Classical Theory of Gauge Fields
1st Edition
Published on 9. February 2009
456 pages
978-1-4008-2509-7 (ISBN)
Description
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.
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.
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05/2002
Princeton University Press
€164.50
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Persons
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.
Content
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
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
