Gravitation, Gauge Theories and the Early Universe

I: Gravitation and Cosmology.- 1. Introduction to General Relativity.- 1. From Special Theory to General Theory.- 2. Einstein’s Thought Experiment.- 3. Geometry or Geometries?.- 4. Riemannian Geometry and Geodesics.- 5. Geometry and Gravitation.- 6. The Line Element.- 7. Summation Convention.- 8. Vectors and Tensors.- 9. Quotient Law.- 10. The Fundamental Tensor.- 11. Raising and Lowering the Suffixes (Indices).- 12. Length of a Vector.- 13. Addition of Vectors at a Point.- 14. Covariant Derivative of a Contravariant Vector.- 15. Covariant Derivative of a Covariant Vector.- 16. The Christoffel Symbols.- 17. Geodesics.- 18. The Curvature Tensor.- 19. Natural Coordinates at a Point.- 20. Symmetry Properties of the Curvature Tensor.- 21. Bianchi Identities and the Ricci Tensor.- 22. The Einstein Tensor and the Field Equations of Gravitation.- 23. Matter Tensor for a Perfect Fluid.- 24. Exercises.- 2. Introduction to Black Holes.- 1. Preamble.- 2. The Schwarzschild Black Hole.- 3. Properties of the Schwarzschild Black Hole.- 4. The Kerr Black Hole.- 5. The Black Hole and the Ergosphere.- 6. The Penrose Process.- 7. Charged Black Holes.- 8. Conclusion.- References.- 3. Black-Hole Thermodynamics and Hawking Radiation.- 4. Introduction to Relativistic Cosmology.- 1. Preamble.- 2. The Cosmic Spacetime.- 3. Cosmological Models.- 4. Dust Models.- 5. Radiation Models.- 6. Models with Nonzero Cosmological Constants.- 7. Observational Contacts.- 8. Conclusion.- References.- 5. Relics of the big Bang.- 1. The Early Universe.- 2. Thermodynamics of the Early Universe.- 3. Primordial Neutrinos.- 4. The Neutron/Proton Ratio.- 5. The Synthesis of Helium and Other Nuclei.- 6. The Microwave Background.- 7. Anisotropies of the Microwave Background.- 8. Cosmology and Particle Physics.- 9. Survival of Massive Particles.- 10. Problems of the Very Early Universe.- References.- 6. An Approach to Anisotropic Cosmologies.- 1. Motivation.- 2. Killing Vectors and Bianchi Types.- 3. Kinematics — Analysis of the Velocity Field.- 4. Perfect Fluid Solutions Classified According to Kinematic Properties.- 5. Some Anisotropic Cosmological Solutions.- 6. Problems.- 7. Topics in Spacetime Structure.- 1. Introduction.- 2. The Manifold Model.- 3. Spacetime Diffeomorphisms.- 4. Killing Vector Fields.- 5. Boundary Attachment and Conformal Campactification for Spacetimes.- References.- 8. Differential Forms and Einstein-Cartan Theory.- 1. Basic Definitions.- 2. Algebra and Calculus of Forms.- 3. Connection and Curvature Forms.- 4. Einstein-Cartan Theory — The Gauge Theory of Gravity.- 5. Gravitation in the Presence of Fermionic Matter.- References.- II: Introduction to Particle Physics and Gauge Field Theories.- 9. Introduction to Classical and Quantum Lagrangian Field Theory.- 1. Classical Lagrangian Field Theory.- 2. Canonical Quantization.- 3. Discrete Symmetries.- 4. Interacting Fields.- 5. Invariant Perturbation Theory.- 6. Primitive Divergences in QED.- 7. QED as a Renormalizable Theory.- 8. V-A as a Nonrenormalizable Theory.- 9. Dimensional Regularization.- Further Reading.- 10. Introduction to Particle Physics, Symmetries and Conservation Laws.- 1. Introduction.- 2. Charge Independence of Nuclear Forces — Isotopic Spin.- 3. Strange Particles.- 4. Nucleon Number Conservation.- 5. Lepton Number Conservation.- 6. Discrete Symmetries.- 7. ?5-Invariance and Weak Interactions.- 8. Strong Interactions: Quarks and Gluons.- 9. Need for Colour.- 10. Gauge Invariance.- Further Reading.- 11. Building up the Standard Gauge Model of High-Energy Physics.- 1. Introduction.- 2. U(1) Gauge Theory.- 3. Spontaneous Breakdown of Symmetry — Goldstone Model.- 4. Higgs Model.- 5. SU(2) Gauge Theory.- 6. Spontaneous Breakdown of SU(2) Symmetry.- 7. One More Model.- 8. General Case of Non-Abelian Symmetry Breakdown.- 9. SU(2) x U(1) Model.- 10. ‘Standard Model’ before Gauge Theory.- 11. Current Algebra and SU(2) x U(1) Charges of the Fermions.- 12. The Electroweak Gauge Theory.- 13. Consequences of the Electroweak Theory.- 14. Renormalizability.- 15. Spontaneous Symmetry Breaking and Phase Transitions.- 16. Deep Inelastic Scatterng, Asymptotic Freedom and Colour SU(3).- 17. The Renormalization Group Equation.