Structural aspects of quantum field theory and noncommutative geometry - Grensing, Gerhard (World Scientific, 2013)
Includes bibliographical references and index
1. Relativistic free fields: bosons. 1.1. Maxwell's equations in relativistic notation. 1.2. Klein-Gordon equation. 1.3. Group velocity and special relativity. 1.4. Nonrelativistic limit and the Schrödinger equation. 1.5. Classical interpretation of fields. 1.6. Normal coordinates. 1.7. Quantized harmonic oscillator. 1.8. Quantization of the Klein-Gordon field -- 2. Lagrange formalism for fields. 2.1. Functionals. 2.2. Euler-Lagrange equations for fields. 2.3. Variational principle. 2.4. Appendix: improper bases -- 3. Relativistic invariance. 3.1. Minkowski space and its symmetry group. 3.2. Transformation law of fields. 3.3. Appendix: Lie groups and Lie algebras -- 4. Special relativity. 4.1. Inertial frames and causality. 4.2. Lengths and time intervals. 4.3. Addition theorem for velocities. 4.4. Rotating frames. 4.5. Accelerated inertial frames. 4.6. Appendix: Product integral --
- 5. Relativistic free fields: fermions. 5.1. Dirac's equation. 5.2. Relativistic invariance of Dirac's equation. 5.3. Variational principle for the Dirac equation. 5.4. On the origin of gauge invariance. 5.5. Nonrelativistic limit. 5.6. 'Classical' interpretation of fermions. 5.7. Clifford algebras and spin groups -- 6. Relativistic free fields and spin. 6.1. Scalar field. 6.2. Dirac field. 6.3. Maxwell field. 6.4. Spin. 6.5. Transformation law of fields and induced representations -- 7. Neutral fermions. 7.1. Charge conjugation. 7.2. Majorana spinors. 7.3. Neutrinos -- 8. Symmetries and conservation laws -- 9. Differential and integral calculus for anticommuting variables. 9.1. Real Grassmann variables. 9.2. Fourier transformation. 9.3. Complex Grassmann variables. 9.4. Appendix: Pfaffians -- 10. Dynamical principles: internal symmetries. 10.1. Internal gauge theories. 10.2. Yang-Mills theory. 10.3. Gauge theories and elementary particle physics --
- 11. Dynamical principles: external symmetries. 11.1 Introduction. 11.2 Dynamics of a relativistic point particle in a gravitational field. 11.3. Differential geometry: a first course. 11.4. Einstein's theory of gravity. 11.5. Differential geometry: a second course. 11.6. Differential geometry: a third course. 11.7. Accelerated observers and inertial systems. 11.8. Gravity as a gauge theory of the Poincaré group -- 12. Supergravity. 12.1. Super Poincaré group. 12.2. Supersymmetry and differential geometry: Cartan connexions. 12.3. Rarita-Schwinger fermions. 12.4. Supergravity as a gauge theory of the super Poincaré group. 12.5. Summary. 12.6. Appendix: Majorana spinors in higher dimensions
13. Cosmology. 13.1. Gauss' normal coordinates. 13.2. Symmetric spaces. 13.3. Realization of maximally symmetric spaces. 13.4. Robertson-Walker metric. 13.5. General relativistic hydrodynamics. 13.6. Friedmann equations. 13.7. Models of the universe. 13.8. Present status of the universe. 13.9. Observational astronomy and cosmological parameters. 13.10. Cosmological constant problem. 13.11. Quintessence. 13.12. Appendix: Geometric optics in the presence of gravity. 13.13. Appendix: Local scale invariance and Weyl geometry -- 14. Quantization of free fields. 14.1. Scalar field. 14.2. Dirac field. 14.3. Maxwell field -- 15. Quantum mechanical perturbation theory. 15.1. Interaction picture. 15.2. Time independent perturbation theory. 15.3. Formal theory of scattering. 15.4. In and out picture. 15.5. Gell-Mann & Low formula. 15.6. Transcription to quantum field theory. 15.7. Reduction formulae --