.6379917832 g/4 =200 4.0084608812 3.9365586048 3.9325538203 3.9307488127 3.9307857892 3.9309286743 3.9309316283 3.9309315732 3.9309313396 3.9309313391

g/4 =0.5 0.7016616429 0.6963769499 0.6962536326 0.6961684978 0.6961717475 0.6961757782 0.6961758231 0.6961743059 0.6961758208 0.6961758208 g/4 =1000 6.8279533136 6.7040032606 6.6970328638 6.6939036178 6.6939667971 6.6942161680 6.6942213631 6.6942212659 6.6942208522 6.6942208505

g/4 = 1.0 0.8125000000 0.8041900946 0.8039140528 0.8037563457 0.8037615232 0.8037705329 0.8037706596 0.8037706575 0.8037706513 0.8037706514 9/4 =8000 13.635282593 13.386598486 13.372561189 13.366269038 13.366395347 13.366898079 13.366908583 13.366908387 13.366907551 13.366907544

g/4 = 2.0 0.9644035598 0.9522936298 0.9517997694 0.9515444198 0.9515517450 0.9515682249 0.9515684933 0.9515684887 0.9515584121 0.9515684727 9/4 =20000 18.501658712 18.163979967 18.144908389 18.136361642 18.136533060 18.137216200 18.137230481 18.137230214 18.137230022 18.137229073

Table 5.8 Comparison of the variational approximations W N at T = 0 for increasing N with the exact ground state energy at various coupling constants g.

was earlier set equal to unity). Then the re-expanded energy WN in Eq. (5.206) has the general scaling form (5.212) where WN is a dimensionless function of the reduced coupling constant and frequency , g

g

== ~V'

, W

W

== n'

(5.213)

respectively. When differentiating (5.212), d WN dO

d - 2,2 d] = [ 1 - 3'g dg W c1W 2

WN

('g, W'2) ,

(5.214)

we discover that the right-hand side can be written as a product of gN and a dimensionless polynomial of order N depending only on a = 0(0 2 - w 2 )/g: d Wrl

dO

N

= g'N PN (a) .

(5.215)

A proof of this will be given in Appendix 5B for any interaction x p . The universal optimal aN-values are obtained from the zeros of pN(a). It is possible to achieve the same universality for the optimal frequencies of the even approximations WN by determining them from the extrema of pN(a) rather than from the turning points of W N as a function of o. The universal functions pN(a) are found most easily by replacing the variable a in the coefficients of the re-expansion (5.206) by its W = 0 -limit alw=o = 0 3 / g = l/g . This yields the simpler expression

dO)

W~ = OWN(g, 0) = 0 f. E}O) (~) 1, l=O

(5.216)

511

5.15 Convergence of Variational Perturbation Expansion

with c}O) =

t E?) ( (1 ~~j!/2 ) (-4/flY-j .

(5.217)

J

)=0

The derivative of WN with respect to

n yields (5.218)

In Section 17.10, we show the re-expansion coefficients cia) in (5.207) to be for large k proportional to EkO): e(O) ~ e-2(7NE(0) Ck ~ k ,

(5.219)

[see Eq. (17.396)]. Thus, at any fixed n, the re-expanded series has the same asymptotic growth as the original series with the same vanishing radius of convergence. The behavior (5.219) can be seen in Fig. 5.20(a) where we have plotted the logarithm of the absolute value of the kth term

Sk = cia)

(~r

(5.220)

of the re-expanded perturbation series (5.202) for various optimal values

nN

and

9 = 40. All curves show a growth ex: kk . The terms in the original series start growing immediately (precocious growth). Those in the re-expanded series, on the other hand, decrease initially and go through a minimum before they start growing (re-

tarded growth) . The dashed curves indicate the analytically calculated asymptotic behavior (5.219).

30

b)

Figure 5.20 Logarithmic plot of kth terms in re-expanded perturbation series at a coupling constant 9 / 4 = 1: (a) Frequencies ON extremizing the approximation WN. The dashed curves indicate the theoretical asymptotic behavior (5.219). (b) Frequencies ON corresponding to the dashed curve in Fig. 5.17. The minima lie for each N precisely at k = N, producing the fastest convergence. The curves labeled 0 = W indicate the kth term in the original perturbation series.

5 Variational Perturbation Theory

512

The increasingly retarded growth is the reason why energies obtained from the variational expansion converge towards the exact result. Consider the terms Sk of the resummed series with frequencies ON taken from the theoretical curve of optimal convergence in Fig. 5.17 (or 5.18). In Fig. 5.20(b) we see that the terms Sk are minimal at k = N, i.e., at the last term contained in the approximation W N . In general, a divergent series yields an optimal result if it is truncated after the smallest term Sk. The size of the last term gives the order of magnitude of the error in the truncated evaluation. The re-expansion makes it possible to find , for every N, a frequency ON which makes the truncation optimal in this sense.

5.16

Variational Perturbation Theory for Strong-Coupling Expansion

From the w -+ 0 -limit of (5.206) , we obtain directly the strong-coupling behavior of WN. Since 0 = (g/g)I /3, we can write (5.221) and evaluate this at the optimal value therefore

g=

l /uN . The large-g behavior of WN is

(5.222) with the coefficient (5.223)

exp( 6A I · 9A2Nl/3) 10.5

,

e.xp( 6.40 - 9.05 N' /3 )

o'~'::< eXp(8.2 - 9. 7 N!/3)

10-10

0

0

10- 10

.... ' "

°000: " ,

"

10- 15

10- 15

10-20

10- 20

10-30 ll---~--~---'o---~_--'o----'

8 1/ 3

Figure 5.21 bo and bl·

27 1 / 3 64 1 / 3 N ' /3

125 1 / 3

216 1 / 3

10-30 ll---~--~_--'o-_-~_--'o----' 8 /3 27 / 3 64 / 3 125 / 3 216 / 3

'

'

N ' /3

'

'

'

Logarithmic plot of N-behavior of strong-coupling expansion coefficients

513

5.16 Variational Perturbation Theory for Strong-Coupling Expansion

o

-1

2

3

5

6

Figure 5.22 Oscillations of approximate strong-coupling expansion coefficient bo as a function of N when approaching exponentially fast the exact limit. The exponential behavior has been factored out. The upper and lower points show the odd-N and even-N approximations, respectively.

The higher corrections to the leading behavior (5.222) are found just as easily. By expanding WN(g ,CP) in powers of 2 ,

w

(5.224) and inserting ,2 W

g2/3

(5.225)

= -;-(g--'/;:'-w""'3)""""2/"'3 '

we obtain the expansion

( g ) - 4/3 ] g)1/3 [ ( g ) - 2/3 WN = ( 4 bo + bi 4w 3 + b2 4w3 + . ..

,

(5.226)

with the coefficients (5.227)

The derivatives on the right-hand side have the expansions

d ( dW 2

)n

, ,2 _ WN(g , W) -

N (,)1 (0)

g

~Enl 4 '

(5.228)

514

5 Variational Perturbation Theory

Table 5.9 Coefficients bn of strong-coupling expansion of ground state energy of anharmonic oscillator obtained from a perturbation expansion of order 251. An extremely precise value for bo was given by F. Vinette and J. Cizek, J. Math. Phys. 32, 3392 (1991): bo = 0.66798625915577710827096201619860199430404936 .... n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

-

bn 0.66798625915577710827096 0.1436687833808649100203 0.008627565680802279128 0.000818208905756349543 0.000082429217130077221 0.000008069494235040966 0.000000727977005945775 0.000000056145997222354 0.000000002949562732712 0.000000000064215331954 0.000000000048214263787 0.000000000008940319867 0.000000000001205637215 0.000000000000130347650 0.000000000000010 760 089 0.0000000000000004458901 0.0000000000000000589898 0.00000000000000001919600 0.00000000000000000328813 0.00000000000000000042962 0.000000000000000000044 438 0.0000000000000000000032305 0.0000000000000000000000314

with the re-expansion coefficients

(5.229)

For increasing N, the coefficients bo, bl, ... converge rapidly against the values shown in Table 5.9. From the logarithmic plot in Fig. 5.21 we extract a convergence

I 8.2- 9.7 NI/3 , Ib0 - bex 0 rve

(5.230)

This behavior will be derived in Subsection 17.10.5, where we shall find that for any 9 > 0, the error decreases at large N roughly like e -[9.7+(cg) - 2/ 3]N I /3 [see Eq. (17.409)]. The approach to this limiting behavior is oscillatory, as seen in Fig. 5.22 where we have removed the exponential falloff and plotted e- 6 .5+9.42N I/ 3 (bo - b1n against N [16].

515

5.17 General Strong-Coupling Expansions

For the proof of the convergence of the variational perturbation expansion in Subsection 17.10.5. it will be important to know that the strong-coupling expansion for the ground state energy o (g)1 /3 [ ( g E( ) = 4 bo + b1 4w3

)-2/3 + b (4w3g )-4/3 + . ..], 2

(5.231)

converges for large enough g > gs·

(5.232)

The same is true for the excited energies.

5.17

General Strong-Coupling Expansions

The coefficients of the strong-coupling expansion can be derived for any divergent perturbation series (5.233) n=O

for which we know that it behaves at large couplings g like M

E(g)

= gp/q

L

bm(g-2/ q)m.

(5.234)

m=O

The series (5.233) can trivially be rewritten as (5.235) with w = 1. We now apply the square-root trick (5.188) and replace w by the identical expression (5.236) containing a dummy scaling parameter O. The series (5.235) is then re-expanded in powers of g up to the order N, thereby treating w 2 - 0 2 as a quantity of order g. The result is most conveniently expressed in terms of dimensionless parameters 9 == gloq and a == (1 - C;2)/fJ, where C; == wlO. Then the replacement (5.236) amounts to w ---+ 0(1- ag)1/2, (5.237) so that the re-expanded series reads explicitly N

WN(g, a)

=

Op

L

cn(a) (gt,

(5.238)

n=O

with the coefficients: (5.239)

516

5 Variational Perturbation Theory

For any fixed g, we form the first and second derivatives of W}J(g) with respect to 0, calculate the O-values of the extrema and the turning points, and select the smallest of these as the optimal scaling parameter ON. The function W N (g) == W N(g, ON) constitutes the Nth variational approximation EN(g) to the function E(g) . We now take this approximation to the strong-coupling limit g --+ 00 . For this we observe that (5.238) has the general scaling form (5 .240)

For dimensional reasons, the optimal ON increases with g for large g like ON ~ gl/qCN, so that g = c"}/ and (J = l/g = c'fv remain finite in the strong-coupling limit, whereas C;2 goes to zero like l/[CN(g/W q)l/qj2. Hence (5 .241)

Here CN plays the role of the variational parameter to be determined by the lowest extremum or turning point of d;.wN(c"ii, 0). The full strong-coupling expansion is obtained by expanding WN(g , C;2) in powers of C;2 = (g/w qg) - 2/ q at a fixed g. The result is

/ [_ WN(g) = gP q bo(g)

_ ( g ) -2/q _ + b1(g) wq + b2(g)

( g ) -4/q wq + ... ] ,

(5 .242)

with

bn (') g

=

~'d'2 (~)n WN(n)(,g, w'2)1 n.

w

w2 =0

g'(2n- p)/q ,

(5 .243)

with respect to C;2. Explicitly:

Since g = c"ii, the coefficients bn(g) may be written as functions of the parameter c: (5 .245)

The values of c which optimize WN(g) for fixed g yield the desired values of CN. The optimization may be performed stepwise using directly the expansion coefficients bn ( c). First we optimize the leading coefficient bo(c) as a function of c and identifying the smallest of them as CN. Next we have to take into account that for large but finite C¥, the trial frequency 0 has corrections to the behavior gl/qC. The coefficient c will depend on g like

5.17 General Strong-Coupling Expansions

C(g) = C+ 1'1

~q A

(

)-2/

517 q

(

+ 1'2 ~q A

)-4/

q

+ ...,

(5.246)

requiring a re-expansion of c-dependent coefficients bn in (5.242). The expansion coefficients c and 1'n for n = 1,2,... are determined by extremizing b2n (c). The final result can again be written in the form (5.242) with b( c)n replaced by the final bn : g) -2/q ( g ) -4/q ] (5.247) + b2 wq + .... WN(g) = gp/q [bo + b1 ( wq The final bn are determined by the equations shown in Table 5.10. The two leading coefficients receive no correction and are omitted. The extremal values of g will have a strong-coupling expansion corresponding to (5.246):

g = c"i/ [1 +

g) - 2/q

=:;:j 2%, see Eq. (5.167)] . In Fig. 5.23 we have plotted the relative deviation of the variational approximation from the exact one in percent. The strong-coupling behavior is known from (5.226). It starts out like l/3, followed by powers of g- 1/3, g- l, g- 5/3. Comparison with (5.234) shows that this

519

5.18 Variational Interpolation between Weak and Strong-Coupling Expansions 1.02

1.0

-2

-3

3

-1

4

logg/4

0.995 Figure 5.23 Ratio of approximate and exact ground state energy of anharmonic oscillator from lowest-order variational interpolation. Dashed curve shows first-order FeynmanKleinert approximation WI (g). The accuracy is everywhere better than 99.5 %. For comparison, we also display the much worse (although quite good) variational perturbation result using the exact aix = 3/4.

corresponds to p = 1 and q = 3. The leading coefficient is given in Table 5.9 with extreme accuracy: bo = 0.667986259155777108270962016919860 . ... In a variational interpolation, this value is used to determine an approximate al (forgetting that we know the exact value a!X = 3/4). The energy (5.238) reads for N = 1 (with a = g/4 instead of g): W1(a,O) =

("20+ 201) ao + 02 a . al

(5.250)

Equation (5.245) yields, for n = 0: c al (5.251) bo = -ao +-. 2 c2 Minimizing bo with respect to c we find c = Cl == 2(aI/2ao)1/3 with bo = 3aocI/4 = 3(a5aI/2)1/3/2. Inserting this into (5.251) fixes al = 2(2/3bo)3/ a5 = 0.773970 ... , quite close to the exact value 3/4. With our approximate al we calculate W1(a , 0) at its minimum, where r.

_

~~-

{

1wcosh [~acosh(g/g(O))l 1w cos [~arccos(g/ g(O)) 1

fur

9 9

> g(O), < g(O),

(5.252)

with g(O) == 2w3ao/3v'3al' The result is shown in Fig. 5.23. Since the difference with respect to the exact solution would be too small to be visible on a direct plot ofthe energy, we display the ratio with respect to the exact energy WI (g) / E2x. The accuracy is everywhere better than 99.5 %.

520

5.19

5 Variational Perturbation Theory

Systematic Improvement of Excited Energies

The variational method for the energies of excited states developed in Section 5.12 can also be improved systematically. Recall the n-dependent level shift formulas (3.515) and (3.516), according to which

(5.253)

By applying the substitution rule (5.188) to the total energies

E(n) =

hJ),(n + 1/2)

and expanding each term in powers of 9 up to level shift

+ ilE(n) ,

l,

we find the contributions to the

~

[3(2n2

_

(~) 2 [2(34n 3 + 51n2 + 59n + 21)a8

+ 2n + 1)a4 + (2n + 1)a2r2],

+4· 3(2n2

1

+ 2n + 1)a6r2 + (2n + 1)a4 r4lnn'

(~) 3 [4. 3(125n4 + 250n3 + 472n2 + 347n + 111)a12 + 51n2 + 59n + 21)a10r2 1 +16· 3(2n2 + 2n + 1)a8r 4 + 2· (2n + 1)a6r6ln2n2 '

+4· 5(34n 3

which for n

(5.254)

= 0 reduce to the corresponding terms in (5.192). The extremization in

n leads to energies which lie only very little above the exact values for all n.

This is illustrated in Table 5.11 for n = 8 (compare with the energies in Table 5.5). A sum over the Boltzmann factors e-j3E~n) produces an approximate partition function Z3 which deviates from the exact one by less than 50.1%. It will be interesting to use the improved variational approach for the calculation of density matrices, particle distributions, and magnetization curves.

521

5.20 Variational 'neatment of Double-Well Potential

Table 5.11 Higher approximations to excited energy with n = 8 of anharmonic oscillator at various coupling constants g. The third-order approximation E~8l(g) is compared with the exact values E~~l(g), with the approximation Ei8 l (g) of the last section, and with the lower approximation of even order E~8l (g) (all in units of nw).

g/4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 50 100 500 1000

5.20

E~~J(g) 13.3790 15.8222 17.6224 19.0889 20.3452 21.4542 22.4530 23.3658 24.2091 24.9950 51.9865 88.3143 111.128 189.756 239.Q12

Ei8 J(g)

E~8J(g)

13.3235257 15.7327929 17.5099190 18.9591071 20.2009502 21.2974258 22.2851972 23.1879959 24.0221820 24.7995745 51.5221384 87.5058600 110.105819 188.001018 236.799221

13.3766211 15.8135994 17.6099785 19.0742800 20.3287326 21.4361207 22.4335694 23.3451009 24.1872711 24.9720376 51.9301030 88.2154879 111.002842 189.540577 238.740320

E~8J(g) 13.3847643 15.8275802 17.6281810 19.0958388 20.3531080 21.4629384 22.4625543 23.3760415 24.2199988 25.0064145 51.9986710 88.3500454 111.173183 189.833415 239.109584

Variational Treatment of Double-Well Potential

Let us also calculate the approximate effective classical potential of third order W 3 (xo) for the double-well potential w2 9 M 2 w4 V(x) = M _x 2 + _x4 + - w2 = -1. (5.255) 2 4 4g' In the expression (5.197), the sign change of w 2 affects only the coupling g2(XO), which becomes (5.256)

[recall (5.176)]. Note the constant energy M 2 w4 /4g in V(x) which shifts the minima of the potential to zero [compare (5.78)] . To see the improved accuracy of W3 with respect to the first approximation W I (xo) discussed in Section 5.7 [corresponding to the first line of (5.197)], we study the limit of zero temperature where the accuracy is expected to be the worst. In this limit, W 3 (xo) reduces to (5.201) and is easily minimized in Xo and O. At larger coupling constants 9 > ge ~ 0.3, the energy has a minimum at Xo = O. For 9 ::::; ge, there is an additional symmetric pair of minima at Xo = ±xm oF 0 (recall Figs. 5.5 and 5.6) . The resulting W3(0) is plotted in Fig. 5.24 together with WI(O). The figure also contains the first excited energy which is obtained by setting w 2 = - 1 in r2 = 2M(w 2 - 02)/g of Eqs. (5.253)- (5.255). For small couplings g , the energies WI (O), W3(0) , .. . diverge and the minima at x = ±x m of Eq. (5.82) become relevant. Moreover, there is quantum tunneling across the central barrier from one minimum to the other which takes place for 9 ::::; ge ~ 0.3 and is unaccounted for by W3(0) and W3 (xm). Tunneling leads to a level splitting to be calculated in Chapter 17. In this chapter, we test the accuracy of WI (X m ) and W 3(x m ) by comparing them with the averages of the two lowest energies. Figure 5.24 shows that the accuracy of the approximation W 3 (x m ) is quite good.

522

5 Variational Perturbation Theory 0.5

l. 8

1.5

:1 1.4

II II II

II II II '\--

E

\

.....1.... • .~

0.6

O.9t-----..,----.,--------~ \ I \

\

',-\.. \

0.8

\

\

E

0.4

0.1

0.2

9

0.3

0.4

Figure 5.24 Lowest two energies in double-well potential as function of coupling strength 9. The approximations are WI(O) (dashed line) and W3(O) (solid line). The dots indicate numeric results of the Schriidinger equation. The lower part of the figure shows WI(x m ) and W3(O) in comparison with the average of the Schriidinger energies (small dots) . Note that WI misses the slope by 25%. Thnneling causes a level splitting to be calculated in Chapter 17 (dotted curves). Note that the approximation WI(x m ) does not possess the correct slope in 9, which is missed by 25%. In fact, a Taylor expansion of WI (Xm) reads (5.257) whereas the true expansion starts out with (5.258) The optimal frequency associated with (5.257) has the expansion !/1(Xm

In

)

= v2 -

3

27V2

4: 9 - 64 9

2

27 3 - 32 9

+ ...

Let us also compare the xo-behavior of W 3 (xo) with that of the true effective classical potential calculated numerically by Monte Carlo simulations. The curves are plotted in Fig. 5.5, and the

5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions

523

agreement is seen to be excellent. There are significant deviations only for low temperatures with f3 ;:: 20. At zero temperature, there exists a simple way of recovering the effective classical potential from the classical potential calculated up to two loops in Eq. (3.767). As we learned in Eq. (3.830), Xo coincides at zero temperature. with X in Eq. (3.767) Thus we merely have to employ the square-root trick (5.186) to the effective potential (3.767) and interchange X by Xo to obtain the variational approximation W1(xo) to the effective classical potantial to be varied. Explicitly, we replace, as in (5.188) , the one-loop contribtions, ground state energies nwT,L in (3.767) by

Vrl~,L + (W~,L(X) - rI~,L) "':; rlT,L + (W~,L(X) - rI~,d/2r1T,L' and exchange WL ,T(X) of the remaining two-loop terms by rlT,L. This yields for the D-dimensional rotated double-well potential (the Mexical hat potential in Fig. 3.13]: _ { W(xo) T='O v(X)

+ Ii

[rlL 4

_ + Wl,(X)] 4r1L + (D

D2 - 1 [VII(X) _ V1(X)]

+ rI~

li 2 - 6(2M)3

X2

{Ii

X3

+

1)1i

[rlT 4

~{...!.... (4) + Wf(X)] 4r1T + 8(2M)2 rli v (X)

2(D - 1) [VIII (X) _ 2V"(X) rlLrlT X X2

+

2V1(X)]} X3

2+ 2r1T 3(D-l) 1 [VII(X) VI(X)]2} + O(Ii)3} x~·xo(5.259) + rlL rlLrlf ---x- - --x'2

III

3 r1 [v (X) ]

This has to be extremized in rlT,L at fixed Xo .

5.21

Higher-Order Effective Classical Potential for Nonpolynomial Interactions

The systematic improvement of the Feynman-Kleinert approximation in Section 5.13 was based on Feynman diagrams and therefore applicable only to polynomial potentials. If we want to calculate higher-order effective classical potentials for nonpolynomial interactions such as the Coulomb interaction, we need a generalization of the smearing rule (5 .30) to the correlation functions of interaction potentials which occur in the expansion (5.180). The second-order term, for example, requires the calculation of

~oc = ,

l0

lif3 llif3 dT dT' (Vj~f(X(T))Vj~nX(T')))~Oc' 0

(5.260)

'

where 1 Vj~nx) = V(x) - ZMrl 2(xo)(x - XO)2 .

(5.261)

Thus we need an efficient smearing formula for local expectations of the form

(Fl(X(Tl))" X

f

.Fn(X(Tn)))~O =

z:'o n

'Dx(T)Fl(X(Tl))'" Fn(x(Tn))8(x - xo) exp { -~Ano [X(T)]} ,

(5.262)

where AnD [X(T)] and z~o are the local action and partition function of Eqs. (5.3) and (5.4). After rearranging the correlation functions to connected ones according to Eqs. (5.182) we find the cumulant expansion for the effective classical potential [see (5.180)]

1if3

Veff,cl(x o )=F.nxo

+~ )))XO lif3 JdT1 (v,xO(X(T mt 1 n ,c o

524

5 Variational Perturbation Theory h{3

- 2:2(3

h{3

J J (V;~~(X(T1))V;~~(X(T2)))~~c J J J (V;~~(X(T1))V;~~(X(T2))V;~~(xh)))~~c + ... . dT1

dT2

o

h{3

+ 6:3(3

(5 .263)

0

h{3

dT1

o

h{3

dT2

0

dT3

0

It differs from the previous expansion (5.180) for polynomial interactions by the potential V(x)

not being expanded around Xo. The first term on the right-hand side is the local free energy (5.6).

5.21.1

Evaluation of Path Integrals

The local pair correlation function was given in Eq. (5.19):

(8X(T)8x(T'))~O == ([X(T) -

XO][X(T') -

xo])~O

=

~G~)XO(T' T') =

a;T'(xo),

(5.264)

with [recall (5.19)- (5.24)]

n cosh[O(xo)(T-T' - n(3/2)] 2 aTT,(xo) = 2MO(xo) sinh[O(xo)n(3/2]

1 T

E (0, n(3) .

(5.265)

Higher correlation functions are expanded in products of these according to Wick's rule (3.302). For an even number of 8X(T)'S one has

(8X(T1)8x(T2) '" 8X(Tn))~O =

2: a;p(1)Tp

(2)

(5.266)

(xo) · ·· a;p(n _l)Tp(n) (XO),

pairs

where the sum runs over all (n - I)!! pair contractions. For an exponential, Wick's rule implies /

[

\ exp i

1

Inserting j(T) =

h{3

dTj(T) 8X(T)

2:::1 ki8(T -

]

) XO

n = exp

[

-~

1 1 h{3

dT

h{3

]

dT'j(T)a;T,(xo)j(T') .

(5.267)

Ti) , this gives for the expectation value of a sum of exponentials (5.268)

By Fourier-decomposing the functions F(X(T)) = J(dk/211')F(k) expik [xo obtain from (5.268) the new smearing formula

+ 8X(T)]

in (5.262), we

(5.269)

where a.;:;~,(xo) is the inverse of the n x n -matrix a~iT,(xo), This smearing formula determines the harmonic expectation values in the variational perturbation expansion (5.263) as convolutions with Gaussian functions. For n = 1 and only the diagonal elements a2(xo) = a;'T(xo) appear in the smearing formula (5.269), which reduces to the previous one in Eq. (5.30) [F(X(T)) = V(X(T)) ].

