Path Integrals and Quantum Anomalies (The International Series of Monographs on Physics)

INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS No. 122 (ISMP) Series Editors

J. Birman S. F. Edwards R. Friend M. Rees D. Sherrington G. Veneziano

City University of New York University of Cambridge University of Cambridge University of Cambridge University of Oxford CERN, Geneva

INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS 124. C. Kiefer: Quantum gravity 123. T. Fujimoto: Plasma, spectroscopy 122. K. Fujikawa, H. Suzuki: Path integrals and quantum anomaiies 121. T. Giamarchi: Quantum physics in one dimension 120. M. Warner, E. Tercntjev: Liquid crystal elastomers 119. L. Jacak, P. Sitko, K. Wieczorek, A. Wojs: Quantum Hall systems 118. J. Wesson: Tokamaks, Third edition 117. G. Volovik: The Universe in a helium droplet 116. L. Pitaevskii. S. Stringari: Bose-Einstein condensation 115. G. Dissertori, I.G. Knowles, M. Schmelling: Quantum chromodynamics 114. B. DeWitt; The global approach to quantum field theory 113. J. Zinn-Justin: Quantum field theory and critical phenomena, Fourth edition 112. R.M. Mazo: Brownian motion - fluctuations, dynamics, and applications 111. H. Nishimori: Statistical physics of spin glasses and information processing - an introduction 110. N.B. Kopnin: Theory of nonequilibrium superconductivity 109. A. Aharoni: Introduction to the theory of ferromagnelism, Second edition 108. H. Dobbs: Helium three 107. R. Wigmans: Calorimetry 106. J. KÜbler: Theory of itinerant electron magnetism 105. Y. Kuramolo, Y. Kitaoka: Dynamics of heavy electrons 104. D. Bardin, G. Passarino: The Standard Model in the making 103. G.C. Branco, L. Lavoura, J.P. Silva: OP Violation 102. T.C. Choy: Effective medium theory 101. H. Araki: Mathematical theory of quantum fields 100. L. M. Pismen: Vortices in nonlinear fields 99. L. Mestel: Stellar magnetism 98. K. H, Bennernann: Nonlinear optics in metals 97. D. Salzmann: Atomic physics in hot plasmas 96. M. Brambilla: Kinetic theory of plasma waves 95. M. Wakatani: Stellarat.or and heliotron devices 91. S. Chikazumi: Physics of ferramagneiisrn 91. R. A. Bertlmann: Anomalies in quantum field theory 90. P. K. Gosh: Ion traps 89. E. Simanek: Inhomoaeneous superconductors 88. S. L. Adler: Quaternionic quantum mechanics and quantum, fields 87. P. S. Joshi: Global aspects in gravitation and cosmology 86. E. R. Pike, S. Sarkar: The. quantum theory of radiation 84. V. Z. Kresin, II. Morawitz, S. A. Wolf: Mechanisms of conventional and high Tc super-conductivity 83. P. G. de Gennes, J. Prost: The physics of liquid crystals 82. B. H. Bransden. M. R. C. McDowell: Charge exchange and the theory of ion-atom collision 81. J. Jensen, A. R. Mackintosh: Rare earth magnetism 80. R. Gastmaiis. T. T. Wu: The ubiquitous photon 79. P. Luchini, H. Motz: Undulators and free-electron lasers 78. P. Weinberger: Electron scattering theory 76. H. Aoki, H. Kamimura: The physics of interacting electrons in disordered systems 75. .J. D. Lawson: The physics of charged particle beams 73. M. Doi, S. F. Edwards: The theory of polymer dynamics 71. R. L. Wolf: Principles of electron tunneling spectroscopy 70. H. K. Henisch: Semiconductor contacts 69. S. Chandrasekhar: The mathematical theory of black holes 68. G. R. Satchler: Direct nuclear reactions 51. C. Mo11er: The theory of relativity 46. H. E. Stanley: Introduction to phase transitions and critical phenomena 32. A. Abragam: Principles of nuclear magnetism 27. P. A. M. Dirac: Principles of quantum mechanics 23. R. E. Pcierls: Quantum theory of solids

Path Integrals and Quantum Anomalies KAZUO FUJIKAWA Department of Physics, University of Tokyo and

HIROSHI SUZUKI Department of Mathematical Sciences, Ibaraki University

CLARENDON PRESS • OXFORD 2004

OXFORD UNIVERSITY PRESS

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chermai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press, 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2004 KE1RO SEKIBUN TO TAISIIOSEI NO RYOSHI-TEKI YABURE (Path Integrals and Quantum Anomalies) by Kazuo Fujikawa © 2001 by Kazuo Fujikawa Originally published in Japanese in 2001 by Iwanami Shoteu, Publishers, Tokyo. This English language edition published in 2004 by Oxford University Press, Oxford by arrangement with the author c/o Iwanami Shoten. Publishers, Tokyo All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue for this title is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 852913 9 (hbk) 10 9 8 7 6 5 4 3 2 1 Printed and bound in India by Thomson Press (India) Ltd.

PREFACE The main purpose of this book is to provide an introduction to the path integral formulation of quantum field theory and its applications to the analyses of symmetry breaking by the quantization procedure. This symmetry breaking is commonly called the "quantum anomaly" or simply the "anomaly". and this naming shows that the effect first appeared as an exceptional phenomenon in field theory. However, as is explained in this book, this effect has turned out to be very fundamental in modern field theory. In the path integral formulation, it has been recognized that this effect arises from a non-trivial Jacobian in the change of path integral variables. Namely, the path integral measure breaks certain symmetries. The study of the quantum anomaly is one of the attempts to understand the basis of quantum theory better and, consequently, it is a basic notion which could influence the entire quantum theory beyond field theory. The quantum anomaly is located at the border of divergence and convergence, though the quantum anomaly itself is perfectly finite, and thus closely related to the presence of an infinite number of degrees of freedom. As such an example, we discuss the phase operator of the photon which involves an infinite number of degrees of freedom related to Bose statistics. The notion of the quantum anomaly has been mainly developed in the applications of field theory to elementary particle physics. We however believe that the notion of the anomaly and the machinery developed to understand this subtle phenomenon will have important implications on other fields such as condensed matter physics. To illustrate these points, we analyze field theory in two-dimensional space-time in some detail. It is shown that the phenomenon called bosonization characteristic to two-dimensions, namely, the description of fermion theory in terms of bosons, is simply understood as a result of the change of path integral variables. Also, the basic notions such as the central charges of Kac-Moody and Virasoro algebras in conformal field theory are nothing but the manifestation of the anomalies. We show explicitly how to derive the central charges from the anomalies in the path integral method. Another important aspect of quantum anomalies is that they are closely related to topological properties. For example, the chiral anomaly when combined with instanton solutions in Yang-Mills theory is a quantum manifestation of the Atiyah-Singer index theorem in mathematics. Similarly, the ghost number anomaly which appears in the first quantization of string theory is a field theoretical manifestation of the Ricmann-Roch theorem in the theory of Riemann surfaces. As for the history of quantum anomalies, which will be explained in Chapter 1, the original indication of the anomaly appeared immediately after the modern formulation of field theory, namely, renormalization theory. It was, howv

