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Classical field theory predicts how physical fields interact with matter, and is a logical precursor to quantum field theory. This introduction focuses purely on modern classical field theory, helping graduates and researchers build an understanding of classical field theory methods before embarking on future studies in quantum field theory. It describes various classical methods for fields with negligible quantum effects, for instance electromagnetism and gravitational fields. It focuses on solutions that take advantage of classical field theory methods as opposed to applications or geometric…mehr
Classical field theory predicts how physical fields interact with matter, and is a logical precursor to quantum field theory. This introduction focuses purely on modern classical field theory, helping graduates and researchers build an understanding of classical field theory methods before embarking on future studies in quantum field theory. It describes various classical methods for fields with negligible quantum effects, for instance electromagnetism and gravitational fields. It focuses on solutions that take advantage of classical field theory methods as opposed to applications or geometric properties. Other fields covered includes fermionic fields, scalar fields and Chern-Simons fields. Methods such as symmetries, global and local methods, Noether theorem and energy momentum tensor are also discussed, as well as important solutions of the classical equations, in particular soliton solutions.
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Autorenporträt
Horäiu N¿stase is a Researcher at the Institute for Theoretical Physics at the Universidade Estadual Paulista, São Paulo. To date, his career has spanned four continents. As an undergraduate he studied at the Universitatea din Bucure¿ti and Københavns Universitet. He later completed his Ph.D. at the State University of New York, Stony Brook, before moving to the Institute for Advanced Study, Princeton University, New Jersey, where his collaboration with David Berenstein and Juan Maldacena defined the pp-wave correspondence. He has also held research and teaching positions at Brown University, Rhode Island and the Tokyo Institute of Technology.
Inhaltsangabe
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 ø4 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.
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 ø4 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.
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