Quantum field theory

Title ......Page 3
Copyright ......Page 4
Contents ......Page 5
Preface ......Page 11
Notes ......Page 13
1.1 Particles and Fields ......Page 15
1.2.1 The classical field ......Page 16
1.2.2 Harmonic oscillator ......Page 19
1.2.3 The quantized radiation field ......Page 21
1.3 The Electric Dipole Interaction ......Page 23
1.4.1 Classical electrodynamics ......Page 28
1.4.2 Quantum electrodynamics ......Page 30
1.4.3 Radiative transitions in atoms ......Page 31
1.4.4 Thomson scattering ......Page 32
1.5 Appendix: The Schrцdinger, Heisenberg and Interaction Pictures ......Page 34
Problems ......Page 36
2. Lagrangian Field Theory ......Page 39
2.1 Relativistic Notation ......Page 40
2.2 Classical Lagrangian Field Theory ......Page 41
2.3 Quantized Lagrangian Field Theory ......Page 44
2.4 Symmetries and Conservation Laws ......Page 45
Problems ......Page 51
3.1 The Real Klein-Gordon Field ......Page 53
3.2 The Complex Klein-Gordon Field ......Page 57
3.3 Covariant Commutation Relations ......Page 60
3.4 The Meson Propagator ......Page 62
Problems ......Page 67
4.1 The Number Representation for Fermions ......Page 69
4.2 The Dirac Equation ......Page 71
4.3 Second Quantization ......Page 75
4.3.1 The spin-statistics theorem ......Page 79
4.4 The Fermion Propagator ......Page 80
4.5 The Electromagnetic Interaction and Gauge Invariance ......Page 84
Problems ......Page 85
5.1 The Classical Fields ......Page 87
5.2 Covariant Quantization ......Page 91
5.3 The Photon Propagator ......Page 95
Problems ......Page 98
6. The S-Matrix Expansion ......Page 101
6.1 Natural Dimensions and Units ......Page 102
6.2 The S-Matrix Expansion ......Page 104
6.3 Wick's Theorem ......Page 108
7. Feynman Diagrams and Rules in QED ......Page 113
7.1 Feynman Diagrams in Configuration Space ......Page 114
7.2 Feynman Diagrams in Momentum Space ......Page 124
7.2.1 The first-order terms S(1) ......Page 126
7.2.2 Compton scattering ......Page 127
7.2.3 Electron-electron scattering ......Page 130
7.2.4 Closed loops ......Page 131
7.3 Feynman Rules for QED ......Page 132
7.4 Leptons ......Page 135
Problems ......Page 138
8. QED Processes in Lowest Order ......Page 141
8.1 The Cross-Section ......Page 142
8.2 Spin Sums ......Page 145
8.3 Photon Polarization Sums ......Page 147
8.4 Lepton Pair Production in (e+e-) Collisions ......Page 149
8.5 Bhabha Scattering ......Page 153
8.6 Compton Scattering ......Page 156
8.7 Scattering by an External Field ......Page 161
8.8 Bremsstrahlung ......Page 167
8.9 The Infrared Divergence ......Page 169
Problems ......Page 172
9. Radiative Corrections ......Page 175
9.1 The Second-Order Radiative Corrections of QED ......Page 176
9.2 The Photon Self-Energy ......Page 181
9.3 The Electron Self-Energy ......Page 186
9.4 External Line Renormalization ......Page 190
9.5 The Vertex Modification ......Page 192
9.6.1 The anomalous magnetic moments ......Page 197
9.6.2 The Lamb shift ......Page 201
9.7 The Infrared Divergence ......Page 205
9.8 High-Order Radiative Corrections ......Page 207
9.9 Renormalizability ......Page 212
Problems ......Page 214
10. Regularization ......Page 217
10.1.1 Some standard integrals ......Page 218
10.1.2 Feynman parameterization ......Page 219
10.2 Cut-Off Regularization: The Electron Mass Shift ......Page 220
10.3.1 Introduction ......Page 222
10.3.2 General results ......Page 224
10.4 Vacuum Polarization ......Page 225
10.5 The Anomalous Magnetic Moment ......Page 228
Problems ......Page 231
11. Gauge Theories ......Page 233
11.1 The Simplest Gauge Theory: QED ......Page 234
11.2.1 Colour and confinement ......Page 236
11.2.2 Global phase invariance and colour conservation ......Page 239
11.2.3 SU(3) gauge invariance ......Page 241
11.2.4 Quantum chromodynamics ......Page 243
11.3.1 Non-minimal interactions ......Page 244
11.3.2 Renormalizability ......Page 247
11.4 Appendix: Two Gauge Transformation Results ......Page 249
11.4.1 The transformation law (11.26b) ......Page 250
11.4.2 The SU(3) gauge invariance of Eq. (11.34) ......Page 251
Problems ......Page 252
12.1 Green Functions ......Page 255
12.2.1 The perturbation expansion ......Page 260
12.2.2 The vacuum amplitude ......Page 262
12.2.3 The photon propagator ......Page 263
12.2.4 Connected Green functions ......Page 266