vi

PREFACE

ever, only in 1969 when the true significance of the quantum anomaly was clearly recognized. On the other hand, Feynman introduced the path integrals in his formulation of renormalization theory. We now briefly describe the contents of this book. Chapter 2 covers the basics of quantum theory. We explain the basic ideas of the Feynman path integral and Sclrwinger's action principle which are required to understand modern field theory and the present book, without assuming knowledge of more than standard undergraduate quantum mechanics. In Chapter 3 an elementary aspect of the quantum theory of the photon initiated by Dirac is explained. The basic idea of the path integral formulation of gauge theory is explained and the notion of the BRST symmetry is introduced. At the same time, the basic property of the photon phase operator is explained by using the notion of index. The notion of index reappears in the later discussions of quantum anomalies. The essence of quantum electrodynamics is summarized in Appendix A to further supplement the basic path integral formulation of gauge theory. This appendix will help understand what is going on in this book better. The phenomenon called the vacuum polarization is explained in Chapter 4. The evaluation of this apparently simple diagram in a sense has been the most difficult calculation in the entire renormalization program of quantum electrodynamics. By using the gauge invariant regularization introduced to control the vacuum polarization diagram, we explain the simplest example of the quantum anomaly, the chiral anomaly, by performing the calculation corresponding to triangle diagrams but actually without relying on Feynman diagrams. Chapter 5 is the main chapter in this book. We explain that the symmetry breaking by quantization (the chiral anomaly) is understood as a non-trivial Jacobian associated with a change of path integral variables. We present several supporting arguments for the use of the gauge invariant mode cut-off to evaluate the Jacobians throughout the present, book. The instanton solution is briefly summarized, and the relation of the chiral anomaly with the Atiyah-Singer index theorem is explained. In this connection, we note a problematic aspect of the unitary transformation to the interaction picture and that the Nambu-Goldstone theorem does not hold in general in the presence of the quantum anomaly in the relevant symmetry. The developments and various applications of quantum anomalies are described in Chapter 6 and subsequent chapters. The descriptions are somewhat condensed in these chapters. Readers may choose appropriate topical subjects in these chapters. The description of each chapter is arranged so that readers can read each chapter without referring to other chapters as much as possible. In Chapter 6, the applications of quantum anomalies to modern gauge theory such as the Standard Model of elementary particles are summarized. Almost all the detailed calculations are explicitly performed. The Weyl transformation is the symmetry which keeps the local angle invariant but the length of space-time is changed. An explanation of the Weyl anomaly is given in Chapter 7. The trace of the energy-momentum tensor is

PREFACE

vii

also involved in the analysis. The renormalization group equation is regarded as a representation of the Weyl anomaly in terms of Green's functions. We thus present a calculation of the Q function in QED and QCD from the viewpoint of the Weyl anomaly. This anomaly is also related to the basis of conformal field theory in two dimensions. To formulate the Weyl anomaly in the path integral, some basic knowledge of field theory in curved space-time is necessary and it is briefly summarized in Appendix B. In Chapter 8 we explain the anomaly-related topics in two-dimensional field theory. This chapter is aimed at possible applications to condensed matter theory also. We thus repeat the elementary calculations of chiral anomalies and provide several representative examples of the bosonization of two-dimensional ferrniori theory. The conceptual basis of the bosonization. in particular, the issue related to local counter-terms, is clarified. As a basis to understand conformal field theory, we formulate the Kac-Moody and Virasoro algebras from the viewpoint of quantum anomalies. The central extensions in these algebras are the manifestations of chiral and general coordinate anomalies, respectively. This formulation is also compared to the operator product expansion method in conformal field theory. The Liouville action is derived in connection with quantized bosonic string theory and the ghost number anomaly is explained in connection with the Riemann-Roch theorem. Chapter 9 covers the interesting developments in the treatment of chiral symmetry in lattice gauge theory which took place over the past several years. The remarkable fact that one can treat the notions of index and chiral anomaly in lattice theory as the Jacobians just as in continuum theory is explained. We do not discuss the practical aspects of lattice gauge theory and its numerical simulations, but rather we concentrate on the conceptual aspects of chiral anomalies and their implications in lattice gauge theory. The lattice theory which is based on finite quantities only consolidates some aspects of the treatment of chiral anomalies in continuum theory where the notion of index is better defined. It is possible to formulate the contents of this chapter with certain mathematical rigor. We present a general calculation of chiral anomalies in arbitrary even-dimensional curved space-time in Chapter 10, and their relations to the Chern character and the Dirac genus in mathematics are explained. We also explain the possible quantum breaking of Einstein's general coordinate transformations in the theory with chiral fermions, and some simple examples are given. In Chapter 11 we briefly comment on a possible intuitive explanation of quantum breaking of symmetries and also on the subjects which are not discussed in the main chapters such as the descent formula and the global SU(2) anomaly. As is clear from the descriptions so far we concentrate on the rather classical and basic aspects of quantum anomalies in the present book, which can be explicitly calculated by an elementary method in the path integral. Advanced subjects such as the anomaly cancellation in superstring theory and supersymmetric theory in general are not discussed, but several references to these subjects are given.

viii

PREFACE

In Appendix C, some basic references directly related to the presentation of this book are given with brief comments. The literature in this subject is vast, and we present only a small fraction of references related to path integrals and anomalies. We apologize to those authors whose contributions are not mentioned. We quote some well-written reviews and textbooks which emphasize different viewpoints on quantum anomalies to remedy the shortcomings. One of the authors (KF) started the study of quantum anomalies in the mid1970s. He thanks Professor C.N. Yang and all the members of his institute at Stony Brook, in particular. Professor P. van Nicuwcnhuizen. for encouragement in the early developments of the path integral formulation of quantum anomalies and for later hospitality at Stony Brook. He also thanks the late Professor B. Sakita at CCNY who continuously encouraged him from the very beginning of the investigation. He thanks Professor K. Nishijima, whose research group stimulated the studies of the chiral anomaly in Japan, and Professor Y. Yamaguchi for encouragement. The present book is an English translation of the book originally written in Japanese by KF, and he thanks Professor K. Kikkawa and Mr. U. Yoshida of the Iwanami Publishing House for the encouragement to write the book. In this translation, we added some remarks and explanations though the basic materials are the same as in the original book. The original Chapter 10 on two-dimensional theory has been moved to Chapter 8 and the analysis of bosonization has been expanded. One of us (HS) thanks Professor T. Fukui for discussions on the bosonization. Last, but not least, we thank Mr. Sonke Adlung, Senior Editor of Oxford University Press, for his enthusiasm and support for this translation. August 2003 Kazuo Fujikawa and Hiroshi Suzuki

CONTENTS 1

2

Genesis of quantum anomalies

1

1.1 Introduction 1.2 Is the photon massless? 1.3 The discovery of the quantum anomaly

1 2 3

The Feynman path integral and Schwinger's action principle

7

2.1 2.2 2.3 2.4 2.5 2.6 3

4

5

6

Quantum theory of a harmonic oscillator Path integral for the harmonic oscillator Quantization of a sca Path integral for fermions Path integral for Dirac particles Feynman path integral and Schwinger's action principle

7 8 11

15 20 23

Quantum theory of photons and the phase operator

31

3.1 3.2 3.3 3.4 3.5

31 35

Canonical quantization of the electromagnet Path integral quantization of the electromagnetic field Photon phase operator and the notion of index Is there a hermitian phase operator? Index theorem for a harmonic oscillator

40 42 44

Regularization of field theory and chiral anomalies

47

4.1 4.2 4.3 4.4

Current conservation and Ward Takahashi identities Self-energy of the photon Quantum breaking of chiral symmetry Adler-Bardeen theorem

47 49 57 62

The Jacobian in path integrals and quantum anomalies

65

5.1 5.2 5.3 5.4 5.5 •5.6

65 71 74 79 84 86

The chiral Jacobian in quantum electrodynamics Ward-Takahashi identities in quantum electrodynamics Chiral anomaly in QCD-type theory Instantons Atiyah-Singer index theorem Nambu-Goldstone theorem

Quantum breaking of gauge symmetry

92

6.1 Gauge theory with axial-vector gauge fields 6.2 Pauli-Villars regularization 6.3 Chiral gauge theory and the quantum anomaly 6.4 Covariant anomaly 6.5 Anomaly cancellation in Weinbcrg-Salam theory

92

ix

96 98 102 109

x

CONTENTS 6.6 6.7

The Wess-Zumino integrability condition Quantum anomalies and anomalous commutators

112 121

7

The 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Weyl anomaly and renormalization group Scale transformation in field theory Identities for the Weyl transformation and Weyl anomalies Identities related to coordinate transformations Weyl anomalies and functions in QED and QCD The Weyl anomaly in curved space-time The Weyl anomaly in two-dimensional space-time Other applications of Weyl anomalies

