Classical Theory of Gauge Fields

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.

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