12.3 Relation to S-Matrix Elements ......Page 268
12.3.1 Crossing ......Page 269
12.4 Functionals and Grassmann Fields ......Page 270
12.4.1 Functionals ......Page 271
12.4.2 Grassmann algebras and Grassmann fields ......Page 273
12.5 The Generating Functional ......Page 277
12.5.1 The free-field case ......Page 281
12.5.2 The perturbation expansion ......Page 284
Problems ......Page 286
13.1 Functional Integration ......Page 289
13.1.1 Classical fields ......Page 290
13.1.2 Grassmann generators ......Page 295
13.1.3 Grassmann fields ......Page 297
13.2 Path Integrals ......Page 299
13.2.1 The generating functional ......Page 300
13.2.2 Free and interacting fields ......Page 301
13.2.3 The free electromagnetic field ......Page 303
13.2.4 The free spinor fields ......Page 305
13.3.1 Wick's theorem ......Page 306
13.3.2 Interactions ......Page 308
13.4 Gauge Independent Quantization? ......Page 311
Problems ......Page 312
14.1 Gluon Fields ......Page 313
14.1.1 The generating functional ......Page 314
14.1.2 A mathematical analogy ......Page 315
14.1.3 The Faddeev-Popov Method ......Page 317
14.1.4 Gauge fixing and ghosts ......Page 318
14.1.5 The electromagnetic field revisited ......Page 320
14.2.1 The QCD Lagrangian ......Page 321
14.2.2 The generating functional ......Page 323
14.2.3 Free fields ......Page 324
14.3.1 Wick's theorem and propagators ......Page 326
14.3.3 The vertex factors ......Page 327
14.4 Feynman Rules for QCD ......Page 332
14.5 Renormalizability of QCD ......Page 335
Problems ......Page 337
15.1 Electron-Positron Annihilation ......Page 339
15.1.1 Two-jet events ......Page 340
15.1.2 Three-jet events ......Page 342
15.2 The Renormalization Scheme ......Page 344
15.2.1 The electron propagator ......Page 345
15.2.2 The photon propagator ......Page 347
15.2.3 Charge renormalization ......Page 349
15.3 The Renormalization Group ......Page 350
15.3.1 The renormalization group equations ......Page 351
15.3.2 Scale transformations ......Page 353
15.3.3 The running charge ......Page 355
15.4 The Strong Coupling Constant ......Page 357
15.4.1 Colour factors ......Page 358
15.4.2 Null diagrams ......Page 359
15.4.3 Renormalization of the coupling constant ......Page 360
15.4.4 The running coupling ......Page 365
15.5 Applications ......Page 366
15.6.1 The gluon self-energy graphs ......Page 371
15.6.2 The quark-gluon vertex corrections ......Page 374
Problems ......Page 376
16.1 Introduction ......Page 377
16.2 Leptonic Weak Interactions ......Page 379
16.3 The Free Vector Boson Field ......Page 383
16.4 The Feynman Rules for the IVB Theory ......Page 385
16.5 Decay Rates ......Page 386
16.6.1 Muon decay ......Page 387
16.6.2 Neutrino scattering ......Page 393
16.6.3 The leptonic decay of the W boson ......Page 394
16.7.1 Neutrino oscillations ......Page 395
16.7.2 Dirac or Majorana neutrinos? ......Page 397
16.8 Difficulties with the IVB Theory ......Page 399
Problems ......Page 401
17.1 QED Revisited ......Page 403
17.2 Global Phase Transformations and Conserved Weak Currents ......Page 405
17.3 The Gauge-Invariant Electroweak Interaction ......Page 409
17.4 Properties of the Gauge Bosons ......Page 413
17.5 Lepton and Gauge Boson Masses ......Page 415
18. Spontaneous Symmetry Breaking ......Page 417
18.1 The Goldstone Model ......Page 418
18.2 The Higgs Model ......Page 422
18.3 The Standard Electroweak Theory ......Page 426
19. The Standard Electroweak Theory ......Page 433
19.1 The Lagrangian Density in the Unitary Gauge ......Page 434
19.2 Feynman Rules ......Page 438
19.3 Elastic Neutrino-Electron Scattering ......Page 446
19.4 Electron-Positron Annihilation ......Page 449
19.5 The Higgs Boson ......Page 456
19.5.1 Higgs boson decays ......Page 458
19.5.2 Higgs boson searches ......Page 460
Problems ......Page 462
A.1 The Dirac Equation ......Page 465
A.3 Traces ......Page 467
A.4 Plane Wave Solutions ......Page 469
A.6 Helicity and Spin Projection Operators ......Page 470
A.7 Relativistic Properties ......Page 472
A.8 Particular Representations of the Gamma-Matrices ......Page 474
Problems ......Page 476
Appendix B: Feynman Rules and Formulae for Perturbation Theory ......Page 477
A - C ......Page 487
D - F ......Page 488
G - K ......Page 489
L - Q ......Page 490
R - S ......Page 491
T - Z ......Page 492