5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions

525

For polynomials F(X(T)) , we set X(T) = Xo + 8X(T) and expand in powers of 8X(T), and see that the smearing formula (5.269) reproduces the Wick expansion (5.266) . For two functions, the smearing formula (5.269) reads explicitly

(5.270) Specializing F2(x(T2)) to quadratic functions in X(T), we obtain from this

(F1 (x (T1 ))) flXO a2( Xo )

+

[1 -

a~'T2(XO)] a4 (XO)

IF ( ( )) [ ( ) l2)XO a~'T2(xO) 1 X T1 X T1 - Xo fl a4 (XO) ,

\

(5.271)

and

\I [x(Td - xol2 [xh) - xol 2)XO fl

5.21.2

=

a4 (Xo)

4 + 2aT1T2 (xo).

(5.272)

Higher-Order Smearing Formula in D Dimensions

The smearing formula can easily be generalized to D-dimensional systems, where the local pair correlation function (5.264) becomes a D x D -dimensional matrix: (5.273) For rotationally-invariant systems, the matrix can be decomposed in the same way as the trial frequency fI;j(xo) in (5.95) into longitudinal and transversal components with respect to Xo: (5.274) where PL;ij(XO) and PT;ij(XO) are longitudinal and transversal projection matrices introduced in (5.96). Denoting the matrix (5.274) by a; T' (xo), we can write the D-dimensional generalization of the smearing formula (5.275) as '

(5.275)

The inverse D x D -matrix a;k2,.k' (xo) is formed by simply inverting the n x n -matrices aL- ?Tk T k' (To), ";::?T T (ro) in the projection formula (5.274) with projection matrices Pdxo) and -1", k k' PT(xo): (5.276) a;k2,.k' (Xo) = aL;~kTk' (ro)pdxo) + aT~TkTk' (ro)PT(xo), 1

In D dimensions, the trial potential contains a D x D frequency matrix and reads

526

5 Variational Perturbation Theory

with the analogous decomposition 2 ) 0ij ( Xo

XOiXOJ'

2

= 0L(XO) - 2-

rO

+ 0T2 ( Xo )

(

8ij

-

XOiXOJ' ) -2-

.

(5.277)

XOj).

(5.278)

ro

The interaction potential (5.261) becomes

V;~~(x) =

V(x) -

~O~j(XO)(Xi -

XOi)(Xj -

To first order, the anisotropic smearing formula (5.275) reads (5.279) with the special cases =

2a},

(5.280)

Inserting this into formula (5.263) we obtain the first-order approximation for the effective classical potential 1i/3

Wl(xo)=F~o + n~

J

dTl

(V;~~(xh)))~~c'

(5.281)

o

in agreement with the earlier result (5.97) . To second-order, the smearing formula (5.275) yields [33]

(5.282) so that rule (5.271) for expectation values generalizes to

(5.283)

(5.284)

(5.285)

2a}ai ,

(5.286) (5.287)

5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions

5.21.3

527

Isotropic Second-Order Approximation to Coulomb Problem

To demonstrate the use of the higher-order smearing formula (5.275), we calculate the effective classical potential of the three-dimensional Coulomb potential e2 V(x) = -~

(5.288)

to second order in variational perturbation theory, thus going beyond the earlier results in Eq. (5.53) and Section 5.10. The interaction potential corresponding to (5.278) is

V;~~(x) =

~ f!2(XO)(x -

-1:1 -

XO)2 .

(5.289)

For simplicity, we consider only the isotropic approximation with only a single trial frequency. Then all formulas derived in the beginning of this section have a trivial extension to three dimensions. Better results will, of course be obtained with two trial frequencies f!i(ro) and f!~(ro) of Section 5.9. The Fourier transform 47re 2flkl2 of the Coulomb potential e2fixi is most conveniently written in a proper-time type of representation as V(k)

4~e21°O dae - ak2/2- ikx,

=

(5.290)

where a has the dimension length square. The lowest-order smeared potentials were calculated before in Section 5.10. For brevity, we consider here only the isotropic approximation in which longitudinal and transverse trial frequencies are identified [compare (5.112)] : \ [xh) - xof)

~o

=

3a2(xo) ,

(IX(~l)I):o

=

1:01 erf [

~] .

(5.291)

The first-order variational approximation to the effective classical potential (5.281) is then given by the earlier-calculated expression (5.115). To second order in variational perturbation theory we calculate expectation values

+ 6a;,T2 (xo) ,

(5.292)

2[3a (xo) - a;,T2 (xo)] .J7ra6 (xo)

(5.293)

9a4 (xo) 4

which follow from the obvious generalization of (5.271) , (5.272) to three dimensions. More involved is the Coulomb-Coulomb correlation function / 1 1 \ IX(Tl)llx(T2)1

)XO fl

(5.294)

Using these smearing results we calculate the second connected correlation functions of the interaction potential (5.289) appearing in (5.263) and find the effective classical potential to second order in variational perturbation theory

W1(xo)

+

[Me f!(Xo) 2

1i.J27ra6 (xo)

4

e Jr.f3

21i

o

_

3M f!3(XO)] 2

[4(xO)

41i

/ 1 1 dT \ IX(T) l lx(O)1

)XO fl

(5.295)

528

5 Variational Perturbation Theory

with the abbreviation

4(

) _ /i [4 + /i 2,B2f!2(xO) - 4 cosh /i,Bf!(xo)

+ /i,Bf!(xo) sinh /i,Bf!(xo) J 8,BM2f!3(xo) sinh[/i,Bf!(xo)j2] ,

I Xo =

(5.296)

the symbol indicating that this is a quantity of dimension length to the forth power. After an extremization of (5.295) with respect to the trial frequency f!(xo) , which has to be done numerically, we obtain the second-order approximation for the effective classical potential of the Coulomb system plotted in Fig. 5.25 for various temperatures. The curves lie all below the first-order ones, and the difference between the two decreases with increasing temperature and increasing distance from the origin.

:: oo\c:,cc~co:;>8fg0 ·0.6

/:;>"/,/ / .

/./:. ,/

/

-1

/

/~'/~

·0.8

/

-" .

.

.

V(ro)

,/ f

! 0.5

1.5

2.5

3.5

ro

Figure 5.25 Isotropic approximation to effective classical potential of Coulomb system in t he first (lines) and second order (dots). The temperatures are 10- 4 , 10- 3, 10- 2,10- 1, and 00 from top to bottom in atomic units. Compare also Fig. 5.9.

5.21.4

Anisotropic Second-Order Approximation to Coulomb Problem

The first-order effective classical potential W 1 (xo) was derived in Eqs. (5.97) and (5.117)- (5.121) . To obtain the second-order approximation W 2 (xo), we insert the Coulomb potential in the representation (5.290) into the second-order smearing formula (5.282) , and find

These are special cases of the more general expectation value

5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions

529

which furthermore leads to

(5.299)

From these smearing results we calculate the second-order approximation to the effective classical potential n{3

W2(xo) = F~o + h~

J

dTl

n{3

(Vi~~(X(Td))~~c -

2;2/3

o

n{3

JJ dTl

0

dT2

(Vi~~(X(Td)Vi~~(X(T2)))~~c . (5.300)

0

The result is

with the abbreviation

1

4 h [4 + h2/32n~,L - 4 cosh h/3nT,L + h/3nT,L sinh h/3nT,L IT,L = 8/3M2n~,L sinh[h/3nT,L/2] ,

(5.302)

which is a quantity of dimension (length)4. After an extremization of (5.115) and (5.301) with respect to the trial frequencies nT, n L which has to be done numerically, we obtain the second-order approximation for the effective classical potential of the Coulomb system plotted in Fig. 5.26 for various temperatures. The second order curves lie all below the first-order ones, and the difference between the two decreases with increasing temperature and increasing distance from the origin.

5.21.5

Zero-Temperature Limit

As a cross check of our result we take (5.301) to the limit T --t O. Just as in the lowest-order discussion in Sect. (5.4), the xo-integral can be evaluated in the saddle-point approximation which becomes exact in this limit, so that the minimum of WN(xo) in xo yields the nth approximation to the free energy at T = 0 and thus the nth approximations EYP the ground state energy E(O) of the Coulomb system. In this limit, the results should coincide with those derived from a direct variational treatment of the Rayleigh-Schrodinger perturbation expansion in Section 3.18. With the help of such a treatment, we shall also carry the approximation to the next order, thereby illustrating the convergence of the variational perturbation expansions. For symmetry reasons, the minimum of the effective classical potential occurs for all temperatures at the origin, such that we may restrict (5.115) and (5.301) to this point. Recalling the zero-temperature limit of the two-point correlations (5.19) from (3.246), lim a;T'(xo) =

{3~=

2

M~( Xo ) exp{ -n(xo) IT -

Til} ,

(5.303)

530

5 Variational Perturbation Theory

we immediately deduce for the first order approximation (5.115) with n = ncO) the limit

~nn - ~JMn e2 . 4 ,.fir n

E(O)(n) = lim Wfl(O) = {3-->oo

1

1

(5.304)

In the second-order expression (5.301) , the zero-temperature limit is more tedious to take. Performing the integrals over a1 and a2 , we obtain the connected correlation function (5.305) Inserting (5.303) , setting T1 = 0 and integrating over the imaginary times T = T2 E [0, n,8], we find

J/ 1i{3

1 1 )XO 4M { n2,82n2 dT \ IX(T) llx(O)1 fl c{3!";oo n2,8n eli{3fl - 1 - n,8n o '

x[e

li{3fl

2

-'Tr- - ;

arcsin

viI -

e-

21i{3fl +

~ In 0«,8) - ~ [In 0«,8)]2 - ~ J1 2

8

du In u

1}

l+u'

2

(5.306)

a({3)

with the abbreviation 10«,8) = 1 +

VI VI -

Inserting this into (5.301) and going to the limit ,8 (0)

.

fl

9

3

E2 (n)=J~~w2(0)=16nn-2,.fir -0.2

e 21i{3fl e 21i{3fl --> 00

JMn

(5.307)

we obtain

'Tr)

2 4 ( M 4 ----,;-e - ; 1+ln2-'2 n2e .

(5.308)

r------r-----.-----,------,---=~

-0.4

-0.6

-0.8

WfT=flL(ro) Wfn'flL(ro)

-1

V(ro) -1.2

L -_ _ _-'---'---_ _ _ _-'--_ _ _ _---"-_ _ _ _---'_ _ _ _- - - '

o

2

3

4

5

ro Figure 5.26 Isotropic and anisotropic approximations to effective classical potential of Coulomb system in first and second order at temperature 0.1 in atomic units. The lowest line represents the high temperature limit in which all isotropic and anisotropic approximations coincide.

5.21 Higher-Order Effective Classical Potential for Nonpolynomial Interactions

531

Postponing for a moment the extremization of (5.304) and (5.308) with respect to the trial frequency n, let us first rederive this result from a variational treatment of the ordinary RayleighSchrodinger perturbation expansion for the ground state energy in Section 3.18. According to the replacement rule (5.186), we must first calculate the ground state energy for a Coulomb potential in the presence of a harmonic potential of frequency w: M

2

e2

2

V(x) = 2W x - ~.

(5.309)

After this, we make the trivial replacement w -> v'n 2 + w2 - n2 and re-expand the energy in powers of w2 - n2, considering this quantity as being of the order e2 and truncating the reexpansion accordingly. At the end we go to w = 0, since the original Coulomb system contains no oscillator potential. The result of this treatment will be precisely the expansions (5.304) and (5.308). The Rayleigh-Schrodinger perturbation expansion of the ground state energy Er;p(w) for the potential (5.309) in Section 3.18 requires knowledge of the matrix elements of the Coulomb potential (5.288) with respect to the eigenfunctions of the harmonic oscillator with the frequency w : 271'

Vn,l,m ;n',l',m' =

7!'

00

JJ

Jdrr21j;~,I,m(r,19, T' > 0,

(5.384)

such that

Gr2(T T') = ~ (e-r(T-T') - 1- e-r(T+T') + ~e-2rT + ~e-2rT') ,

2r

2

2

'

for T > T' > O.

(5.385) In the limit r ---. 0, this becomes -!IT - T'I. We therefore obtain at zero temperature , n G-fl 'r (T, T) T~O 2M

[0r

{0

2 n -_ - - 1T-T 'I 2M

f2

2- ( , Go T, T)

2

_0

+r

r 2- -2 ( 1 +-

f3

0

2- 2 ')] f2 GO(T, T

(5 .386)

1 - 2rT - -e 1 - 2rT')} e- r IT- T'1+ e - r(T+T') - -e 2 2 '

to be inserted into (5.378) to get the expectation value (eik[X(T)-X(T')l) . The last three terms can be avoided by shifting the time interval under consideration and thus the Fourier expansion (5.352) from (0, n(3) to (-nf3 /2, nf3 /2) , which changes Green function (5.387) to

Gw2(T, T') =

L: sin Vn(T + nf3/2) sin Vn(T' + nf3/2) n=1 v; + w oo

2

sinhw(nf3/2 - T) sinhw(T' + nf3/2) £ T > T' > O. W sinh wnf3 ' or

(5.387)

541

5.22 Polarons

We have seen before at the end of Section 3.20 that such a shift is important when discussing the limit T ---> 0 which we want to do in the sequel. With the symmetric limits of integration, the Green function (5.384) looses its last term [compare with (3.147) for real times] and (5.386) simplifies to

n { -IT [22 Co,r(T T') ~ - - T'I ,

T=O

2M

r2

+r

2 - [22

f3

(1- e-r 1r - r , I) } .

At any temperature, we have the complicated expression for

Co,r(T, T') = -

2~ {~;

[T - T' -

n~ (T -

T

> T':

T')2]

r~::n~~;rf1 [sinhrf1(nN2-T) sinhrf1 (T' + n/3/2)-(T'

(5.388)

(5 .389) --->

T)-(T

--->

T')] }.

With the help of the Fourier integral (5.346) we find from this the expectation value of the interaction in (5.347) :

For zero temperature, this leads directly to the expectation value of the interaction in (5.347):

The expectation value of the harmonic trial interaction in (5.356), on the other hand, is simply found from the correlation function (5.376) [or equivalently from the second derivative of 1°,G (k, T, T') with respect to the momenta] : / \[X(T) - X(T')] 2)0,C = -6G- 0 'r (T, T').

(5.392)

At low temperatures, this leads to an integral (5.393) This expectation value contributes to the ground state energy a term (5.394)

542

5 Variational Perturbation Theory

Note that this term can be derived from the derivative of the ground state energy (5.368) as C8cE~,c. Together with -a/2V2 times the result of (5.391), the inequality (5.370) for the ground state energy becomes Eo

311,

0 W

2

< - (f - n) - liw--

10

y'7rW

- 4f

00

dt17

0

e-W~T

V-2Cfl,['(t17,0)

.

(5.395)

This has to be minimized in nand C, or equivalently, in nand f. Considering the low-temperature limit, we have taken the upper limit of integration to infinity (the frequency w corresponds usually to temperatures of the order of 1000 K). For small 0 , the optimal parameters nand f differ by terms of order o . We can therefore expand the integral in (5.395) and find that the minimum lies, in natural units with 11, = w = 1, at n = 3 and f = 3[1 + 20(1 - P)/3f, where P = 2[(1 - r)1/2 - 1]. From this we obtain the upper bound Eo :S

02 -0 -

81

+ ... r::::; - 0 -

0.01230 2

+ ...

(5.396)

This agrees well with the perturbative result [21] E!X

=

-0 -

0.01591962200 2

-

0.0008060700480 3

-

0(0 4 ).

(5.397)

The second term has the exact value (5.398)

In the strong-coupling region, the best parameters are n = 1, f = 40 2 /97T - [4 (log 2+ 1/2) - 1], where I r::::; 0.5773156649 is the Euler-Mascheroni constant (2.467). At these values, we obtain the upper bound 2

0 Eo :S - 37T - 3

(14 + log 2) + 0(0-

2 ) r::::;

-0.10610 - 2.8294 + 0(0- 2 ).

(5.399)

This agrees reasonably well with the precise strong-coupling expansion [23]. (5.400)

The numerical results for variational parameters and energy are shown in Table 5.12.

5.22.3

Effective Mass

By performing a shift in the velocity of the path integral (5.347) , Feynman calculated also an effective mass for the polaron. The result is (5.401)

543

5.22 Polarons

Table 5.12 Numerical results for variational parameters and energy.

a

r

n

1 3 5 7 9 11 15

3.110 3.421 4.034 5.S10 9.S50 15.41 30.0S

2.S71 2.560 2.140 1.604 1.2S2 1.162 1.076

~Eci2)

Eo -1.01 -3.13 -5.44 -S.l1 -11.5 -15.7 -26.7

E tot

-0.0035 -0.031 -0.OS3 -0.13 -0.17 -0.22 -0.39

-1.02 -3.16 -5.52 -S.24 -11.7 -15.9 -27.1

correction 0.35% 1.0% 1.5% 1.6% 1.4% 1.4% 1.5%

The reduced effective mass m == M eff / M has the weak-coupling expansion mw =

1+

a

"6 + 2.469136 x

1O- 2a 2 + 3.566719 x 1O-3a 3 +

...

(5.402)

and behaves for strong couplings like ms ~ ~

16 4 4 Sl7r2a - 31f (1

0.0201410: 4 -

2

+ log 4) a + 11.S5579 + .. . 1.0127750?+ 11.85579 + ... .

(5.403)

The exact expansions are [31]

a

1 +"6

+ 2.362763 x

0.0227019a4

5.22.4

1O- 2a 2 + O(a 4 ),

(5.404)

+ O(a 2 ).

(5.405)

Second-Order Correction

With some effort, also the second-order contribution to the variational energy has been calculated at zero temperature [32]. It gives a contribution to the ground state energy (2) _ ~Eo -

1 1 I - 21if31i2 \ (Ant

2)n,c -

Aint,harm)

c

(5.406)

.

Recall the definitions of the interactions in Eqs. (5.371) and (5.372). There are three terms

~E(2,1) = __1_~ o

21if31i2

{I\

A2 )n,c "'-'nt

_ [(A )n,C]2} mt

,

(5.407)

544

5 Variational Perturbation Theory

and (2,3)

6.Eo

_ -

1 1 - 2fi{3 fi2

{I\Aint,harm 2 )n,c -

[

(Aint,harm)

n,c]2}

.

(5.409)

The second term can be written as 1

(2,2) _

6.Eo

1 {

- - 2fi{3 fi2 -2Coc (Aint)

n,c}

the third as (2,3) _

6.Eo

1

1 {

- - 2fi{3 fi2 {I - Coel (Aint,harm)

,

n,c} .

(5.410)

(5.411)

The final expression is rather involved and given in Appendix 5C. The second-order correction leads to the second term (5.398) found in perturbation theory. In the strong coupling limit, it changes the leading term _0: 2 /37r ~ -0.1061 in (5.399) into 1 2 - 47r -;:

(2n)! 24n(n!)2n(2n + 1)

E 00

-17 +64 arcsin( \h;v'3) - 321og(4 Y2;v'3) ( ) 47r ' 5.412

which is approximately equal to -0.1078. The corrections are shown numerically in the previous Table 5.12.

5.22.5

Polaron in Magnetic Field, Bipolarons, Small Polarons, Polaronic Excitons, and More

Feynman's solution of the polaron problem has instigated a great deal of research on this subject [34]. There are many publications dealing with a polaron in a magnetic field. In particular, there was considerable discussion on the validity of the JensenPeierls inequality (5.10) in the presence of a magnetic field until it was shown by Larsen in 1985 that the variational energy does indeed lie below the exact energy for sufficiently strong magnetic fields . On the basis of this result he criticized the entire approach. The problem was, however, solved by Devreese and collaborators who determined the range of variational parameters for which the inequality remained valid. In the light of the systematic higher-order variational perturbation theory developed in this chapter we do not consider problems with the inequality any more as an obstacle to variational procedures. The optimization procedure introduced in Section 5.13 for even and odd approximations does not require an inequality. We have seen that for higher orders, the exact result will be approached rapidly with exponential convergence. The inequality is useful only in Feynman's original lowest-order variational approach where it is important to know the direction of the error. For higher orders, the importance of this information decreases rapidly since the convergence behavior allows us to estimate the limiting value quantitatively, whereas the inequality tells us merely the sign of the error which is often quite large in the lowest-order variational approach, for instance in the Coulomb system.

545

5.22 Polarons

There is also considerable interest in bound states of two polarons called bipolarons. Such investigations have become popular since the discovery of hightemperature superconductivity. 3

5.22.6

Variational Interpolation for Polaron Energy and Mass

Let us apply the method of variational interpolation developed in Section 5.18 to the polaron. Starting from the presently known weak-coupling expansions (5.397) and (5.404) we fix a few more expansion coefficients such that the curves fit also the strong-coupling expansions (5.400) and (5.405) . We find it convenient to make the series start out with a O by removing an overall factor -a from E and deal with the quantity -E!x/ a . Then we see from (5.400) that the correct leading power in the strong-coupling expansion requires taking p = 1, q = 1. The knowledge of bo and bl allows us to extend the known weak coupling expansion (5.397) by two further expansion terms. Their coefficients a3, a4 are solutions of the equations [recall (5.245)]

35 15 a2 128 aoc + al + 8-;;35ao 5a2 a3 - - - -- - 32 c 4 c3 c3 '

bo

2a3

35

15 a2 c2

8

4a3 -

~ -

(5.413) (5.414)

The constant c governing the growth of ON for a bo in c, which yields the equation 128 ao -

a4

+ --;} + c3'

---> 00

4a4

is obtained by extremizing

0

7 = .

(5.415)

The simultaneous solution of (5.413)- (5.415) renders

0.09819868, 6.43047343 x 10- 4 , -8.4505836 X 10-5 .

(5.416)

The re-expanded energy (5.238) reads explicitly as a function of a and 0 (for E including the earlier-removed factor -a)

(5.417) Extremizing this we find 0 4 as a function of a [it turns out to be quite well approximated by the simple function 0 4 ~ c4a+1/(1+0.07a)]. This is to be compared with the optimal frequency obtained from minimizing the lower approximation W 2 (a, 0):

O~ = 1 + 4a2x2 + 3ao

1(1

Y

+ 4a2x2 )2 -1, 3ao

(5.418)

546

5 Variational Perturbation Theory

which behaves like C2o+1+ . . . with C2 = V8a2/3ao ~ 0.120154. The resulting energy is shown in Fig. 5.28, where it is compared with the Feynman variational energy. For completeness, we have also plotted the weak-coupling expansion, the strongcoupling expansion, the lower approximation W2 (a) , and two Pade approximants given in Ref. [22] as upper and lower bounds to the energy.

-5

- 10 -15 -20 -25

-30

E

Figure 5.28 Variational interpolation of polaron energy (solid line) between the weakcoupling expansion (dashed) and the strong-coupling expansion (short-dashed) shown in comparison with Feynman's variational approximation (fat dots) , which is an upper bound to the energy. The dotted curves are upper and lower bounds coming from Pade approximants [22] . The dot-dashed curve shows the variational perturbation theory W 2 (a) which does not make use of the strong-coupling information.

Consider now the effective mass of the polaron, where the strong-coupling behavior (5.405) fixes p = 4, q = 1. The coefficient bo allows us to determine an approximate coefficient a3 and to calculate the variational perturbation expansion W3 (a) . From (5 .245) we find the equation (5.419)

whose minimum lies at C3 = V8a2/3ao where bo = V32aV27al' Equating bo of Eq. (5.419) with the leading coefficient in the strong-coupling expansion (5.405), we obtain a3 = [27a1b5 /32j1/3 ~ 0.0416929. The variational expression for the polaron mass is from (5.238) (5.420)

This is extremal at (5.421)

547

5.22 Polarons

2.5

2 1.5

\. 0. 5

~.

o

2

4

~

•

•

•

•

•

- .-.~ --.

6

8

10

12

14

Figure 5.29 Variational interpolation of polaron effective mass between the weak(dashed) and strong-coupling expansions (short-dashed). To see better the differences between the strongly rising functions, we have divided out the asymptotic behavior mas = 1 + bo0:4 before plotting the curves. The fat dots show Feynman's variational approximation. The dotted curves are upper and lower bounds coming from Pade approximants [22].

From this we may find once more c3 = V8a2/3ao . The approximation W3 (0:) = W3 (0: ,D3 ) for the polaron mass is shown in Fig. 5.29, where it is compared with the weak and strong-coupling expansions and with Feynman's variational result . To see better the differences between the curves which all grow fast with 0:, we have divided out the asymptotic behavior mas = 1 + b0 0: 4 before plotting the data. As for the energy, we have again displayed two Pade approximants given in Ref. [22] as upper and lower bounds to the energy. Note that our interpolation differs considerably from Feynman's and higher order expansion coefficients in the weak- or the strongcoupling expansions will be necessary to find out which is the true behavior of the model. Our curve has, incidentally, the strong-coupling expansion

m S = 0.02270190:4 + 0.1257220: 2 + 1.15304 + 0(0:- 2),

(5.422)

the second term ex: 0:2-term being in sharp contrast with Feynman's expression (5.403). On the weak-coupling side, a comparison of our expansion with Feynman's in Eq. (5.402) shows that our coefficient a3 ;:::; 0.0416929 is about 10 times larger than his. Both differences are the reason for our curve forming a positive arch in Fig. 2, whereas Feynman's has a valley. It will be interesting to find out how the polaron mass really behaves. This would be possible by calculating a few more terms in either the weak- or the strong-coupling expansion.

