Classical Field Theory

Preface; Introduction; 1. Short review of classical mechanics; 2. Symmetries, groups and Lie algebras. Representations; 3. Examples: the rotation group and SU(2); 4. Review of special relativity. Lorentz tensors; 5. Lagrangeans and the notion of field; electromagnetism as a field theory; 6. Scalar field theory, origins and applications; 7. Nonrelativistic examples; water waves, surface growth; 8. Classical integrability. Continuum limit of discrete, lattice and spin systems; 9. Poisson brackets for field theory and equations of motion. Applications; 10. Classical perturbation theory and formal solutions to the equations of motion; 11. Representations of the Lorentz group; 12. Statistics, symmetry, and the spin-statistics theorem; 13. Electromagnetism and the Maxwell equation; Abelian vector fields; Proca field; 14. The energy-momentum tensor; 15. Motion of charged particles and electromagnetic waves; Maxwell duality; 16. The Hopfion solution and the Hopf map; 17. Complex scalar field and electric current. Gauging a global symmetry; 18. The Noether theorem and applications; 19. Nonrelativistic and relativistic fluid dynamics. Fluid vortices and knots; 20. Kink solutions in o4 and sine-Gordon, domain walls and topology; 21. The Skyrmion scalar field solution and topology; 22. Field theory solitons for condensed matter: the XY and rotor model, spins, superconductivity and the KT transition; 23. Radiation of a classical scalar field. The Heisenberg model; 24. Derrick's theorem, Bogomolnyi bound, the Abelian-Higgs system and symmetry breaking; 25. The Nielsen-Olesen vortex, topology and applications; 26. Nonabelian gauge theory and the Yang-Mills equation; 27. The Dirac monopole and Dirac quantization; 28. The 't Hooft-Polyakov monopole solution and topology; 29. The BPST-'t Hooft instanton solution and topology; 30. General topology and reduction on an ansatz; 31. Other soliton types. Nontopological solitons: Q-balls; unstable solitons: sphalerons; 32. Moduli space; soliton scattering in moduli space approximation; collective coordinates; 33. Chern-Simons terms: emergent gauge fields, the Quantum Hall Effect (integer and fractional), anyonic statistics; 34. Chern-Simons and self-duality in odd dimensions, its duality to topologically massive theory and dualities in general; 35. Particle-vortex duality in 3 dimensions, particle-string duality in 4 dimensions, and p-form fields in 4 dimensions; 36. Fermions and Dirac spinors; 37. The Dirac equation at its solutions; 38. General relativity: metric and general coordinate invariance; 39. The Einstein action and the Einstein equation; 40. Perturbative gravity: Fierz-Pauli action, de Donder gauge and other gauges, gravitational waves; 41. Nonperturbative gravity: the vacuum Schwarzschild solution; 42. Deflection of light by the Sun and comparison with general relativity; 43. Fully linear gravity: parallel plane (pp) waves and gravitational shockwave solutions; 44. Dimensional reduction: the domain wall, cosmic string and BTZ black hole solutions; 45. Time dependent gravity: the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological solution; 46. Vielbein-spin connection formulation of general relativity and gravitational instantons; References; Index.