- 18. Formal Derivation of the Renormalization Group Equation.- 19. Solution of the Renormalization Group Equation.- 20. Hydrodynamic Analogy.- 21. Fixed Points and Asymptotic Freedom.- 22. Asymptotic Freedom of QCD.- 23. Infrared Problem and Colour Confinement.- 24. Tests of QCD.- 25. The Standard Model of High Energy Physics.- 26. Beyond the Standard Model.- References.- 12. Introduction to Grand Unification Theories.- 1. Grand Unification — A Survey of Basic Ideas.- 2. Grand Unified Theory Based on G = SU(5).- 3. Spontaneous Symmetry Breaking.- 4. Predictions of Minimal SU(5).- 5. Baryon Asymmetry.- 6. Phase Transitions in the Early Universe.- 13. Topology and Homotopy.- 1. What is Topology?.- 2. Why the Recent Interest in Topology?.- 3. Homotopy Theory.- 4. Chern Classes.- References.- 14. Introduction to Compact Simple Lie Groups.- III: Quantum Effects in the Early Universe and Approaches to the Unification of Fundamental Forces.- 15. Quantum Field Theory in Curved Spacetime: Canonical Quantization.- 1. Quantum Field Theory in Curved Spacetime.- 2. Canonical Quantization of the Scalar Field in CST.- 3. The Conformal Vacuum.- 4. A Toy Model with Particle Creation.- 5. The Adiabatic Vacuum.- References.- 16. Zeta Function Regularisation and Effective Action in Curved Spacetime.- 1. The Riemann Zeta Function.- 2. Applications.- 3. Path Integral Formulation for QFT in CST.- 4. Conformal Anomalies.- 5. Phase Transition in a De Sitter Universe.- References.- 17. Inflationary Cosmology and Quantum Effects in the Early Universe.- 1. Quantum Field Theory in Curved Spacetime: A Short History.- 2. Problems in Standard Cosmology.- 3. Inflation.- 4. Free Lunch.- 5. The ‘New’ Model.- 6. Evolution of the Scalar Field.- 7. Linde’s Chaotic Inflation.- 8. Hawking’s Limits on Inflationary Models.- 9. Quantum Effects in the Early Universe.- 10. The Fundamental Problem.- 11. De Witt-Schwinger Expansion of Green’s Function.- 12. Renormalization.- 13. Other Methods.- 14. Example of Back Reaction.- 15. Applications.- References.- 18. Quantum Cosmology — The Story So Far.- 1. Introduction.- 2. Minisuperspace of Conformal Degree of Freedom.- 3. Quantized FRW Universes.- 4. Applications of Quantum Gravity.- 5. Critique, Comparison and Open Questions.- Appendix 1: Schrödinger Approach to Field Theory.- Appendix 2: The Wheeler-De Witt Equation.- Notes and References.- 19. The Photon, the Graviton and the Gravitino.- 1. The Photon.- 2. The Graviton.- 3. The Gravitino.- 4. The Rarita-Schwinger Lagrangian.- 20. The Vierbein, Vielbeins and Spinors in Higher Dimensions.- 1. The Vierbein.- 2. Vielbeins.- 3. Spinors in d-dimensions.- 21. Kaluza-Klein Theories.- 1. Kaluza-Klein Theories.- 2. Spontaneous Compactification and Isometry Groups.- 3. Harmonic Expansions, Chiral Fermions and All That.- References.- 22. Kaluza-Klein Cosmology.- 1. Introduction.- 2. Five-Dimensional Kaluza-Klein Theory.- 3. Remarks.- 4. Dimensional Reduction.- 5. Cosmology.- References.- 23. An Elementary Introduction to the Gauge Theory Approach to Gravity.- 1. Introduction.- 2. The Yang-Mills Construction.- 3. Gauging a Special Relativistic Matter Lagrangian.- 4. Kinematics of the Gravitational Variables.- 5. The Gravitational Action.- 6. Translational Gauge Potentials.- References.- 24. Graded Lie Algebras.- 1. Introduction.- 2. Examples of Graded Lie Algebras (GLAs).- 3. Maps of GLAs.- 4. Classification of GLAs.- References.- 25. Supersymmetry and Supergravity.- 1. Introduction.- 2. Coleman-Mandula Theorem and Supersymmetry Algebra.- 3. Representation of the Supersymmetry Algebra on One-Particle States.- 4. Representations of the Supersymmetry Algebra on Fields and Invariant Lagrangians.- 5. Spontaneous Breakdown of Supersymmetry.- 6. Pure N = 1 Supergravity in Four Dimensions.- 7. N = 1, D = 11 Supergravity.- 8. N = 1, D = 10 Supergravity.- 9. Concluding Remarks.- References.- 26. An Overview of Superstring Theory.- 1. Introduction.- 2. Duality.- 3. The Veneziano Formula.- 4. Free Relativistic String.- 5. Orthonormal Gauge.- 6. Quantization.- 7. Light Cone Quantization.- 8. Hamiltonian Formalism.- 9. Quantization.- 10. Lorentz Covariance.- 11. Spectrum.- 12. Closed Strings.- 13. Interacting Strings.- 14. Field Theory Limit.- 15. Superstrings.- 16. Problems and Prospects.- References.