123 123 124 127 132 140 144 147

8

Two-dimensional field theory and bosonization 8.1 Chiral anomalies in two-dimensional theory 8.2 Abelian bosonization of fermions 8.3 Non-Abelian bosonization of fermion theory 8.4 Kac Moody algebra and Virasoro algebra 8.5 Quantum theory of strings and Liouville action 8.6 Ghost number anomaly and the Riemann-Roch theorem

149 149 157 168 174 188 194

9

Index theorem on the lattice and chiral anomalies 9.1 Lattice gauge theory 9.2 Lattice Dirac fields and species doubling 9.3 Representation of the Ginsparg-Wilson algebra 9.4 Atiyah-Singer index theorem on the lattice and the chiral anomaly 9.5 The operator D satisfying the Ginsparg-Wilson relation 9.6 Some characteristic features of lattice chiral theory

196 196 198 202 206 210 218

10 Gravitational anomalies 10.1 Chiral U(l) gravitational anomalies 10.2 Evaluation by a quantum mechanical path integral 10.3 Chern character and Dirac genus 10.4 Anomaly in general coordinate transformations 10.5 General properties of gravitational anomalies 10.6 Explicit examples of gravitational anomalies

223 223 228 230 231 234 236

11 Concluding remarks

240

A Basics of quantum electrodynamics A.1 Quantum electrodynamics A.2 Interaction representation and perturbation formulas

246 246 249

B

253 253 258

Field theory in curved space-time B.1 Coordinate transformation and energy-momentum tensor B.2 Path integral measure in gravitational theory

CONTENTS C

xi

References with brief comments

262

C.1 Genesis of quantum anomalies C.2 The Feynman path integral and Schwinger's action principle C.3 Quantum theory of photons and the phase operator C.4 Regularization of field theory and chiral anomalies C.5 The Jacobian in path integrals and quantum anomalies C.6 Quantum breaking of gauge symmetry C.7 The Weyl anomaly and renormalization group C.8 Two-dimensional field theory and bosonization C.9 Index theorem on the lattice and chiral anomalies C.10 Gravitational anomalies C.ll Concluding remarks

262

Index

263 263 264 265 266 270 272 275 277 279 282

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1 GENESIS OF QUANTUM ANOMALIES 1.1

Introduction

The central dogma in modern physics is the principle of quantum theory. All dynamics, cither classical or relativistic dynamics, or all the forces such as electromagnetic and nuclear forces need to be formulated to conform to the principle of quantum theory when applied to microscopic systems. The general theory of relativity is not exceptional, and quantum gravity needs to be formulated in microscopic domains. Even classical dynamics is realized as a vanishing limit of the Planck constant H in quantum theory. The basic equation of quantum theory is given by the Schrodingcr equation

and the wave function y> has a meaning as the probability amplitude. The wave function is an element of infinite-dimensional Hilbert space, as is expected from the fact that the wave function of a hydrogen atom has an infinite number of components. To emphasize this aspect, the wave function is written as the state vector \ip). From the definition of probability, the square of the state vector needs to be non-negative, namely, we postulate that the norm of the wave function be positive definite The time development of the Schrodinger equation is described by a hermitian Hamiltonian, and thus the time development induces a unitary transformation of the wave function

By definition, the unitary transformation does not change the magnitude (total probability) of the wave function. What changes is, for example, that the first component is transformed to the second component of an infinite-dimensional state vector. To apply quantum theory to general phenomena including photons, it is necessary to quantize the field variable itself and thus formulate field theory. Intuitively, one distributes oscillators at each point of four-dimensional space-time in field theory, and quantizes all these oscillators. The excitation or de-excitation of those oscillators are interpreted as the creation or annihilation of associated particles. There arc an infinite number of points and thus one needs to handle an infinite number of oscillators, or if one formulates the problem suitably, one 1

2

GENESIS OF QUANTUM ANOMALIES

deals with a system where an infinite number of harmonic oscillators interact with each other. For this reason, the field theory is called a quantum theory of an infinite number of degrees of freedom. We have an infinite-dimensional Hilbert space even for a single harmonic oscillator, and now we have an infinite number of oscillators. A representative field theory is quantum electrodynamics which deals with the electron and the photon and their interactions. As is expected, researchers encountered various subtle problems when they first attempted to describe the creation and annihilation of particles. Among them, the divergence problem in field theory is the best known. A careful examination of the divergence problem associated with creation and annihilation led to the discovery of the phenomenon called the quantum anomaly or simply anomaly, which is the main subject of this book. The essential difference between the quantum anomaly and divergences is that the quantum anomaly does not diverge, though both of them are related to an infinite number of degrees of freedom. It is proper to understand the quantum anomaly as a symmetry breaking by the quantization procedure. However, the quantum anomaly usually appears in a form closely related to divergences in interaction picture perturbation theory. For this reason, it is often said that the quantum anomaly is associated with divergences and their regularization. This way of characterizing the anomaly correctly reflects the historical origin of the quantum anomaly (and its certain aspects), but one should rigorously distinguish the quantum anomaly from divergences.

1.2 Is the photon massless? We here briefly describe the history of the study of the quantum anomaly. Quantum electrodynamics (QED) and its renormalization formulated in the late 1940s indicated that the applicability domain of the principle of quantum theory is quite broad and possibly covers all the energy ranges we can think of. This quantum electrodynamics, as is well known, was formulated by S. Tomonaga, J. Schwinger, R. Feynman, F. Dyson and others. In particular, the relativistic formulation of quantum electrodynamics by Tomonaga and his associates in isolated Japan after the Second World War was a surprise to physicists in the United States, in addition to the meson, theory of H. Yukawa. J. Oppenheimer, who was director of the Institute for Advanced Study at Princeton at that time, asked Tomonaga to send a summary of the research of his group to him, which was later published as a Letter to the Editor in the Physical Review. In this letter, Tomonaga explained that covariant quantum electrodynamics and its renormalization prescription can deal successfully with all the problems associated with the electron mass, the electric charge, and the field variables of the electron and the photon, but he encountered a difficulty with the photon mass. He stated, "But for this subtraction we cannot find a reasoning so natural and plausible as that used in the case of mass-type and charge-type infinities, where the subtraction was considered as an amalgamation. This is because it would necessarily result in a drastic change of the Maxwell equation for the radiation".

THE DISCOVERY OF THE QUANTUM ANOMALY

3

If this difficulty should persist, it suggested that we may not be able to handle the experiment all y established vanishing photon mass and the gauge principle, which ensures the vanishing photon mass, in interaction picture perturbation theory. In retrospect, this problem of how to ensure the vanishing photon mass or the gauge symmetry in field theory was the starting point of the study of the quantum anomaly, the main subject of the present book. It is not obvious that one can maintain gauge symmetry or justify the unitary transformation to the interaction picture in field theory which deals with an infinite number of degrees of freedom. These are the essential aspects of symmetry breaking by the quantization procedure we are going to discuss. 1.3

The discovery of the quantum anomaly

We briefly explain the problem associated with the photon self-energy on the basis of relativistic quantum mechanics which can describe the creation and annihilation of particles in perturbation theory. A more detailed analysis will be given later in this book. We denote the interaction Hamiltonian, which describes the interaction between the photon (usually denoted by 7) and the electron (denoted by e) and the positron (denoted e), by HI. The photon self-energy in second-order perturbation theory is given by

The sum over the energy of the intermediate states En = Ee + Es generally diverges. This is because the rapid increase in the number of states allowed for the electron-positron pair for increasing energy En and the allowed energy is unlimited for a relativistic theory in flat, Minkowski space-time. This is a manifestation of the infinite number of degrees of freedom in momentum space for the creation of an electron- positron pair. The vanishing photon mass is ensured if the energy correction A.E(k) as a function of the momentum k satisfies

In terms of the more intuitive Feynman diagram, this perturbation formula corresponds to a calculation of the Feynman diagram in Fig. 1.1. The photon, which is virtually converted to a pair of an electron and a positron, recornbines after a certain time, and the quantum correction to the photon self-energy arises in this process. This diagram diverges (to be precise, a quadratic divergence), and the photon mass which should be zero may become indeterminate if the renormalization prescription should not be unique. As is explained later, the photon mass is in fact kept to be zero up to any finite order in perturbation in the modern formulation of quantum electrodynamics. In 1949, however, this problem of the photon self-energy was the fundamental issue of renormalization theory, and the two members of Tomonaga's group,