548

5 Variational Perturbation Theory

Note that our interpolation algorithm is much more powerful than Pade's. First, we can account for an arbitrary fractional leading power behavior o;P as 0; ---> 00. Second, the successive lower powers in the strong-coupling expansion can be spaced by an arbitrary 2/ q. Third, our functions have in general a cut in the complex 0;plane approximating the cuts in the function to be interpolated (see the discussion in Subsection 17.10.4). Pade approximants, in contrast, have always an integer power behavior in the strong-coupling limit, a unit spacing in the strong-coupling expansion, and poles to approximate cuts.

5.23

Density Matrices

In path integrals with fixed end points, the separate treatment of the path average Xo (l / n!3)Jon/3 dr x( r) looses its special virtues. Recall that the success of this separation in the variational approach was based on the fact that for fixed Xo, the fluctuation square width a2(xo) shrinks to zero for large temperatures like n2 / 12MkB T [recall (5.25)] . A similar shrinking occurs for paths whose endpoints held fixed, which is the case in path integrals for the density. Thus there is no need for a separate treatment of Xo , and one may develop a variational perturbation theory for fixed endpoints instead. These may, moreover, be taken to be different from one another Xb # Xa , thus allowing us to calculate directly density matrices.4 The density matrix is defined by the normalized expression

=

P(Xb, xa) =

~P(Xb' xa) ,

(5.423)

where P(Xb , xa) is the unnormalized transition amplitude given by the path integral

J

P(Xb' xa) = (Xb n!3lxaO) =

'Dx exp {-A[x]/n} ,

(5.424)

(Xa ,O),,-,,(x b,n/ kBT)

summing all paths with the fixed endpoints x(O) = Xa and x(n/kBT) = Xb. The diagonal matrix elements of the density matrix in the integrand yield, of course, the particle density (5.90) . The diagonal elements coincide with the partition function density z(x) introduced in Eq. (2.331). The partition function divided out in (5.423) is found from the trace 00

z=

Jdxp(x,x).

(5.425)

-00

5.23.1

Harmonic Oscillator

As usual in the variational approach, we shall base the approximations to be developed on the exactly solvable density matrix of the harmonic oscillator. For the sake of generality, this will be assumed to be centered around X m , with an action

An,xm [x]

=

r

Jo

n / k BT

dr

{ I} "2 Mi?(r)I + "2M02[x(r) - xm]2 .

(5.426)

4H. Kleinert, M. Bachmann, and A. Pelster, Phys. Rev. A 60 , 3429 (1999) (quant-ph/ 9812063).

549

5.23 Density Matrices

Its unnormalized density matrix is [see (2.409)]

with the abbreviation (5.428) At fixed endpoints Xb, Xa and oscillation center Xm , the quantum-mechanical correlation functions are given by the path integral 1 (Ol(x(rl)) 02(x(r2))'" )~~~x: = _ n,Xm ( ) Po Xb, Xa

J

x

VxOl(x(rl)) O2 (x(r2)) .. . exp{ -An,xm[xJlfi}. (5.429)

(Xa,O)""'(Xb,li/kBT)

The path x(r) at a fixed imaginary time r has a distribution p(x,r)

where

Xci (r)

_

n,Xm b, a

1 [ (x - XcI(r))2] expb2() , V 27rb2(r) 2 r

= (b(x-x(r)))x X = I

(5.430)

is the classical path of a particle in the harmonic potential Xci

( ) _ xbsinhnr+xasinhn(fi/kBT-r) r sinhfin/kBT '

(5.431)

and b2 (r) is the square width b2 (r)

= _fi_ {coth fin _ cosh[n(2r -fi/kBT)]} . kBT

2Mn

sinhfin/kBT

(5.432)

In contrast to the square width a2 (xo) in Eq. (5.24) this depends on the Euclidean time r, which makes calculations more cumbersome than before. Since the r lies in the interval 0 ::::: r ::::: fi/kBT, the width (5.432) is bounded by 2 fi fin b (r) ::::: 2Mn tanh 2k BT'

(5.433

)

thus sharing with a2 (xo) the property of remaining finite at all temperatures. The temporal average of (5.432) is -2 kBT b = T

rli/ kBT drb (r) =

io

2

fi ( fin kBT) 2Mn coth kBT - fin .

(5.434)

Just as a2 (xo), this goes to zero for T --+ 00. Note however, that the asymptotic behavior is b2 ---+ fin/6k BT, (5.435) T -->oo

which is twice as big as that of a 2 (xo) in Eq. (5.25) (see Fig. 5.30).

550

5 Variational Perturbation Theory

1.0

" " " " " I

I

a~ot

I

I

,,'

,,"

0.5

""

/

/

/

/

/

/

/

/

fluctuation width

/

/

//

II

2 atot,cl

/

\

/ \

/

0.0 1.0

0.0

2.0

Figure 5.30 Temperature dependence of fluctuation widths of any point X(T) on the path in a harmonic oscillator (l2 is the generic square length in units of 'hI MQ). The quantity a2 (dashed) is the quantum-mechanical width, whereas a2 (xo) (dash-dotted) shares the width after separating out the fluctuations around the path average Xo. The quantity a~l (long-dashed) is the width of the classical distribution, and b2 (solid curve) is the fluctuation width at fixed ends which is relevant for the calculation of the density matrix by variational perturbation theory (compare Fig. 3.14).

5.23.2

Variational Perturbation Theory for Density Matrices

To obtain a variational approximation for the density matrix, we separate the full action into the harmonic trial action and a remainder (5.436) with an interaction

Andx(T)] =

rn

io

{3

(5.437)

dTVint(X(T)) ,

where the interaction potential is the difference between the original one V(x) and the inserted displaced harmonic oscillator: (5.438) The path integral (5.424) is then expanded perturbatively around the harmonic expression (5.427) as

P(Xb,X a) = pofl,Xm(Xb,Xa) [1-

~(Aindx] )~,xxm + ~/ Arnt[x] )fl,Xm - ... J, 21i \ It

b, a

Xb,Xa

(5.439)

551

5.23 Density Matrices

with the harmonic expectation values defined in (5.429). The sum can be evaluated as an exponential of its connected parts, going over to the cumulant expansion:

P(Xb, xa) =pofl,xm (Xb, xa) exp

[-~( Aindx] )~,Xxma, + ~/ Afnt [x] \ fl ,x m 217, \ / C

b,

II,

-

. ..

Xb,Xa,C

J,(5.440)

where the cumulants are defined as usual [see (3.482), (3.483)]. The series (5.440) is truncated after the N-th term, resulting in the N-th order approximant for the quantum statistical density matrix ,. " f2,Xm

PN

_

,.., f2,Xm

(Xb, xa) - Po

(Xb, xa) exp

[

E n!lin N

(-

l)n

n

f2 ,Xm

]

(Aint[x] )Xb,Xa,C ,

(5.441)

which explicitly depends on the two variational parameters 0 and X m . By analogy with classical statistics, where the Boltzmann distribution in configuration space is controlled by the classical potential V (x) according to [recall (2.352)]

-fi T exp [-,8V(x)] , Y~

Pcb) =

(5.442)

we shall now work with the alternative type of effective classical potential Veff,cl(x a, Xb) introduced in Subsection 3.25.4. It governs the unnormalized density matrix [see Eq. (3.834)]

P(Xb, xa) =

J27rli~ ,8 exp [_,8Veff,cl(Xb, Xa)].

(5.443)

Variational approximations to Veff,cl(Xb, xa) of Nth order are obtained from (5.427) , (5.441), and (5.443) as a cumulant expansion

Wfl ,xm( ) 1 I sinh li,80 N Xb, Xa = 2,8 n li,80

MO + 2li,8 sinh li,80

{( -2

xb

-2) h li,80 + xa cos -

T - }

XbXa

(5.444) They have to be optimized in the variational parameters 0 and Xm for a pair of endpoints Xb, Xa . The result is denoted by W N(Xb, xa). The optimal values O(xa, Xb) and xm(x a, Xb) are denoted by ON(X a, Xb), x;;,(xa, Xb). The Nth-order approximation for the normalized density matrix is then given by

PN ( Xb,Xa) = ZN- 1-PNn~,X~( Xb,Xa) ,

(5.445)

where the corresponding partition function reads (5.446)

In principle, one could also optimize the entire ratio (5.445), but this would be harder to do in practice. Moreover, the optimization of the unnormalized density matrix is the only option, if the normalization diverges due to singularities of the potential, for example in the hydrogen atom

552

5.23.3

5 Variational Perturbation Theory

Smearing Formula for Density Matrices

In order to calculate the connected correlation functions in the variational perturbation expansion (5.441), we must find efficient formulas for evaluating expectation values (5.429) of any power of the interaction (5.437)

This can be done by an extension of the smearing formula (5.30) . For this we rewrite the interaction potential as

V'int(X(Tl)+X m )= !dZ1V'int(ZI+X m )! -00

~;ei'\IZlexp[_ fo hf3 dTiAI8(T-Tl)X(T)] '

-00

(5.448) and introduce a current n

J(T)

=

L

inA 18(T - Tl),

(5.449)

1=1

so that (5.447) becomes

The kernel Kfl,Xm bl represents the generating functional for all correlation functions of the displaced harmonic oscillator

For zero current j, this generating functional reduces to the Euclidean harmonic propagator (5.427): (5.452) and the solution of the functional integral (5.451) is given by (recall Section 3.1)

(5.453)

553

5.23 Density Matrices

where Xcl (T) denotes the classical path (5.431) and cgl (T, T') the harmonic Green function with Dirichlet boundary conditions (3.386), to be written here as

d 2l (

') _ _ n_coshO(IT - T'I- n(3) - coshO(T + T' n2 T, T - 2MO sinh nf30

-

n(3)

.

(5.454)

The expression (5.453) can be simplified by using the explicit expression (5.449) for the current j. This leads to a generating functional (5.455)

where we have introduced the n-dimensional vectors A = (AI, ... , An) and Xci = (Xcl(T1) , .. ' , Xcl(Tn))T with the superscript T denoting transposition, and the symmetric n x n-matrix C whose elements are C k1 = cgl(Tk,Tl)' Inserting (5.455) into (5.450), and performing the integrals with respect to AI, ... , An , we obtain the n-th order smearing formula for the density matrix

The integrand contains an n-dimensional Gaussian distribution describing both thermal and quantum fluctuations around the harmonic classical path Xcl(T) of Eq. (5.431) in a trial oscillator centered at X m , whose width is governed by the Green functions (5.454). For closed paths with coinciding endpoints (Xb = x a ), formula (5.456) leads to the n-th order smearing formula for particle densities

P(Xa) =

~P(Xa, xa) = ~

f DXO

(5.468)

The second term is determined by inserting the high-temperature limit of the total fluctuation width (5.26): 2 kBT (5.469) atot,cl = M02 ' and of the polynomials (5.464) into the expansion (5.461), leading to ) ~ Van( 2 Xa T~ ->00

~ _1_H ~ 2n I n

n=O

n.

(J

M02j3 ) Joo 2 Xa - 00

J

dz lJ; ( ) - Mn 2{3z2/ 2H (M02j3 ) Vint z e n 2 Z. M02 j3

21f /

(5.470)

Then we make use of the completeness relation for Hermite polynomials 1r:;;e _ x2 ~ 1 Hn () ~ ~ x Hn (') x n=O 2 n.

y1f

= is (x - x ') ,

(5.471)

which may be derived from Mehler's formula (2.295) in the limit b --+ 1-, to reduce the smeared interaction potential Va~(xa) to the pure interaction potential (5.438): (5.472)

Recalling (5.438) we see that the first-order effective classical potential (5.468) approaches the classical one: (5.473)

This is a consequence of the vanishing fluctuation width b2 of the paths around the classical orbits. This property is universal to all higher-order approximations to the effective classical potential (5.444). Thus all correction terms with n > 1 must disappear in the limit j3 --+ 0,

r

{3~

-1~(-I)n(An[l)n

---;3 f:2

n!1in

"'int X

Xa,Xa,C

0

= .

(5.474)

b) Zero-Temperature Limit

At low temperatures, the first-order effective classical potential (5.466) becomes

W- 1n,qm( Xa ) -_ -11,0 2

+

l'1m Va2n (Xa ) . {3->00

(5.475)

557

5.23 Density Matrices

The zero-temperature limit of the smeared potential in the second term defined in (5.467) follows from Eq. (5.461) by taking into account the limiting procedure for the polynomials C~n) in (5.465) and the zero-temperature limit of the total fluctuation width (5.26), which is equal to the zero-temperature limit of a2 (xo): a;ot ° = a;ot = , T=O

n/2Mn. Thus we obtain with Ho(x) = 1 and the inverse length K, == l/'xn = vMn/n [recall (2.301)]:

J~~ Va~(xa) =

1 ~Ho(K,Z)2 dz

exp{ _K,2 Z2} Vint(z).

(5.476)

- 00

Introducing the harmonic eigenvalues

E~=nn(n+D,

(5.477)

and the harmonic eigenfunctions [recall (2.299) and (2.300)] (5.478) we can re-express the zero-temperature limit of the first-order effective classical potential (5.475) with (5.476) by (5.479) This is recognized as the first-order Rayleigh-Schrodinger perturbative result for the ground state energy. For the discussion of the quantum-mechanical limit of the first-order normalized density, (5.480) we proceed as follows. First we expand (5.480) up to first order in the interaction, leading to

Inserting (5.428) and (5.461) into the third term in (5.481), and assuming n not to depend explicitly on X a , the xa-integral reduces to the orthonormality relation for Hermite polynomials

2nn~.j1r

JdXaHn(xa)Ho(xa)e-X~ 00

-00

=

8no ,

(5.482)

558

5 Variational Perturbation Theory

so that the third term in (5.481) eventually becomes

But this is just the n = 0 -term of (5.461) with an opposite sign, thus canceling the zeroth component of the second term in (5.481), which would have been divergent for j3 --+ 00. The resulting expression for the first-order normalized density is

The zero-temperature limit of C~n) is from (5.465) and (5.477) 1·

1m

(3->oo

j3C(n) _ {3

-

En n

2 -

(5.485)

En ' 0

so that we obtain from (5.484) the limit

(5.486)

Taking into account the harmonic eigenfunctions (5.478) , we can rewrite (5.486) as

. ( )1 2 = ["I.n( "I.n( ) ('t/J~ 1't/J~) PIn( Xa ) = 1"1 % Xa % Xa )]2 - 2"%I.n( Xa )"'"' ~ o/n Xa En1Vint _ En ' n>O

n

(5.487)

0

which is just equivalent to the harmonic first-order Rayleigh-Schrodinger result for particle densities. Summarizing the results of this section, we have shown that our method has properly reproduced the high- and low-temperature limits. Due to relation (5.487) , the variational approach for particle densities can be used to determine approximately the ground state wave function 't/JO (xa) for the system of interest.

5.23.5

Smearing Formula in Higher Spatial Dimensions

Most physical systems possess many degrees of freedom. This requires an extension of our method to higher spatial dimensions. In general, we must consider anisotropic harmonic trial systems, where the previous variational parameter n2 becomes a D x D -matrix n~v with j.t, v = 1, 2, ... , D.

559

5.23 Density Matrices

a) Isotropic Approximation An isotropic trial ansatz

(5.488) can give rough initial estimates for the properties of the system. In this case, the n-th order smearing formula (5.458) generalizes directly to

x

(5.489)

with the D-dimensional vectors Zl = (Zll' Z21, •.• , ZDl)T . Note, that Greek labels J.L, v, ... = 1, 2, . .. , D specify spatial indices and Latin labels k, I, ... = 0,1,2, ... , n refer to the different imaginary times. The vector Zo denotes r a , the matrix a2 is the same as in Subsection 5.23.3. The harmonic density reads

p~(r) =)27raoo 12

D

exp [-2 \

aoo

t x~] .

(5.490)

1'=1

b) Anisotropic Approximation In the discussion of the anisotropic approximation, we shall only consider radially-symmetric potentials V(r) = V(lr l) because of their simplicity and their major occurrence in physics. The trial frequencies decompose naturally into a radial frequency n£ and a transverse one nT as in (5.95) :

(5.491) with T a = 1r a I· For practical reasons we rotate the coordinate system by points along the first coordinate axis, Ta ,

(ra)1' == 21'0 = { 0, and

Xn

=

U Xn so that

J.L = 1, 2~J.L~D ,

ra

(5.492)

n2-matrix is diagonal: 0

nT2

n2 =

U

0 0

0

nT2

0

0

! )~un'u-' nT2

After this rotation, the anisotropic n-th order smearing formula in D dimensions reads

(5.493)

560

5 Variational Perturbation Theory

The components of the longit udinal and transversal matrices aL and a~ are aLkl = aL(Tk,Tz) ,

ahl = a~(Tk,Tz) ,

(5.495)

where the frequency n in (5.459) must be substituted by the new variational parameters nL , nT, respectively. For the harmonic density in the rotated system p~L ,T (1') which is used to normalize (5.494), we find Po{k ,T( r- ) -_

Appendix 5A

1 12- D-l exp -~~ - -221fa LOO

21fa TOO

[

- - 12-

2 a LOO

-2 X l

-

D ] " xp. -2 - 12- 'L.J . 2 a TOO 1"=2

(5.496)

Feynman Integrals for T=fo 0 without Zero Frequency

The Feynman integrals needed in variational perturbation theory of the anharmonic oscillator at nonzero temperature can be calculated in close analogy to those of ordinary perturbation theory in Section 3.20. The calculation proceeds as explained in Appendix 3D, except that the lines represent now the t hermal correlation function (5.19) with the zero-frequency subtracted from the spectral decomposition:

With the dimensionless variable x == li{3w , the results for the quantities a'fl defined of each Feynman diagram with L lines and V vertices as in (3.547) , but now without the zero Matsubara frequency, are [compare with the results (3D.3)-(3D.11)]

~!

wx

a~

(::coth:: -1) 2 2 '

(5A.1)

~ (4+ x2 -4 cosh x + x sinh x ) , ( ~)2 81xsinh2-

(5A.2)

W

2

~ (-3 x cosh:: + 2 x ( ~) 3 _1_ 64x sinh 2 3 _

W

3

cosh:: + 3 x cosh 3x 2 2

2

+48 sinh ~ + 6 x 2 sinh ~ - 16 sinh 3;) ,

a~

(5A.3)

~ (-864 + 18 X4 + 1152 cosh x + 32 x 2 cosh x ( ~) 4 ~ 768x sinh4W

2 - 288 cosh 2x - 32 x 2 cosh 2x - 288 x sinh x + 24 x 3 sinh x

+ 144 x sinh 2x + 3 x 3 sinh 2x) ,

(5A.4)

5 __1_ _ _1_ (672 xcosh:: _ 8 x 3 cosh:: + 24 x5cosh:: ( ~) w 4096x sinh5 :: 2 2 2 3

2

3x 3 3x 5x 3 5x - 1008 x cosh 2 + 3 x cosh 2 + 336 x cosh 2 + 5 x cosh 2 -7680 sinh

~-

352 x 2 sinh ~ + 72 x4 sinh

~ + 3840

sinh 3;

+ 224 x 2 sinh 3; + 12 x4 sinh 3; - 768 sinh 5; _ 64 x 2 sinh 5;) ,

(5A.5)

561

Appendix 5A Feynman Integrals for T =1= 0 without Zero Frequency

4~ (-107520 -7360 x 2 + 624 x4 + 96 x ( w~)6 49112 5 x smh

6

6 -

2

+ 161280 cosh x + 12000 x 2 cosh x - 777 X4 cosh x + 24 x 6 cosh x - 64512 cosh 2x - 5952 x 2 cosh 2x + 144 x4 cosh 2x + 10752 cosh 3x -28800 x sinh x + 1312 x 2 cosh 3x + 9 X4 cosh 3x + 1120 x 3 sinh x +324 x 5 sinh x

+ 23040 x sinh 2x -

320 x 3 sinh 2x - 5760 x sinh 3x

-160 x3 sinh3x),

(5A.6)

~ (- 24 - 4x2+24coshx+ x 2 cosh x- 9xsinhx) , ( w~) 3 24x1 sinh 2

2 -

2

(~) 28~X2 Si~3:: 4

(45 x cosh

~-

6 x 3 cosh

~-

45 x cosh 3;

2 -432 sinh ~ - 54 x 2 sinh ~ lO a 3'

(~

) 5

23014x 3

Si~

4::

+ 144

sinh 3;

+ 4 x 2 sinh 3;) ,

(-3456 - 414 x 2 - 6 x4

+ 4608

(5A.7)

cosh x+

2

496 x 2 cosh x - 1152 cosh 2x - 82 x 2 cosh 2x - 1008 x sinh x 16 x 3 sinh x

+ 504 x sinh 2x + 5 x 3 sinh2x) .

(5A.8)

Six of these integrals are the analogs of those in Eqs. (3.547). In addition there are the three integrals a~, a~, and ali? , corresponding to the three diagrams (5A.9)

respectively, which are needed in Subsection 5.14.2. They have been calculated with zero Matsubara frequency in Eqs. (3D.8)- (3D.11). In the low-temperature limit where x = nh(3 --t 00, the x-dependent factors in Eqs. (5A.l)(5A.8) converge towards the same constants (3D.13) as those with zero Matsubara frequency, and the same limiting relations hold as in Eqs. (3.550) and (3D.14). The high-temperature limits x --t 0, however, are quite different from those in Eq. (3D.16) . The present Feynman integrals all vanish rapidly for increasing temperatures. For L lines and V vertices, h(3(I/w) v -la?l goes to zero like (3v ((3/12) L. The first V factors are due to the V-integrals over T, the second are the consequence of the product of n/2 factors a2 Thus a?l behaves like (5A.1O) Indeed, the x-dependent factors in (5A.l)- (5A.8) vanish now like x/12, x 3 /720 , x 5 /30240, x 5 /241920, x 7 /11404800 , 193x8 /47551795200, x 4/30240, x 6 /1814400, x 7 /59875200,

(5A.11)

respectively. When expanding (5A.l)- (5A.8) into a power series, the lowest powers cancel each other. For the temperature behavior of these Feynman integrals see Fig. 5.32. We have plotted the reduced Feynman integrals a~L(x) in which the low-temperature behaviors (3.550) and (3D.14) have been divided out of a~L.

562

5 Variational Perturbation Theory

0.5

1

1.5

2

L/x Figure 5.32 Plot of the reduced Feynman integrals a~L(x) as functions of L/x = LkBTj'hw. The integrals (3D.4)- (3D.ll) are indicated by decreasing dash-Ienghts. Compare Fig. 3.16. The integrals (5A.2) and (5A.3) for a~ and ag can be obtained from the integral (5A.l) for a 2 via the operation 1 (5A.12) 2

'hn ( o)n 'hn ( n!

-

ow

= n!

-

o)n

2w ow

'

with n = 1 and n = 2, respectively. This is derived following the same steps as in Eqs. (3D.18)(3D.20). The absence of the zero Matsubara frequency does not change the argument. Also, as in Eqs. (5.195)- (3D.21), the same type of expansion allows us to derive the three integrals from the one-loop diagram (3.546) .

Appendix 5B

Proof of Scaling Relation for Extrema of W N

Here we prove the scaling relation (5.215), according to which the derivative of the Nth approximation W N to the ground state energy can be written as [14] (5B.l) where pN(a) is a polynomial of order N in the scaling variable a = fl(fl2 - 1)/g. For the sake of generality, we consider an anharmonic oscillator with a potential gx P whose power P is arbitrary. The ubiquitous factor 1/4 accompanying 9 is omitted, for convenience. The energy eigenvalue of the ground state (or any excited state) has an Nth order perturbation expansion N

EN(g) =

1

Wl:EI CC P!2)/J '

(5B .2)

1=0

where EI are rational numbers. After the replacement (5.188), the series is re-expanded at fixed r in powers of 9 up to order N, and we obtain (5B.3) with the re-expansion coefficients [compare (5.207)] (5B.4)

563

Appendix 5B Proof of Scaling Relation for the Extrema of W N

Here a is a scaling variable for the potential gx P generalizing (5.208) (note that it is four times as big as the previous a , due to the different normalization of g): _ n(P-2)/2(n 2 _ w2)

a=

(5B.5)

.

9

We now show that the derivative dWN(g,n)/dn has the following scaling form generalizing (5B.l): dWjJ ( 9 )N dn = n(P+2)/2 PN(a),

(5B.6)

where pN(a) is the following polynomial of order N in the scaling variable a: _2 dcN + 1 (a) da

2 ~ E ( (1 - If-j)/2 ) (N ~ J N+l-j

+ 1- j) (_a)N - j

j=O

The proof starts by differentiating (5B.3) with respect to dWjJ ~[ P+2 dn = ~ cl(a) - - 2 - lcl (a)

.

(5B.7)

n, yielding

dCl] (

+ n dn

9 )l n(p+2)/2 .

