Introduction
1 Lorentz and Poincare Invariance 1
1.1 Lorentz Invariance 1
1.2 Structure of the Lorentz Group 7
1.3 Poincare Invariance 10
1.4 Maxwell Equations 10
1.5 Notations and Conventions 12
2 Relativistic Quantum Mechanics of Free Particles 15
2.1 Hilbert Space 15
2.2 Matrices in Hilbert Space 22
2.3 Fields 25
2.4 Structure of Hilbert Space 29
3 Interacting Fields 32
3.1 Physical System 32
3.2 Hilbert Space 33
3.3 Magnitude of Hilbert Space 34
3.4 U-matrix, S-matrix 35
3.5 Interpolating Fields 39
3.6 Feynman Rules 46
3.7 Feynman Propagator 54
3.8 Scattering Cross Section 55
3.9 Lifetime 60
3.10 Numerical Evaluation 62
3.11 Schrodinger Equation, Bound States 62
4 Particles with Spin 68
Representations of the Lorentz Group 68
The Dirac Eauation 76
4.3 Fermion Fields 79
4.4 The E.M. Field 85
4.5 Quantum Electrodynamics 87
4.6 Charged Vector Boson Fields 91
4.7 Electron-Proton Scattering. The Rutherford Formula 92
5 Explorations 98
5.1 Scattering Cross Section for e+e~ —> ^ pT 98
5.2 Pion Decay. Two Body Phase Space. Cabibbo Angle 101
5.3 Vector Boson Decay 105
5.4 Muon Decay. Fiertz Transformation 108
5.5 Hyperon Leptonic Decay 117
5.6 Pion Decay and PCAC 124
5.7 Neutral Pion Decay and PCAC 131
6 Renormalization 137
6.1 Introduction 137
6.2 Loop Integrals 137
6.3 Self Energy 145
6.4 Power Counting 148
6.5 Quantum Electrodynamics 151
6.6 Renormalizable Theories 153
6.7 Radiative Corrections: Lamb Shift 154
6.8 Radiative Corrections: Top Correction to p-Parameter 157
6.9 Neutral Pion Decay and the Anomaly 162
7 Massive and Massless Vector Fields 169
7.1 Subsidiary Condition Massive Vector Fields 169
7.2 Subsidiary Condition Massless Vector Fields 171
7.3 Photon Helicities 173
7.4 Propagator and Polarization Vectors of Massive
Vector Particles 174
7.5 Photon Propagator 177
7.6 Left Handed Photons 180
8 Unitarity 183
8.1 [/-matrix 183
8.2 Largest Time Equation 185
8.3 Cutting Equations 187
8.4 Unitarity and Cutting Equation 191
8.5 Unitarity: General Case
8.6 Kallen-Lehmann Representation, Dispersion
Relation 200
8.7 Momenta in Propagators 204
9 Quantum Electrodynamics: Finally 207
9.1 Unitarity 207
9.2 Ward Identities 208
Appendix A Complex Spaces, Matrices, CBH
Equation 213
A.I Basics 213
A.2 Differentiation of Matrices 217
A.3 Functions of Matrices 218
A.4 The CBH Equation 220
Appendix B Traces 224
B.I General 224
B.2 Multi-Dimensional 7-Matrices 230
B.3 Frequently Used Equations 231
Appendix C Dimensional Regularization 233
Appendix D Summary. Combinatorial Factors 243
D.I Summary 243
D.2 External Lines, Spin Sums, Propagators 245
D.3 Combinatorial Factors 247
Appendix E Standard Model 249
E.I Lagrangian 249
E.2 Feynman Rules 258
Appendix F Metric and Conventions 273
F.I General Considerations 273
F.2 Translation Examples 279
F.3 Translation Dictionary 281
Index 282
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