Part 1: Path Integrals .......................................... 1
1 Operator calculus ............................................ 3
1.1 Free propagator ......................................... 3
1.2 Euclidean formulation ................................... 6
1.3 Path-ordering of operators ............................. 10
1.4 Feynman disentangling .................................. 13
1.5 Calculation of the Gaussian path integral .............. 18
1.6 Transition amplitudes .................................. 20
1.7 Propagators in external field .......................... 29
2 Second quantization ......................................... 35
2.1 Integration over fields ................................ 35
2.2 Grassmann variables .................................... 37
2.3 Perturbation theory .................................... 38
2.4 Schwinger-Dyson equations .............................. 40
2.5 Commutator terms ....................................... 40
2.6 Schwinger-Dyson equations (continued) .................. 41
2.7 Regularization ......................................... 45
3 Quantum anomalies from path integral ........................ 47
3.1 QED via path integral .................................. 47
3.2 Chiral Ward identity ................................... 48
3.3 Chiral anomaly ......................................... 51
3.4 Chiral anomaly (calculation) ........................... 55
3.5 Scale anomaly .......................................... 59
4 Instantons in quantum mechanics ............................. 65
4.1 Double-well potential .................................. 65
4.2 The instanton solution ................................. 68
4.3 Instanton contribution to path integral ................ 70
4.4 Symmetry restoration by instantons ..................... 75
4.5 Topological charge and θ-vacua ......................... 76
Bibliography to Part 1 ...................................... 79
Part 2: Lattice Gauge Theories ................................. 83
5 Observables in gauge theories ............................... 85
5.1 Gauge invariance ....................................... 85
5.2 Phase factors (definition) ............................. 88
5.3 Phase factors (properties) ............................. 93
5.4 Aharonov-Bohm effect ................................... 95
6 Gauge fields on a lattice ................................... 99
6.1 Sites, links, plaquettes and all that ................. 100
6.2 Lattice formulation ................................... 102
6.3 The Haar measure ...................................... 107
6.4 Wilson loops .......................................... 110
6.5 Strong-coupling expansion ............................. 113
6.6 Area law and confinement .............................. 117
6.7 Asymptotic scaling .................................... 119
7 Lattice methods ............................................ 123
7.1 Phase transitions ..................................... 124
7.2 Mean-field method ..................................... 128
7.3 Mean-field method (variational) ....................... 131
7.4 Lattice renormalization group ......................... 133
7.5 Monte Carlo method .................................... 136
7.6 Some Monte Carlo results .............................. 140
8 Fermions on a lattice ...................................... 143
8.1 Chiral fermions ....................................... 143
8.2 Fermion doubling ...................................... 145
8.3 Kogut-Susskind fermions ............................... 151
8.4 Wilson fermions ....................................... 152
8.5 Quark condensate ...................................... 156
9 Finite temperatures ........................................ 159
9.1 Feynman-Kac formula ................................... 160
9.2 QCD at finite temperature ............................. 166
9.3 Confinement criterion at finite temperature ........... 168
9.4 Deconfining transition ................................ 170
9.5 Restoration of chiral symmetry ........................ 175
Bibliography to Part 2 ..................................... 179
Part 3: 1/N Expansion ......................................... 185
10 O(N) vector models ......................................... 187
10.1 Four-Fermi theory ..................................... 188
10.2 Bubble graphs as the zeroth order in 1/N .............. 191
10.3 Functional methods for φ4 theory ...................... 200
10.4 Nonlinear sigma model ................................. 208
10.5 Large-N factorization in vector models ................ 211
11 Multicolor QCD.............................................. 213
11.1 Index or ribbon graphs ................................ 214
11.2 Planar and nonplanar graphs ........................... 218
11.3 Planar and nonplanar graphs (the boundaries) .......... 224
11.4 Topological expansion and quark loops ................. 230
11.5 't Hooft versus Veneziano limits ...................... 233
11.6 Large-N factorization ................................. 237
11.7 The master field ...................................... 243
11.8 1/N as semiclassical expansion ........................ 246
12 QCD in loop space .......................................... 249
12.1 Observables in terms of Wilson loops .................. 249
12.2 Schwinger-Dyson equations for Wilson loop ............. 255
12.3 Path and area derivatives ............................. 258
12.4 Loop equations ........................................ 263
12.5 Relation to planar diagrams ........................... 267
12.6 Loop-space Laplacian and regularization ............... 269
12.7 Survey of nonperturbative solutions ................... 274
12.8 Wilson loops in QCD2 .................................. 275
12.9 Gross-Witten transition in lattice QCD2 ............... 282
13 Matrix models .............................................. 287
13.1 Hermitian one-matrix model ............................ 288
13.2 Hermitian one-matrix model (solution at N = ∞) ........ 294
13.3 The loop equation ..................................... 297
13.4 Solution in 1/N ....................................... 300
13.5 Continuum limit ....................................... 303
13.6 Hermitian multimatrix models .......................... 311
Bibliography to Part 3 ..................................... 315
Part 4: Reduced Models ........................................ 323
14 Eguchi-Kawai model ......................................... 325
14.1 Reduction of the scalar field (lattice) ............... 325
14.2 Reduction of the scalar field (continuum) ............. 330
14.3 Reduction of the Yang-Mills field ..................... 332
14.4 The continuum Eguchi-Kawai model ...................... 336
14.5 Rd symmetry in perturbation theory .................... 340
14.6 Quenched Eguchi-Kawai model ........................... 342
15 Twisted reduced models ..................................... 351
15.1 Twisting prescription ................................. 351
15.2 Twisted reduced model for scalars ..................... 355
15.3 Twisted Eguchi-Kawai model ............................ 362
15.4 Twisting prescription in the continuum ................ 368
15.5 Continuum version of ТЕК .............................. 372
16 Noncommutative gauge theories .............................. 377
16.1 The noncommutative space .............................. 378
16.2 The Uθ(1) gauge theory ................................ 383
16.3 One-loop renormalization .............................. 386
16.4 Noncommutative quantum electrodynamics ................ 389
16.5 Wilson loops and observables .......................... 391
16.6 Compactification to tori .............................. 396
16.7 Morita equivalence .................................... 401
Bibliography to Part 4 ..................................... 405
Index ......................................................... 411
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