4

GENESIS OF QUANTUM ANOMALIES

FIG. 1.1. Photon self-energy correction

FIG. 1.2. (a) Two-photon decay of the neutral TT meson; (b) two-photon decay of a neutral axial-vector meson H. Fukuda and Y. Miyamoto, analyzed the next simplest Feyiiman diagrams in Fig. 1.2. (a) and (b) for the two-photon decay of the neutral TV meson, which is commonly denoted by TT°, namely 7r° —> 77. In terms of modern language, they compared two processes: In one of them the neutral spinless TT meson virtually splits into a quark q and anti-quark q pair in the vacuum, and those two quarks emit two photons before pair annihilation. In the other process, the spin 1 (axial-)vector meson Ati dissociates into a quark q and anti-quark q pair and the pair of quarks emit two photons before pair annihilation. The prediction of renormalization theory is that there exists a simple relation (a symmetry called "Dyson's symmetry" at that time) between two processes. However, an explicit calculation indicated that the gauge invariance in the second graph is spoiled and that the relation between the two graphs docs not hold. The latter graph diverges (to be precise a linear divergence) and thus a careful calculation is required. But the linear divergence already appeared in the self-energy of the electron in renormalization theory, and it is known that renormalization works without any difficulty for the electron self-energy. Consequently, the calculation of the triangle diagrams should also work. The appearance of those discrepancies from the predictions of renormalization theory was a very serious issue for the renormalization program itself. In fact, Tomonaga, together with his collaborators, analyzed the triangle diagrams by using the Pauli Villars regulator, of which a preprint was sent to Tomonaga from Pauli. They concluded that the issue of gauge invariance can be successfully handled by the Pauli-Villars regulator but the relation between the two

THE DISCOVERY OF THE QUANTUM ANOMALY

5

graphs was not uniquely resolved. They stated that we have to wait for future experimental results to resolve the issue. A similar arid detailed analysis was performed by J. Steinberger at Princeton, who learned of the calculation of Fukuda arid Miyamoto through Yukawa staying at Princeton at that time.1 He also used the Pauli-Villars regulator and arrived at a conclusion similar to that of Tomonaga. The issues related to the gauge invariance of the photon self-energy and the triangle diagrams were analyzed in greater detail by J. Schwinger in 1951. He handled the photon self-energy by showing that the gauge invariance can be consistently imposed but concluded anomalous behavior of the triangle diagrams. The distinction between the ambiguity related to divergences and the quantum breaking of symmetry was not clear at that time, and as a result a fundamental understanding of this strange behavior was not achieved. This problem of the triangle diagrams for the pion (i.e., n meson) decay came up as a major issue again in the late 1960s. At that time, the understanding of the pion as a Nambu- Goldstoiie particle associated with spontaneous breaking of chiral symmetry (or PCAC) was established. If one combines this interpretation of the pion with the (naive) calculation of the triangle diagrams, it was concluded that the neutral pion cannot decay into two photons in the ideal limit of the Nambu Goldstone particle. This contradicted the experimental fact that the neutral pion predominantly decays into two photons. This difficulty was analyzed in great detail by J. Bell and R. Jackiw at CERN. Bell and Jackiw noticed the inevitable deviation from PCAC if one applies the conventional Pauli-Villars regularization to the a model which incorporates PCAC. They then showed that one can preserve both PCAC and gauge invariance if one uses a modification of Gupta's implementation of the Pauli-Villars regularization, but this spoils renormalizability. On the other hand, S. Adler at Princeton performed a general analysis of the triangle diagrams in spinor electrodynamics and discussed the issue of the neutral pion decay in the appendix of his paper. The final conclusion was that a proper understanding of the anomalous behavior discussed by Fukuda, Miyamoto, Steinberger and Tomonaga resolves the discrepancy between theory and experiment. At the same time, it was concluded that the anomalous behavior of the triangle diagrams is unavoidable in relativistic local field theory with gauge symmetry. Namely, it was established that the Feynman diagrams can exhibit behavior different from a naive manipulation of canonical field theory and that the anomalous behavior is consistent with the basic postulate of local field theory and explains the experimental results well. In effect, these analyses in 1969 marked the discovery of the breaking of certain symmetries by the quantization procedure, namely, the quantum anomalies. 1

Footnote 11 in J. Steinberger, Phys. Rev. 76 (1949) 1180, reads "Fukuda and Miyamoto. Prog. Theor. Phys. (in press), were the first to notice that the old results were not gauge invariant. Their work formed the starting point of this research. I wish to thank H. Yukawa for making their results available to me before publication". Incidentally, Steinberger later turned to experimental physics and received the Nobel Prize for the discovery of two neutrinos.

6

GENESIS OF QUANTUM ANOMALIES

These papers influenced the entire subsequent developments of the subject. For example, motivated by these papers T. Kimura evaluated the triangle anomaly in the presence of the background gravitational field in the same year of 1969. It is known that there are two main classes of symmetries which are broken by the quantization procedure. The first is the chiral symmetry associated with Dirac's 75 and it is related to the triangle diagrams we have discussed so far, and it is called the chiral anomaly. The other is the Weyl transformation, which changes the length scale of space-time, keeping the local angle invariant; this is called the Weyl anomaly or conformed anomaly. On the other hand, in the formulation of renormalization theory Feynman invented the path integral methods of quantum mechanics arid quantum field theory. The path integral and the canonical operator formulation of quantum theory arc formally equivalent, but the path integral later found many applications in practical calculations. Combined with an intuitive understanding of quantum processes, the path integral is becoming increasingly important in the modern formulation of quantum theory and, in particular, quantum field theory. It has been recognized that the quantum anomalies are understood as arising from non-trivial Jacobians associated with the change of integration variables in the path integral formulation. The path integral measure breaks those symmetries. This path integral method provides a conceptually more satisfactory derivation of anomalous identities with the anomaly terms present from the beginning instead of discovering the anomalies after the evaluation of current divergences. The purpose of this book is to provide an introduction to the path integral method of quantization and its applications to the analyses of quantum anomalies. We show that quantum anomalies are phenomena basic to the entire field theory, in particular, to gauge theory in general on the basis of the path integral formulation. We thus start with an introductory account of the path integral formulation of quantum field theory in the next chapter.

2

THE FEYNMAN PATH INTEGRAL AND SCHWINGER'S ACTION PRINCIPLE 2.1

Quantum theory of a harmonic oscillator

In this section, we study the quantization of the most important harmonic oscillator as a review of quantum mechanics. The Hamiltonian of the harmonic oscillator is written in terms of the coordinate q and momentum p as

where m is the mass arid uj is the frequency. The quantization is defined by the Hciscnberg commutation relation

where H is the Planck constant divided by 2?r. The time-independent Schrodiriger equation is given by

and the energy eigenvalue is given by

with a non-negative integer n. The eigenfunction un(q) for the nth eigenstate is written in terms of a Hermite polynomial. The description of the same problem in terms of creation and annihilation operators is important not only in the present context but also for second quantization discussed later. In this approach, we define

By using the commutation relations of p and q, we can show 7

PATH INTEGRAL AND THE ACTION PRINCIPLE

8

and the Hamiltoriian is written as

We also have When we write the state \n) which satisfies

the absence of negative energy states requires the lowest energy state |0) to satisfy From the view point of quantum theory, this condition has a more fundamental meaning as the absence of negative norm states

Starting with the ground state |0), we can generate the general state n) by repeatedly multiplying the operator «t

The time development of the Schrodinger wave function

is written as

and it shows that the time development operator (evolution operator) is very fundamental. 2.2

Path integral for the harmonic oscillator

The above evolution operator is written in terms of the path integral as

The right-hand side represents a sum over all the paths in phase space starting from (ft at t = ii and ending at q/ at I — £/. In the left-hand side, the operators q

