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Classical Theory of Gauge Fields - Valery Rubakov

Classical Theory of Gauge Fields

By: Valery Rubakov, Stephen S. Wilson (Translator)


Published: 1st May 2002
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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.

"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

Prefacep. ix
p. 1
Gauge Principle in Electrodynamicsp. 3
Electromagnetic-field action in vacuump. 3
Gauge invariancep. 5
General solution of Maxwell's equations in vacuump. 6
Choice of gaugep. 8
Scalar and Vector Fieldsp. 11
System of units h = c = 1p. 11
Scalarfield actionp. 12
Massive vectorfieldp. 15
Complex scalarfieldp. 17
Degrees of freedomp. 18
Interaction offields with external sourcesp. 19
Interactingfields. Gauge-invariant interaction in scalar electrodynamicsp. 21
Noether's theoremp. 26
Elements of the Theory of Lie Groups and Algebrasp. 33
Groupsp. 33
Lie groups and algebrasp. 41
Representations of Lie groups and Lie algebrasp. 48
Compact Lie groups and algebrasp. 53
Non-Abelian Gauge Fieldsp. 57
Non-Abelian global symmetriesp. 57
Non-Abelian gauge invariance and gaugefields: the group SU(2)p. 63
Generalization to other groupsp. 69
Field equationsp. 75
Cauchy problem and gauge conditionsp. 81
Spontaneous Breaking of Global Symmetryp. 85
Spontaneous breaking of discrete symmetryp. 86
Spontaneous breaking of global U(1) symmetry. Nambu-Goldstone bosonsp. 91
Partial symmetry breaking: the SO(3) modelp. 94
General case. Goldstone's theoremp. 99
Higgs Mechanismp. 105
Example of an Abelian modelp. 105
Non-Abelian case: model with complete breaking of SU(2) symmetryp. 112
Example of partial breaking of gauge symmetry: bosonic sector of standard electroweak theoryp. 116
Supplementary Problems for
p. 127
p. 135
The Simplest Topological Solitonsp. 137
Kinkp. 138
Scale transformations and theorems on the absence of solitonsp. 149
The vortexp. 155
Soliton in a model of n-field in (2 + 1)-dimensional space-timep. 165
Elements of Homotopy Theoryp. 173
Homotopy of mappingsp. 173
The fundamental groupp. 176
Homotopy groupsp. 179
Fiber bundles and homotopy groupsp. 184
Summary of the resultsp. 189
Magnetic Monopolesp. 193
The soliton in a model with gauge group SU(2)p. 193
Magnetic chargep. 200
Generalization to other modelsp. 207
The limit mh/mv 0p. 208
Dyonsp. 212
Non-Topological Solitonsp. 215
Tunneling and Euclidean Classical Solutions in Quantum Mechanicsp. 225
Decay of a metastable state in quantum mechanics of one variablep. 226
Generalization to the case of many variablesp. 232
Tunneling in potentials with classical degeneracyp. 240
Decay of a False Vacuum in Scalar Field Theoryp. 249
Preliminary considerationsp. 249
Decay probability: Euclidean bubble (bounce)p. 253
Thin-wall approximationp. 259
Instantons and Sphalerons in Gauge Theoriesp. 263
Euclidean gauge theoriesp. 263
Instantons in Yang-Mills theoryp. 265
Classical vacua and 0-vacuap. 272
Sphalerons in four-dimensional models with the Higgs mechanismp. 280
Supplementary Problems for
p. 287
p. 293
Fermions in Background Fieldsp. 295
Free Dirac equationp. 295
Solutions of the free Dirac equation. Dirac seap. 302
Fermions in background bosonicfieldsp. 308
Fermionic sector of the Standard Modelp. 318
Fermions and Topological External Fields in Two-dimensional Modelsp. 329
Charge fractionalizationp. 329
Level crossing and non-conservation of fermion quantum numbersp. 336
Fermions in Background Fields of Solitons and Strings in Four-Dimensional Space-Timep. 351
Fermions in a monopole backgroundfield: integer angular momentum and fermion number fractionalizationp. 352
Scattering of fermions off a monopole: non-conservation of fermion numbersp. 359
Zero modes in a backgroundfield of a vortex: superconducting stringsp. 364<br
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691059273
ISBN-10: 0691059276
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 456
Published: 1st May 2002
Country of Publication: US
Dimensions (cm): 24.64 x 16.61  x 3.4
Weight (kg): 0.77