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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 (cid:1)c2002 by Princeton University Press Original title: Klassic(cid:1)heskie Kalibrovoc(cid:1)hnye Poli(cid:1)a (cid:1)cV.A. Rubakov, 1999 (cid:1)cE´ditorial 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 10 9 8 7 6 5 4 3 2 1 Contents Preface ix Part I 1 1 Gauge Principle in Electrodynamics 3 1.1 Electromagnetic-field action in vacuum . . . . . . . . . . . . 3 1.2 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 General solution of Maxwell’s equations in vacuum . . . . . 6 1.4 Choice of gauge . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Scalar and Vector Fields 11 2.1 System of units (cid:1)=c=1 . . . . . . . . . . . . . . . . . . . 11 2.2 Scalar field action. . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Massive vector field. . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Complex scalar field . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Interaction of fields with external sources . . . . . . . . . . 19 2.7 Interacting fields. Gauge-invariant interaction in scalar electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.8 Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . 26 3 Elements of the Theory of Lie Groups and Algebras 33 3.1 Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Lie groups and algebras . . . . . . . . . . . . . . . . . . . . 41 3.3 Representations of Lie groups and Lie algebras . . . . . . . 48 3.4 Compact Lie groups and algebras . . . . . . . . . . . . . . . 53 4 Non-Abelian Gauge Fields 57 4.1 Non-Abelian global symmetries . . . . . . . . . . . . . . . . 57 4.2 Non-Abelian gauge invariance and gauge fields: the group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Generalization to other groups . . . . . . . . . . . . . . . . 69 vi 4.4 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Cauchy problem and gauge conditions . . . . . . . . . . . . 81 5 Spontaneous Breaking of Global Symmetry 85 5.1 Spontaneous breaking of discrete symmetry . . . . . . . . . 86 5.2 Spontaneous breaking of global U(1) symmetry. Nambu– Goldstone bosons . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Partial symmetry breaking: the SO(3) model . . . . . . . . 94 5.4 General case. Goldstone’s theorem . . . . . . . . . . . . . . 99 6 Higgs Mechanism 105 6.1 Example of an Abelian model . . . . . . . . . . . . . . . . . 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 sector of standard electroweak theory. . . . . . . . . . . . . 116 Supplementary Problems for Part I 127 Part II 135 7 The Simplest Topological Solitons 137 7.1 Kink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2 Scale transformations and theorems on the absence of solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.3 The vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.4 Soliton in a model of n-field in (2+1)-dimensional space–time . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8 Elements of Homotopy Theory 173 8.1 Homotopy of mappings. . . . . . . . . . . . . . . . . . . . . 173 8.2 The fundamental group . . . . . . . . . . . . . . . . . . . . 176 8.3 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . 179 8.4 Fiber bundles and homotopy groups . . . . . . . . . . . . . 184 8.5 Summary of the results . . . . . . . . . . . . . . . . . . . . 189 9 Magnetic Monopoles 193 9.1 The soliton in a model with gauge group SU(2) . . . . . . . 193 9.2 Magnetic charge . . . . . . . . . . . . . . . . . . . . . . . . 200 9.3 Generalization to other models . . . . . . . . . . . . . . . . 207 9.4 The limit m /m →0 . . . . . . . . . . . . . . . . . . . . 208 H V 9.5 Dyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 vii 10 Non-Topological Solitons 215 11 Tunneling and Euclidean Classical Solutions in Quantum Mechanics 225 11.1 Decay of a metastable state in quantum mechanics of one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 11.2 Generalization to the case of many variables . . . . . . . . . 232 11.3 Tunneling in potentials with classical degeneracy . . . . . . 240 12 Decay of a False Vacuum in Scalar Field Theory 249 12.1 Preliminary considerations . . . . . . . . . . . . . . . . . . . 249 12.2 Decay probability: Euclidean bubble (bounce) . . . . . . . . 253 12.3 Thin-wall approximation . . . . . . . . . . . . . . . . . . . . 259 13 Instantons and Sphalerons in Gauge Theories 263 13.1 Euclidean gauge theories . . . . . . . . . . . . . . . . . . . . 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 14.1 Free Dirac equation . . . . . . . . . . . . . . . . . . . . . . 295 14.2 Solutions of the free Dirac equation. Dirac sea . . . . . . . 302 14.3 Fermions in background bosonic fields . . . . . . . . . . . . 308 14.4 Fermionic sector of the Standard Model . . . . . . . . . . . 318 15 Fermions and Topological External Fields in Two-dimensional Models 329 15.1 Charge fractionalization . . . . . . . . . . . . . . . . . . . . 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 momentum and fermion number fractionalization . . . . . . 352 viii 16.2 Scattering of fermions off a monopole: non-conservation of fermion numbers . . . . . . . . . . . . . . . . . . . . . . . . 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 17.1 Level crossing and Euclidean fermion zero modes . . . . . . 374 17.2 Fermion zero mode in an instanton field . . . . . . . . . . . 378 17.3 Selection rules. . . . . . . . . . . . . . . . . . . . . . . . . . 385 17.4 Electroweaknon-conservationofbaryonandleptonnumbers at high temperatures . . . . . . . . . . . . . . . . . . . . . . 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. Integration along valleys . . . . . . . . . . . . . . . . . . . . 418 A.4 Growing instanton cross sections . . . . . . . . . . . . . . . 423 Bibliography 429 Index 441 Preface Thisbookisbasedonalecturecoursetaughtoveranumberofyearsinthe 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,thetheoryofgaugefieldsiscoveredincoursesonquantum 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 bookdoesnotrequireaknowledgeofquantummechanics, Chapters11–13 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, attheverystart, itisassumedthatthereaderhasaknowledgeofclassical 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. ThebookhasanAppendix,whichbrieflydiscussestheroleofinstantons assaddlepointsoftheEuclideanfunctionalintegralinquantumfieldtheory 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