PATH INTEGRAL FOR THE HARMONIC OSCILLATOR

9

and p arc defined in the time-independent Schrodinger representation, but in the right-hand side q(t) and p(t) stand for the time-dependent classiml coordinate and momentum, respectively. We now want to understand what this path integration means in more detail. By noting the completeness of the coordinate and also momentum representations

we divide the time interval into N segments with e = (tf — tj)/7V. We then obtain

We then note for smalle

and

We thus obtain

In this expression, the path integral measure is denned by

If we consider the limit AT -*• oc by denoting the time derivative of q by q which is defined for small e by

we obtain the final expression of the path integral

10

PATH INTEGRAL AND THE ACTION PRINCIPLE

The crucial property to be noted here is that one takes a sum over connected paths in the path integral. Even in quantum theory, the path of a particle does not become discontinuous. The above derivation of the path integral formula shows that the path integral representation is valid for a more general class of Hamiltonians of the form

We can also perform the integral over the momentum variables explicitly for general V(q). To be more concrete, we first note (the integral here is known as the Fresnel integral)

and then the Feynman path integral formula becomes in the limit

We here defined go = Qi, Q.N = Qf, and the path integral measure including the normalization factor by

The last expression in equ (2.26) can be readily generalized to particle motion in three-dimensional space whose coordinates are given by q, and it is written as

This expression shows that the time development in quantum mechanics is given by a sum over all the paths in four-dimensional space-time starting from (t,x) by

This equation is derived by applying the variational principle to the action integral S = f d4x C, written in terms of the Lagrangian density

as

Our notational conventions arc

namely, we take a sum over indices which appear twice, and the metric of fourdimensional space-time is given by g^ = (1, —1, —1, -1) = g^v The Harniltonian density is thus given by

where the momentum (density) conjugate to the field variable is defined by

12

PATH INTEGRAL AND THE ACTION PRINCIPLE If one considers an orthoiiormal complete system of real plane waves

inside a box with volume V, (wg(af)} which satisfies can expand the field variables as

one

The Hamiltoriian is then written as

with ui(k) — c\lkz + (mc/H)2, namely, it, is written as a sum of an infinite number of harmonic oscillators with frequency w(fe). When one defines annihilation and creation operators by

the Harniltonian is written as

and the quantization condition is given by In a specific representation of state vectors of the Hilbert space, which is called the Fock representation, the state vector of a scalar particle is given by2

This state vector represents a state with n-\ particles with momentum tiki, n^ particles with momentum Kk-2 - n^ particles with momentum hk%, and so on. 2 We used the real basis set in eqn (2.35) to make the later transformation to the path integral in terms of the field variable <j>(l, x) easier. To define the cigenstates of the momentum, one needs to make a unitary transformation

QUANTIZATION OF A SCALAR FIELD

13

A crucial difference of the physical interpretation of the particle number representation (Fock space representation) from the state vector in the first quantization of the harmonic oscillator is that the state vector with n particles with momentum hk, for example, is given by

and this state is not interpreted as the nth excitation of the harmonic oscillator as in the quantum mechanics of the harmonic oscillator. The path integral representation of the evolution operator is given as a generalization of a single harmonic oscillator as

One can convert this path integral into one in terms of the field variable as follows: First one remembers

arid thus transforms the variables q%(t) into the field variable (t, x) in the sense of Dirac as

arid

One also recalls that the specification of all the variables {ql} at t = ti is equivK alent to the specification of \(f>i(x}} at all the space points at t = ti. We thus

which preserves the form of the Hamillonian 77 = J^g ^'(^(clcg + 1/2) and the canonical quantization condition

14

PATH INTEGRAL AND THE ACTION PRINCIPLE

denote the state vector by |0j,*j}. The path integral is thus written in a relativistic invariant form

with a suitable normalization factor N, which includes the Jacobian for the above change of variables. Namely, if one performs the path integral with the measure

with a weight factor which is given by an action integral multiplied by i/h, the matrix element of the quantum mechanical evolution operator is given by

So far we discussed a free scalar field without any interaction but. for example, the interaction Lagrangian density, which describes the interaction among four scalar fields when they come together at a coincident point in four-dimensional space-time, is given by

where the coupling constant g determines the strength of the interaction. The interaction Hamiltonian is then given by the last term in

In this case, if one repeats a similar analysis as above the path integral is given by a Lorentz invariant form as

This last expression is again given by a sum over all the possible field configurations in four-dimensional space-time with the exponential weight factor, which

PATH INTEGRAL FOR FERMIONS

15

is given by the classical action multiplied by i/h, and the final expression is manifestly Lorentz covariant. As a classical field, the two scalar fields commute with each other at all the space-time points, [4>(t, of), (t, x),(t, f )] = (H/i)8^(x~-x') in quantum theory, and it represents Bose particles. The path integral formula (2.53) gives a basis of the path integral of all the Lorentz invariant Bose fields. 2.4

Path integral for fermions

To define the path integral for fermions, we first discuss a fermion with a single degree of freedom which is described by the Hamiltonian

arid then extend the formulation to field theory later. Here Fiu stands for the energy carried by the fcrmion and a^ and a stand for creation and annihilation operators with anti-commutation relations

The commutation relations with the Hamiltonian are the same as in the case of bosons

The physical states are limited to the vacuum state |0) and the one-particle state |1), and these states are specified by using a^ arid the Hamiltonian H as

Because of a^ a t |Q) = Q, one gees that there are no states such as |2) which contain more than one particle. Namely, the Pauli exclusion principle is satisfied. The fact that the fermion is quantized by the anti-commutation relation suggests that the fermionic particle at the classical level is described by the Grassmami numbers which always anti-commute. The Grassmann numbers, for example, £ and 77, satisfy the anti-commutation relations

by definition. Consequently, we have, for example, ry2 = 0 and the Grassmann numbers have no notion of magnitude. This fact explains why the notion of the fermion did not exist in classical physics, and one may say that the fermion is a purely quantum mechanical notion.

16

PATH INTEGRAL AND THE ACTION PRINCIPLE

The integral over the Grassmann numbers, namely, the linear projection from the Grassmann numbers to complex numbers, is defined by the left derivative. To be specific, a general function of £ and f* is written as

with complex coefficients cn ~ c3 by recalling £2 = (£*) 2 = 0. The integral is then defined by

by noting that the complex number and the Grassmann number commute with each other by definition. The definition of the left derivative means that we move the relevant variable to the left-most position before integration, such as in

The integral thus defined satisfies the crucial property that the integration measure is invariant under "translation" of the integration variable. Namely, if one defines £,'=£ + n with e another Grassmann number, one can confirm

The first equality in this relation comes from the fact that the naming of the integration variable is arbitrary (in the conventional integral. f dx f ( x ) = j dy f ( y ) ) , and the second equality can be confirmed by an explicit calculation. A similar property is also satisfied by dt;*. and we have the fundamental relations

From the viewpoint of the path integral, the existence of the "translation invariant measure" implies the existence of the path integral measure which ensures the equation of motion, and thus it is the most basic property of the path integral measure. We here use the so-called coherent states which are convenient to avoid the complications related to the change of orders of various fermionic operators and Grassmann numbers. The coherent state for the fermion is defined by

PATH INTEGRAL FOR FERMIONS

17

and it is written as a linear combination of |0) and |1)

by expanding into powers of a and o,t by noting a|0) = 0. We used the anticommuting properties among the Grassmann numbers and the fcrmionic operators By noting one can confirm that the coherent states thus defined satisfy

One can also confirm the completeness relations for the coherent states by using the integral for Grassmann numbers

In the path integral we deal with the matrix elements of the evolution operator where the state na) stands for the eigenstate of the particle number. Physically, the number representation (Fock representation) is important, but one can represent the number states \na) by using the completeness relation of coherent states as

and thus we first consider the evolution operator between coherent states

When one divides the time interval into N small intervals

18

PATH INTEGRAL AND THE ACTION PRINCIPLE

the above evolution operator is written by using the completeness of the coherent states as

We here defined the path integral measure by

The matrix element for an infinitesimal time interval is written by using the explicit form of the Hamiltonian H = fojjcfia and the properties of the coherent states as

to the accuracy of linear order in e. See eqn (2.68). Namely, we neglect all the terms of order O(e 2 ) or higher. Those terms of the order O(e 2 ) or higher are shown to be neglected in the limit N —> oo. To the same linear accuracy in e, we can write, by using eqn (2.68),

where we defined £j ~ (£j+i ™ £j)/ e - Irl the limit N —> oc by noting eN = tb — ta, we obtain the path integral formula

Here we defined £0 = £a and £^ = ££. An important feature of this path integral is that we obtain the time development operator of quantum mechanics by exponentiating the i/h times the classical action integral obtained from (symmetrized with respect to the time derivative)

by the replacement a —> £, aJ —i £* and integrating over the variables £ and £*. The action integral is defined such that the variational principle gives rise to the equation of motion.

