Quantum Field Theory (eBook)

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Michael V. Sadovskii, Institute for Electrophysics, Russian Academy of Sciences, Russia.

Preface 5
1 Basics of elementary particles 13
1.1 Fundamental particles 13
1.1.1 Fermions 14
1.1.2 Vector bosons 15
1.2 Fundamental interactions 16
1.3 The Standard Model and perspectives 17
2 Lagrange formalism. Symmetries and gauge fields 21
2.1 Lagrange mechanics of a particle 21
2.2 Real scalar field. Lagrange equations 23
2.3 The Noether theorem 27
2.4 Complex scalar and electromagnetic fields 30
2.5 Yang-Mills fields 36
2.6 The geometry of gauge fields 42
2.7 A realistic example - chromodynamics 50
3 Canonical quantization, symmetries in quantum field theory 52
3.1 Photons 52
3.1.1 Quantization of the electromagnetic field 52
3.1.2 Remarks on gauge invariance and Bose statistics 57
3.1.3 Vacuum fluctuations and Casimir effect 60
3.2 Bosons 62
3.2.1 Scalar particles 62
3.2.2 Truly neutral particles 66
3.2.3 CPT-transformations 69
3.2.4 Vector bosons 73
3.3 Fermions 75
3.3.1 Three-dimensional spinors 75
3.3.2 Spinors of the Lorentz group 79
3.3.3 The Dirac equation 86
3.3.4 The algebra of Dirac’s matrices 91
3.3.5 Plane waves 93
3.3.6 Spin and statistics 95
3.3.7 C, P, T transformations for fermions 97
3.3.8 Bilinear forms 98
3.3.9 The neutrino 99
4 The Feynman theory of positron and elementary quantum electrodynamics 105
4.1 Nonrelativistic theory. Green’s functions 105
4.2 Relativistic theory 108
4.3 Momentum representation 112
4.4 The electron in an external electromagnetic field 115
4.5 The two-particle problem 122
5 Scattering matrix 127
5.1 Scattering amplitude 127
5.2 Kinematic invariants 130
5.3 Unitarity 133
6 Invariant perturbation theory 136
6.1 Schroedinger and Heisenberg representations 136
6.2 Interaction representation 137
6.3 S-matrix expansion 140
6.4 Feynman diagrams for electron scattering in quantum electrodynamics 147
6.5 Feynman diagrams for photon scattering 152
6.6 Electron propagator 154
6.7 The photon propagator 158
6.8 The Wick theorem and general diagram rules 161
7 Exact propagators and vertices 168
7.1 Field operators in the Heisenberg representation and interaction representation 168
7.2 The exact propagator of photons 170
7.3 The exact propagator of electrons 176
7.4 Vertex parts 180
7.5 Dyson equations 184
7.6 Ward identity 185
8 Some applications of quantum electrodynamics 187
8.1 Electron scattering by static charge: higher order corrections 187
8.2 The Lamb shift and the anomalous magnetic moment 192
8.3 Renormalization - how it works 197
8.4 “Running” the coupling constant 201
8.5 Annihilation of e+e~ into hadrons. Proof of the existence of quarks 203
8.6 The physical conditions for renormalization 204
8.7 The classification and elimination of divergences 208
8.8 The asymptotic behavior of a photon propagator at large momenta . 212
8.9 Relation between the “bare” and “true” charges 215
8.10 The renormalization group in QED 219
8.11 The asymptotic nature of a perturbation series 221
9 Path integrals and quantum mechanics 223
9.1 Quantum mechanics and path integrals 223
9.2 Perturbation theory 231
9.3 Functional derivatives 237
9.4 Some properties of functional integrals 238
10 Functional integrals: scalars and spinors 244
10.1 Generating the functional for scalar fields 244
10.2 Functional integration 249
10.3 Free particle Green’s functions 252
10.4 Generating the functional for interacting fields 259
10.5 f4 theory 262
10.6 The generating functional for connected diagrams 269
10.7 Self-energy and vertex functions 272
10.8 The theory of critical phenomena 276
10.9 Functional methods for fermions 289
10.10 Propagators and gauge conditions in QED 297
11 Functional integrals: gauge fields 299
11.1 Non-Abelian gauge fields and Faddeev-Popov quantization 299
11.2 Feynman diagrams for non-Abelian theory 305
12 The Weinberg-Salam model 314
12.1 Spontaneous symmetry-breaking and the Goldstone theorem 314
12.2 Gauge fields and the Higgs phenomenon 320
12.3 Yang-Mills fields and spontaneous symmetry-breaking 323
12.4 The Weinberg-Salam model 329
13 Renormalization 338
13.1 Divergences in f4 338
13.2 Dimensional regularization of f4-theory 342
13.3 Renormalization of f4-theory 347
13.4 The renormalization group 354
13.5 Asymptotic freedom of the Yang-Mills theory 360
13.6 “Running” coupling constants and the “grand unification” 367
14 Nonperturbative approaches 373
14.1 The lattice field theory 373
14.2 Effective potential and loop expansion 385
14.3 Instantons in quantum mechanics 390
14.4 Instantons and the unstable vacuum in field theory 401
14.5 The Lipatov asymptotics of a perturbation series 407
14.6 The end of the “zero-charge” story? 409
Bibliography 414
Index 418

lt;P>"In fact, it is an excellent textbook for a first encounter with the basic structure of QFT […]. […] for those who wish to explore the connections with condensed matter physics this book is ideal." Zentralblatt für Mathematik