Classical Theory of Gauge Fields This Page Intentionally Left Blank Classical Theory of Gauge Fields Valery Rubakov Translated by Stephen S Wilson PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright c 2002 by Princeton University Press Original title: Klassicheskie Kalibrovochnye Polia c V.A. Rubakov, 1999 c Editorial´ URSS, 1999 Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Control Number 2002102049 ISBN 0-691-05927-6 (hardcover : alk. paper) The publisher would like to acknowledge the translator of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. ∞ www.pupress.princeton.edu Printed in the United States of America 10987654321 Contents Preface ix Part I 1 1 Gauge Principle in Electrodynamics 3 1.1Electromagnetic-fieldactioninvacuum............ 3 1.2Gaugeinvariance........................ 5 1.3 General solution of Maxwell’s equations in vacuum ..... 6 1.4Choiceofgauge......................... 8 2 Scalar and Vector Fields 11 2.1 System of units = c =1................... 11 2.2Scalarfieldaction........................ 12 2.3Massivevectorfield....................... 15 2.4Complexscalarfield...................... 17 2.5Degreesoffreedom....................... 18 2.6Interactionoffieldswithexternalsources.......... 19 2.7 Interacting fields. Gauge-invariant interaction in scalar electrodynamics......................... 21 2.8Noether’stheorem....................... 26 3 Elements of the Theory of Lie Groups and Algebras 33 3.1Groups.............................. 33 3.2Liegroupsandalgebras.................... 41 3.3RepresentationsofLiegroupsandLiealgebras....... 48 3.4CompactLiegroupsandalgebras............... 53 4 Non-Abelian Gauge Fields 57 4.1Non-Abelianglobalsymmetries................ 57 4.2 Non-Abelian gauge invariance and gauge fields: the group SU(2).............................. 63 4.3Generalizationtoothergroups................ 69 vi 4.4Fieldequations......................... 75 4.5Cauchyproblemandgaugeconditions............ 81 5 Spontaneous Breaking of Global Symmetry 85 5.1Spontaneousbreakingofdiscretesymmetry......... 86 5.2 Spontaneous breaking of global U(1) symmetry. Nambu– Goldstonebosons........................ 91 5.3 Partial symmetry breaking: the SO(3)model........ 94 5.4Generalcase.Goldstone’stheorem.............. 99 6 Higgs Mechanism 105 6.1ExampleofanAbelianmodel................. 105 6.2 Non-Abelian case: model with complete breaking of SU(2) symmetry............................ 112 6.3 Example of partial breaking of gauge symmetry: bosonic sectorofstandardelectroweaktheory............. 116 Supplementary Problems for Part I 127 Part II 135 7 The Simplest Topological Solitons 137 7.1Kink............................... 138 7.2 Scale transformations and theorems on the absence of solitons............................. 149 7.3Thevortex........................... 155 7.4 Soliton in a model of n-field in (2 + 1)-dimensional space–time........................... 165 8 Elements of Homotopy Theory 173 8.1Homotopyofmappings..................... 173 8.2 The fundamental group . .................. 176 8.3Homotopygroups........................ 179 8.4 Fiber bundles and homotopy groups ............. 184 8.5Summaryoftheresults.................... 189 9 Magnetic Monopoles 193 9.1 The soliton in a model with gauge group SU(2)....... 193 9.2Magneticcharge........................ 200 9.3Generalizationtoothermodels................ 207 9.4 The limit mH /mV → 0 .................... 208 9.5Dyons.............................. 212 vii 10 Non-Topological Solitons 215 11 Tunneling and Euclidean Classical Solutions in Quantum Mechanics 225 11.1Decayofametastablestateinquantummechanicsofone variable............................. 226 11.2Generalizationtothecaseofmanyvariables......... 232 11.3 Tunneling in potentials with classical degeneracy . ..... 240 12 Decay of a False Vacuum in Scalar Field Theory 249 12.1Preliminaryconsiderations................... 249 12.2 Decay probability: Euclidean bubble (bounce) ........ 253 12.3Thin-wallapproximation.................... 259 13 Instantons and Sphalerons in Gauge Theories 263 13.1Euclideangaugetheories.................... 263 13.2 Instantons in Yang–Mills theory ................ 265 13.3 Classical vacua and θ-vacua.................. 272 13.4 Sphalerons in four-dimensional models with the Higgs mechanism........................... 280 Supplementary Problems for Part II 287 Part III 293 14 Fermions in Background Fields 295 14.1FreeDiracequation...................... 295 14.2SolutionsofthefreeDiracequation.Diracsea....... 302 14.3Fermionsinbackgroundbosonicfields............ 308 14.4FermionicsectoroftheStandardModel........... 