PATH INTEGRAL FOR FERMIONS

19

The partition function Z(/3) in statistical mechanics at temperature T is given

by where /3 = l/(kT). This partition function is written by using the completeness relation of the coherent states as

Here we used the relation (n|£&){£a n) = (£a\n)(n - £&} which is confirmed by an explicit calculation for the state vectors {n|£t,} and (£a n}. In contrast, the coherent states by definition contain an even number of Grassmann variables (counting a and a^ also) and thus commute with other Grassmann numbers. The partition function is thus obtained by the path integral with the replacement tb — ta —> —ififi and an anti-symmetric boundary condition with respect to the time variable. To be explicit

where we expanded e ^H in powers of J3H and we used (a^o) 2 = of a valid for the fermionic oscillator. We also used cqn (2.68). The result of the operator formalism is thus recovered. Incidentally, if one performs the path integral with a periodic boundary condition, one obtains

20

PATH INTEGRAL AND THE ACTION PRINCIPLE

where F stands for the operator counting the number of fermions in the state vector, .F|l) = |1) and FjO) = 0. This path integral plays an important role in the analysis of theories with supersyrnmetry. 2.5

Path integral for Dirac particles

We now apply the path integral in the previous section to field theory. The classical action for a fermion satisfying the Dirac equation is given by

The 4 x 4 Dirac matrices 7^, fj, = 0, 1, 2, 3, are defined by

Their explicit forms arc given in terms of 2 x 2 Pauli matrices

by thefollowing4 x 4 matrices

In 7°, 1 stands for a 2 x 2 unit matrix. The hcrmitian 75 matrix which describes the fundamental chiral property in Dirac theory is defined by

and it is given in the present explicit representation of 7 matrices as

The variables ip(t,x)a, a = I ~ 4, stand for a four-component spinor and these components describe the physical four degrees of freedom corresponding to a particle and an anti-particle with spin up and down, respectively. We defined the Dirac conjugate by When one rewrites the above Lagrangian density (2.84) as

one sees that the canonical momentum conjugate to i,h(t,x) is given by

PATH INTEGRAL FOR DIRAC PARTICLES

21

and the Hamiltonian density is given by

In the present case, we choose a complete set of four-component functions {un(x)} defined for the hermitian operator h

arid expand the field variable as

We then obtain

We fix the canonical quantization condition so that the relation

derived from the equation motion ' m — ze. 3 3 This ie prescription is regarded as imposing a positive energy condition, which is basic in the proof of the spin-statistics theorem in the path integral framework, cf. K. Fujikawa, Int. J. Mod. Phys. A 16 (2001) 4025.

22

PATH INTEGRAL AND THE ACTION PRINCIPLE

The path integral is thus defined as a generalization of the case of a single oscillator as

On the other hand, one may expand the classical Grassmann variable ib(t,x) in terms of the Grassmann coefficients an (t) as

If one uses the notation un(S) = (x n), the relation

holds, and one can confirm

Also, the relation

is confirmed. In the path integral, the specification of an infinite number of {a n (t/)} is equivalent to the specification of 'ip(if,x) for all x, and thus we write ({<xn(tf)},tf\ = ( t l > f , t f . The path integral of a Dirac particle is thus written by using the classical action integral

In the present derivation, we have the action which is symmetrized with respect to the time derivative in the exponential. In the actual application of the

FEYNMAN PATH INTEGRAL AND SCHWINGER'S ACTION PRINCIPLE 23 path integral explained in the next section, we define general Green's functions by choosing tf —^ oo. t, —*• —oo and we add Schwinger's source terms, which are localized in space-time. The S-matrix element is then defined as a suitable limit of Green's functions. In such a formulation, one can choose a symmetric ip(tf —» oo,of) = ip(ti —$• —oo,x) boundary condition (or anti-symmetric boundary condition) with respect to the time variable. We may then perform a partial integral with respect to time, and the path integral is written as

where we used the integration variable tt>(t,x) = if>(t,x)^j° instead of y(t, x ) ^ . As a more fundamental formulation, one may consider a Euclidean theory with Euclidean time r defined by t —l —IT. One may then impose symmetric boundary conditions both on bosons and fermions in the path integral. One thus defines the path integral as a generalization of the partition function in statistical mechanics and considers the limit Tf — Ti —>• oo. In this case, only the lowest energy state survives

in the above limit. In this way, one can define a general path integral which starts with a ground state in the infinite past and ends with a ground state in the infinite future. Here, F stands for the operator counting the number of fermions in the state \n). See eqn (2.83). In field theory, the ground state is a result of very complicated interactions, and thus this definition of path integral simultaneously defines the physical ground state. In this formulation, one recovers the path integral with the ordinary action integral with Feynman's ie prescription if one applies the inverse Wick rotation T —>• — it to the Minkowski theory after performing the path integral. 2.6

Feynman path integral and Schwinger's action principle

In this section we explain Schwinger's action principle, which provides a convenient means to relate the path integral to the operator formalism of field theory. Historically, motivated by a paper by Dirac written in 1933,4 Feynman formulated the path integral arid Schwinger formulated the action principle as a basis of quantum theory. ^Incidentally, this paper also commented on a priori probability, which strongly influenced Yukawa and through him indirectly TOmonaga when he formulated the Lorentz invariant Schrodinger equation denned on a space-like surface.

24

PATH INTEGRAL AND THE ACTION PRINCIPLE

In the following discussion we use the Lagrangian (2.51) for the scalar field A(x) for definitcness. By noting the relation

derived from the Heisenberg equation in the operator formalism, one can derive

Since |0) is a generalization of the coordinate representation \q) in quantum mechanics, we have a relation such as

Consequently, one can write the path integral representation

by remembering the results in Section 2.3. When one divides the time interval into infinitesimal time intervals, the time t is assigned to the field i(x) —^ 0(*,zO in the integrand. This relation means that a time evolution specified by the path integral takes place from the initial state at tj to a state at t, where the path integral includes the factor 4>(t,x), and then the time evolution continues to the final state at i f . From this relation, one can establish

where the time ordering operation of bosonic fields is defined by using a step function as5 5 To be precise, the T* product in the path integral avoids the coincident point t\ = i g j and one needs to consider the ordinary T product in the canonical formalism to treat the case ti = fe- The T product is obtained from the T* product by the Bjorkeii-Johnson-Low (BJL) prescription, which is discussed later when we need the T product.

