Classical theory of gauge fields

457 Pages·2002·2.208 MB·English

by  Valery Rubakov, Stephen S. Wilson

Classical Theory of Gauge Fields
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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