318 15 Fermions and Topological External Fields in Two-dimensional Models 329 15.1Chargefractionalization.................... 329 15.2 Level crossing and non-conservation of fermion quantum numbers............................. 336 16 Fermions in Background Fields of Solitons and Strings in Four-Dimensional Space–Time 351 16.1 Fermions in a monopole background field: integer angular momentumandfermionnumberfractionalization...... 352 viii 16.2 Scattering of fermions off a monopole: non-conservation of fermionnumbers........................ 359 16.3 Zero modes in a background field of a vortex: superconducting strings . ................... 364 17 Non-Conservation of Fermion Quantum Numbers in Four-dimensional Non-Abelian Theories 373 17.1LevelcrossingandEuclideanfermionzeromodes...... 374 17.2Fermionzeromodeinaninstantonfield........... 378 17.3Selectionrules.......................... 385 17.4 Electroweak non-conservation of baryon and lepton numbers athightemperatures...................... 392 Supplementary Problems for Part III 397 Appendix. Classical Solutions and the Functional Integral 403 A.1 Decay of the false vacuum in the functional integral formalism............................ 404 A.2 Instanton contributions to the fermion Green’s functions . 411 A.3 Instantons in theories with the Higgs mechanism. Integrationalongvalleys.................... 418 A.4Growinginstantoncrosssections............... 423 Bibliography 429 Index 441 Preface This book is based on a lecture course taught over a number of years in the Department of Quantum Statistics and Field Theory of the Moscow State University, Faculty of Physics, to students in the third and fourth years, specializing in theoretical physics. Traditionally, the theory of gauge fields is covered in courses on quantum field theory. However, many concepts and results of gauge theories are already in evidence at the level of classical field theory, which makes it possible and useful to study them in parallel with a study of quantum mechanics. Accordingly, the reading of the first ten chapters of this book does not require a knowledge of quantum mechanics, Chapters 11–13 employ the concepts and methods conventionally presented at the start of a course on quantum mechanics, and a thorough knowledge of quantum mechanics, including the Dirac equation, is only required in order to read the subsequent chapters. A detailed acquaintance with quantum field theory is not mandatory in order to read this text. At the same time, at the very start, it is assumed that the reader has a knowledge of classical mechanics, special relativity and classical electrodynamics. Part I of the book contains a study of the basic ideas of the theory of gauge fields, the construction of gauge-invariant Lagrangians and an analysis of spectra of linear perturbations, including perturbations about a non-trivial ground state. Part II is devoted to the construction and interpretation of solutions, whose existence is entirely due to the nonlinearity of the field equations, namely, solitons, bounces and instantons. In Part III, we consider certain interesting effects, arising in the interactions of fermions with topological scalar and gauge fields. The book has an Appendix, which briefly discusses the role of instantons as saddle points of the Euclidean functional integral in quantum field theory and several related topics. The purpose of the Appendix is to give an initial idea about this rather complex aspect of quantum field theory; the presentation there is schematic and makes no claims to completeness (for example, we do not touch at all upon the important questions relating to supersymmetric gauge theories). To read the Appendix, one needs to be x Preface acquainted with the quantum theory of gauge fields. Of course, the majority of questions touched upon in this book are considered in one way or another in other existing books and reviews on quantum field theory, a far from complete list of which is given at the end of the book. In a sense, this book may serve as an introduction to the subject. The book contains two mathematical digressions, where elements of the theory of Lie groups and algebras and of homotopy theory are studied, without any claim to completeness or mathematical rigor. This should make it possible to read the book without constantly resorting to the specialized literature in these areas. We should like to thank our colleagues F.L. Bezrukov, D.Yu. Grigoriev, M.V. Libanov, D.V. Semikoz, D.T. Son, P.G. Tinyakov and S.V. Troitsky for their great help in the preparation and teaching of the lecture course, attentive reading of the manuscript and preparation for its publication. Classical Theory of Gauge Fields This Page Intentionally