1. Results in Relativistic Quantum Mechanics ................... 1
1.1. Conventions ........................................... 1
1.2. Spin-zero particle .................................... 1
1.3. Dirac equation ........................................ 3
2. The Construction of Fields .................................. 7
2.1. The correspondence of particles and fields ............ 7
2.2. Spin-zero bosons ...................................... 8
2.3. Lagrangian and Hamiltonian ........................... 11
2.4. Functional derivatives ............................... 13
2.5. The field operator for fermions ...................... 14
3. Canonical Quantization ..................................... 17
3.1. Lagrangian, phase space, and Poisson brackets ........ 17
3.2. Rules of quantization ................................ 23
3.3. Quantization of a free scalar field .................. 25
3.4. Quantization of the Dirac field ...................... 28
3.5. Symmetries and conservation laws ..................... 32
3.6. The energy-momentum tensor ........................... 34
3.7. The electromagnetic field ............................ 36
3.8. Energy-momentum and general relativity ............... 37
3.9. Light-cone quantization of a scalar field ............ 38
3.10. Conformal invariance of Maxwell equations ............ 39
4. Commutators and Propagators ................................ 43
4.1. Scalar field propagators ............................. 43
4.2. Propagator for fermions .............................. 50
4.3. Grassman variables and fermions ...................... 51
5. Interactions and the S-matrix .............................. 55
5.1. A general formula for the S-matrix ................... 55
5.2. Wick's theorem ....................................... 61
5.3. Perturbative expansion of the 5-matrix ............... 62
5.4. Decay rates and cross sections ....................... 67
5.5. Generalization to other fields ....................... 69
5.6. Operator formula for the N-point functions ........... 72
6. The Electromagnetic Field .................................. 77
6.1. Quantization and photons ............................. 77
6.2. Interaction with charged particles ................... 81
6.3. Quantum electrodynamics (QED) ........................ 83
7. Examples of Scattering Processes ........................... 85
7.1. Photon-scalar charged particle scattering ............ 85
7.2. Electron scattering in an external Coulomb field ..... 87
7.3. Slow neutron scattering from a medium ................ 89
7.4. Compton scattering ................................... 92
7.5. Decay of the π0 meson ................................ 95
7.6. Cerenkov radiation ................................... 97
7.7. Decay of the p-meson ................................. 99
8. Functional Integral Representations ....................... 103
8.1. Functional integration for bosonic fields ........... 103
8.2. Green's functions as functional integrals ........... 105
8.3. Fermionic functional integral ....................... 108
8.4. The 5-matrix functional ............................. 111
8.5. Euclidean integral and QED .......................... 112
8.6. Nonlinear sigma models .............................. 114
8.7. The connected Green's functions ..................... 119
8.8. The quantum effective action ........................ 122
8.9. The S-matrix in terms of Г .......................... 126
8.10. The loop expansion ................................. 127
9. Renormalization ........................................... 133
9.1. The general procedure of renormalization ............ 133
9.2. One-loop renormalization ............................ 135
9.3. The renormalized effective potential ................ 144
9.4. Power-counting rules ................................ 145
9.5. One-loop renormalization of QED ..................... 147
9.6. Renormalization to higher orders .................... 157
9.7. Counterterms and renormalizability .................. 162
9.8. RG equation for the scalar field .................... 169
9.9. Solution to the RG equation and critical behavior ... 173
10. Gauge Theories ............................................ 179
10.1. The gauge principle ................................. 179
10.2. Parallel transport .................................. 183
10.3. Charges and gauge transformations ................... 185
10.4. Functional quantization of gauge theories ........... 188
10.5. Examples ............................................ 194
10.6. BRST symmetry and physical states ................... 195
10.7. Ward-Takahashi identities for Q-symmetry ............ 200
10.8. Renormalization of nonabelian theories .............. 203
10.9. The fermionic action and QED again .................. 206
10.10.The propagator and the effective charge ............. 206
11. Symmetry .................................................. 219
11.1. Realizations of symmetry ............................ 219
11.2. Ward-Takahashi identities ........................... 221
11.3. Ward-Takahashi identities for electrodynamics ....... 223
11.4. Discrete symmetries ................................. 226
11.5. Low-energy theorem for Compton scattering ........... 232
12. Spontaneous symmetry breaking ............................. 237
12.1. Continuous global symmetry .......................... 237
12.2. Orthogonality of different ground states ............ 242
12.3. Goldstone's theorem ................................. 244
12.4. Coset manifolds ..................................... 247
12.5. Nonlinear sigma models .............................. 249
12.6. The dynamics of Goldstone bosons .................... 249
12.7. Summary of results .................................. 253
12.8. Spin waves .......................................... 254
12.9. Chiral symmetry breaking in QCD ..................... 255
12.10.The effective action ................................ 258
12.11.Effective Lagrangians, unitarity of the S-matrix .... 263
12.12.Gauge symmetry and the Higgs mechanism .............. 266
12.13.The standard model .................................. 270
13. Anomalies I ............................................... 281
13.1. Introduction ........................................ 281
13.2. Computation of anomalies ............................ 282
13.3. Anomaly structure: why it cannot be removed ......... 289