(5B.8)

Using the chain rule of differentiation we see from (5B.4) that del _ [n(P+2)/2 n dn - 2 9

+

P - 2 ] dCl 2

(5B.9)

ada'

and (5B.8) can be rewritten as dWjJ dn

N [(

~

P +2 ) 1- -2- I cl(a)

+

(n(p+2)/2 2 9

P - 2 ) del] ( 9 )l n(P+2)/2 . da

+ -2-a

(5B.IO) After rearranging the sum, this becomes

(5B.11) The first term vanishes trivially since co happens to be independent of a. The sum in the second line vanishes term by term: (5B.12) To see this we form the derivative

2 del + 1 = 2 ~ E- ( (1- Pt 2j)/2 ) ( . -I-I) (_ )l-j d

a

~

j=O

J

1+I-j

J

a

,

(5B.13)

564

5 Variational Perturbation Theory

and use the identity

2( (1-~j)/2) I+l-j

=

yj+21-1 ((I-~j)/2 ) j-l-l l-j

(5B.14)

to rewrite (5B.13) as

2 dct +1 da

=~E-( (1 - l-j ~j)/2) L..JJ

(P - 2 j +21_1) (-a)l-j

j=O

implying

P - 2 adct = ~ 2 da L..J

j=O

2

E- ( (1 - l-J Efj)/2 ) P-

'

2(1_ j)(-a)l-j.

2

J

(5B.15)

(5B.16)

By combining this with (5B.4), (5B.13) , we obtain Eq. (5B.12) which proves that the second line in (5B.ll) vanishes. Thus we are left with the last term on the right-hand side of Eq. (5B.ll). Using (5B .12) for 1= N leads to

dWjJ

=

9

-2 (

dO.

n(p+2)/2

) N dcN+1(a) da'

(5B.17)

When expressing d€N+l (a)/da with the help of (5B.4), we arrive at

dWjJ dO.

_

-

(

9

2 n(P+2)/2

{;EJ. (

)N N

(1 - ~j)/2 ) _. _ N-j N+l-j (N + l J)( a) . (5B.18)

This proves the scaling relation (5B.6) with the polynomial (5B.7) . The proof can easily be extended to physical quantities QN(9) with a different physical dimension n , which have an expansion (5B.19) rather than (5B.2). In this case the quantity [Q N(gW/" has again an expansion like (5B.2). By rewriting QN(g) as {[QN(gW /" }" and forming the derivative using the chain rule we see that the derivative vanishes whenever the polynomial PN(a) vanishes, which is formed from [QN(9W/a as in Eq. (5B.7).

Appendix 5C

Second-Order Shift of Polaron Energy

For brevity, we introduce the dimensionless variable p == wf!.r and

(5C.l) Going to natural units with 1i = M = w = 1, Feynman's variational energy (5.395) takes the form

Eo =

~(r 4r

0.)2 - _1_ r [= dpe- PF- 1/ 2(p). .,;:IT

Jo

(5C .2)

The second-order correction (5.406) reads

f!.E(2) = _ 412 r2J + _1_(r2 _ 0. 2 ) [= dpe-PF-1/2(p) _ _ 3_(r2 _ 0.2)2 o 7r 2r.,;:IT Jo 16r3 _ _ I_(r2 _ n2) [= dpe-PF-3/2(p) {(1+ 0. 2 ) (l -e-rp) + r2_n2 pe-rp}, 4.,;:IT Jo r r

(5C.3)

565

Notes and References where I denotes the integral

(5C.4) with

+ PI 2': 0 + PI 2': 0 P2 + PI < 0

Q = Ql for P3 - P2

and P3 - P2 2': 0,

Q = Q2 for P3 - P2

and P3 - P2 < 0,

Q = Q3 for P3 -

and P3 - P2 < 0,

(5C.5)

and Ql =

~F-l/2(pDF-l/2(p~) r2; 0 2e- rp3 (1 _ e- rp1 )(erp -

Q2 =

~F-l/2(pdF-l/2(P2)

x

Q3 = X

1),

(5C.6)

2

{r2; 2 0

[e - r (P2 - P3) _ e- rp3 (1

+ e- r (P' - P2) _

e- rP1 )] - 202(P2 - P3) } ,

(5C.7)

~F-l/2(pdF-l/2(P2)

{r2; 0 2

[ _e-rp3 (1

_ e- rp1 ) _

e- r (P2-P3) (e rP1 -

1)] - 20 2PI} .

(5C.8)

Notes and References The first-order variational approximation to the effective classical partition function V eff cl (xo) presented in this chapter was developed in 1983 by RP. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (http://www.physik.iu-berlin . derkleinert/159) . For further development see: H. Kleinert, Phys. Lett. A 118, 195 (1986) (ibid.http/148) ; B 181, 324 (1986) (ibid.http/151) ; W. Janke and B.K. Chang, Phys. Lett. B 129, 140 (1988); W . Janke, in Path Integrals from meV to MeV, ed. by V. Sa-yakanit et ai., World Scientific, Singapore, 1990. A detailed discussion of the accuracy of the approach in comparison with several other approximation schemes is given by S. Srivastava and Vishwamittar, Phys. Rev. A 44, 8006 (1991) . For a similar, independent development containing applications to simple quantum field theories, see R Giachetti and V. Tognetti, Phys. Rev. Lett. 55, 912 (1985); Int. J. Magn. Mater. 54-57, 861 (1986) ; R Giachetti, V. Tognetti, and R Vaia, Phys. Rev. B 33, 7647 (1986); Phys. Rev. A 37, 2165 (1988) ; Phys. Rev. A 38, 1521, 1638 (1988) ; Physica Scripta 40, 451 (1989). R Giachetti, V. Tognetti, A. Cuccoli, and R Vaia, lecture presented at the XXVI Karpacz School of Theoretical Physics, Karpacz, Poland, 1990. See also R Vaia and V. Tognetti, Int. J . Mod. Phys. B 4,2005 (1990); A. Cuccoli, V. Tognetti, and R Vaia, Phys. Rev. B 41 , 9588 (1990); A 44, 2743 (1991); A. Cuccoli, A. Maradudin, A.R McGurn, V. Tognetti, and R Vaia, Phys. Rev. D 46, 8839 (1992) .

566

5 Variational Perturbation Theory

The variational approach has solved some old problems in quantum crystals by extending in a simple way the classical methods into the quantum regime. See V.1. Yukalov, Mosc. Univ. Phys. Bull. 31, 10-15 (1976); S. Liu, G.K. Horton, and E .R. Cowley, Phys. Lett. A 152, 79 (1991) ; A. Cuccoli, A. Macchi, M. Neumann, V. Tognetti, and R. Vaia, Phys. Rev. B 45, 2088 (1992). The systematic extension of the variational approach was developed by H. Kleinert, Phys. Lett. A 173, 332 (1993) (quant-ph/9511020) . See also J. Jaenicke and H. Kleinert, Phys. Lett. A 176, 409 (1993) (ibid.http/217) ; H. Kleinert and H. Meyer, Phys. Lett. A 184, 319 (1994) (hep-th/9504048). A similar convergence mechanism was first observed within an order-dependent mapping technique in the seminal paper by R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). For an introduction into various resummation procedures see C.M. Bender and S.A. Orszag, Advanced Mathematical Methods fo r Scientists and Engineers , McGraw-Hill, New York, 1978. The proof of the convergence of the variational perturbation expansion to be given in Subsection 17.10.5 went through the following stages: First a weak estimate was found for the anharmonic integral: I.R.C. Buckley, A. Duncan, H.F. Jones, Phys. Rev. D 47, 2554 (1993); C.M. Bender, A. Duncan, H.F. Jones, Phys. Rev. D 49, 4219 (1994). This was followed by a similar extension to the quantum-mechanical case: A. Duncan and H.F. Jones, Phys. Rev. D 47, 2560 (1993) ; C. Arvanitis, H.F. Jones, and C.S. Parker, Phys.Rev. D 52, 3704 (1995) (hep-ph/9502386); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. 241, 152 (1995) (hep-th/9407027). The exponentially fast convergence observed in the calculation of the strong-coupling coefficients of Table 5.9 was, however, not explained. The accuracy in the table was reached by working up to the order 251 with 200 digits in Ref. [13]. The analytic properties of the strong-coupling expansion were studied by C.M. Bender and T .T . Wu, Phys. Rev. 184, 1231 (1969); Phys. Rev. Lett. 27, 461 (1971) ; Phys. Rev. D 7, 1620 (1973) ; ibid. D 7, 1620 (1973); C.M. Bender, J . Math. Phys. 11, 796 (1970); T. Banks and C.M. Bender, J. Math. Phys. 13, 1320 (1972) ; J.J . Loeffel and A. Martin, Cargese Lectures on Physics (1970) ; D. Bessis ed., Gordon and Breach, New York 1972, Vol. 5, p.415; B. Simon, Ann. Phys. (N.Y.) 58, 76 (1970) ; Cargese Lectures on Physics (1970) , D. Bessis ed., Gordon and Breach, New York 1972, Vol. 5, p. 383. The problem of tunneling at low barriers (sliding) was solved by H. Kleinert , Phys. Lett. B 300, 261 (1993) (ibid.http/214) . See also Chapter 17. Some of the present results are contained in H. Kleinert, Pfadintegrale in Quantenmechanik, Statistik und Polymerphysik , B.-I. Wissenschaftsverlag, Mannheim, 1993. A variational approach to tunneling is also used in chemical physics: M.J. Gillan, J. Phys. C 20, 362 (1987) ; G.A. Voth, D. Chandler, and W.H. Miller, J. Chern. Phys. 91 , 7749 (1990); G.A. Voth and E.V. OGorman, J. Chern. Phys. 94, 7342 (1991) ; G.A. Voth, Phys. Rev. A 44,5302 (1991). Variational approaches without the separate treatment of Xo have been around in the literature for some time:

Notes and References

567

T. Barnes and G.1. Ghandour, Phys. Rev. D 22, 924 (1980) ; B.S. Shaverdyan and A.G. Usherveridze, Phys. Lett. B 123, 316 (1983) ; K. Yamazaki, J. Phys. A 17, 345 (1984) ; H. Mitter and K. Yamazaki, J. Phys. A 17, 1215 (1984); P.M. Stevenson, Phys. Rev. D 30,1712 (1985); D 32,1389 (1985); P .M. Stevenson and R Tarrach, Phys. Lett. B 176, 436 (1986); A. Okopinska, Phys. Rev. D 35, 1835 (1987); D 36, 2415 (1987) ; W. Namgung, P.M. Stevenson, and J.F. Reed, Z. Phys. C 45, 47 (1989) ; U. Ritschel, Phys. Lett. B 227, 44 (1989); Z. Phys. C 51 , 469 (1991) ; M.H. Thoma, Z. Phys. C 44,343 (1991); 1. Stancu and P.M. Stevenson, Phys. Rev. D 42, 2710 (1991); R Tarrach, Phys. Lett. B 262, 294 (1991); H. Haugerud and F. Raunda, Phys. Rev. D 43, 2736 (1991); A.N. Sissakian, 1.1. Solovtsov, and O.Y. Shevchenko, Phys. Lett. B 313, 367 (1993). Different applications of variational methods to density matrices are given in V.B. Magalinsky, M. Hayashi, and H.V. Mendoza, J. Phys. Soc. Jap. 63,2930 (1994); V.B. Magalinsky, M. Hayashi, G.M. Martinez Pena, and R Reyes Sanchez, Nuovo Cimento B 109, 1049 (1994). The particular citations in this chapter refer to the publications [1] H. Kleinert, Phys. Lett. A 173, 332 (1993) (quant-ph/9511020) . [2] The energy eigenvalues of the anharmonic oscillator are taken from F .T . Hioe, D. MacMillan, and E .W. Montroll, Phys. Rep. 43, 305 (1978); W. Caswell, Ann. Phys. (N.Y.) 123, 153 (1979) ; R1. Somorjai, and D.F. Hornig, J. Chern. Phys. 36, 1980 (1962) . See also K. Banerjee, Proc. Roy. Soc. A 364, 265 (1978) ; R Balsa, M. Plo, J.G. Esteve, A.F. Pacheco, Phys. Rev. D 28, 1945 (1983); and most accurately F. Vinette and J . Cizek, J. Math. Phys. 32, 3392 (1991) ; E.J . Weniger, J. Cizek, J . Math. Phys. 34, 571 (1993). [3] RP. Feynman, Statistical Mechanics , Benjamin, Reading, 1972, Section 3.5. [4] M. Hillary, RF. O'Connell, M.O . Scully, and E.P. Wigner, Phys. Rep. 106, 122 (1984). [5] H. Kleinert, Phys. Lett. A 118, 267 (1986) (ibid.http/145). [6] For a detailed discussion of the effective classical potential of the Coulomb system see W. Janke and H. Kleinert, Phys. Lett. A 118, 371 (1986) (ibid.http/153). [7] C. Kouveliotou et al., Nature 393, 235 (1998); Astroph. J. 510, L115 (1999) ; K. Hurley et al. , Astroph. J. 510, L111 (1999); V.M. Kaspi, D. Chakrabarty, and J. Steinberger, Astroph. J. 525, L33 (1999) ; B. Zhang and A.K. Harding, (astro-ph/0004067). [8] The perturbation expansion of the ground state energy in powers of the magnetic field B was driven to high orders in J.E. Avron, B.G. Adams, J. Cizek, M. Clay, M.L. Glasser, P. Otto, J. Paldus, and E. Vrscay, Phys. Rev. Lett. 43, 691 (1979); B.G. Adams, J.E. Avron, J. Cizek, P. Otto, J . Paldus, RK. Moats, and H.J. Silverstone, Phys. Rev. A 21, 1914 (1980). This was possible on the basis of the dynamical group 0(4,1) and the tilting operator (13.181) found by the author in his Ph.D. thesis. See H. Kleinert, Group Dynamics of Elementary Particles, Fortschr. Physik 6, 1 (1968)

568

5 Variational Perturbation Theory (ibid.http/i); H. Kleinert, Group Dynamics of the Hydrogen Atom, Lectures in Theoretical Physics, edited by W.E. Brittin and A.O. Barut, Gordon and Breach, N.Y. 1968, pp. 427-482 (ibid.http/4).

[9] Precise numeric calculations of the ground state energy of the hydrogen atom in a magnetic field were made by H. Ruder, G. Wunner, H. Herold, and F. Geyer, Atoms in Strong Magnetic Fields (SpringerVerlag, Berlin, 1994). [10] M. Bachmann, H. Kleinert, and A. Pelster, Phys. Rev. A 62, 52509 (2000) (quantph/0005074), Phys. Lett. A 279,23 (2001) (quant-ph/00051O). [11] L.D. Landau and E .M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965. [12] J .C. LeGuillou and J. Zinn-Justin, Ann. Phys. (N.Y.) 147, 57 (1983). [13] W. Janke and H. Kleinert, Phys. Rev. Lett. 75, 2787 (1995) (quant-ph/9502019) . [14] The high accuracy became possible due to a scaling relation found in W. Janke and H. Kleinert, Phys. Lett. A 199, 287 (1995) (quant-ph/9502018). [15] For the proof of the exponentially fast convergence see H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) (quant-ph/9502019); R. Guida, K. Konishi, and H. Suzuki, Ann. Phys. 249, 109 (1996) (hep-th/9505084) . The proof will be given in Subsection 17.10.5. [16] The oscillatory behavior around the exponential convergence shown in Fig. 5.22 was explained in H. Kleinert and W. Janke, Phys. Lett. A 206, 283 (1995) (quant-ph/9502019) in terms of the convergence radius of the strong-coupling expansion (see Section 5.15) . [17] H. Kleinert, Phys. Rev. D 60 , 085001 (1999) (hep-th/9812197); Phys. Lett. B 463, 69 (1999) (cond-mat/9906359). See also Chapters 19-20 in the textbook H. Kleinert and V. Schulte-Frohlinde, Critical Properties of ip4-Theories, World Scientific, Singapore 2001 (ibid.http/b8) [18] H. Kleinert, Phys. Lett. A 207, 133 (1995) . [19] See for example the textbooks H. Kleinert, Gauge Fields in Condensed Matter, Vol. I Superflow and Vortex Lines, Vol. II Stresses and Defects, World Scientific, Singapore, 1989 (ibid.http/bi). [20] R.P. Feynman, Phys. Rev. 97, 660 (1955). [21] S. Hohler and A. Miillensiefen, Z. Phys. 157, 159 (1959) ; M.A. Smondyrev, Theor. Math. Fiz. 68, 29 (1986); O.V. Selyugin and M.A. Smondyrev, Phys. Stat. Sol. (b) 155, 155 (1989) . [22] N.N. Bogoliubov (jun) and V.N. Plechko, Teor. Mat. Fiz. [Sov. Phys.-Theor. Math. Phys.]' 65, 423 (1985); illv. Nuovo Cimento 11, 1 (1988) . [23] S.J. Miyake, J. Phys. Soc. Japan, 38, 81 (1975). [24] J.E. Avron, LW. Herbst, B. Simon, Phys. Rev. A 20, 2287 (1979). [25] LD. Feranshuk and L.L Komarov, J . Phys. A: Math. Gen. 17, 3111 (1984) . [26] H. Kleinert, W. Kiirzinger, and A. Pelster, J. Phys. A: Math. Gen. 31, 8307 (1998) (quantph/9806016). [27] H. Kleinert, Phys. Lett. A 173, 332 (1993) (quant-ph/9511020).

Notes and References

569

[28] H. Kleinert, Phys. Rev. D 57, 2264 (1998) and Addendum: Phys. Rev. D 58, 107702 (1998). [29] F.J. Wegner, Phys. Rev. B 5, 4529 (1972); B 6, 1891 (1972). [30] H. Kleinert, Phys. Lett. A 207, 133 (1995) (quant-ph/9507005) . [31] J. Rossler, J. Phys. Stat. Sol. 25, 311 (1968). [32] J.T. Marshall and L.R. Mills, Phys. Rev. B 2, 3143 (1970). [33] Higher-order smearing formulas for nonpolynomial interactions were derived in H. Kleinert, W. Kiirzinger and A. Pelster, J. Phys. A 31 , 8307 (1998) (quant-ph/9806016). [34] The polaron problem is solved in detail in the textbook R.P. Feynman, Statistical Mechanics, Benjamin, New York, 1972, Chapter 8. Extensive numerical evaluations are found in T.D. Schultz, Phys. Rev. 116, 526 (1959); See also M. Dineykhan, G.V. Efimov, G. Ganbold, and S.N. Nedelko, Oscillator Representation in Quantum Physics, Springer, Berlin, 1995. An excellent review article is J.T. Devreese, Polarons, Review article in Encyclopedia of Applied Physics, 14, 383 (1996) (cond-mat /0004497). This article contains ample references on work concerning polarons in magnetic fields, for instance F .M. Peeters, J.T. Devreese, Phys. Stat. Sol. B 110, 631 (1982); Phys. Rev. B 25, 7281, 7302 (1982) ; Xiaoguang Wu, F.M. Peeters, J .T. Devreese, Phys. Rev. B 32, 7964 (1985); F. Brosens and J.T. Devreese, Phys. Stat. Sol. B 145, 517 (1988). For discussion of the validity of the Jensen-Peierls inequality (5.10) in the presence of a magnetic field, see J.T. Devreese and F . Brosens, Solid State Communs. 79, 819 (1991) ; Phys. Rev. B 45, 6459 (1992) ; Solid State Communs. 87, 593 (1993); D. Larsen in Landau Level Spectroscopy , Vol. 1, G. Landwehr and E. Rashba (eds.), North Holland, Amsterdam, 1991, p. 109. The paper D. Larsen, Phys. Rev. B 32, 2657 (1985) shows that the variational energy can lie lower than the exact energy. The review article by Devreese contains numerous references on bipolarons, small polarons, and polaronic excitations. For instance: J.T. Devreese, J . De Sitter, M.J. Goovaerts, Phys. Rev. B 5, 2367 (1972) ; L.F. Lemmens, J. De Sitter, J.T. Devreese, Phys. Rev. B 8, 2717 (1973) ; J.T. Devreese, L.F. Lemmens, J. Van Royen, Phys. Rev. B 15, 1212 (1977); J. Thomchick, L.F. Lemmens, J.T. Devreese, Phys. Rev. B 14, 1777 (1976) ; F .M. Peeters, Xiaoguang Wu, J .T . Devreese, Phys. Rev. B 34, 1160 (1986); F .M. Peeters, J .T. Devreese, Phys. Rev. B 34, 7246 (1986); B 35, 3745 (1987); J .T . Devreese, S.N. Klimin, V.M. Fomin, F . Brosens, Solid State Communs. 114, 305 (2000) . There exists also a broad collection of articles in E .K.H. Salje, A.S. Alexandrov, W .Y. Liang (eds.), Polarons and Bipolarons in High-Tc Superconductors and Related Materials , Cambridge University Press, Cambridge, 1995. A generalization of the harmonic trial path integral (5.356), in which the exponential function e-OIT-T' 1 at zero temperature is replaced by I( IT - T'I), has been proposed by M. Saitoh, J . Phys. Soc. Japan. 49, 878 (1980), and further studied by R. Rosenfelder and A.W. Schreiber, Phys. Lett. A 284, 63 (2001) (cond-mat/ 0011332).

570

5 Variational Perturbation Theory In spite of a much higher numerical effort, this generalization improves the ground state energy only by at most 0.1 % (the weak-coupling expansion coefficient -0.012346 in (5.396) is changed to -0.012598, while the strong-coupling coefficients in (5.399) remain unchanged at this level of accuracy. For the effective mass, the lowest nontrivial weak-coupling coefficient of 12 in (5.402) is changed by 0.0252 % while the strong-coupling coefficients in (5.403) remain unchanged at this level of accuracy.

Aevo rarissima nostro, simplicitas.

Simplicity, a very rare thing in our age. OVID, Ars Amatoria, Book 1, 241

6 Path Integra Is with T opologica I Constra ints The path integral representations of the time evolution amplitudes considered so far were derived for orbits x(t) fluctuating in Euclidean space with Cartesian coordinates. Each coordinate runs from minus infinity to plus infinity. In many physical systems, however, orbits are confined to a topologically restricted part of a Cartesian coordinate system. This changes the quantum-mechanical completeness relation and with it the derivation of the path integral from the time-sliced time evolution operator in Section 2.1. We shall consider here only a point particle moving on a circle, in a half-space, or in a box. The path integral treatment of these systems is the prototype for any extension to more general topologies.

6.1

Point Particle on Circle

For a point particle on a circle, the orbits are specified in terms of an angular variable 'P(t) E [0,271'] subject to the topological constraint that 'P = 0 and 'P = 271' be identical points. The initial step in the derivation of the path integral for such a system is the same as before: The time evolution operator is decomposed into a product .

('Pbtbl'Pata ) = ('Pb lexp

[-~(tb -

II

N+l

ta)iI] l'Pa) == ('Pb l

'

exp ( -~EiI) l'Pa }.

(6.1)

The restricted geometry shows up in the completeness relations to be inserted between the factors on the right-hand side for n = 1, . .. , N:

(6.2) If the integrand is singular at 'P = 0, the integrations must end at an infinitesimal piece below 271'. Otherwise there is the danger of double-counting the contributions from the identical points 'P = 0 and 'P = 271'. The orthogonality relations on these intervals are

(6.3) 571

572

6 Path Integrals with Topological Constraints

The O.

(6.26)

Given a free particle moving in such a geometry, we want to calculate N+l

(rbtblrata) =

.

(rbllI exp ( -~iiE)

Ira) .

(6.27)

576

6 Path Integrals with Topological Constraints

As usual, we insert N completeness relations between the N of a vanishing Hamiltonian, the amplitude (6.27) becomes

+ 1 factors.

In the case

(6.28) For each scalar product (rnlrn-l) = lS(rn - rn-l) , we substitute its spectral representation appropriate to the infinite-wall boundary at r = O. It consists of a superposition of the free-particle wave functions vanishing at r = 0:

(r lrl )

2 rOOdksinkrsinkr'

Jo

1 dk 00

-00

(6.29)

7r

27r

[expik(r - rl) - expik(r + r l)] = lS(r - rl) - lS(r + rl).

This Fourier representation does a bit more than what we need. In addition to the IS-function at r = r' , there is also a IS-function at the unphysical reflected point r = -r'. The reflected point plays a similar role as the periodically repeated points in the representation (6.11) . For the same reason as before, we retain the reflected points in the formula as though rl were permitted to become zero or negative. Thus we rewrite the Fourier representation (6.29) as

(r lrl)

=

x~r

1

00 -00

dp

[i

]

27r1i exp "h,p(x - Xl) + i7r(u(x) - U(XI)) x'=r"

where

u(x)

= 8(-x)

(6.30)

(6.31)

with the Heaviside function 8(x) of Eq. (1.309). For symmetry reasons, it is convenient to liberate both the initial and final positions rand rl from their physical half-space and to introduce the localized states Ix) whose scalar product exists on the entire x-axis:

(x lx l)

1

L 00 dP1i exp [~p(XIl - Xl) x"=±x -00 27r lS(x - Xl) - lS(x + Xl).

+ i7r(U(X") -

u(xl))] (6.32)

With these states, we write

(r lrl ) = (x lxl) lx=r,x'=r"

(6.33)

We now take the trivial transition amplitude with zero Hamiltonian (6.34) extend it with no harm by the reflected IS-function (6.35)

577

6.2 Infinite Wall

and factorize it into many time slices: (6.36)

(rb = r N+1, r a = ro), where the trivial amplitude of a single slice is (xnflxn-10)0

=

(XnIXn-l),

x E (-00,00).

