Theoretical and Mathematical Physics

0. Preliminaries.- 0.0 Historical Introduction.- 0.1 Minkowski Space and the Lorentz Group.- 0.2 Tensors in Minkowski Space.- 0.3 Some Topics of Scattering Theory.- 0.4 Problems.- 1. Relativistic Quantum Mechanics.- 1.1 Spinor Representations of the Lorentz Group.- 1.2 Invariant Field Equations.- 1.3 Algebraic Properties of the Dirac Equation.- 1.4 Discussion of the Free Dirac Equation.- 1.5 Gauge Invariance and Electromagnetic Fields.- 1.6 The Hydrogen Atom.- 1.7 Problems.- 2. Field Quantization.- 2.1 Second Quantization in Fock Space.- 2.2 Quantization of the Dirac Field.- 2.3 Discussion of the Commutation Functions.- 2.4 The Scattering Operator (S-Matrix) in Fock Space.- 2.5 Perturbation Theory.- 2.6 Electron Scattering.- 2.7 Pair Production.- 2.8 The Causal Phase of the S-Matrix.- 2.9 Non-Perturbative Construction of the Causal Phase.- 2.10 Vacuum Polarization.- 2.11 Quantization of the Radiation Field.- 2.12 Problems.- 3. Causal Perturbation Theory.- 3.1 The Method of Epstein and Glaser.- 3.2 Splitting of Causal Distributions.- 3.3 Application to QED.- 3.4 Electron Scattering (Moeller Scattering).- 3.5 Electron-Photon Scattering (Compton Scattering).- 3.6 Vacuum Polarization.- 3.7 Self-Energy.- 3.8 Vertex Function: Causal Distribution.- 3.9 Vertex Function: Retarded Distribution.- 3.10 Form Factors.- 3.11 Adiabatic Limit.- 3.12 Charged Particles in Perturbative QED.- 3.13 Charge Normalization.- 3.14 Problems.- 4. Properties of the S-Matrix.- 4.1 Vacuum Graphs.- 4.2 Operator Character of the S-Matrix.- 4.3 Normalizability of QED.- 4.4 Discrete Symmetries.- 4.5 Poincare Covariance.- 4.6 Gauge Invariance and Ward Identities.- 4.7 Unitarity.- 4.8 Renormalization Group.- 4.9 Interacting Fields and Operator Products.- 4.10 Field Equations.- 4.11 Problems.- 5. Other Electromagnetic Couplings.- 5.1 Scalar QED: Basic Properties.- 5.2 Scalar QED: Gauge Invariance.- 5.3 Axial Anomalies.- 5.4 (2+1)-Dimensional QED: Vacuum Polarization.- 5.5 (2+1)-Dimensional QED: Mass Generation.- 5.6 Problems.- 6. Epilogue: Non-Abelian Gauge Theories.- Appendices.- A: The Hydrogen Atom According to the Schrodinger Equation.- B: Regularly Varying Functions.- C: Spence Functions.- D: Grassmann Test Functions.- Bibliographical Notes.