13.4. Anomalies in the standard model ..................... 290
13.5. The Lagrangian for π0 decay ......................... 294
13.6. The axial U{1) problem .............................. 295
14. Elements of differential geometry ......................... 299
14.1. Manifolds, vector fields, and forms ................. 299
14.2. Geometrical structures on manifolds and gravity ..... 310
14.2.1. Riemannian structures and gravity ........... 310
14.2.2. Complex manifolds ........................... 313
14.3. Cohomology groups ................................... 315
14.4. Homotopy ............................................ 319
14.5. Gauge fields ........................................ 324
14.5.1 Electrodynamics .............................. 324
14.5.2. The Dirac monopole: A first look ............ 326
14.5.3. Nonabelian gauge fields ..................... 327
14.6. Fiber bundles ....................................... 329
14.7. Applications of the idea of fiber bundles ........... 333
14.7.1. Scalar fields around a magnetic monopole .... 333
14.7.2. Gribov ambiguity ........................... 334
14.8. Characteristic classes .............................. 336
15. Path Integrals ............................................ 341
15.1. The evolution kernel as a path integral ............. 341
15.2. The Schrodinger equation ............................ 344
15.3. Generalization to fields ............................ 345
15.4. Interpretation of the path integral ................. 350
15.5. Nontrivial fundamental group for С .................. 351
15.6. The case of H2(C) ≠ 0 ............................... 353
16. Gauge theory: configuration space ......................... 359
16.1. The configuration space ............................. 359
16.2. The path integral in QCD ............................ 364
16.3. Instantons .......................................... 366
16.4. Fermions and index theorem .......................... 369
16.5. Baryon number violation in the standard model ....... 373
17. Anomalies II .............................................. 377
17.1. Anomalies and the functional integral ............... 377
17.2. Anomalies and the index theorem ..................... 379
17.3. The mixed anomaly in the standard model ............. 383
17.4. Effective action for flavor anomalies of QCD ........ 384
17.5. The global or nonperturbative anomaly ............... 386
17.6. The Wess-Zumino-Witten (WZW) action ................. 390
17.7. The Dirac determinant in two dimensions ............. 392
18. Finite temperature and density ............................ 399
18.1. Density matrix and ensemble averages ................ 399
18.2. Scalar field theory ................................. 402
18.3. Fermions at finite temperature and density .......... 404
18.4. A condition on thermal averages ..................... 405
18.5. Radiation from a heated source ...................... 406
18.6. Screening of gauge fields: Abelian case ............. 409
18.7. Screening of gauge fields: Nonabelian case .......... 415
18.8. Retarded and time-ordered functions ................. 419
18.9. Physical significance of Im ΠRμ ..................... 422
18.10.Nonequilibrium phenomena ............................ 424
18.11.The imaginary time formalism ........................ 430
18.12.Symmetry restoration at high temperatures ........... 435
18.13.Symmetry restoration in the standard model .......... 439
19. Gauge theory: Nonperturbative questions ................... 445
19.1. Confinement and dual superconductivity .............. 445
19.1.1. The general picture of confinement .......... 445
19.1.2. The area law for the Wilson loop ............ 447
19.1.3. Topological vortices ........................ 449
19.1.4. The nonabelian dual superconductivity ....... 454
19.2.'t Hooft-Polyakov magnetic monopoles ................. 457
19.3. The 1/TV-expansion .................................. 462
19.4. Mesons and baryons in the 1/N expansion ............. 465
19.4.1. Chiral symmetry breaking and mesons ......... 466
19.4.2. Baryons ..................................... 468
19.4.3. Baryon number of the skyrmion ............... 470
19.4.4. Spin and flavor for skyrmions ............... 472
19.5. Lattice gauge theory ................................ 475
19.5.1. The reason for a lattice formulation ........ 475
19.5.2. Plaquettes and the Wilson action ............ 476
19.5.3. The fermion doubling problem ................ 479
20. Elements of Geometric Quantization ........................ 485
20.1. General structures .................................. 485
20.2. Classical dynamics .................................. 491
20.3. Geometric quantization .............................. 492
20.4. Topological features of quantization ................ 496
20.5. A brief summary of quantization ..................... 499
20.6. Examples ............................................ 500
20.6.1. Coherent states ............................. 500
20.6.2. Quantizing the two-sphere ................... 501
20.6.3. Compact Kahler spaces of the G/H-type ....... 506
20.6.4. Charged particle in a monopole field ........ 508
20.6.5. Anyons or particles of fractional spin ...... 510
20.6.6. Field quantization, equal-time, and
light-cone .................................. 513
20.6.7. The Chern-Simons theory in 2+1 dimensions ... 515
20.6.8. #-vacua in a nonabelian gauge theory ........ 522
20.6.9. Current algebra for the
Wess-Zumino-Witten (WZW) model .............. 525
Appendix:Relativistic Invariance .............................. 533
A-l Poincare algebra ........................................ 533
A-2 Unitary representations of the Poincare algebra .......... 537
A-3 Massive particles ........................................ 538
A-4 Wave functions for spin-zero particles ................... 540
A-5 Wave functions for spin-| particles ...................... 542
A-6 Spin-1 particles ......................................... 543
A-7 Massless particles ....................................... 544
A-8 Position operators ....................................... 545
A-9 Isometries, anyons ....................................... 545
General References ............................................ 549
Index ......................................................... 551
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