(6.37)

With the help of (6.32), this can be written as

(6.38) The sum over the reflected points Xn = ±rn is now combined, at each n, with the integral Jooo drn to form an integral over the entire x-axis, including the unphysical half-space x < O. Only the last sum cannot be accommodated in this way, so that we obtain the path integral representation for the trivial amplitude

(6.39) The measure of this path integral is now of the conventional type, integrating over all paths which fluctuate through the entire space. The only special feature is the final symmetrization in Xb = ±rb. It is instructive to see in which way the final symmetrization together with the phase factor exp[i7ra(x)] = ±1 eliminates all the wrong paths in the extended space, i.e., those which cross the origin into the unphysical subspace. This is illustrated in Fig. 6.2. Note that having assumed Xa = ra > 0, the initial phase a(x a ) can be omitted. We have kept it merely for symmetry reasons. In the continuum limit, the exponent corresponds to an action (6.40)

The first term is the usual canonical expression in the absence of a Hamiltonian. The second t erm is new. It is a pure boundary term: (6.41)

which keeps track of the topology of the half space x > 0 embedded in the full space x E (-00,00). This is why the action carries the subscript "topol" .

578

6 Path Integrals with Topological Constraints

Xa

x

Figure 6.2 Illustration of path counting near reflecting wall. Each path touching the wall once is canceled by a corresponding path of equal action crossing the wall once into the unphysical regime (the path is mirror-reflected after the crossing). The phase factor exp[i11'0'(Xb) ] provides for the opposite sign in the path integral. Only paths not touching the wall at all cannot be canceled in the path integral.

The topological action (6.41) can be written formally as a local coupling of the velocity at the origin:

Afopol[xl = -11'n

l

tb

ta

dtx(t)8(x(t)).

(6.42)

This follows directly from (6.43) Consider now a free point particle in the right half-space with the usual Hamiltonian p2 H= 2M

(6.44)

The action reads

AlP, xl =

l

tb

dtlPx - p2/2M - n1fx(t)8(x(t))],

(6.45)

ta

and the time-sliced path integral looks like (6.39) , except for additional energy terms -p;'/2M in the action. Since the new topological term is a pure boundary term, all the extended integrals in (6.39) can be evaluated right away in the same way as for a free particle in the absence of an infinite wall. The result is

6.3 Point Particle in Box

579

with Xa = ra' This is indeed the correct result: Inserting the Fourier transform of the Gaussian (Fresnel) distribution we see that

(6.47) which is the usual spectral representation of the time evolution amplitude. Note that the first part of (6.46) may be written more symmetrically as

In this form, the phase factors e i7r 0 corresponds to a strong-coupling limit in the reduced coupling constant g. According to the general theory of variational perturbation theory and its strong-coupling limit in Sections 5.14 and 5.17, the Nth order approximation to the strong-coupling limit of p(g), to be denoted by p' , is found by replacing, in the series truncated after the 3The programs can be downloaded from www.physik.fu-berlin.derkleinert/b5/programs

586

6 Path Integrals with Topological Constraints

Nth term, FNCg/w), the frequency w by the identical expression where

Vn2 - gr /2M, 2

C6.81) For a moment , this is treated as an independent variable, whereas n is a dummy parameter. Then the square root is expanded binomially in powers of g, and FNCg/Vn 2 - gr 2 /2M) is re-expanded up to order gN. After that, r is replaced by its proper value. In this way we obtain a function FNCg, n) which depends on n, which thus becomes a variational parameter. The best approximation is obtained by extremizing FNCg , n) with respect to w. Setting w = 0, we go to the strong-coupling limit g -+ 00 . There the optimal n grows proportionally to g, so that gin = e- 1 is finite, and the variational expression FNCg, n) becomes a function of fNCe) . In this limit, the above re-expansion amounts simply to replacing each power w n in each expansion terms of FNCg) by the binomial expansion of C1 - 1) -n/2 truncated after the CN - n )th term, and replacing 9 by c 1 . The first nine variational functions fNCe) are listed in Table 6.1. The functions fNCe) are minimized starting from hCe) and searching the minimum of each successive hCe), hCe), .. . nearest to the previous one. The functions fNCe) together with their minima are plotted in Fig. 6.5. The minima lie at

Table 6.1

First eight variational functions fN(C),

hCe) = ~ + l~ c + ~~ hCe) = ~ + 3~C + ~ 1 1 15 35c f 4 C) e = 4: - 256c3 + 128c + 256 C) e C) J6 e f 7 C) e f C) J8 e f C) f

J5 f

J9

5 35 = 4:1 - 512c3 + 256c + 63c 512 1 1 35 315 231 c = 4: + 2048c5 - 2048c3 + 2048c + 2048 1 7 105 693 429c = 4: + 4096c5 - 4096c3 + 4096c + 4096 1 5 63 1155 3003 6435c = 4: - 65536c7 + 16384c5 - 32768c3 + 16384c + 65536 1 45 231 3003 6435 12155 c e = 4: - 131072 c7 + 32768 c5 - 65536 c 3 + 32768 c + 131072

c 0.8

0.494 0.492 0.49

Figure 6.5 Variational functions fN(C) for particle between walls up to N = 16 are shown

together with their minima whose y-coordinates approach rapidly the correct limiting value 1/2.

587

6.4 Strong-Coupling Theory for Particle in Box

(N, J'lJin ) = (2,0.466506), (3,0.492061), (4,0.497701), (5,0.499253), (6,0.499738), (7,0.499903), (8,0.499963), (9,0.499985), (10,0.499994), (11,0.499998), (12,0.499999), (13,0.5000), (14,0.50000), (15,0.50000), (16,0.5000).

(6.82)

They converge exponentially fast against the known result 1/2, as shown in Fig. 6.6.

6.4.4

Special Properties of Expansion

The alert reader will have noted that the expansion coefficients (6.80) possess two special properties: First, they lack the factorial growth at large orders which would be found for a single power [u 2 (r)]k+1 of the interaction potential, as mentioned in Eq.(3C.27) and will be proved in Eq. (17.323) . The factorial growth is canceled by the specific combination of the different powers in the interaction (6.72), making the series (6.71) convergent inside a certain circle. Still, since this circle has a finite radius (the ratio test shows that it is unity) , this convergent series cannot be evaluated in the limit of large 9 which we want to do, so that variational strong-coupling theory is not superfluous. However, there is a second remarkable property of the coefficients (6.80): They contain an infinite number of zeros in the sequence of coefficients for each odd number, except for the first one. We may take advantage of this property by separating off the irregular term a19 = 9/4 = n: 2/4d 2 , setting a = 9 2 / 4w 2 , and rewriting F(g) as N

F(a)

1 ] , 4"1 [ 1 + foh(a)

=

h(a) ==

L 22n+la2n a n .

(6.83)

n=O

Inserting the numbers (6.80), the expansion of h(a) reads

h(a) = 1

~ _ a2 a3 + 2 8 + 16

_

~ a4 128

+

..!....- a 5 _ ~a6 ~ a 7 _ 256

1024

+

2048

429

32768 a

8

+ .... (6.84)

We now realize that this is the binomial power series expansion of VI + a. Substituting this into (6.83), we find the exact ground state energy for the Euclidean action (6.67) (6.85)

0.5 0.499998 0.499996 0.499994

• •

0.00005

•

!'!Jin

•

0.499992

Figure 6.6 exact value.

Exponentially fast convergence of strong-coupling approximations towards

588

6 Path Integrals with Topological Constraints

Here we can go directly to the strong-coupling limit a -. 00 to recover the exact ground state energy E(O) = 7r 2 /2d 2 . The energy (6.85) can of course be obtained directly by solving the Schrodinger equation associated with the potential (6.72), 2

.\) -1 { -8-2 + [,\(1 - -22

8x

cos

d2 1] } 'l/J(x) = -E'l/J(x), 7r 2

X

where we have replaced u -. dX/7r and set w 2 d 4 /7r 4

(6.86)

== .\(.\ - 1), so that (6.87)

Equation (6.86) is of the Piischl-Teller type [see Subsection 14.4.5], and has the ground state wave function, to be derived in Eq. (14.163),

'l/JO(x) = const x cos'" x ,

(6.88)

with the eigenvalue 7r 2E(O) /d 2 = (.\2 - 1)/2, which agrees of course with Eq. (6.85). If we were to apply the variational procedure to the series h(a)/fo in F of Eq. (6.85), by replacing the factor l/w 2n contained in each power an by fl = y'fl2 - ra and re-expanding now in powers of a rather than g, we would find that all approximation hN(C) would possess a minimum with unit value, such that the corresponding extremal functions fN(C) yield the correct final energy in each order N .

6.4.5

Exponentially Fast Convergence

With the exact result being known, let us calculate the exponential approach of the variational approximations observed in Fig. (6.6). Let us write the exact energy (6.85) as 1

E(O) -- _(g 4 + ...)g2 + 4w2) .

(6.89)

After the replacement w -. ...)fl2 - pg, this becomes (6.90)

where g == g/fl2. The Nth-order approximant fN(9) of E(O) is obtained by expanding (6.91) in powers of g up to order N, N

(6.91) fN(g) = fl"Lhk(p)gk, o and substituting p by 2Mr2 = (1- C;P)/g [compare (6.81)], with 12>2 == w2/fl2 . The resulting function of g is then optimized. It is straightforward to find an integral representation for FN(g) . Setting rg == z, we have 1

FN =

-2· 7rZ

i

coz

dz N+1

1- zN+l 1 f(z),

- z

(6.92)

where the contour Co refers to small circle around the origin and

F(z)

=

~4 (=r + Jr2Z2 -4Z+4) (6.93)

Notes and References with branch points at

589

Zl,2

= 2r2 (1

±

VI - l/r2).

For z < 1, we rewrite

1 - zN+l = (1 - z)(1 + z + ... + zN) = (1 - z)(N + 1) -(1- z)2 [N + (N -1)z + .. . + zN- l] and estimate this for z

~

(6.94)

1 as (6.95)

Dividing the approximant (6.92) by 11, and indicating this by a hat, we use (6.94) to write FN as a sum over the discontinuities across the two branch cuts:

(N + 1) 27ri

(N + 1)

1 dzF(z) F(z) = leo zN+l

L1

00

2

i=l

Zi

(N + 1) F(N)(O) N!

d F(z). N:l

(6.96)

Z

The integrals yield a constant plus a product (6.97) which for large N can be approximated using Stirling's formula (5.204) by (6.98) In the strong-coupling limit of interest here, w2 = 0, and r = 1/9 = l1/g = c. In Fig. 6.5 we see that the optimal c-values tend to unity for N --+ 00, so that l!J.jN goes to zero like e- N , as observed in Fig. 6.6.

Notes and References There exists a large body of literature on this subject, for example L.S. Schulman, J. Math. Phys. 12, 304 (1971); M.G.G. Laidlaw and C. DeWitt-Morette, Phys. Rev. D 3, 1375 (1971); J.S. Dowker, J. Phys. A 5, 936 (1972); P.A. Horvathy, Phys. Lett. A 76, 11 (1980) and in Differential Geometric Methods in Math . Phys. , Lecture Notes in Mathematics 905, Springer, Berlin, 1982; J.J. Leinaas and J . Myrheim, Nuovo Cimento 37, 1, (1977). The latter paper is reprinted in the textbook F . Wilczek, Fractional Statistics and Anyon Superconductivity, World Scientific, 1990. See further P.A. Horvathy, G. Morandi, and E .C.G. Sudarshan, Nuovo Cimento D 11, 201 (1989) , and the textbook L.S. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1981. It is possible to account for the presence of hard walls using infinitely high 8-functions: C. Grosche, Phys. Rev. Lett. 71, 1 (1993); Ann. Phys. 2, 557 (1993); (hep-th/9308081); (hep-

th/9308082) ; (hep-th/9402110); M.J. Goovaerts, A. Babcenco, and J.T. Devreese, J . Math. Phys. 14, 554 (1973); C. Grosche, J . Phys. A Math. Gen. 17,375 (1984) .

590

6 Path Integrals with Topological Constraints

The physically important problem of membranes between walls has been discussed in W . Helfrich, Z. Naturforsch. A 33, 305 (1978); W. Helfrich and RM. Servuss, Nuovo Cimento D 3, 137 (1984); W . Janke and H. Kleinert, Phys. Lett. 58, 144 (1987) (http://www . physik. fu-berlin. del -kleinert/143); W. Janke, H. Kleinert, and H. Meinhardt, Phys. Lett. B 217, 525 (1989) (ibid.http/184) ; G. Gompper and D.M. Kroll, Europhys. Lett. 9, 58 (1989) ; RR Netz and R Lipowski, Europhys. Lett. 29. 345 (1995); F. David, J. de Phys. 51 , C7-115 (1990); H. Kleinert, Phys. Lett. A 257 ,269 (1999) (cond-mat/9811308); M. Bachmann, H. Kleinert, A. Pelster, Phys. Lett. A 261 , 127 (1999) (cond-mat/9905397). The problem has been solved with the help of the strong-coupling variational perturbation theory developed in Chapter 5 by H. Kleinert, Phys. Lett. A 257, 269 (1999) (cond-mat/9811308); M. Bachmann, H. Kleinert, and A. Pelster, Phys. Lett. A 261, 127 (1999) (cond-mat/9905397). The quantum-mechanical calculation presented in Section 6.4 is taken from H. Kleinert, A. Chervyakov, and B. Hamprecht, Phys. Lett. A 260, 182 (1999) (condmat/9906241).

Mirum, quod divina natura dedit agros. It's wonderful that divine nature has given us fields. VARRO, 82 B.C .

7 Many Particle Orbits Statistics and Second Quantization Realistic physical systems usually contain groups of identical particles such as specific atoms or electrons. Focusing on a single group, we shall label their orbits by x(v)(t) with // = 1,2, 3, ... , N. Their Hamiltonian is invariant under the group of all N! permutations of the orbital indices //. Their Schrodinger wave functions can then be classified according to the irreducible representations of the permutation group. Not all possible representations occur in nature. In more than two space dimensions, there exists a super-selection rule, whose origin is yet to be explained, which eliminates all complicated representations and allows only for the two simplest ones to be realized: those with complete symmetry and those with complete antisymmetry. Particles which appear always with symmetric wave functions are called bosons . They all carry an integer-valued spin. Particles with antisymmetric wave functions are called fermions 1 and carry a spin whose value is half-integer. The symmetric and antisymmetric wave functions give rise to the characteristic statistical behavior of fermions and bosons. Electrons, for example, being spin-1/2 particles, appear only in antisymmetric wave functions. The antisymmetry is the origin of the famous Pauli exclusion principle, allowing only a single particle of a definite spin orientation in a quantum state, which is the principal reason for the existence of the periodic system of elements, and thus of matter in general. The atoms in a gas of helium, on the other hand, have zero spin and are described by symmetric wave functions . These can accommodate an infinite number of particles in a single quantum state giving rise to the famous phenomenon of Bose-Einstein condensation. This phenomenon is observable in its purest form in the absence of interactions, where at zero temperature all particles condense in the ground state. In interacting systems, Bose-Einstein statistics can lead to the stunning quantum state of superfluidity. The particular association of symmetry and spin can be explained within relativistic quantum field theories in spaces with more than two dimensions where it is shown to be intimately linked with the locality and causality of the theory. 1 Had M. Born as editor of Zeitschrift fiir Physik not kept a paper by P. Jordan in his suitcase for half a year in 1925, they would be called jardanons . See the bibliographical notes by B. Schroer (hep-thj0303241) .

591

592

7 Many Particle Orbits - Statistics and Second Quantization

In two dimensions there can be particles with an exceptional statistical behavior. Their properties will be discussed in Section 7.5. In Chapter 16, such particles will serve to explain the fractional quantum Hall effect. The problem to be solved in this chapter is how to incorporate the statistical properties into a path integral description of the orbits of a many-particle system. Afterwards we describe the formalism of second quantization or field quantization in which the path integral of many identical particle orbits is abandoned in favor of a path integral over a single fluctuating field which is able to account for the statistical properties in a most natural way.

7.1

Ensembles of Bose and Fermi Particle Orbits

For bosons, the incorporation of the statistical properties into the orbital path integrals is quite easy. Consider, for the moment, distinguishable particles. Their many-particle time evolution amplitude is given by the path integral

II [J VDx(V)] N

t IX(l) X(N). t ) = (Xb(l) " . " X(Nl. b ,b a "." a ,a

eiA(N)

Iii ,

(7.1)

v=l

with an action of the typical form

(7.2) where V(x(v)) is some common background potential for all particles interacting via the pair potential Vint(x(v) - X(VI)). We shall ignore interactions involving more than two particles at the same time, for simplicity. If we want to apply the path integral (7.1) to indistinguishable particles of spin zero, we merely have to add to the sum over all paths x(v)(t) running to the final positions x~v) the sum of all paths running to the indistinguishable permuted final positions x~(v)) . The amplitude for n bosons reads therefore (1) (N).t)_"( (p(l)) , . .. ,X(p(N))·tl (1) (N) · t ) ( Xb(1) , ... , Xb(N).tl ,b xa , .. . , xa , a - L... Xb ,b Xa , ... , Xa , a , b

(73) .

p(v)

where p(v) denotes the N! permutations of the indices v. For bosons of higher spin, the same procedure applies to each subset of particles with equal spin orientation. A similar discussion holds for fermions. Their Schrodinger wave function requires complete antisymmetrization in the final positions. Correspondingly, the amplitude (7.1) has to be summed over all permuted final positions x~p(v)), with an extra minus sign for each odd permutation p(v). Thus, the path integral involves both sums and differences of paths. So far, the measure of path integration has always been a true sum over paths. For this reason it will be preferable to attribute the alternating sign to an interaction between the orbits, to be called a statistics interaction . This interaction will be derived in Section 7.4.

7.1 Ensembles of Bose and Fermi Particle Orbits

593

For the statistical mechanics of Bose- and Fermi systems consider the imaginarytime version of the amplitude (7.3): ( Xb(1), ... , Xb(N).,

"f31 Xa"' (1) "

IL

(N). Xa ,

0) -_ '" ((P(l)) (P(N)). "f31 (1) L..J fp(v) Xb , ... , Xb , IL Xa"'"

(N). Xa ,

p(v)

0) ,

(7.4)

where fp(v) = ±1 is the parity of even and odd permutations p(v) , to be used for Bosons and Fermions, respectively. Its spatial trace integral yields the partition function of N-particle orbits:

A factor liN! accounts for the indistinguishability of the permuted final configurations. For free particles, each term in the sum (7.4) factorizes:

where each factor has a path integral representation (xip(v))nf3lx~) 0)0 =

111i~ M ] VDx(v) exp - dT - X(v)2(T), l X(V)(Ii~)=x(P(V)) (O)=xl:') n0 2 [

b

(7.7)

X(V)

which is solved by the imaginary-time version of (2.130):

(7.8) The partition function can therefore be rewritten in the form Z oeN) --

l I ND

V27rn2f31M

N'

'

J

dD X (1)

...

d D X (N)", L..Jfp(v) p(v)

N II exp

{I

M [x(p(v)) _xCv)] 2 nf3

2}

----=-----:-::----''-

n

.

v=l

(7.9) This is a product of Gaussian convolution integrals which can easily be performed as before when deriving the time evolution amplitude (2.70) for free particles with the help of Formula (2.69). Each convolution integral simply extends the temporal length in the fluctuation factor by nf3 . Due to the indistinguishability of the particles, only a few paths will have their end points connected to their own initial points, i.e., they satisfy periodic boundary conditions in the interval (0, n(3) . The sum over permutations connects the final point of some paths to the initial point of a different path, as illustrated in Fig. 7.1. Such paths satisfy periodic boundary conditions on an interval (0, wn(3), where w is some integer number. This is seen most clearly by drawing the paths in Fig. 7.1 in an extended zone scheme shown in Fig. 7.2, which is reminiscent of Fig. 6.1. The extended zone scheme can, moreover , be placed on a

594

7 Many Particle Orbits - Statistics and Second Quantization

Ii~

T

Figure 7.1 Paths summed in partition function (7.9). Due to indistinguishability of particles, final points of one path may connect to initial points of another.

1

3 1

2

1

I-

-I

I-

-I

Ii~

,

,

31i_3

21if3

T

Figure 7.2 Periodic representation of paths summed in partition function (7.9), once in extended zone scheme, and once on D-dimensional hypercylinder embedded in D + 1 dimensions. The paths are shown in Fig. 7.1. There is now only one closed path on the cylinder. In general there are various disconnected parts of closed paths.

hypercylinder, illustrated in the right-hand part of Fig. 7.2. In this way, all paths decompose into mutually disconnected groups of closed paths winding around the cylinder, each with a different winding number w [1]. An example for a connected path which winds three times 3 around the D-dimensional cylinder contributes to the partition function a factor [using Formula (2.69)]: to!.Ztl3

=

1 3D jdDX(lldDX(2ldDX(3l exp .J21fn2(J/M

x exp

{-.!. M [X(3LX(2lt} n 2

l_X(1)t} {_.!.M[X(1)-X(3f}_ {_.!.M[X(2 n 2 n(J exp n 2 n(J -

n(J

VD D' (7.10) .J21fn23fJ/M

For cycles of length w the contribution is to!.zt lw

=

Zo(w(J) ,

(7.11)

where Zo(w(J) is the partition function of a free particle in a D-dimensional volume VD for an imaginary-time interval wn(J: (7.12)

595

7.1 Ensembles of Bose and Fermi Particle Orbits

In terms of the thermal de Broglie length leChj3) == V27rn 2 j3 / M associated with the temperature T = 1/kBj3 [recall (2.351)]' this can be written as

Zo(wj3)

=

Vv If (wnj3) .

(7.13)

There is an additional factor 1/w in Eq. (7.11) , since the number of connected windings of the total w! closed paths is (w - 1)!' In group theoretic language, it is the number of cycles of length w, usually denoted by (1,2 , 3, .. . , w), plus the (w -1)! permutations of the numbers 2, 3, . .. , w. They are illustrated in Fig. 7.3 for w = 2, 3, 4. In a decomposition of all N! permutations as products of cycles, the number of elements consisting of C1, C2 , C3 , ... cycles oflength 1, 2, 3, .. . contains (7.14) elements [2] . (12)

~= ~ !~2I (23)

(12 )

(31)

(123 )

(132 )

~= ~3 ~X~ ~X~3 ~~~ T~TiT~'f 333 3 3 3 ;.i! ........il.il3. ........3.; (12 ) 11 lxl 2 22 2 3 33 344 44 (132) (123 )

1;z:1

(13 )

(14 )

21~12 3 3 4-4 (134 )

1~(

l~f

(23 )

1-

1

(2 4) 11

2 2 2X2 3 3 3 3 4 4 4- 4 (14 3) (1 24)

32~23 4 4 (142 )

1~1 21~12

1 ~{

12 ~12 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4- 4 4- 4 4 4 4 (1342) (13241 (1234 ) (124 3) '1 1

2 2 3 3 3 3 4 4 4 4 (1423) (14321

(34)

1-1 22 3X 3 4 4 (234)

1-1 32~23 4

4

(24 3) 1-1 2 2

3;Z:34 4

(12 )(34) lX l 2 2 3X 3 4 4

(13 )(24 ) (1 4)( 23) 12 3 4

~12 1)(1 2 2 3 3 4 4

3 4

I ~I! If~I! 2 ~2 I :~I! II§I! II~I! 3

3

.4

4

Figure 7.3 Among the w! permutations of the different windings around the cylinder, (w -I)! are connected. They are marked by dotted frames. In the cycle notation for permutation group elements, these are (12) for two elements, (123) , (132) for three elements, (1234) , (1243), (1324) , (1342), (1423), (1432) for four elements. The cycles are shown on top of each graph, with trivial cycles of unit length omitted. The graphs are ordered according to a decreasing number of cycles.

596

7 Many Particle Orbits - Statistics and Second Quantization

With the knowledge of these combinatorial factors we can immediately write down the canonical partition function (7.9) of N bosons or fermions as the sum of all orbits around the cylinder, decomposed into cycles: N

1 'L..Jtp(v)M(Cl, " Zo(N) (13) = N' . .. , C N ) • plY)

II

(7.15)

w=l N=~wwCw

The sum can be reordered as follows: N

Z(N)_~ o - N!

M(Cl, .. . , CN)tw ,C" .. . ,cn

II

[Zo(wJ3)tw

.

(7.16)

w=l

C" "" CN

N=~wwCw

The parity tw ,c"""cn of permutations is equal to (±1)~w (W+l)Cw . Inserting (7.14), the sum (7.16) can further be regrouped to

L

Cl,,,, ,CN N=~wwCw

(7.17)

For N = 0, this formula yields the trivial partition function Z~O) (13) = 1 of the noparticle state, the vacuum. For N = 1, i.e., a single particle, we find Z~l)(J3) = Zo(J3). The higher zt) can be written down most efficiently if we introduce a characteristic temperature T(O) C

[ ] 2/D N kBM VD((D /2) ,

= 27r1i2 -

(7.18)

and measure the temperature T in units of T~O), defining a reduced temperature t == T/T~O). Then we can rewrite Z~l)(J3) as t D/ 2VD. Introducing further the Ndependent variable N ]2/D [ (7.19) TN == ((D/2) t, we find Z~l) = Tf/2. A few low-N examples are for bosons and fermions: Z o(2)

±2-1-D/2 T2D/2 + T2D,

Z~3)

±3- 1- D/ 2Tf/ 2 + 2- 1- D/ 2Tf ± T12-1TtD/2, ±T2- DTf/ 2 + (T 3 - D + 3- 1- D/ 2) Tf±T2- ~T1D/2

Z~4)

(7.20)

+ 3- 1T 3Ti D.