FEYNMAN PATH INTEGRAL AND SCHWINGER'S ACTION PRINCIPLE 25

The step function Q(ti —12) is defined to be 1 for ti > t-2 and 0 for *2 > *i- This time ordering is generalized for a product of n fields, and the fields are arranged in ascending order of time starting from the right-most field. The time ordering specified by T* in the operator formalism is realized in the path integral if one performs the path integral with c-number fields arranged in an arbitrary order. This is because the path integral is performed starting with a past state to a future state following the time evolution, and the time ordering is automatically incorporated. In the case of fcrmionic fields, the time ordering is defined by

by taking into account the anti-commuting property of field variables. In the path integral, this ordering with a signature factor is automatically realized if one uses Grassrnann variables. We next introduce a c-number source field J(x) and define

The path integral with this source field is then defined by

where the index J on the left-hand side indicates that the time evolution is described by the Lagrangian C j . Schwiriger's action principle states that the change of the transition amplitude is given by

when one changes the Lagrangian slightly, which describes the time development, by keeping the initial and final states fixed. It is possible to show that this action principle is equivalent to the Schrodinger equation, though we do not discuss the equivalence here. If one applies this action principle to the case where the source term J(x) in the Lagrangian C, is slightly changed at the space-time point x, one obtains

It is important to recognize that one has a c-numbcr operation in the left-hand side of this relation, while one deals with an operator in the right-hand side. This

26

PATH INTEGRAL AND THE ACTION PRINCIPLE

property is common to the path integral, and the precise relation between the two is clarified later. The field equation in the operator formalism is obtained by a variation of the action with respect to (f> as

The matrix element of this operator equation should vanish, arid when combined with the action principle described above gives rise to

If one can solve the functional differential equation for {(/>/,t/|j,t;)j in eqn (2.120), one can evaluate all the transition amplitudes in quantum theory. In this respect, the path integral corresponds to solving the functional differential equation by performing a Fourier transformation in a functional space from J ( x ) to 6(x). To be explicit, one may set

and insert this expression to the above equation (2.120). One then obtains

where Sj = / d4x jCj. If one sets F[• ip4 and the metric g/^, = (I, — 1, —1, —1) is transformed to g^ = (—1, —1, — 1, —1). In Euclidean theory, the integral becomes a Gaussian integral instead of a Fresncl integral and the path integral of the time evolution operator is defined in a more reliable way. If one associates the operator formalism with the path integral by means of Schwinger's action principle, the it prescription is automatically incorporated as a result of the operator formalism. However, it is important to understand that the if. prescription arises from the physical postulate that the negative energy states do not propagate in the forward time direction. In the applications of field theory, the vacuum to vacuum transition amplitude

with Schwinger's source function J(x), which has a value only in the localized region of space-time, is fundamental. Since the source function has a value only in the localized region of space-time, the asymptotic vacua in eqn (2.134) coincide with the vacua in eqn (2.131). As a physical picture, one deals with the

FEYNMAN PATH INTEGRAL AND SCHWINGER'S ACTION PRINCIPLE 29 probability amplitude (or general Green's functions) for the process where the vacuum state at t = — oo evolves into states with many particles generated by the source function J(x) in intermediate time and then those particles are reabsorbed into the source function J ( x ) and finally ending at the vacuum state at t = oc. The Green's function is denned by

The transition amplitude from a general state (different from the vacuum) to another general state is constructed from the Green's function by the so-called LSZ prescription. In this prescription, one sets the time coordinates of some of the field variables appearing in the Green's function either at — oo or oo and those field variables are associated with either initial or final states, respectively. For the details of this prescription, readers are referred to standard textbooks listed at the end of this book. We however note that the LSZ construction of the physical scattering amplitude starting with Green's functions, which incorporate all the effects of interactions, gives rise to a more natural picture of the physical process compared to the specification of asymptotic states by imposing a priori boundary conditions on field variables. From a technical view point, the rcnormalization prescription, for example, is more naturally formulated in terms of Green's functions. We have not specified a precise normalization condition of the path integral in field theory. Here we would like to give a normalization which is useful in practical applications. Without a precise specification of the path integral measure, the left- and right-hand sides of eqn (2.131) approach for t; — ti —> oo

with a constant factor N. A simple way to take care of this normalization factor N is to define

where Z is given by

30

PATH INTEGRAL AND THE ACTION PRINCIPLE

This prescription corresponds to the normalization of the exact ground state eigenvalue of H at EQ = 0. In fact we have

This normalization is also consistent with the normalization of the Green's function in the operator formalism. The Green's function in the Heisenberg picture is given by (Q\T*$(xi)4>(xi) • • • 6(xn)\0}. By noting the relation H\0} = E0\0) = 0, one can write

Our normalization of the path integral (2.137) and the definition of Green's functions in eqn (2.135) are consistent with this operator expression in the limit I/ —> oo, ti —> —oo. In practical applications of the path integral, it is convenient to tentatively include the factor Z into the path integral measure in the process of calculations and fix the normalization factor as above only when it is required.

3 QUANTUM THEORY OF PHOTONS AND THE PHASE OPERATOR In this section, we first present the essence of the quantization of the electromagnetic field and its path integral representation. The electromagnetic field is the simplest example of a gauge field and its quantization nicely illustrates the technical problems associated with the quantization of gauge fields in general. We next discuss the problem associated with the phase operator of the photon, which appears as a result of quantizing the electromagnetic field. This problem is analyzed on the basis of the notion of index and the postulate of positive definite Hilbert space. We explain that this problem of the phase operator is closely related, in technical terms, to the chiral anomaly to be discussed later. 3.1

Canonical quantization of the electromagnetic field

We describe the essence of the canonical quantization of the electromagnetic field. This quantization is basically given as a generalization of the quantization of the harmonic oscillator, but the analysis is slightly more involved due to the appearance of the notion of gauge invariance. The Lagrangian density which describes Maxwell's electromagnetic field in the vacuum is written in terms of the four electromagnetic potentials A^(x) = Ap(t,x), ,u = 0, 1, 2. 3, as

We adopt in this book the convention that we take a sum over indices which appear twice in the same equation. Our metric convention of space-time is g^v = (1,-1.-1, —1). The electromagnetic tensor F!JjV = d^Av - dvA^. — —Fvti is related to the electric field E and magnetic field B in the following way

The electromagnetic tensor F^v does not change under the gauge transformation (i.e., change of variables)

namely. F'^v = F^v: the tensor is invariant under the gauge transformation. Here <jj(x) = u(t,x) is an arbitrary function of space-time, and the freedom related 31

32

QUANTUM THEORY OF PHOTONS AND THE PHASE OPERATOR

to 'jj(x) is called the gauge freedom. This gauge iiivariance shows that one of Afl(x) can be chosen to be an arbitrary function by suitably choosing w(x). This reduction of freedom by restricting A^ (x) is called gauge fixing. This restriction on Ay (x) itself is called the gauge condition, and the choice of the gauge condition is rather arbitrary. The gauge condition on the spatial components of A^x). which is called the Coulomb gauge,

is fundamental. The gauge condition called the Landau (or Lorentz) gauge, which preserves Lorentz invariance,

is also commonly used. The gauge condition which is sometimes called the Weyl gauge is also convenient when one analyzes the theoretical aspects of quantization. We here tentatively adopt the Coulomb gauge (3.4). In this gauge condition, the action for Maxwell theory, which is defined by the space-time integral of the Lagrangian density, is given after partial integration by

The equation of motion in the vacuum for the time component AQ becomes

where A = Y?k-i (^k)'2 is called the Laplacian. From this equation, we conclude AQ(X) = 0 by imposing the boundary condition AQ(X) = 0 at spatial infinity. Consequently, by remembering the space-time metric convention g^ = (1, — 1, — 1, — 1), the Lagrangian density is reduced to

and the equation of motion is given by

CANONICAL QUANTIZATION OF THE ELECTROMAGNETIC FIELD

33

The momentum variable canonically conjugate to A& is given by

and the Hamiltoriian is given by

Here we used the notation Ak = (d/dt)AkWe expand the field variables in terms of a complete set of plane waves as

by keeping the Coulomb gauge condition dkAk = dkHk = 0 in mind. Here we have k0 = \k\ as a result of the equation of motion (3.10), and as the polarization vector e^A' (k) which satisfies kfP*> (k) — 0 we choose

when the momentum is in the positive direction of the z axis. The polarization for a general momentum k is defined as a suitable rotation of this expression. The canonically conjugate momentum is then given by

and the Hamiltonian (3.12) is written as

with the notation ui(k) = c\k\ and tentatively assuming that the momentum is discrete, as in a box normalization. This Hamiltonian shows that the wave motion of light is equivalent to an assembly of an infinite number of harmonic oscillators. Because of the condition

