Classical Theory of Gauge Fields

by Valery Rubakov, Stephen S. Wilson

Overview

Based on a highly regarded lecture course at Moscow State University, this is a clear and systematic introduction to gauge field theory. It is unique in providing the means to master gauge field theory prior to the advanced study of quantum mechanics. Though gauge field theory is typically included in courses on quantum field theory, many of its ideas and results can be understood at the classical or semi-classical level. Accordingly, this book is organized so that its early chapters require no special knowledge of quantum mechanics. Aspects of gauge field theory relying on quantum mechanics are introduced only later and in a graduated fashion—making the text ideal for students studying gauge field theory and quantum mechanics simultaneously.


The book begins with the basic concepts on which gauge field theory is built. It introduces gauge-invariant Lagrangians and describes the spectra of linear perturbations, including perturbations above nontrivial ground states. The second part focuses on the construction and interpretation of classical solutions that exist entirely due to the nonlinearity of field equations: solitons, bounces, instantons, and sphalerons. The third section considers some of the interesting effects that appear due to interactions of fermions with topological scalar and gauge fields. Mathematical digressions and numerous problems are included throughout. An appendix sketches the role of instantons as saddle points of Euclidean functional integral and related topics.


Perfectly suited as an advanced undergraduate or beginning graduate text, this book is an excellent starting point for anyone seeking to understand gauge fields.

Product Details

ISBN-13:	9780691059273
Publisher:	Princeton University Press
Publication date:	05/26/2002
Series:	Mathematical Sciences Ser.
Edition description:	New Edition
Pages:	456
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Valery Rubakov is Professor of Physics at Moscow State University. He is a Member of the Russian Academy of Sciences and was awarded its A. A. Friedmann Prize in 1999.

Table of Contents

Preface ix

Part I 1

Chapter 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

Chapter 2: Scalar and Vector Fields 11

2.1 System of units h = c = 1 11

2.2 Scalarfield action 12

2.3 Massive vectorfield 15

2.4 Complex scalarfield 17

2.5 Degrees of freedom 18

2.6 Interaction offields with external sources 19

2.7 Interactingfields. Gauge-invariant interaction in scalar electrodynamics 21

2.8 Noether's theorem 26

Chapter 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

Chapter 4: Non-Abelian Gauge Fields 57

4.1 Non-Abelian global symmetries 57

4.2 Non-Abelian gauge invariance and gaugefields: the group SU(2) 63

4.3 Generalization to other groups 69

4.4 Field equations 75

4.5 Cauchy problem and gauge conditions 81

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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 mh/mv 0 208

9.5 Dyons 212

Chapter 10: Non-Topological Solitons 215

Chapter 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

Chapter 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

Chapter 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 0-vacua 272

13.4 Sphalerons in four-dimensional models with the Higgs mechanism 280

Supplementary Problems for Part II 287

Part III 293

Chapter 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 bosonicfields 308

14.4 Fermionic sector of the Standard Model 318

Chapter 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

Chapter 16: Fermions in Background Fields of Solitons and Strings in Four-Dimensional Space-Time 351

16.1 Fermions in a monopole backgroundfield: integer angular momentum and fermion number fractionalization 352

16.2 Scattering of fermions off a monopole: non-conservation of fermion numbers 359

16.3 Zero modes in a backgroundfield of a vortex: superconducting strings 364

Chapter 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 instantonfield 378

17.3 Selection rules 385

17.4 Electroweak non-conservation of baryon and lepton numbers 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

What People are Saying About This

John Bahcall

Professor Rubakov is an outstanding researcher and an exceptionally clear lecturer, an unusual combination that shines through in this illuminating text. Students and active researchers can all learn something from this well-organized and insightful text, which is written so as to be widely accessible but authoritative.
— John Bahcall, Institute for Advanced Study

Igor Klebanov

This thorough, clear, and readable book is an important addition to the available literature on solitons in field theory. The inclusion of materials on semiclassical quantization of field theories and on the relevant mathematics, in addition to the sections covering classical gauge fields, broadens its appeal. The book will be very useful In advanced undergraduate as well as graduate courses on field theory. It will also serve as a modern review and reference for working theoretical physicists.
— Igor Klebanov, Princeton University

Dirk ter Haar

This is an excellent text on field theory. The material is well thought out, well organized, well presented, and amply supplemented with problems.
— Dirk ter Haar, author of "Master of Modern Physics"

From the Publisher

"This thorough, clear, and readable book is an important addition to the available literature on solitons in field theory. The inclusion of materials on semiclassical quantization of field theories and on the relevant mathematics, in addition to the sections covering classical gauge fields, broadens its appeal. The book will be very useful In advanced undergraduate as well as graduate courses on field theory. It will also serve as a modern review and reference for working theoretical physicists."—Igor Klebanov, Princeton University

"This is an excellent text on field theory. The material is well thought out, well organized, well presented, and amply supplemented with problems."—Dirk ter Haar, author of Master of Modern Physics

"Professor Rubakov is an outstanding researcher and an exceptionally clear lecturer, an unusual combination that shines through in this illuminating text. Students and active researchers can all learn something from this well-organized and insightful text, which is written so as to be widely accessible but authoritative."—John Bahcall, Institute for Advanced Study

Recipe

"This thorough, clear, and readable book is an important addition to the available literature on solitons in field theory. The inclusion of materials on semiclassical quantization of field theories and on the relevant mathematics, in addition to the sections covering classical gauge fields, broadens its appeal. The book will be very useful In advanced undergraduate as well as graduate courses on field theory. It will also serve as a modern review and reference for working theoretical physicists."—Igor Klebanov, Princeton University

"This is an excellent text on field theory. The material is well thought out, well organized, well presented, and amply supplemented with problems."—Dirk ter Haar, author of Master of Modern Physics

"Professor Rubakov is an outstanding researcher and an exceptionally clear lecturer, an unusual combination that shines through in this illuminating text. Students and active researchers can all learn something from this well-organized and insightful text, which is written so as to be widely accessible but authoritative."—John Bahcall, Institute for Advanced Study

Editorial Reviews

"Classical Theory of Gauge Fields is indeed . . . unique . . . and without alternative for all those who want to immerse themselves in this particular area of theoretical physics."—-H. Hogreve, Mathematical Reviews

From the Publisher

Classical Theory of Gauge Fields is indeed . . . unique . . . and without alternative for all those who want to immerse themselves in this particular area of theoretical physics.

Mathematical Reviews

Classical Theory of Gauge Fields is indeed . . . unique . . . and without alternative for all those who want to immerse themselves in this particular area of theoretical physics.

" Mathematical Reviews Hogreve