From zt)(J3) we calculate the specific heat [recall (2.600)] of the free canonical ensemble: (N) _ d2 [ ~ [ (N)] (N)] _ (7.21 ) CO - T dT2 Tlog Zo - TN dTj, TN log Zo ,

597

7.1 Ensembles of Bose and Fermi Particle Orbits

and plot it [3] in Fig. 7.4 against t for increasing particle number N. In the limit ---> 00, the curves approach a limiting form with a phase transition at T = TJO) , which will be derived from a grand-canonical ensemble in Eqs. (7.67) and (7.70). The partition functions can most easily be calculated with the help of a recursion relation [4, 5], starting from Z~O) == 1:

N

zt)({)) =

~ fJ±1)n-1 Zo(n{))zt- n)({)) .

(7.22)

n=1

2 1.75 1.5

C~N)/NkB

1.25

1 0.75 0.5 '/

0.25

'/

Figure 7.4 Plot of the specific heat of free Bose gas with N = 10, 20, 50, 100, 500, 00 particles. The curve approaches for large T the Dulong-Petit limit 3kBN12 corresponding to the three harmonic kinetic degrees of freedom in the classical Hamiltonian p2/2M. There are no harmonic potential degrees of freedom.

This relation is proved with the help of the grand-canonical partition function which is obtained by forming the sum over all canonical partition functions zt) ({)) with a weight factor zN: 00

Zao({)) ==

The parameter z

L

zt)({) ) zN . (7.23) N=O is the Boltzmann factor of one particle with the chemical potential

f.1:

z = z ({)) == e(3Jl.

(7.24)

It is called the fugacity of the ensemble. Inserting the cycle decompositions (7.17),

the sum becomes

L C 1 , ... , C N

N=EwwCw

IT ~ [(±1)W_1 Z0 (W{))ew(3Jl]Cw . -1 w-

CWo

W

The right-hand side may be rearranged to

Zao({)) =

j}1C~O C~! [(±1)W_1Z0(w~eW(3Jlrw

(7.25)

598

7 Many Particle Orbits - Statistics and Second Quantization

(7.26) From this we read off the grand-canonical free energy [recall (1.542)] of noninteracting identical particles (7.27) This is simply the sum of the contributions (7.11) of connected paths to the canonical partition function which wind w = I, 2, 3, ... times around the cylinder [1 , 6] . Thus we encounter the same situation as observed before in Section 3.20: the free energy of any quantum-mechanical system can be obtained from the perturbation expansion of the partition function by keeping only the connected diagrams. The canonical partition function is obviously obtained from (7.27) by forming the derivative: (7.28) It is now easy to derive the recursion relation (7.22) . From the explicit form (7.27) , we see that

~Zao = - (~(3Fa) 8z 8z

(7.29)

Zao .

Applying to this N - 1 more derivatives yields

8 N- 1 [8 ] N- 1 (N _ I)! ( 81+1 ) 8 N- I- 1 8ZN- 1 8z Zao = - ~ l!(N -l-l)! 8z1+1(3Fa 8ZN- I- 1Zao . To obtain from this Ztl we must divide this equation by N! and evaluate the derivatives at z = O. From (7.27) we see that the l + 1st derivative of the grandcanonical free energy is

81+1 I 8 z 1+ 1(3Fa z=o = _(±l)ll! Zo((l + 1)(3).

(7.30)

Thus we obtain 1 8N I 1 N-1 N! 8z NZao z=o = N ~ (±1)1 Zo((l

1

+ 1)(3) (N -l-l)!

8 N- I- 1 I 8ZN- I- 1Zao z=o

Inserting here (7.28) and replacing l --+ n-1 we obtain directly the recursion relation (7.22). The grand-canonical free energy (7.27) may be simplified by using the property 1

Zo(w(3 ) = Zo((3) W D / 2

(7.31)

599

7.2 Bose-Einstein Condensation

of the free-particle partition function (7.12), to remove a factor 1/ ,juP from Zo(w(3). This brings (7.27) to the form 1

Fa

ew {31'

00

= - (3-Zo((3) L(±I)W-l w=l

D/2+1'

(7.32)

W

The average number of particles is found from the derivative with respect to the chemical potential2

fJ

N

00

= -7lFa = Zo((3) up,

ew {31'

L (±I)W- l W D/2' w=l

(7.33)

The sums over w converge certainly for negative or vanishing chemical potentialp" i.e. , for fugacities smaller than unity. In Section 7.3 we shall see that for fermions , the convergence extends also to positive p,. If the particles have a nonzero spin 8, the above expressions carry a multiplicity factor gs = 28 + 1, which has the value 2 for electrons. The grand-canonical free energy (7.32) will now be studied in detail thereby revealing the interesting properties of many-boson and many-fermion orbits, the ability of the former to undergo Bose-Einstein condensation, and of the latter to form a Fermi sphere in momentum space.

7.2

Bose-Einstein Condensation

We shall now discuss the most interesting phenomenon observable in systems containing a large number of bosons, the Bose-Einstein condensation process.

7.2.1

Free Bose Gas

For bosons, the above thermodynamic functions (7.32) and (7.33) contain the functions 00

(,,(z) ==

ZW

L-' w=l wI'

(7.34)

These start out for small z like z, and increase for z -+ 1 to ((v) , where ((z) is Riemann's zeta function (2.519). The functions (,,(z) are called Polylogarithmic functions in the mathematical literature [7] , where they are denoted by Li"(z). They are related to the Hurwitz zeta function ((v, a, z) == 'L'::=ozw/(w + a)" as (,,(z) = z((v, 1, z). The functions ¢(z , v, a) = ((v, a, z) are also known as Lerch functions . In terms of the functions (.,(z) , and the explicit form (7.12) of Zo((3), we may write Fa and N of Eqs. (7.32) and (7.33) simply as 2In grand-canonical ensembles, one always deals with the average particle number (N) for which one writes N in all thermodynamic equations [recall (1.548)]. This should be no lead to confusion.

7 Many Particle Orbits - Statistics and Second Quantization

600

Fa

(7.35)

N

(7.36)

The most interesting range where we want to know the functions (,..,(z) is for negative small chemical potential J.L . There the convergence is very slow and it is useful to find a faster-convergent representation. As in Subsection 2.15.6 we rewrite the sum over w for z = ef31-' as an integral plus a difference between sum and integral

(7.37) The integral yields f(l- v)( -f3J.LY-l, and the remainder may be expanded sloppily in powers of J.L to yield the Robinson expansion (2.579):

There exists a useful integral representation for the functions (v(z):

(7.39) where iv(a) denotes the integral v-I

00

iv(a)

= io(

dE

E

ee-a - 1

,

(7.40)

containing the Bose distribution function (3.93):

(7.41) Indeed, by expanding the denominator in the integrand in a power series

(7.42) and performing the integrals over E, we obtain directly the series (7.34). It is instructive to express the grand-canonical free energy Fa in terms of the functions i,..,(a). Combining Eqs. (7.35) with (7.39) and (7.40), we obtain

F. =-~Z ( )i D/ 2+l(f3J.L) =_ 1 VD {oodE ED/2 a f3 0 f3 f(D/2+1) f3r(D/2+1) V27rn2f3/MD io ee- f31-' _ l'

(

) 7.43

601

7.2 Bose-Einstein Condensation

The integral can be brought to another form by partial integration, using the fact that 1 0 --::--= -log(1 -

OE

e,,-,6JL - 1

e-,,+,6JL).

(7.44)

The boundary terms vanish, and we find immediately: (7.45)

This expression is obviously equal to the sum over momentum states of oscillators with energy fiwp == p2/2M, evaluated in the thermodynamic limit N --+ 00 with fixed particle density N /V, where the momentum states become continuous: (7.46) This is easily verified if we rewrite the sum with the help of formula (1.555) for the surface of a unit sphere in D dimensions and a change of variables to the reduced particle energy E = fJp 2/2M as an integral

"~

--+

VD

J

dDp 1 (27rfi)D - VDSD (27rfi)D

1

VD

f(D/2) V27rfi2fJ/MD

roo d

Jo

J

dpp

D- l (2M/fJ)D/21 - VDSD (27rfi)D 2

dEE

D/2- 1

(7.47)

D/2-1

EE

J

.

Another way of expressing this limit is

L p

--+

roo dE N",

Jo

(7.48)

where N" is the reduced density of states per unit energy interval: (7.49)

The free energy of each oscillator (7.46) differs from the usual harmonic oscillator expression (2.483) by a missing ground-state energy fiw p /2. The origin of this difference will be explained in Sections 7.7 and 7.14. The particle number corresponding to the integral representations (7.45) and (7.46) is

602

7 Many Particle Orbits - Statistics and Second Quantization

1 .__.__.__._.__.__.__.__._-_.__.__.

0.8 0.6 0.4 0.2 0.2

0.4

Figure 7.5 Plot of functions (y(z) for thermodynamics.

0.6

l/

0.8

1

z

= 3/2 and 5/2 appearing in Bose-Einstein

For a given particle number N, Eq. (7.36) allows us to calculate the fugacity as a function of the inverse temperature, z(f3) , and from this the chemical potential J.L(f3) = f3- 1 log z(f3) . This is most simply done by solving Eq. (7.36) for f3 as a function of z, and inverting the resulting function f3(z) . The required functions (y(z) are shown in Fig. 7.5. There exists a solution z(f3) only if the total particle number N is smaller than the characteristic function defined by the right-hand side of (7.36) at unit fugacity z(f3) = 1, or zero chemical potential J.L = 0:

VD

N < W(nf3) ((D/2).

(7.51)

Since le(nf3) decreases with increasing temperature, this condition certainly holds at sufficiently high T. For decreasing temperature, the solution exists only as long as the temperature is higher than the critical temperature Tc = l/kBf3c , determined by (7.52) This determines the critical density of the atoms. The de Broglie length at the critical temperature will appear so frequently that we shall abbreviate it by Pc: (7.53) The critical density is reached at the characteristic temperature T~O) introduced in Eq. (7.18). Note that for a two-dimensional system, Eq. (7.18) yields T~O) = 0, due to ((1) = 00, implying the nonexistence of a condensate. One can observe, however , definite experimental signals for the vicinity of a transition. In fact , we have neglected so far the interaction between the atoms, which is usually repulsive. This will give rise to a special type of phase transition called Kosterlitz-Thouless transition. For a discussion of this transition see other textbooks [8].

603

7.2 Bose-Einstein Condensation

By combining (7.18) with (7.36) we obtain an equation for the temperature dependence of z above Te: 1

= (T)D/2 (D/2(Z(T)) Te

((D/2) '

T > Te·

(7.54)

This is solved most easily by calculating T /Te as a function of z = ef3/L. If the temperature drops below Te , the system can no longer accommodate all particles N in a normal state. A certain fraction of them, say Neond(T), is forced to condense in the ground state of zero momentum, forming the so-called Bose-Einstein condensate. The condensate acts like a particle reservoir with a chemical potential zero. Both phases can be described by the single equation for the number of normal particles, i.e., those outside the condensate: (7.55) For T > Te , all particles are normal and the relation between fJ and the temperature is found from the equation Nn(T) = N, where (7.55) reduces to (7.36). For T < Te, however, the chemical potential vanishes so that z = 1 and (7.55) reduces to (7.56) which yields the temperature dependence of the number of normal particles: Nn(T) N

= (T)D/2 Te

, T < Te·

(7.57)

The density of particles in the condensate is therefore given by Neond(T) N

= 1- Nn(T) = 1- (T)D/2 Te

N

(7.58)

We now calculate the internal energy which is, according to the general thermodynamic relation (1.550), given by E

= Fa + TS + fJN = Fa - TfJrFa + fJN = 8f3(f3Fa) + fJN.

(7.59)

Expressing N as -8Fa/8fJ, we can also write (7.60) Inserting (7.35) we see that only the f3-derivative of the prefactor contributes since (f38f3 - fJ8/L) applied to any function of z = ef3/L vanishes. Thus we obtain directly D

E = --F.a

2

'

(7.61)

604

7 Many Particle Orbits - Statistics and Second Quantization

which becomes with (7.35) and (7.36):

D E = - 2{3Zo({3KD/2+l(Z) =

D (D/2+l(Z) (D/2(Z) NkBT.

(7.62)

-"2

The entropy is found using the thermodynamic relation (1.571): 1 (E - JlN - FG) = T 1 --2-FG ( D +- 2 S =T JlN) ,

(7.63)

or, more explicitly,

S = k [D + 2 Z ({3)1" ( )_ B 2 0 ,>D/2+1 Z

{3Jl N]

{3] Jl.

= k BN [D 2+ 2 (D/2+l(Z) (D/2(Z)

For T < Te , the entropy is given by (7.64) with Jl the number N n of normal particles of Eq. (7.57):

(7.64)

= 0, Z = 1 and N replaced by

(7.65) The particles in the condensate do not contribute since they are in a unique state. They do not contribute to E and FG either since they have zero energy and Jl = O. Similarly we find from (7.62):

E = D Nk T (T)D/2 (D/2+l(1) = ~TS < 2 B Te (D/2(1) D+2 T e , the chemical potential at fixed N satisfies the equation (7.68) This follows directly from the vanishing derivative (3o{3N = 0 implied by the fixed particle number N. Applying the derivative to Eq. (7.36) and using the relation zOz(v(z) = (v-l(Z), as well as (3o{3f(z) = zOzf(z) (30{3({3Jl) , we obtain

(7.69) thus proving (7.68).

605

7.2 Bose-Einstein Condensation

The specific heat C at a constant volume in units of kB is found from the derivative C = T8rSIN = -/32o(3EIN' using once more (7.68):

C = kBN [(D

+ 2)D (D/2+1 (Z) 4

(D/2(Z)

T> Te.

_ D2 (D/2(Z)] 4 (D/2-1(Z) ,

(7.70)

At high temperatures, C tends to the Dulong-Petit limit DkBN/2 since for small Z all (v(z) behave like z. Consider now the physical case D = 3, where the second denominator in (7.70) contains (1/2(Z). As the temperature approaches the critical point from above, Z tends to unity from below and (1/2(Z) diverges. Thus 1/(1/2(1) = 0 and the second term in (7.70) disappears, yielding a maximal value in three dimensions 15 (5/2(1) 4 1. Inserting (7.142) back into (7.162) , we may re-express the normal fraction as a function of the temperature as follows : (7.143) and the condensate fraction as (7.144) Including the next term in (7.162), the condensate fraction becomes (7.145) It is interesting to re-express the critical temperature (7.142) in terms of the particle density at the origin which is at the critical point, according to Eq. (7.134),

(7.146) where we have shortened the notation on an obvious way. From this we obtain

k

T(O) = B

e

27r1i 2

M

[

no(O) ] 2/D ((D/2)

(7.147)

Comparing this with the critical temperature without a trap in Eq. (7.18) we see that both expressions agree if we replace N /V by the uniform density no (0). As an

619

7.2 Bose-Einstein Condensation

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0. 4

0.6

00

0.2 0.4

T ITc

0.6

0.8

1

1.2

1.4

1.6

1.8

T I Tc

Figure 7.9 Condensate fraction Ncond/N == 1 - Nn/N as function of temperature for total number of N particles. The long- and short-dashed curves on the left-hand show the zeroth and first-order approximations (7.144) and (7.163). The dotted curve displays the free-boson behavior (7.58). The right-hand figure shows experimental data for 40000 87Rb atoms near Tc by Ensher et. al., Phys. Rev. Lett. 77, 4984 (1996). The solid curve describes noninteracting Bose gas in a harmonic trap [ef. (7.144)]. The dotted curve corrects for finite-N effects [cf. (7.163)]. The dashed curve is the best fit to the data. The transition lies at Tc ,:::; 280 nK, about 3% below the dotted curve.

obvious generalization, we conclude that Bose condensation in any trap will set in when the density at the lowest point reaches the critical value determined by (7.147) [and (7.18)]. Note the different power D and argument in ((D) in Eq. (7.142) in comparison with the free-boson argument D/2 in Eq. (7.18). This has the consequence that in contrast to the free Bose gas, the gas in a trap can form a condensate in two dimensions. There is now, however, a problem in D = 1 dimension where (7.142) gives a vanishing transition temperature. The leading-order expression (7.142) for critical temperature can also be calculated from a simple statistical consideration. For small w , the density of states available to t he bosons is given by the classical expression (4.216):

- (-)

pcl(E) -

M 27rn?

D/2

1

f(D/2)

J

dD x [E

-

V(x)] D / 2-1 .

(7.148)

The number of normal particles is given by the equation Nn

=

roo

JEmin

dE

pcl(E)

eE / kBT

-

1'

(7.149)

where E min is the classical ground state energy. For a harmonic trap, the spatial integral in (7.148) can be done and is proportional to ED , after which the integral over E in (7.149) yields [kBT/nw ]D((D) = (T/TJol)DN , in agreement with (7.143).

620

7 Many Particle Orbits - Statistics and Second Quantization

Alternatively we may use the phase space formula dDp 1 _ ~ JdD dDp - n,B[p2/2M+V(xl ] X (27rn)D e,B[P2/2M+V(xl] _ 1 - ~ X (27rn)D e

N - JdD n -

=

f

1 D J dDxe-n,BV(x l , 2 n=1 V27rn nf3 / M

where the spatial integration produces a factor side becomes again (T /TJol)D N.

7.2.7

(7 .150)

V27r /M w nf3 so that the right-hand 2

More General Anisotropic Trap

The equation (7.150) for the particle number can be easily calculated for a more general trap where the potential has the anisotropic power behavior V(x) = M (;?a?

f (EJ)Pi ,

i=1 ai . f d _ . . average ah W IS some requency parameter an a IS t he geometnc were 2

(7.151)

== [rrDi=1 ai ]1/D.

Inserting (7.151) into (7.150) we encounter a product of integrals (7.152) so that the right-hand side of (7.150) becomes (T /TJol/j N , with the critical temperature (7.153) where jj is the dimensionless parameter _ D D 1 D== -+ 2 i=1 Pi

L-'

(7.154)

A harmonic which takes over the role of D in the harmonic formula (7.142). trap with different oscillator frequencies WI, . . . , WD along the D Cartesian axes, is a special case of (7.151) with Pi == 2, = (;;2a,2ja; and jj = D, and formula (7.153) reduces to (7.142) with W replaced by the geometric average of the frequencies (;; == (WI '" WD)I/D. The parameter ji disappears from the formula. A free Bose gas in a box of size VD = np:'I(2ai) = 2DjiD is described by (7.151) in the limit Pi ---> 00 where jj = D/2. Then Eq. (7.153) reduces to

w;

7rn2 [~] 2/D _ 27rn2 [ N ] 2/D - 2Mji2 ((D/2) - M VD((D/2) ,

(Ol _

kBTc

(7.155)

621

7.2 Bose-Einstein Condensation

in agreement with (7.53) and (7.18). Another interesting limiting case is that of a box of length L = 2al in the xdirection with PI = 00, and two different oscillators of frequency W2 and W3 in the other two directions. To find T~O) for such a Bose gas we identify c; 2 0,2 / a~ 3 = W~ 3 in the potential (7.151), so that C;4/ii 2 = w~wUai, and obtain " k B

T(O) c

7.2.8

=

nC; ( ~ )

1/5 [

2MC;

] 2/5 = n - ( 27r AWl AW2 ) N al((5/2) w L2

1/5 [ ~ ] 2/5

((5/2)

(7.156)

Rotating Bose-Einstein Gas

Another interesting potential can be prepared in the laboratory by rotating a Bose condensate [48] with an angular velocity 0 around the z-axis. The vertical trapping frequencies is W z ~ 27r x 11.0Hz ~ 0.58x nK , the horizontal one is W.l ~ 6 x W z . The centrifugal forces create an additional repulsive harmonic potential, bringing the rotating potential to the form V(x)

M w;

(

2

2

K

ri )

nwz ( :\2 Z2 rl K ri ) + 361]:\2 + 2'-:\4 '

= -2- z + 361]r .l + 2':\2 = 2 Wz

Wz

Wz

(7.157)

Wz

where rl = x 2 + y2, 1] == 1 - 0 2 / wi, K ~ 0.4, and Awz == 3.24511,ID ~ 1.42 x 10- 3 K. For 0 > W.l, 1] turns negative and the potential takes the form of a Mexican hat as shown in Fig. 3.13, with a circular minimum at r~ = -361]A~j K. For large rotation speed, the potential may be approximated by a circular harmonic well, so that we may apply formula (7.156) with al = 27rrm , to obtain the 1]-independent critical temperature k

T(O) B

c

~ n (K)I/5 [ N ]2/5 wz;;: ((5 / 2)

~

(7.158)

K = 0.4 and N = 300000, this yields Tc ~ 53nK. At the critical rotation speed 0 = W.l, the potential is purely quartic r.l /(X2+y2). To estimate TiO) we approximate it for a moment by the slightly different potential (7.151) with the powers PI = 2, P2 = 4, P3 = 4, al = A wz , a2 = a3 = Awz (K/2)1 /4, so that formula (7.153) becomes

For

(0) _

kBTc

7r 2 K

-nwz [ 16f4 (5/4)

] 1/5 [

N ] 2/5 ((5/2)

(7.159)

It is easy to change this result so that it holds for the potential ex r 4 = (x + y)4 rather than X4 + y4: we multiply the right-hand side of equation (7.149) for N by a factor 27r J rdrdxdy e- r4 7r 3/ 2 (7.160) xcy J dxdy e- ' r[5 /4]2 .

622

7 Many Particle Orbits - Statistics and Second Quantization

This factor arrives inversely in front of N in Eq. (7.161), so that we obtain the critical temperature in the critically rotating Bose gas 4 r;,

(0) _

kBTc

- nwz

1/5 [

( -:; )

N ] 2/5 ((5/2)

(7.161)

The critical temperature at 0 = W.L is therefore by a factor 41 / 5 ;:::; 1.32 larger than at infinite O. Actually, this limit is somewhat academic in a semiclassical approximation since the quantum nature of the oscillator should be accounted for.

7.2.9

Finite-Size Corrections

Experiments never take place in the thermodynamic limit. The particle number is finite and for comparison with the data we must calculate finite-size corrections coming from finite N where 11 w ;:::; 1/N1/D is small but nonzero. The transition is no longer sharp and the definition of the critical temperature is not precise. As in the thermodynamic limit, we shall identify it by the place where ZD = 1 in Eq. (7.139) for N n = N . For D > 3, the corrections are obtained by expanding the first term in the sum (7.138) in powers of wand performing the sums over wand subtracting ((0) for the sum E~=11: 1 [ Nn(Tc) = ((3nw)D ((D)

2

] (3nw D (wn(3) + -2((D-1) + ----u- D (3D-1)((D-2) + ...

((0). (7.162) The higher expansion terms contain logarithmically divergent expressions, for instance in one dimension the first term ((1), and in three dimensions ((D-2) = ((1) . These indicate that the expansion powers of (3nw is has been done improperly at a singular point. Only the terms whose (-function have a positive argument can be trusted. A careful discussion along the lines of Subsection 2.15.6 reveals that ((1) must be replaced by (reg(l) = -log((3nw) + const., similar to the replacement (2.588). The expansion is derived in Appendix 7A. For D > 1, the expansion (7.162) can be used up to the ((D - 1) terms and yields the finite-size correction to the number of normal particles Nn

N Setting N n

=

(T)D TJO)

D ((D - 1)

1

+ "2 (l - l/D(D) N1/D ·

(7.163)

= N, we obtain a shifted critical temperature by a relative amount bTc TJO) =

1 ((D - 1)

1

-"2 ((D- 1)/D(D) N1/D + ....

(7.164)

In three dimensions, the first correction shifts the critical temperature TJO) downwards by 2% for 40000 atoms. Note that correction (7.164) has no direct w-dependence whose size enters only implicitly via TJO) of Eq. (7.142). In an anisotropic harmonic trap, the temperature

623

7.2 Bose-Einstein Condensation

shift would carry a dimensionless factor w/w where w is the arithmetic mean w == (Lp:'l Wi) / D. The higher finite-size corrections for smaller particle numbers are all calculated in Appendix 7A. The result can quite simply be deduced by recalling that according to the Robinson expansion (2.581), the first term in the naive, wrong power series expansion of (v(e - b ) = L~=l e- wb /w v = Lk::o( -b)k((v - k)/k! is corrected by changing the leading term ((v) to r(1 - v)( _b)v- l + ((v) which remains finite for all positive integer v. Hence we may expect that the correct equation for the critical temperature is obtained by performing this change in Eq. (7.162) on each ((v). This expectation is confirmed in Appendix 7A. It yields for D = 3,2,1 the equations for the number of particles in the excited states 2

+]

=

Nn

=

(,8liw)2 ((2) -,8liw log,8nw - 'Y + 2

Nn

=

[ ,8liw 1 (,8liw)3 ((3)3 + -2-((2) - (,8liw) 2 ( log,8nw - 'Y + 1 24 9 +)...] , (7.167)

_ 1_

,8nw

[_ (log,8nw _ 'Y) _ ,8nw ((0) 2

+(,8liw)

Nn

1 [

(

12

(( -1)

... ,

1) - 7(,8nw? 12 ((0) + ... ] ,

(7.165) (7.166)

where'Y = 0.5772 . .. is the Euler-Mascheroni number (2.467). Note that all nonlogarithmic expansion terms coincide with those of the naive expansion (7.162). These equations may be solved for ,8 at N n = N to obtain the critical temperature of the would-be phase transition. Once we study the position of would-be transitions at finite size, it makes sense to include also the case D = 1 where the thermodynamic limit has no transition at all. There is a strong increase of number of particles in the ground state at a "critical temperature" determined by equating Eq. (7.165) with the total particle number N , which yields (0) _

1

~

1

kBTc - nwN (- l,8n N' og w + 'Y ) ~ liwN og1

D=l.

