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Methods of Contemporary Gauge Theory
     

Methods of Contemporary Gauge Theory

by Yuri Makeenko, P. V. Landshoff, Makeenko Yuri
 

ISBN-10: 0521022150

ISBN-13: 9780521022156

Pub. Date: 08/28/2005

Publisher: Cambridge University Press

This book provides a thorough introduction to quantum theory of gauge fields. Emphasis is placed on four non-perturbative methods: path integrals, lattice gauge theories, the 1/N expansion, and reduced matrix models, which have important contemporary applications. The book uses a novel approach to gauge theories based on path-dependent phase factors known as

Overview

This book provides a thorough introduction to quantum theory of gauge fields. Emphasis is placed on four non-perturbative methods: path integrals, lattice gauge theories, the 1/N expansion, and reduced matrix models, which have important contemporary applications. The book uses a novel approach to gauge theories based on path-dependent phase factors known as the Wilson loops, and provides comprehensive coverage of large-N Yang-Mills theory.

Product Details

ISBN-13:
9780521022156
Publisher:
Cambridge University Press
Publication date:
08/28/2005
Series:
Cambridge Monographs on Mathematical Physics Series
Edition description:
New Edition
Pages:
432
Product dimensions:
6.85(w) x 9.72(h) x 0.87(d)

Table of Contents

Prefacexi
Part 1Path Integrals1
1Operator calculus3
1.1Free propagator3
1.2Euclidean formulation6
1.3Path-ordering of operators10
1.4Feynman disentangling13
1.5Calculation of the Gaussian path integral18
1.6Transition amplitudes20
1.7Propagators in external field29
2Second quantization35
2.1Integration over fields35
2.2Grassmann variables37
2.3Perturbation theory38
2.4Schwinger-Dyson equations40
2.5Commutator terms40
2.6Schwinger-Dyson equations (continued)41
2.7Regularization45
3Quantum anomalies from path integral47
3.1QED via path integral47
3.2Chiral Ward identity48
3.3Chiral anomaly51
3.4Chiral anomaly (calculation)55
3.5Scale anomaly59
4Instantons in quantum mechanics65
4.1Double-well potential65
4.2The instanton solution68
4.3Instanton contribution to path integral70
4.4Symmetry restoration by instantons75
4.5Topological charge and [theta]-vacua76
Bibliography to Part 179
Part 2Lattice Gauge Theories83
5Observables in gauge theories85
5.1Gauge invariance85
5.2Phase factors (definition)88
5.3Phase factors (properties)93
5.4Aharonov-Bohm effect95
6Gauge fields on a lattice99
6.1Sites, links, plaquettes and all that100
6.2Lattice formulation102
6.3The Haar measure107
6.4Wilson loops110
6.5Strong-coupling expansion113
6.6Area law and confinement117
6.7Asymptotic scaling119
7Lattice methods123
7.1Phase transitions124
7.2Mean-field method128
7.3Mean-field method (variational)131
7.4Lattice renormalization group133
7.5Monte Carlo method136
7.6Some Monte Carlo results140
8Fermions on a lattice143
8.1Chiral fermions143
8.2Fermion doubling145
8.3Kogut-Susskind fermions151
8.4Wilson fermions152
8.5Quark condensate156
9Finite temperatures159
9.1Feynman-Kac formula160
9.2QCD at finite temperature166
9.3Confinement criterion at finite temperature168
9.4Deconfining transition170
9.5Restoration of chiral symmetry175
Bibliography to Part 2179
Part 31/N Expansion185
10O(N) vector models187
10.1Four-Fermi theory188
10.2Bubble graphs as the zeroth order in 1/N191
10.3Functional methods for [open phi superscript 4] theory200
10.4Nonlinear sigma model208
10.5Large-N factorization in vector models211
11Multicolor QCD213
11.1Index or ribbon graphs214
11.2Planar and nonplanar graphs218
11.3Planar and nonplanar graphs (the boundaries)224
11.4Topological expansion and quark loops230
11.5't Hooft versus Veneziano limits233
11.6Large-N factorization237
11.7The master field243
11.81/N as semiclassical expansion246
12QCD in loop space249
12.1Observables in terms of Wilson loops249
12.2Schwinger-Dyson equations for Wilson loop255
12.3Path and area derivatives258
12.4Loop equations263
12.5Relation to planar diagrams267
12.6Loop-space Laplacian and regularization269
12.7Survey of nonperturbative solutions274
12.8Wilson loops in QCD[subscript 2]275
12.9Gross-Witten transition in lattice QCD[subscript 2]282
13Matrix models287
13.1Hermitian one-matrix model288
13.2Hermitian one-matrix model (solution at N = [infinity])294
13.3The loop equation297
13.4Solution in 1/N300
13.5Continuum limit303
13.6Hermitian multimatrix models311
Bibliography to Part 3315
Part 4Reduced Models323
14Eguchi-Kawai model325
14.1Reduction of the scalar field (lattice)325
14.2Reduction of the scalar field (continuum)330
14.3Reduction of the Yang-Mills field332
14.4The continuum Eguchi-Kawai model336
14.5R[superscript d] symmetry in perturbation theory340
14.6Quenched Eguchi-Kawai model342
15Twisted reduced models351
15.1Twisting prescription351
15.2Twisted reduced model for scalars355
15.3Twisted Eguchi-Kawai model362
15.4Twisting prescription in the continuum368
15.5Continuum version of TEK372
16Noncommutative gauge theories377
16.1The noncommutative space378
16.2The U[subscript [theta](1) gauge theory383
16.3One-loop renormalization386
16.4Noncommutative quantum electrodynamics389
16.5Wilson loops and observables391
16.6Compactification to tori396
16.7Morita equivalence401
Bibliography to Part 4405
Index411

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