Introduction to Relativistic Quantum Field Theory

Quantum Mechanics......Page 11
Choice of the Picture......Page 13
Formal Solution of the Equations of Motion......Page 16
Example: The Free Particle......Page 18
The Feynman-Kac Formula......Page 20
The Path Integral for the Harmonic Oscillator......Page 23
Some Rules for Path Integrals......Page 25
The Schrödinger Wave Equation......Page 26
Potential Scattering......Page 28
Generating functional for Vacuum Expectation Values......Page 34
Bosons and Fermions, and what else?......Page 35
The Fock Space Representation of Quantum Mechanics......Page 37
Canonical Field Quantisation......Page 41
Space and Time in Special Relativity......Page 42
Tensors and Scalar Fields......Page 46
Noether's Theorem (Classical Part)......Page 51
Canonical Quantisation......Page 55
The Most Simple Interacting Field Theory: phi**4......Page 60
The LSZ Reduction Formula......Page 62
The Dyson-Wick Series......Page 64
Wick's Theorem......Page 66
The Feynman Diagrams......Page 68
Relativistic Quantum Fields......Page 75
Causal Massive Fields......Page 76
Massive Vector Fields......Page 77
Massive Spin-1/2 Fields......Page 78
Massless Vector Field......Page 82
Massless Helicity 1/2 Fields......Page 84
Quantisation of the spin-1/2 Dirac Field......Page 85
Discrete Symmetries and the CPT Theorem......Page 89
Charge Conjugation for Dirac spinors......Page 90
Time Reversal......Page 91
Lorentz Classification of Bilinear Forms......Page 94
The CPT Theorem......Page 96
Remark on Strictly Neutral Spin--1/2--Fermions......Page 97
Path Integral Formulation......Page 98
Example: The Free Scalar Field......Page 104
The Feynman Rules for phi**4 revisited......Page 106
LSZ Reduction......Page 108
The equivalence theorem......Page 110
Generating Functional for Connected Green's Functions......Page 111
Effective Action and Vertex Functions......Page 113
Noether's Theorem (Quantum Part)......Page 118
hbar-Expansion......Page 119
A Simple Interacting Field Theory with Fermions......Page 123
Infinities and how to cure them......Page 129
Overview over the renormalisation procedure......Page 133
Wick rotation......Page 135
Dimensional regularisation......Page 139
The Gamma-function......Page 140
Spherical coordinates in d dimensions......Page 147
Standard-integrals for Feynman integrals......Page 148
The 4-point vertex correction at 1-loop order......Page 150
Power counting......Page 152
The setting-sun diagram......Page 155
Weinberg's Theorem......Page 159
Proof of Weinberg's theorem......Page 162
Proof of the Lemma......Page 169
Application of Weinberg's Theorem to Feynman diagrams......Page 170
BPH-Renormalisation......Page 173
Some examples of the method......Page 174
The general BPH-formalism......Page 176
Zimmermann's forest formula......Page 178
Global linear symmetries and renormalisation......Page 181
Example: 1-loop renormalisation......Page 186
Homogeneous RGEs and modified BPHZ renormalisation......Page 189
The homogeneous RGE and dimensional regularisation......Page 192
Solutions to the homogeneous RGE......Page 194
Asymptotic behaviour of vertex functions......Page 195
The Gell-Mann-Low equation......Page 196
The Callan-Symanzik equation......Page 197
Gauge Theory......Page 203
Matter Fields interacting with Photons......Page 209
Canonical Path Integral......Page 211
Invariant Cross Sections......Page 215
Compton Scattering......Page 219
Pair-annihilation......Page 222
The background field method for non-gauge theories......Page 224
Gauge theories and background fields......Page 225
Renormalisability of the effective action in background field gauge......Page 228
The principle of local gauge invariance......Page 233
Quantisation of nonabelian gauge field theories......Page 237
BRST-Invariance......Page 239
S-Matrix......Page 242
The symmetry properties in the background field gauge......Page 244
The BFG Feynman rules......Page 247
The Ward-Takahashi identities......Page 250
The Fundamental Lemma of Variational Calculus......Page 255
Functional Derivatives......Page 257
The Lorentz Group......Page 261
Representations of the Lorentz Group......Page 268
Representations of the Full Lorentz Group......Page 269
Unitary Representations of the Poincare Group......Page 272
The Massive States......Page 277
Massless Particles......Page 278
The Invariant Scalar Product......Page 280
Amplitudes for various free fields......Page 283
Laurent expansion of the Gamma-Function......Page 284
Bibliography......Page 285