(7.168)

Note that this result can also be found also from the divergent naive expansion (7.162) by inserting for the divergent quantity ((1) the dimensionally regularized expression (reg (l) = -log(,8nw) of Eq. (2.588).

7.2.10

Entropy and Specific Heat

By comparing (7.126) with (7.125) we see that the grand-canonical free energy can be obtained from N n of Eq. (7.162) by a simple multiplication with -1/,8 and an increase of the arguments of the zeta-functions ((v) by one unit. Hence we have, up to first order corrections in w

Fa (,8,f.tc) = -

,8(,8~w)D

[ ((D + 1) + ,8liw~ ((D) + ...J .

(7.169)

624

7 Many Particle Orbits - Statistics and Second Quantization

From this we calculate immediately the entropy S = -OTFC = k B{320(3Fc as

S

=

-kB(D + l){3Fc

=

kB(D + 1) ({3~)D [ ((D + 1) + {3nw ~ ((D) + ... J . (7.170)

In terms of the lowest-order critical temperature (7.142), this becomes

T)D((D+1) D[((D)]l/D(T)D- l } S = kBN(D + 1) {( Ti O) ((D) + 2 ---y:{ Ti O) + ... . (7.171) From this we obtain the specific heat C = TOTS below Te:

T )D ((D+1) D(D-1) [((D)] l/D( T )D- l } C=kBN(D+1) { D ( Ti O) ((D) + 2 ---y:{ Ti O) + .... (7.172) At the critical temperature, this has the maximal value

((D+1) { [D-1 e+1/D(D) D ((D-1)] 1 } Cmax';::;jk BN(D+1)D ((D) 1+ - 2- ((D+1) -2 (l- l/D(D) Nl/D . (7.173) In three dimensions, the lowest two approximations have their maximum at (7.174) Above Te, we expand the total particle number (7.126) in powers of w as in (7.162). The fugacity of the ground state is now different from unity: (7.175) The grand-canonical free energy is (7.176) and the entropy

The specific heat C is found from the derivative -(30(3SIN as

C({3,f.1)

=kB({3n~)D +~

[D (D2

{(D+1)D(D+l(ZD)

+ 1) {3nw- D{3f.1- (30(3({3f.1)] (D(ZD)+ . . . }.

(7.178)

625

7.2 Bose-Einstein Condensation

The derivative f3o(3(f3fJ) is found as before from the condition: oN(f3,fJ(f3))/of3 = 0, implying

o=

( f3 :W)D

{- [D(D(ZD)

+ f317,w ~ (D - l)(D-l(ZD)]

+ [(D-l(ZD) + f31iw~ (D-2(ZD)] [f30(3(f3fJ)-f317,W~] + .. .}, so that we obtain

(7.179)

D

(D(ZD) + f317,W 2 (D- 2(ZD) + ... f30(3(f3fJ) = D D (D-l(ZD) + f317,2W(D-2(ZD) + .. .

(7.180)

(7.181)

so that from (7.60) (7.184) The chemical potential at fixed N satisfies the equation (7.185) The specific heat at a constant volume is (7.186) At high temperatures, C tends to the Dulong-Petit limit DNkB since for small

Z

all

(,,(z) behave like z. This is twice a big as the Dulong-Petit limit of the free Bose gas since there are twice as many harmonic modes. As the temperature approaches the critical point from above, from below and we obtain a maximal value in three dimensions

Z

tends to unity

C(O) = k N [12(4(1) - 9(3(1)] ::;:, k N 422785 max

B

(3(1)

(2(1)

B·

.

(7.187)

The specific heat for a fixed large number N of particles in a trap has a much sharper peak than for the free Bose gas. The two curves are compared in Fig. 7.10, where we also show how the peak is rounded for different finite numbers N.3 3p.W. Courteille, V.S. Bagnato, and V.l. Yukalov, Laser Physics 2, 659 (2001).

626

7 Many Particle Orbits - Statistics and Second Quantization 10

10

\

1\\ N

harmonic trap

/ \t-N_=_1000 _

4

/f

'":;..

t

N= 100

/' ,...:,..,.,.,..,...1"''''

free Bose gas 0.2

0.4

0.6

0.8

I

1.2

= 10000

1.4

0.25

0.50

0.75

T /TJO)

1.00

1.25

1.50

T /TJO)

Figure 7.10 Peak of specific heat for infinite (left-hand plot) and various finite numbers 100, 1000, 10000 of particles N (right-hand plots) in harmonic trap. The large-N curve is compared with that of a free Bose gas.

7.2.11

Interactions in Harmonic Trap

Let us now study the effect of interactions on a Bose gas in an isotropic harmonic trap. This is most easily done by adding to the free part (7.132) the interaction (7.101) with n(x) taken from (7.131), to express the grand-canonical free energy by analogy with (7.102) as

Fa =

JdDx { -*Zw(/3) fo

ZD

d: (D({3nw; z; x)

+g

[Zw ({3)(D ({3nw; ZD; X)]2} . (7.188)

Using the relation (7.103), this takes a form more similar to (7.106):

Fa =

J {

dz -j31 Zw({3) dDx JorzD -;(D({3nw;z;x)-2fLl~(n{3)Zw({3) [(D({3nw;zD;X)]

2} .

(7.189) As in Eq. (7.113), we now construct the variational free energy to be extremized with respect to the local parameter a(x). Moreover, we shall find it convenient to express (( z) as foz (dz' / z')( (z') . This leads to the variational expression (7.190) where

ZD

=

e-(DIiw/2-J1.l/3

a ==

= e- D /3Iiw/2 z and

al~ (n{3) Zw ({3) = le(~{3) l~ (n{3)Zw({3).

The extremum lies at a(x) = ~(x)

~(x)

(7.191)

where by analogy with (7.114):

== -4a(D({3nw; zDe~(xl; x) .

(7.192)

627

7.2 Bose-Einstein Condensation

For a small trap frequency w, we use the function (v (f3nw ; Zv; x) in the approximate form (7.133) , written as

(df3nw; zv; x)

=L frf!-00

w 2 Mf3

w=l

21TW

v z'De- Mw {3w

2 2

x

/2.

(7.193)

In this approximation, Eq. (7.192) becomes

~(

u X ) wzo ~ -

4-a

~ ~

J

w=l

w2 Mf3 vzve w wI;(x) e -Mw{3w 2 x 2 /2 . -

(7.194)

21TW

For small ii, ~(x) is also small, so that the factor ewI;(x) on the right-hand side is close to unity and can be omitted. This will be inserted into the equation for the particle number above Tc: (7.195) Recall that in the thermodynamic limit for D > 1 where the phase transition properly exists, (v(f31iw; Zv; x) and c'v(f3nw; Zv; x) coincide, due to (7.139) and (7.141). From (7.195) we may derive the following equation for the critical temperature as a function of zv: 1

=

Zw(f3)

JdV

Zw(f3~O))

c'v(f3nw;zv eI;(x) ;x) (v(f3~O)1iw ; 1) ,

x-=--"::----'~----'~

(7.196)

where T~O) is the critical temperature in the trap without the repulsive interaction. The critical temperature Tc of the interacting system is reached if the second argument of c'df31iw; zVeI;(x); x) hits the boundary of the unit convergence radius of the expansion (7.130) for x = 0, i.e., if zVeI;(O) = 1. Thus we find the equation for Tc: 1=

Z w ((.I) fJc

J

I" (f3 1iw. eI;c(x)-I;c(O). x) dV x 0 dE N n~.

(7.222)

g

The Bose function contains a pole at 0: = 0 which prevents the existence of a solution for positive 0: . In the analytically continued fermionic function (7.215), on the other hand, the point 0: = 0 is completely regular. Consider now a Fermi gas close to zero temperature which is called the degenerate limit. Then the reduced variables E = E/kBT and 0: = l-£/kBT become very large and the distribution function (7.217) reduces to

nf

g

=

{I

0

for EO:

= e (E -

(7.223)

0:).

All states with energy E lower than the chemical potential 1-£ are filled, all higher states are empty. The chemical potential 1-£ at zero temperature is called Fermi energy E F : (7.224) 1-£1T=O =EF · The Fermi energy for a given particle number N in a volume VD is found by performing the integral (7.222) at T = 0: 1

D(PF)D r;: ,

gsV

f(D/2+1) J47TD

(7.225)

where (7.226) is the Fermi momentum associated with the Fermi energy. Equation (7.225) is solved for EF by

EF = 27rn? [r(D/2 + 1)]2/D (.!!.-)2/D, M

gs

VD

(7.227)

and for the Fermi momentum by

_

PF -

2V1in [f(D/2+1)]1/D(.!!.-)1/D v:D n. gs

(7.228)

632

7 Many Particle Orbits - Statistics and Second Quantization

Note that in terms of the particle number N, the density of states per unit energy interval and volume can be written as (7.229)

As the gas is heated slightly, the degeneracy in the particle distribution function in Eq. (7.223) softens. The degree to which this happens is governed by the size of the ratio kT / E F • It is useful to define a characteristic temperature, the so-called Fermi temperature (7.230)

For electrons in a metal, PF is of the order of 'Ii/IA. Inserting M = me = 9.109558 X 10- 28 g, further kB = 1.380622 X 10- 16 erg/K and 'Ii = 6.0545919 X 10- 27 erg sec, we see that TF has the order of magnitude TF

~

44000K.

(7.231)

Hence, even far above room temperatures the relation T /TF « 1 is quite well fulfilled and can be used as an expansion parameter in evaluating the thermodynamic properties of the electron gas at nonzero temperature. Let us calculate the finite-T effects in D = 3 dimensions. From Eq. (7.225) we obtain _ gsV 2 .f (f..L) (7.232) N = N(T, f..L) = 1~('lif3) y'1i 23 / 2 kBT . Expressing the particle number in terms of the Fermi energy with the help of Eq. (7.225), we obtain for the temperature dependence of the chemical potential an equation analogous to (7.54): (7.233)

To evaluate this equation, we write the integral representation (7.216) as i~(a)

=

1

00

- a

1 a

o

where E = x and obtain

+ a.

(a

+ x)n- 1

dx-'-----'--c-e X +l

dx

(a-x)n-1 e- X + 1

+

1

00

0

(a+x)n-1

dx-'-----'--c-eX

+1

In the first integral we substitute 1/(e- X

+ 1)

(7.234) = 1 - 1/(eX

+ 1),

(7.235)

633

7.3 Gas of Free Fermions

In the limit a--->oo, only the first term survives, whereas the last term is exponentially small, so that it can be ignored in a series expansion in powers of l/a. The second term is expanded in such a series:

f

roo dx-f-- = 2 L

(n-1)! a n- 1- k(1 - Tk)((k + 1). (n-1-k)! (7.236) In the last equation we have used the integral formula for Riemann's (-function4 2

k=odd

(n-1) a n- 1- k k Jo

e

+1

k=odd

(7.237)

At even positive and odd negative integer arguments, the zeta function is related to the Bernoulli numbers by Eq. (2.567). The lowest values of ((x) occurring in the expansion (7.236) are ((2) , ((4) , ... , whose values were given in (2.569), so that the expansion of (a) starts out like

i;

i~(a) =

.!.an + 2(n-1).!.((2)an- 2 + 2(n-1)(n-2)(n-3)~((4)an-4 + ... n 2 8

Inserting this into Eq. (7.233) where n 1 = (kBT)3/2 ~ [~ EF 2 3

.

(7.238)

= 3/2, we find the low-temperature expansion

(~)3/2 + 7r2(~)-1/2 + ~ (~)-5/2 ... J ' kBT

12

kBT

3 . 320

kBT

(7.239)

implying for fl the expansion (7.240)

These expansions are asymptotic. They have a zero radius of convergence, diverging for any T. They can, however, be used for calculations if T is sufficiently small or at all T after a variational resummation it la Section 5.18. We now turn to the grand-canonical free energy Fe. In terms of the function (7.216) , this reads 1 gsV 1 .f (7.241) Fe = -73l~(nf3) f(5/2) z5/2(a) . Using again (7.238) , this has the expansion Fc(T, fl , V)

1

57r2 (kBT)2 77r 4 (kBT)4 = Fc(O, fl , V) [1 + 8 ---;;- - 384 ---;;- + . .. ,

where Fc(O ,fl, V) 41.8. Gradshteyn and 1.M. Ryzhik, op. cit. , Formula 3.411.3.

(7.242)

634

7 Many Particle Orbits -

Statistics and Second Quantization

By differentiating Fe with respect to the temperature at fixed /-l, we obtain the low-temperature behavior of the entropy

B

7f2 kBT kB - E - N+ .. . 2 F

=

(7.243)

From this we find a specific heat at constant volume

C = TaB I aT V,N

T~O B = kB 7f2 kBT N + ...

(7.244)

2 EF

This grows linearly with increasing temperature and saturates at the constant value

3kB N /2 which obeys the Dulong-Petit law of Section 2.12 corresponding to three kinetic and no potential harmonic degrees of freedom in the classical Hamiltonian p2/2M. See Fig. 7.11 for the full temperature behavior. The linear behavior is C kEN 3

Dulong-Petit Law

2"

T

Figure 7.11 Temperature behavior of specific heat of free Fermi gas. As in the free Bose gas, the Dulong-Petit rule gives a high-temperature limit 3kBN/2 for the three harmonic kinetic degrees of freedom in the classical Hamiltonian p2/2M. There are no harmonic potential degrees of freedom. due to the progressive softening near the surface of the Fermi distribution which makes more and more electrons thermally excitable. It is det ected experimentally in metals at low temperature where the contribution of lattice vibrations freezes out as (T/TDt Here TD is the Debye temperature which characterizes the elastic stiffness of the crystal and ranges from T D ~ 90K in soft metals like lead over T D ~ 389K for aluminum to T D ~ 1890K for diamond. The experimental size of the slope is usually larger than the free electron gas value in (7.244) . This can be explained mainly by the interactions with the lattice which result in a large effective mass Meff > M . Note that the quantity Fc(O, /-l , V) is temperature dependent via the chemical potential/-l. Inserting (7.240) into (7.242) we find the complete T-dependence 57f2 (kBT)2 7f4 (kBT)4 Fc(T,/-l , V) = Fc(O, EF, V) [ 1 + 12 EF - 16 EF

+ .. .1,

(7.245)

7.4 Statistics Interaction

635

where (7.246) As in the boson gas, we have a relation (7.61) between energy and grandcanonical free energy: 3 E = --F.e (7.247) 2 ' such that equation (7.245) supplies us with the low-temperature behavior of the internal energy: (7.248) The first term is the energy of the zero-temperature Fermi sphere. Using the relation = aE/VaT, the second term yields once more the leading T -+ 0 -behavior (7.244) of specific heat. This behavior of the specific heat can be observed in metals where the conduction electrons behave like a free electron gas. Due to Bloch's theorem , a single electron in a perfect lattice behaves just like a free particle. For many electrons, this is still approximately true, if the mass of the electrons is replaced by an effective mass. Another important macroscopic system where (7.244) can be observed is a liquid consisting of the fermionic isotope 3He. There are two electron spins and an odd number of nucleon spins which make this atom a fermion. Also there the strong interactions in the liquid produce a screening effect which raises to an effective value of the mass to 8 times that of the atom.

Cv

7.4

Statistics Interaction

First, we consider only two identical particles; the generalization to n particles will be obvious. For simplicity, we ignore the one-body potentials V(x(v») in (7.2) since they cause only inessential complications. The total orbital action is then (7.249) The standard change of variables to center-of-mass and relative coordinates (7.250) respectively, separates the action into a free center-of-mass and a relative action

A = ACM+ Arel =

lb dt~ X? + l~b dt [~X2 - Vint(X)] ,

(7.251)

with a total mass M = M(1)+M(2) and a reduced mass f.t = M(l) M(2) /(M(1) +M(2»). Correspondingly, the time evolution amplitude of the two-body system factorizes

7 Many Particle Orbits - Statistics and Second Quantization

636

into that of an ordinary free particle of mass M, (XbtbIXata) , and a relative amplitude (Xbtblxata). The path integral for the center-of-mass motion is solved as in Chapter 2. Only the relative amplitude is influenced by the particle statistics and needs a separate treatment for bosons and fermions. First we work in one dimension only. Many of the formulas arising in this case are the same as those of Section 6.2, where we derived the path integral for a particle moving in a half-space x = r > 0; only the interpretation is different. We take care of the indistinguishability of the particles by restricting x to the positive semiaxis x = r 2 0; the opposite vector -x describes an identical configuration. The completeness relation of local states reads therefore

l)O drlr)(rl = 1.

(7.252)

To write down the orthogonality relation, we must specify the bosonic or fermionic nature of the wave functions. Since these are symmetric or antisymmetric, respectively, we express (rb lra) in terms of the complete set of functions with these symmetry properties:

(rb lra) = 2

roo dp {

Jo

7r'h

c?sprb/h c?spra/'h }.

sm prb/'h sm pra/'h

(7.253)

This may be rewritten as

(rb lra) =

1

00

- 00

dp

27r'h

(eiP(rb - ra)/1i

±

eiP(rb+ ra)/Ii)

= i5(rb - ra) ± i5(rb + ra) . (7 .254)

The infinitesimal time evolution amplitude of relative motion is then, in the canonical formulation,

(rn Elrn- l 0)

=

(rn le- i,k,el/lilrn_l)

(7.255) where H rel (p, x) is the Hamiltonian of relative motion associated with the action Arel in Eq. (7.251). By combining N + 1 factors, we find the time-sliced amplitude (7.256) x { exp

[~ }; Pn(rn -

rn- l)] ± exp

[* };: Pn(rn + rn- l)] } e-K' L,:~11

Hcel(Pn ,rn) ,

valid for bosons and fermions, respectively. By extending the radial integral over the entire space it is possible to remove the term after the ±sign by writing (7.257)

7.4 Statistics Interaction

637

where the function a-(x) vanishes identically for bosons while being equal to

a(x)

=

8(-x)

(7.258)

for fermions, where 8(x) is the Heaviside function (1.309) . As usual, we have identified Xb == XN+ l and Xa == Xo which is equal to ra' The final sum over Xb = ±rb accounts for the indistinguishability of the two orbits. The phase factors ei7r(7(xn) give the necessary minus signs when exchanging two fermion positions. Let us use this formula to calculate explicitly the path integral for a free twoparticle relative amplitude. In the bosonic case with a vanishing a-term, we simply obtain the free-particle amplitude summed over the final positions ±rb:

For fermions, the phases a(xn) in (7.257) cancel each other successively, except for the boundary term (7.260) When summing over Xb = ±rb in (7.257), this causes a sign change of the term with Xb = -rb and leads to the antisymmetric amplitude (7.261) Let us also write down the continuum limit of the time-sliced action (7.257). It reads (7.262) The last term is the desired Fermi statistics interaction. It can also be written as

Ar =

-n;r

l tb dti:(t)J(x(t)) = n;r ltb dt8 8(-x(t)). t

ta

(7.263)

ta

The right-hand expression shows clearly the pure boundary character of A f , which does not change the equations of motion. Such an interaction is called a topological

interaction. Since the integrals in (7.257) over x and p now cover the entire phase space and a(x) enters only at the boundaries of the time axis, it is possible to add to the action any potential \lint (r). As long as the ordinary path integral can be performed, also the path integral with the additional a-terms in (7.257) can be done immediately. It is easy to generalize this result to any number of fermion orbits x(v)(t), 1/ = 1, . . . ,n. The statistics interaction is then I:v .( )"'] 2 00 r J dx-Ixp)(xpl= dr _e-,m-n'Ye- r 211" 211" m,n=O

Jmf JnT

The angular integration enforces m = n, and the integrals over r2 cancel the factorials, as in (7.348), thus proving the resolution of the identity (7.342), which can also be written as J dZ:dZ Iz)(zi = 1. (7.355) This resolution of the identity can now be inserted into a product decomposition of a Boltzmann operator

(zble-.6iL IZa)

=

(zble-.6iL/(N+l)e-.6Hw/(N+l) . . . e-.6Hw /(N+l) IZa),

(7.356)

to arrive at a sliced path integral [compare (2.2)- (2.4)] (

)- rrN [J -11"dZ~dzn] Nrr+l( ) -(3/( N+1 ) . Zb 1e-.6Hwl Za-n=l n=lZn 1e -fHwl Zn-l,ZO-Za,ZN+l-Zb,f= (7.357)

We now calculate the matrix elements (Znle-fHw IZn- l) and find

(Znle-fHwIZn _l) ~ (znI 1 - titlzn- l) = (ZnIZn- l) - f(znljtlzn - l). Using (7.350) we find

(7.358)

7.8 Coherent States

653

The matrix elements of the operator Hamiltonian (7.298) is easily found. The coherent states (7.345) are eigenstates of the annihilation operator a with eigenvalue z: 00 zn 00 zn alz) = e- z' z/ 2 L -aln) = e- z' z/ 2 L In - 1) = zlz). n=O vInf n=l v(n - I)!

Thus we find immediately

(znIHwlzn-l) = 1iw(znl(at a + aat)IZn_l) = nw

(Z~Zn-l +

(7.360)

D.

Inserting this together with (7.359) into (7.358) , we obtain for small integral

(7.361) E

the path (7.362)

(7.363) The gradient terms can be regrouped using formula (2.35) , and rewriting its righthand side as PN+IXN+l - POXo + L.;:~l(Pn - Pn- l)Xn- l. This leads to

I)} .

n N+1{ z~V'zn+w (Z~Zn-l+2 A~[z*'Z]=2(-Z;Zb+Z:Za)+nE];

(7.364)

Except for the surface terms which disappear for periodic paths, this action agrees with the time-sliced Euclidean action (7.310) , except for a trivial change of variables

a ---; z. As a brief check of formula (7.362) we set N = 0 and find

A~[z*, z] = ~(-Z;Zb + z:za) + nz; (Zb -

za)

+ W (Z;Za + ~) ,

(7.365)

D] .

(7.366)

and the short-time amplitude (7.364) becomes

(zbl e- Elfw IZa) = exp

[-~(Zb'Zb + z:za) + zb'za -

Enw (Zb'Za

+

Applying the recovery operations (7.346) we find (Ole-ElfwIO) = [e(ZbZb+z~Za)/2(zbl e-Elfwlza)L.=o z=o = e- fnw / 2,

(1Ie - Elfw 11) = {Oz

[e(ZbZb+z~za)/2(zble-Elfw IZa) ] 8: L.=o,z=o = e- E3nw /2,

(Ole- EHw ll) = (1Ie- EHw IO) = O.

(7 .367) (7.368)

(7.369)

Thus we have shown that for fixed ends, the path integral gives the amplitude for an initial coherent state IZa) to go over to a final coherent state IZb)' The partition function (7.337) is obtained from this amplitude by forming the diagonal integral

Zw =

JdZ:dz (zle-.B Hw

Iz).

(7.370)

654

7 Many Particle Orbits -

7.9

Second-Quantized Fermi Fields

Statistics and Second Quantization

The existence of the periodic system of elements is based on the fact that electrons can occupy each orbital state only once (counting spin-up and -down states separately) . Particles with this statistics are called fermions. In the above Hilbert space in which n-particle states at a point x are represented by oscillator states In, x ), this implies that the particle occupation number n can take only the values n

° (no electron) ,

n

1 (one electron).

It is possible to construct such a restricted many-particle Hilbert space explicitly by subjecting the quantized fields ~t (x) , ~(x) or their Fourier components a~, ap to anticommutation relations, instead of the commutation relations (7.282) , i.e., by postulating

[~(x, t) , ~t(x' , t)l+ At At [1/J (x, t), 1/J (x', t)l+ [~(x, t), ~ (x' , t)l+

0,

(7.371)

0,

or for the Fourier components

[ap(t), a~, (t)l+ [a~(t) , a~, (t)l+ [ap(t) , ap (t)l+ l

Here

[A, Sl + denotes the

Jppl,

0,

0.

anticommutator of the operators

[A,Sl+ == As + sA.

(7.372)

A and S (7.373)

Apart from the anticommutation relations, the second-quantized description of Fermi fields is completely analogous to that of Bose fields in Section 7.6.

7.10

Fluctuating Fermi Fields

The question arises as to whether it is possible to find a path integral formulation which replaces the anticommuting operator structure. The answer is affirmative, but at the expense of a somewhat unconventional algebraic structure. The fluctuating paths can no longer be taken as c-numbers. Instead, they must be described by anticommuting variables.

7.10.1

Grassmann Variables

Mathematically, such objects are known under the name of Grassmann variables. They are defined by the algebraic property (7.374)

655

7.10 Fluctuating Fermi Fields

which makes them nilpotent: (7.375)

These variables have the curious consequence that an arbitrary function of them possesses only two Taylor coefficients, Fa and F1 , (7.376)

They are obtained from F(O) as follows :

F(O), F

, _a

(7.377)

= aoF.