34

QUANTUM THEORY OF PHOTONS AND THE PHASE OPERATOR

fce^A) (fc) = 0 the light is a transverse wave polarized in the two perpendicular directions A = 1, 2 with respect to the motion. The quantization condition is thus given by

and

The energy carried by each plane wave is given by

by neglecting the zero point energy. The Heisenberg equation of motion gives

and the quantized operator at a general time is given by

to be consistent with the expression A^(x} in eqri (3.13). The state vector for the photon in the Fock space representation is given by

and it represents the state which contains n\ photons with momentum arid polarization (Ari.Ai), 112 photons with momentum and polarization (fe,^), and so on. This means that the photons associated with the electromagnetic wave are Bose particles. The probability amplitude, which corresponds to the familiar Schrodiriger amplitude in quantum mechanics, for a single free photon is given by

If one formally introduces the coordinate and momentum variables starting with the annihilation and creation operators by

PATH INTEGRAL QUANTIZATION OF THE ELECTROMAGNETIC FIELD 35

the Hamiltonian for the electromagnetic field is written as

3.2

Path integral quantization of the electromagnetic field

We have shown that the free electromagnetic field is equivalent to an assembly of an infinite number of harmonic oscillators. The electromagnetic field contains two independent transverse components, and thus the path integral representation is equivalent to the case with two free scalar fields in eqn (2.48). We thus tentatively write

where the field variables are actually constrained by the Coulomb gauge condition dkAk(x) = 0, and thus we have only two integral variables. The Coulomb gauge condition is generally understood as a decomposition of the three spatial components of the potential Ak(x) into two transverse components dkA^'(x) —O.i — 1, 2, and the gauge freedom dkw(x) as

In this notation, the "length in functional space" when one varies A/t(x) infinitesimallv is written as

If one recalls that the invariant volume element in general relativity is given by dV ~ i/dett^,, d4x when the length is given by d 2 s = ) in a four-dimensional space whose coordinates are given by (a, b). On the other hand, the function g(x) = x4- + ixf which is used to construct the instanton solution shows that a point on the unit hypersurface in four-dimensional Euclidean space-time described by (x4,x), which is also an 53, and a point on the hypersurface described by g(x) of the gauge group SU(2), are in 1 : 1 cor respondence. Namely, when the coordinates (x4,x) of space-time cover S3 once, the element g(x) covers the hypersurface S3 in the gauge space once. The quantity which describes this topological property is called the winding number. The winding number v for the specific g(x) above is given by

by using the totally anti-symmetric symbol f^"aP normalized by g1234 = 1. The integral here stands for a surface integral over the hypersurface S3 located at infinity of four-dimensional space-time, and dS^ stands for the surface element which is orthogonal to the /u-axis. In fact, near the point x4 = oo, namely, near the point 3 x = 0, for example, the integrand of the integral to define v is written as

which shows that the integrand at any point of the hypersurface is given by the surface clement for a unit hypersurface divided by 2?r 2 . If one recalls that the surface area of the unit hypersurface (which is a volume in the conventional sense) is given by 2?r2, the above integral over the entire hypersurface in fact gives the winding number v = 1. If one uses

instead of g ( x ) in the formula defining the winding number //, it is shown that the winding number // = n is obtained. Also, the anti-instantoii with v = — 1 is obtained by setting n = —1 in this formula, or equivalently by

82

THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES The winding number v is written by using the instantori solution as

The integrand here is written as a total divergence, and the integral is written as a surface integral at space-time infinity by using the Gauss law10

where we used the behavior of the instanton solution at infinity in the last expression. Namely, this expression agrees with the definition of the windingnumber (5.85). Finally, we would like to show that the instanton discussed so far in fact satisfies the Yang-Mills equation. We define FIJtv = (l/2)e /i)ya ^F a 0 and use the following Schwarz inequality

If one uses the formula for the winding number on the right-hand side of this relation, we have

and the equality holds only for the case

where ± corresponds to the signature of the winding number v. If one uses the we have + signature in this relation (5.93) and uses

which is in fact confirmed to be satisfied by /(r 2 ) = r 2 /(r 2 +p 2 ) used to construct the instanton in eqn (5.82). This shows that the instanton gives the winding number v = 1 and at the same time it gives the minimum of the action for v = 1. The field configuration 16

This property is shown by using the Jacobi identity

The quantity appearing in eqn (5.90). K^ = l/(87r 2 ) the Chcrn-Simons form a,nd is very fundamental.

is called

INSTANTONS

83

which gives the stationary point of the action is a solution of the field equation derived from the action, and thus the instanton is in fact the solution of the Euclidean Yang-Mills field equation. The instanton solution

approaches the configuration ig(x)d fj/<j^(x) at infinity, which is gauge equivalen to the vacuum. The gauge function g ( x ) , however, cannot be expressed as a superposition of conventional "small" gauge transformations which satisfy the boundary condition 5(00) = 1. Physically, this gauge function g(x) is interpreted as describing tunneling starting from one vacuum at x4 = — oo to another vacuum at x4 = oo following the Euclidean imaginary time. (For this reason, this solution is called an "instant-on" , indicating that it appears and then disappears instantly in time unlike the ordinary soliton.) The value of the action for this solution SE = —8n2/g2 describes the height of the tunneling barrier, and in the path integral the factor with this action in the exponential

is understood as giving the leading term of the tunneling probability. The vacuum in non-Abelian gauge theory thus has a structure similar to the motion of the electron moving in a periodic (sharp) potential, and it is described by a wave function similar to the Bloch wave. The path integral of Euclidean gauge theory is shown to be written as

though we here forgo the details of this derivation. In this path integral, the sum over v runs over all the integers, and the term v — 1 corresponds to the contribution from the instanton solution we have discussed so far. The general v corresponds to multi-instanton solutions. The real parameter 0 in this formula is an arbitrary constant, arid the vacuum state \9, ±oc) is called the 9 vacuum. The gauge field configuration in the path integral measure described df by v is Afj,(x) = A, L(x}(v) + a^(x) by using the instanton solution A^(x)^ with instanton number (i.e., the winding number) v and the fluctuation a M (x) around the solution.16 In analogy with the Bloch wave, the exponential factor exp(i0f) has the following meaning: v corresponds to the (difference) of the positions of the electron and the real parameter B corresponds to the Bloch momentum. 16

The instanton solution with v = 1 contains eight deformation parameters corresponding to four parameters describing the position of the center and one parameter p describing the size of the instanton and three other gauge parameters. In the path integral, one needs to perform the integral over these parameters also. For general simple groups, it is known that the instanton solution is constructed for each SU(2) sub-group, and no other solutions.

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THE JACOBIAN IN PATH INTEGRALS AND QUANTUM ANOMALIES When one performs a change of the integration variables

with an infinitesimal constant parameter a, the calculation of the Jacobian (5.73) shows that the path integral measure is transformed as

and thus the parameter 0 is changed as

At the same time, the Euclidean action changes as

These properties show that the 0 parameter can be freely changed in the theory with a massless fermion, and the parameter 0 has no physical meaning in such a theory. (In the presence of the source terms / dx (fjiji + iprj), one needs to perform the simultaneous re-definition of sources t] —>• /7exp(—id/2) and 77 —S> cxp(—i6/2)r to completely eliminate the 9 dependence from the generating functional.) The 9 term is written as

and ^""PFpvFap is a generalization of EB in QED and thus breaks CP symmetry. This breaking of CP symmetry for 0 ^ 0 in QCD is known as the strong CP problem. 5.5

Atiyah Singer index theorem

Coming back to the analysis of the quantum anomaly, we re-examine the calculation of the Jacobian factor for the chiral transformation

If one considers a global limit of this relation by choosing a as a constant, both sides of this relation are written as

ATIYAH-SINGER INDEX THEOREM

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If one combines this relation with the relations valid for the eigenfunctions with

which is derived from ^75 + J5$> = 0, one obtains

by rioting /(O) = 1. In this relation, n± respectively represent the number of eigenstates with vanishing eigenvalue for the Dirac operator T/) with positive or negative chirality eigenvalues

and the right-hand side of eqri (5.106) shows the instanton number (or Pontryagin number). For $>n(x) = 0, one can show Jf)^(fn(x) = 0 and thus Jf>\(\ ± 7o)/2](^n(^) = 0. One can thus choose the eigenfunctions of y>