A Modern Introduction to Quantum Field Theory

Cover......Page 1
Title Page......Page 5
Contents......Page 9
Preface......Page 13
Notation......Page 14
1.1 Overview......Page 19
1.2 Typical scales in high-energy physics......Page 22
Further reading......Page 29
Exercises......Page 30
2.1 Lie groups......Page 31
2.2 The Lorentz group......Page 34
2.3 The Lorentz algebra......Page 36
2.4 Tensor representations......Page 38
2.4.1 Decomposition of Lorentz tensors under SO(3)......Page 40
2.5.1 Spinors in non-relativistic quantum......Page 42
2.5.2 Spinors in the relativistic theory......Page 44
2.6.1 Scalar fields......Page 47
2.6.2 Weyl fields......Page 49
2.6.3 Dirac fields......Page 50
2.6.4 Majorana fields......Page 51
2.7 The Poincaré group......Page 52
2.7.1 Representation on fields......Page 53
2.7.2 Representation on one-particle states......Page 54
Summary of chapter......Page 58
Exercises......Page 59
3.1 The action principle......Page 61
3.2 Noether’s theorem......Page 64
3.2.1 The energy–momentum tensor......Page 67
3.3.1 Real scalar fields; Klein–Gordon equation......Page 69
3.3.2 Complex scalar field; U(1) charge......Page 71
3.4.1 The Weyl equation; helicity......Page 72
3.4.2 The Dirac equation......Page 74
3.4.3 Chiral symmetry......Page 80
3.4.4 Majorana mass......Page 81
3.5.1 Covariant form of the free Maxwell equations......Page 83
3.5.2 Gauge invariance; radiation and Lorentz gauges......Page 84
3.5.3 The energy–momentum tensor......Page 85
3.5.4 Minimal and non-minimal coupling to matter......Page 87
3.6 First quantization of relativistic wave equations......Page 91
The fine structure of the hydrogen atom......Page 92
Relativistic energy levels in a magnetic field......Page 97
Summary of chapter......Page 98
Exercises......Page 99
4.1.1 Real scalar fields. Fock space......Page 101
4.1.2 Complex scalar field; antiparticles......Page 104
4.2.1 Dirac field......Page 106
4.2.2 Massless Weyl field......Page 108
4.2.3 C, P, T......Page 109
4.3.1 Quantization in the radiation gauge......Page 114
4.3.2 Covariant quantization......Page 119
Summary of chapter......Page 123
Exercises......Page 124
5.1 The S-matrix......Page 127
5.2 The LSZ reduction formula......Page 129
5.3 Setting up the perturbative expansion......Page 134
5.4 The Feynman propagator......Page 138
5.5 Wick’s theorem and Feynman diagrams......Page 140
5.5.1 A few very explicit computations......Page 141
5.5.2 Loops and divergences......Page 146
5.5.3 Summary of Feynman rules for a scalar field......Page 149
5.5.4 Feynman rules for fermions and gauge bosons......Page 150
5.6 Renormalization......Page 153
5.7 Vacuum energy and the cosmological constant problem......Page 159
5.8 The modern point of view on renormalizability......Page 162
5.9 The running of coupling constants......Page 164
Summary of chapter......Page 170
Further reading......Page 171
Exercises......Page 172
6.1 Relativistic and non-relativistic normalizations......Page 173
6.2 Decay rates......Page 174
6.3 Cross-sections......Page 176
6.4 Two-body final states......Page 178
6.5 Resonances and the Breit–Wigner distribution......Page 181
6.6 Born approximation and non-relativistic scattering......Page 185
Three-body kinematics and phase space......Page 189
Inelastic scattering of non-relativistic electrons on atoms......Page 191
Summary of chapter......Page 195
Exercises......Page 196
7.1 The QED Lagrangian......Page 198
7.2 One-loop divergences......Page 201
e⁺e⁻ → γ → µ⁺µ⁻......Page 204
Electromagnetic form factors......Page 206
Exercises......Page 211
8.1 A four-fermion model......Page 213
8.2 Charged and neutral currents in the Standard Model......Page 215
µ⁻ → e⁻ ν̄_e ν_µ......Page 220
π⁺ → l⁺ ν_l......Page 223
Isospin and flavor SU(3)......Page 227
K⁰ → π⁻ l⁺ ν_l......Page 230
Summary of chapter......Page 234
Exercises......Page 235
9 Path integral quantization......Page 237
9.1 Path integral formulation of quantum mechanics......Page 238
9.2 Path integral quantization of scalar fields......Page 242
9.3 Perturbative evaluation of the path integral......Page 243
9.4 Euclidean formulation......Page 246
9.5 QFT and critical phenomena......Page 249
9.6 QFT at finite temperature......Page 256
Instantons and tunneling......Page 257
Summary of chapter......Page 259
Further reading......Page 260
10.1 Non-abelian gauge transformations......Page 261
10.2 Yang–Mills theory......Page 264
10.3 QCD......Page 266
10.4 Fields in the adjoint representation......Page 268
Further reading......Page 270
11.1 Degenerate vacua in QM and QFT......Page 271
11.2 SSB of global symmetries and Goldstone bosons......Page 274
11.3 Abelian gauge theories: SSB and superconductivity......Page 277
11.4 Non-abelian gauge theories:  the masses of W± and Z⁰......Page 280
Summary of chapter......Page 282
Further reading......Page 283
12.1 Chapter 1......Page 284
12.2 Chapter 2......Page 285
12.3 Chapter 3......Page 288
12.4 Chapter 4......Page 290
12.5 Chapter 5......Page 293
12.6 Chapter 6......Page 294
12.7 Chapter 7......Page 297
12.8 Chapter 8......Page 299
Bibliography......Page 303
Index......Page 305