The existence of only two parameters in F(O) is the reason why such functions naturally collect amplitudes of two local fermion states, Fa for zero occupation, Fl for a single occupation. It is now possible to define integrals over functions of these variables in such a way that the previous path integral formalism remains applicable without a change in the notation, leading to the same results as the second-quantized theory with anticommutators. Recall that for ordinary real functions, integrals are linear functionals. We postulate this property also for integrals with Grassmann variables. Since an arbitrary function of a Grassmann variable F(O) is at most linear in 0, its integral is completely determined by specifying only the two fundamental integrals J dO and J dO O. The values which render the correct physics with a conventional path integral notation are

dO J.j'h

0,

(7.378)

J ~O .j'h

1.

(7.379)

Using the linearity property, an arbitrary function F(O) is found to have the integral = Fl = F'. (7.380) J ~F(O) .j'h Thus, integration of F(O) coincides with differentiation. This must be remembered whenever Grassmann integration variables are to be changed: The integral is transformed with the inverse of the usual Jacobian. The obvious equation

dO ( dO' (') c· ) 0 = c· F , = c· J --F 0 J --F .j'h .j'h

(7.381)

for any complex number c implies the relation

dO' [dO] - lF(O')' J ~F(O'(O))=J .j'h .j'h dO'

(7.382)

656

7 Many Particle Orbits - Statistics and Second Quantization

For ordinary integration variables, the Jacobian dO/dO' would appear without the power -1. When integrating over a product of two functions F(O) and G(O), the rule of integration by parts holds with the opposite sign with respect to that for ordinary integrals:

dO G (O) oOF(O) a J V2ii

JdO[O] V2ii ooG(O) F(O).

=

(7.383)

There exists a simple generalization of the Dirac 8-function to Grassmann variables. We shall define this function by the integral identity

J

~8(0 -

O')F(O') == F(O).

(7.384)

Inserting the general form (7.376) for F(O), we see that the function

8(0-0')=0'-0

(7.385)

satisfies (7.384). Note that the 8-function is a Grassmann variable and, in contrast to Dirac's 8-function, antisymmetric. Its derivative has the property

8'(0 - 0') == 008(0 - 0') = -1.

(7.386)

It is interesting to see that 8' shares with Dirac's 8' the following property:

~8'(0 -

J

O')F(O')

=

-F'(O),

(7.387)

with the opposite sign of the Dirac case. This follows from the above rule of partial integration, or simpler, by inserting (7.386) and the explicit decomposition (7.376) for F(O).

The integration may be extended to complex Grassmann variables which are combinations of two real Grassmann variables 01, O2 :

a*

=

~(Ol -

i02 ),

(7.388)

The measure of integration is defined by

J

da*da == J d0 2 d0 1 == - J dada*. 7r

2m

(7.389)

7r

Using (7.378) and (7.379) we see that the integration rules for complex Grassmann variables are

J da:da

(7.390)

da* da * J --aa

(7.391 )

7r

657

7.10 Fluctuating Fermi Fields

Every function of a* a has at most two terms: F(a*a) = Fo

+ Fl

a*a.

(7.392)

In particular, the exponential exp{ -a* Aa} with a complex number A has the Taylor series expansion e - a'Aa = 1- a* Aa. (7.393) Thus we find the following formula for the Gaussian integral:

J

da: da e- a' Aa

= A.

(7.394)

The integration rule (7.390) can be used directly to calculate the Grassmann version of the product of integrals (7.315) . For a matrix A which can be diagonalized by a unitary transformation, we obtain directly (7.395)

Remarkably, the fermion integration yields precisely the inverse of the boson result (7.315).

7,10,2

Fermionic Functional Determinant

Consider now the time-sliced path integral of the partition function written like (7.309) but with fermionic anticommuting variables. In order to find the same results as in operator quantum mechanics it is necessary to require the anticommuting Grassmann fields a(T), a* (T) to be antiperiodic on the interval T E (0, Ti(3), i.e., a(Ti(3)

= -a(O) ,

(7.396)

or in the sliced form (7.397)

Then the exponent of (7.395) has the same form as in (7.315), except that the matrix A of Eq. (7.398) is replaced by 1 -l+EW f

A =10(1-

-

EW)~T

+ lOW =

0 1

0 0

-l+EW

0

0

0

0 0 1 -1

0 0 0 0

+ EW

0

. ..

-l+EW

1-

lOW

0 0 0 1 (7.398)

where the rows and columns are counted from 1 to N + 1. The element in the upper right corner is positive and thus has the opposite sign of the bosonic matrix

658

7 Many Particle Orbits - Statistics and Second Quantization

in (7.318). This makes an important difference: While for w gave

= 0 the bosonic matrix (7.399)

due to translational invariance in

T,

we now have (7.400)

The determinant of the fermionic matrix (7.398) can be calculated by a repeated expansion along the first row and is found to be (7.401) Hence we obtain the time-sliced fermion partition function (7.402) As in the boson case, we introduce the auxiliary frequency We

1

== --log(l - EW)

(7.403)

= 1 + e-(3liwe .

(7.404)

E

and write Z~N in the form Z:.:

This partition function displays the typical property of Fermi particles. There are only two terms, one for the zero-particle and one for the one-particle state at a point. Their energies are 0 and 'hw e , corresponding to the Hamiltonian operator (7.405) In the continuum limit into

E --+

0, where We

--+ W ,

the partition function Zw N goes over (7.406)

Let us generalize also the fermion partition function to a system with a timedependent frequency O(T), where it reads (7.407) with the sliced action N

A!: = 'h L [a~(an - an-i) n=i

+ EOna~an-il ,

(7.408)

or, expressed in terms of the difference operator V', N

A!:

=

'hE

L

n=i

a~ [(1- EOn)V' + On] an·

(7.409)

7.10 Fluctuating Fermi Fields

659

The result is N

Z~N

= detN+l[E(l - d2)\7 + EW] = 1 -

II (1 -

dl n ).

(7.410)

n=O

As in the bosonic case, it is useful to introduce the auxiliary frequency (7.411) and write

z5

N

in the form (7.412)

If we attempt to write down a path integral formula for fermions directly in the continuum limit, we meet the same phenomenon as in the bosonic case. The difference operator (7.398) turns into the corresponding differential operator

(7.413) which now acts upon periodic complex functions e-iw;"T with the odd Matsubara frequencies (7.414) = w(2m + l)kBT/n, m = 0, ±1, ±2, . . ..

w:"

The continuum partition function can be written as a path integral

f

Va*Va

[ { Iif3

-w-

exp -

Nwdet (aT

+ w),

Jo

dT (a*oTa + wa*a)

]

(7.415)

with some normalization constant N w determined by comparison with the timesliced result. To calculate Z~, we take the eigenvalues of the operator aT + w, which are now -iw;, + w, and evaluate the product of ratios 00

II m=-oo

_iwf +w ~ f

-twm

= cosh(nw;3/2).

(7.416)

This corresponds to the ratio of functional determinants det (aT + w) = h(" /.1/2) det (aT) cos nWfJ .

(7.417)

In contrast to the boson case (7.336), no prime is necessary on the determinant of aT since there is no zero frequency in the product of eigenvalues (7.416). Setting N w = 1/2det (aT) ' the ratio formula produces the correct partition function z~

= 2cosh(nw;3/2).

(7.418)

660

7 Many Particle Orbits - Statistics and Second Quantization

Thus we may write the free-fermion path integral in the continuum form explicitly as follows :

f Va*Va [_11i d

( *!l * )] _ 2 det (OT + w) a UTa + wa a d (!l) et U

i3

exp

7r

0

T

T

2 cosh(1iw,B/2).

(7.419)

The determinant of the operator OT

+ w)

det (OT

+w

= det (-OT

can again be replaced by

+ w)

= Jdet (-0;'

+ w2).

(7.420)

As in the bosonic case, this Fermi analog of the harmonic oscillator partition function agrees with the results of dimensional regularization in Subsection 2.15.4 which will ensure invariance of path integrals under a change of variables, as will be seen in Section 10.6. The proper fermionic time-sliced partition function corresponding to the dimensional regularization in Subsection 2.15.4 is obtained from a fermionic version of the time-sliced oscillator partition function by evaluating

(7.421 ) with a product over the odd Matsubara frequencies

w;". The result is (7.422)

with We given by (7.423)

sinh(we /2) = Ew/2 .

This follows from the Fermi analogs of the product formulas (2.398), (2.400):7

NIt (1 - _.- -,-:i-;: ~_:X" '+"'l)-1l")

cos(N + l)x cos x

N = even,

(7.424)

sin2 x ) . 2 (2m+1)1l" sm 2(N+l)

cos(N + l)x,

N=odd.

(7.425)

sm

m=O

(N - 1)/2 (

II m=O

1-

2(N+ 1)

For odd N , where all frequencies occur twice, we find from (7.425) that N

II m=O

(

sin 2 x ) 1- . 2~ sm 2(N+l)

1/2

=cos(N+1)x,

71.8. Gradshteyn and I.M. Ryzhik, op. cit. , Formulas 1.391.2, 1.391.4.

(7.426)

661

7.10 Fluctuating Fermi Fields

and thus, with (7.423), directly (7.422) . For even N, where the frequency with m = N/2 occurs only once, formula (7.424) gives once more the same answer, thus proving (7.426) for even and odd N . There exists no real fermionic oscillator action since x 2 and j;2 would vanish identically for fermions, due to the nilpotency (7.375) of Grassmann variables. The product of eigenvalues in Eq. (7.421) emerges naturally from a path integral in which the action (7.408) is replaced by a symmetrically sliced action. An important property of the partition function (7.418) of (7.422) is that the ground-state energy is negative: E(O)

= _ 1iw

(7.427)

2'

As discussed at the end of Section 7.7, such a fermionic vacuum energy is required for each bosonic vacuum energy to avoid an infinite vacuum energy of the world, which would produce an infinite cosmological constant, whose experimentally observed value is extremely small.

7.10.3

Coherent States for Fermions

For the bosonic path integral (7.304) we have studied in Section 7.8, the case that the endpoint values aa = a(Ta) and ab = a(Tb) of the paths a(T) are held fixed. The result was found to be the matrix element of the Boltzmann operator e- f3Hw between coherent states la ) = e- a*a/2eaa t 10) [recall (7.345)]. There exists a similar interpretation for the fermion path integral (7.415) if we hold the endpoint values aa = a(Ta) and ab = a(Tb) ofthe Grassmann paths fixed. By analogy with Eq. (7.345) we introduce coherent states [50]

(7.428) The corresponding adjoint states read

(7.429) Note that for consistency of the formalism, the Grassmann elements ( anticommute with the fermionic operators. The states 10) and (11 and their conjugates (0 1 and (1 1can be recovered from the coherent states I() and ((I by the operations:

In) = [1()e(*(/2

an] _1_ , ( (=0 v'nf

(7.430)

These formula simplify here to

10) = [1()e(*(/2L =0, 11) = [1 ()e(*(/2

a(L=o'

(7.431)

(0 1= [e(*(/2(( IL=0' (1 1= [o(*e(*(/2(( I]

(=0

.

(7.432)

662

7 Many Particle Orbits - Statistics and Second Quantization

For an operator agonal elements trO =

0 , the trace can be calculated from the integral over the antidi-

J

d(:d( (-(101()

=

J

d(:d( e-(*«((Ol- (*(11)0(10) - (11)).

(7.433)

Using the integration rules (7.390) and (7.391), this becomes trO = (01010)

+ (11011).

(7.434)

The states Ie) form an overcomplete set in the one-fermion Hilbert space. The scalar products are [compare (7.350)]:

e-

(i (, /2- (2 (2/2+ (i (2

(7.435)

e- (i«, - (2)/2+«i - (. We have shown that as in the Bose case the path integral with fixed ends gives the amplitude for an initial coherent state I(a) to go over to a final coherent state I(b). The fermion partition function (7.419) is obtained from this amplitude by forming the trace of the operator e- (3Hw , which by formula (7.434) is given by the integral over the antidiagonal matrix elements

(7.440) The antidiagonal matrix elements lead to antiperiodic boundary conditions of the fermionic path integral.

663

7.11 Hilbert Space of Quantized Grassmann Variable

7.11

Hilbert Space of Quantized Grassmann Variable

To understand the Hilbert space associated with a path integral over a Grassmann variable we recall that a path integral with zero Hamiltonian serves to define the Hilbert space via all its scalar products as shown in Eq. (2.18):

(Xbtblxata)

=

J1)x J~:n exp [i JdtP(t)±(t)]

=

(xblxa)

=

8(Xb - xa).

(7.441)

A momentum variable inside the integral corresponds to a derivative operator p == -ili8x outside the amplitude, and this operator satisfies with x = x the canonical commutation relation [p, x] = -iii [see (2.19)]. By complete analogy with this it is possible to create the Hilbert space of spinor indices with the help of a path integral over anticommuting Grassmann variables. In order to understand the Hilbert space, we shall consider three different cases.

7.11.1

Single Real Grassmann Variable

First we consider the path integral of a real Grassmann field with zero Hamiltonian

.] . J1)B27r exp [ih, Jdt 2iii B(t)B(t) From the Lagrangian

£(t)

i

.

(7.442)

21iB(t)B(t)

(7.443)

8£ iii Po=-. =--B.

(7.444)

=

we obtain a canonical momentum 8B

2

Note the minus sign in (7.444) arising from the fact that the derivative with resect to Banticommutes with the variable B on its left. The canonical momentum is proportional to the dynamical variable. The system is therefore subject to a constraint X =Po

iii

+ -2 B =

0.

(7.445)

In the Dirac classification this is a second-class constraint, in which case the quantization proceeds by forming the classical Dirac brackets rather than the Poisson brackets (1.21), and replacing them by ± i/Ii times commutation or anticommutation relations, respectively. For n dynamical variables qi and m constraints Xp the Dirac brackets are defined by (7.446) where C pq is the inverse of the matrix (7.447)

664

7 Many Particle Orbits - Statistics and Second Quantization

For Grassmann variables Pi, gi, the Poisson bracket (1.21) carries by definition an overall minus sign if A contains an odd product of Grassmann variables. Applying this rule to the present system we insert A = Po and B = () into the Poisson bracket (1.21) we see that it vanishes. The constraint (7.445), on the other hand, satisfies

{X,X}

=

{ Po + 22'n (),po + 22'n () }

-in{po,()}

=

=

-in.

(7.448)

Hence C = -in with an inverse i/n. The Dirac bracket is therefore (7.449) With the substitution rule {A, B}v ---; (-i/n)[A, B]+, we therefore obtain the canonical equal-time anticommutation relation for this constrained system: A

[P(t) , ()(t)]+ =

in

-2'

(7.450)

or, because of (7.444),

[O(t),O(t)]+ = 1.

(7.451)

The proportionality of Po and () has led to a factor 1/2 on the right-hand side with respect to the usual canonical anticommutation relation. Let 'Ij;(()) be an arbitrary wave function of the general form (7.376): (7.452) The scalar product in the space of all wave functions is defined by the integral (7.453) In the so-defined Hilbert space, the operator given by the differential operator

0 is diagonal,

while the operator p is

p = inoo,

(7.454)

to satisfy (7.450). The matrix elements of the operator pare

('Ij;'lpl'lj;) -=

J

d() 'Ij;'*(())in~'Ij;(()) 2'iT o()

= in'lj;'*'Ij; 1 1·

(7.455)

By calculating (7.456)

7.11 Hilbert Space of Quantized Grassmann Variable

665

we see that the operator fJ is anti-Hermitian, this being in accordance with the opposite sign in the rule (7.383) of integration by parts. Let 19) be the local eigenstates of which the operator 8 is diagonal: 819) = 919).

(7.457)

The operator 8 is Hermitian, such that (918 = (919 = 9(91 ·

(7.458)

The scalar products satisfy therefore the usual relation (9' - 9)(9'19)

=

o.

(7.459)

On the other hand, the general expansion rule (7.376) tells us that the scalar product S = (9'19) must be a linear combination of So + S19 + SW + S299'. Inserting this into (7.459), we find (7.460) (9'19) = -9' + 9 + S299', where the proportionality constants So and SI are fixed by the property (9'19) =

Jd9" (9'10")(9"19). 21f

(7.461)

The constant S2 is an arbitrary real number. Recalling (7.385) we see that Eq. (7.460) implies that the scalar product (9'19) is equal to a

J(27rn)D' dDpV

(7.558)

For infinitely thin time slices, t -> 0 and ,:;\(p) reduces to w(p) and the expressions (7.556), (7.557) turn into the usual free energies of bosons and fermions. They agree completely with the expressions (7.46) and (7.220) derived from the sum over orbits. Next we may introduce a fluctuating field in space and imaginary time

'IjJ(x, T)

=

~L

yV

eipXap(T) ,

(7.559)

p

and rewrite the action (7.553) in the local form

Ae['IjJ*''IjJ]

=

101i,6 dT

JdDx ['IjJ*(X, T)

n8T 'IjJ (x, T)

+ 2~ V 'IjJ*(x , T)V'IjJ(X, T)]

. (7.560)

677

7.14 Spatial Degrees of Freedom

The partition function is given by the functional integral

Z =

f V'l/lD'I/;*e- A.['I/J·,'l/Jl/li .

(7.561)

Thus we see that the functional integral over a fluctuating field yields precisely the same partition function as the sum over a grand-canonical ensemble of fluctuating orbits. Bose or Fermi statistics are naturally accounted for by using complex or Grassmann field variables with periodic or antiperiodic boundary conditions, respectively. The theory based on the action (7.560) is completely equivalent to the second-quantized theory of field operators. In order to distinguish the second-quantized or quantum field description of many particle systems from the former path integral description of many particle orbits, the former is referred to as the first-quantized approach, or also the world-line approach. The action (7.560) can be generalized further to include an external potential V(x, T), i.e. , it may contain a general Schrodinger operator if(T) = p2 + V(x , T) instead of the gradient t erm:

A" ['1/;* , '1/;] = foli/3dT J dDx {'I/;*(x, T)noT'I/; (x, T)+ 'I/;*(X, T) [if(T) -11] 'I/;(x, T)}. (7 .562) For the sake of generality we have also added a chemical potential to enable the study of grand-canonical ensembles. This action can be used for a second-quantized description of the free Bose gas in an external magnetic trap potential V(x) , which would, of course, lead to the same results as the first-quantized approach in Section 7.2.4. The free energy associated with this action 1

A

F = :eTrlog[noT + H(T) -11]

(7 .563)

was calculated in Eq. (3.139) as an expansion F = _l_Tr [rli/3 dT if(T)] 2n(3 Jo

-.!. f='!'Tr

{Te- nJoni3dT" [H(T"l - l-'l/li} ,

(3 n=l n

(7.564)

The sum can be evaluated in the semiclassical expansion developed in Section 4.9. For simplicity, we consider here only time-independent external potentials, where we must calculate Tr [e- n/3(H - l-'l]. Its semiclassical limit was given in Eq. (4.257) continued to imaginary time. From this we obtain F = ~Tr if _ 1 2 (3

1

V21fn2(3/M

D

~ JdD [z(x)]n ~ X D/2+1' n

(7.565)

n=l

where z(x) == e- /3[V(xl - I-'J is the local fugacity (7.24). The sum agrees with the previous first-quantized result in (7.132) and (7.133). The first term is due to the symmetric treatment of the fields in the action (7.562) [recall the discussion after Eq. (7.340)]. One may calculate quantum corrections to this expansion by including the higher gradient terms of the semiclassical expansion (4.257). If the potential is timedependent , the expansion (4.257) must be generalized accordingly.

678

7 Many Particle Orbits -

7.14.2

Statistics and Second Quantization

First versus Second Quantization

There exists a simple set of formulas which illustrates nicely the difference between first and second quantization, i.e., between path and field quantization. Both may be though of as being based on two different representations of the Dirac 8-function. The first-quantized representation is (7.566) the second-quantized representation

=

iM(D) (Xb - Xa)8(tb - ta)

f

V '¢V '¢* '¢(Xb , tb)'¢*(Xa, ta)e(i/Ii) J dDx

roo dt,p*(x,t),p(x,t).

(7.567) The first representation is turned into a transition amplitude by acting upon it with the time-evolution operator e-di(tb-t a), which yields (X

t Ix t )

bb a a

= edi (tb- t a)8(D) (xb - x a) =

JVDx

f

VDp (27rn)D

e(i/Ii)J,~dt(Px-H) .

(7.568)

By multiplying this with the Heaviside function 8(tb - t a), we obtain the solution of the inhomogeneous Schrodinger equation (7.569) This may be expressed as a path integral representation for the resolvent

in

\

Xbtb l . tnat -

1 A

H

Xata

)

-

-

-8(tb - ta)

l

X(t b)=Xb

V

D

x

f

x(ta)=Xa

VDp (i/Ii) J,'b dt(px- H) - ")D ( ea . (7 .570) 27rn

The same quantity is obtained from the second representation (7.567) by changing the integrand in the exponent from '¢*(x , t)'¢(x, t) to '¢*(x, t)(inat - H)'¢( x , t): / Xbtb I . in 1xata\ \ tnat - H / A

=

f

V '¢V'¢* '¢(Xb, tb)'¢*(Xa, ta)e(i/Ii) J dDx

roo dW(x,t)(ilio,-H),p(x,t).

(7.571)

This is the second-quantized functional integral representation of the resolvent.

7.14.3

Interacting Fields

The interaction between particle orbits in a grand-canonical ensemble can be accounted for by anharmonic terms in the particle fields. A pair interaction between orbits, for example, corresponds to a fourth-order self interaction. An example is the interaction in the Bose-Einstein condensate corresponding to the energy in Eq. (7.101) . Expressed in terms of the fields it reads

679

7.14 Spatial Degrees of Freedom

A~nt[7{i*' 7{i] = -l'f3dT flE = -~ l'f3 dT

Jd3x7{i*(x, T + TJ)7{i*(X, T + TJ)7{i(X , T)7{i(X, T) ,

(7.572) where TJ > 0 is an infinitesimal time shift. It is then possible to develop a perturbation theory in terms of Feynman diagrams by complete analogy with the treatment in Section 3.20 of the anharmonic oscillator with a fourth-order self interaction. The free correlation function is the momentum sum of oscillator correlation functions:

(7{i(x, T)7{i*(X', T')) =

L (a

p ( T)a~,

(T'))ei(px- p'x')

=

p,p'

L (a

p ( T)a~( T'))eiP(x-

x').

(7.573)

p

The small TJ > 0 in (7.572) is necessary to specify the side of the jump of the correlation functions (recall Fig. 3.2). The expectation value of flE is given by (7.101) , with a prefactor g rather than gj2 due to the two possible Wick contractions. Inserting the periodic correlation function (7.528), we obtain the Fourier integral

(7{i(x, T) 7{i* (X' , T'))

=

L(1 + nwp)e-Wp(T-T')+iP(X-X').

(7.574)

P

Recalling the representation (3.284) of the periodic Green function in terms of a sum over Matsubara frequencies , this can also be written as

(7{i(x, T) 7{i*(X', T')) = ~

L .

-1

nJ3 Wm ,p ZWm -

e-iWm(T-T')+ip(x-x') .

wp

(7.575)

The terms in the free energies (7.102) and (7.107) with the two parts (7.108) and (7.109) can then be shown to arise from the Feynman diagrams in the first line of Fig. 3.7. In a grand-canonical ensemble, the energy nwp in (7.575) is replaced by liwp - 1-£. The same replacement appears in wp of Eq. (7.574) which brings the distribution function nwp to [recall (7.529)] 1

(7.576)

ef3l' .

The expansion of the Feynman integrals in powers where z is the fugacity z = of z yields directly the expressions (7.102) and (7.107).

7.14.4

Effective Classical Field Theory

For the purpose of studying phase transitions, a functional integral over fields 7{i(x, T) with an interaction (7.572) must usually be performed at a finite temperature. Then is often advisable to introduce a direct three-dimensional extension of the effective classical potential Veffcl (xo) introduced in Section 3.25 and used efficiently in Chapter 5. In a field theory we can set up, by analogy, an effective classical action which is a functional of the three-dimensional field with zero Matsubara frequency ¢>(x) == 7{io(x). The advantages come from the reasons discussed in Section 3.25, that

680

7 Many Particle Orbits - Statistics and Second Quantization

the zero-frequency fluctuations have a linearly diverging fluctuation width at high temperature, following the Dulong-Petit law. Thus only the nonzero-modes can be treated efficiently by the perturbative methods explained in Subsection 3.25.6. By analogy with the splitting of the measure of path integration in Eq. (3.805) , we may factorize the functional integral (7.561) into zero- and nonzero-Matsubara frequency parts as follows: (7.577) and introduce the Boltzmann factor [compare (3.810)] contain the effective classical action (7.578) to express the partition function as afunctional integral over time-independent fields in three dimensions as: (7.579)

In Subsection 3.25.1 we have seen that the full effective classical potential Veffcl(XO) in Eq. (3.809) reduces in the high-temperature limit to the initial potential V(xo). For the same reason, the full effective classical action in the functional integral (7.577) can be approximated at high temperature by the bare effective classical action, which is simply the zero-frequency part of the initial action:

This follows directly from the fact that, at high temperature, the fluctuations in the functional integral (7.578) are strongly suppressed by the large Matsubara frequencies in the kinetic terms. Remarkably, the absence of a shift in the critical temperature in the first-order energy (7.113) deduced from Eq. (7.116) implies that the chemical potential in the effective classical action does not change at this order [34]. For this reason, the lowest-order shift in the critical temperature of a weakly interacting Bose-Einstein condensate can be calculated entirely from the three-dimensional effective classical field theory (7.577) with the bare effective classical action (7.580) in the Boltzmann factor. The action (7.580) may be brought to a more conventional form by introducing the differently normalized two-component fields