Classical field theory

Cover......Page 1
Front Matter......Page 3
Classical Field Theory......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 19
Acknowledgments......Page 20
Introduction......Page 21
Part I: GENERAL PROPERTIES OF FIELDS;SCALARS AND GAUGE FIELDS......Page 23
1 Short Review of ClassicalMechanics......Page 25
2 Symmetries, Groups, and Lie algebras;Representations......Page 36
3 Examples: The Rotation Group and SU(2)......Page 46
4 Review of Special Relativity: Lorentz Tensors......Page 58
5 Lagrangeans and the Notion of Field;Electromagnetism as a Field Theory......Page 68
6 Scalar Field Theory, Origins, and Applications......Page 76
7 Nonrelativistic Examples:WaterWavesand Surface Growth......Page 87
8 Classical Integrability: Continuum Limitof Discrete, Lattice, and Spin Systems......Page 96
9 Poisson Brackets for Field Theory and Equationsof Motion: Applications......Page 103
10 Classical Perturbation Theory and FormalSolutions to the Equations of Motion......Page 111
11 Representations of the Lorentz Group......Page 119
12 Statistics, Symmetry, and theSpin-Statistics Theorem......Page 130
13 Electromagnetism and the Maxwell Equation;Abelian Vector Fields; Proca Field......Page 140
14 The Energy-Momentum Tensor......Page 149
15 Motion of Charged Particles and ElectromagneticWaves; Maxwell Duality......Page 159
16 The Hopfion Solution and the Hopf Map......Page 172
17 Complex Scalar Field and Electric Current:Gauging a Global Symmetry......Page 180
18 The Noether Theorem and Applications......Page 187
19 Nonrelativistic and Relativistic Fluid Dynamics:Fluid Vortices and Knots......Page 194
Part II: SOLITONS AND TOPOLOGY;NON-ABELIAN THEORY......Page 205
20 Kink Solutions in φ4 and Sine-Gordon, DomainWalls and Topology......Page 207
21 The Skyrmion Scalar Field Solution and Topology......Page 217
22 Field Theory Solitons for CondensedMatter:The XY and Rotor Model, Spins,Superconductivity, and the KT Transition......Page 225
23 Radiation of a Classical Scalar Field:The Heisenberg Model......Page 231
24 Derrick’s Theorem, Bogomolnyi Bound, theAbelian-Higgs System, and Symmetry Breaking......Page 239
25 The Nielsen-Olesen Vortex, Topologyand Applications......Page 248
26 Non-Abelian Gauge Theory and the Yang–MillsEquation......Page 258
27 The Dirac Monopole and Dirac Quantization......Page 266
28 The ’t Hooft–Polyakov Monopole Solutionand Topology......Page 278
29 The BPST-’t Hooft InstantonSolution and Topology......Page 289
30 General Topology and Reduction on an Ansatz......Page 300
31 Other Soliton Types. Nontopological Solitons:Q-Balls; Unstable Solitons: Sphalerons......Page 307
32 Moduli Space; Soliton Scattering in Moduli SpaceApproximation; Collective Coordinates......Page 316
Part III: OTHER SPINS OR STATISTICS;GENERAL RELATIVITY......Page 329
33 Chern–Simons Terms: Emergent Gauge Fields, theQuantum Hall Effect (Integer and Fractional),Anyonic statistics......Page 331
34 Chern–Simons and Self-Duality in OddDimensions, Its Duality to TopologicallyMassiveTheory and Dualities in General......Page 341
35 Particle–Vortex Duality in Three Dimensions,Particle–String Duality in Four Dimensions, andp-Form Fields in Four Dimensions......Page 352
36 Fermions and Dirac Spinors......Page 360
37 The Dirac Equation and Its Solutions......Page 369
38 General Relativity: Metric and GeneralCoordinate Invariance......Page 378
39 The Einstein Action and the Einstein Equation......Page 389
40 Perturbative Gravity: Fierz-Pauli Action,de Donder Gauge and Other Gauges,GravitationalWaves......Page 400
41 Nonperturbative Gravity: The VacuumSchwarzschild Solution......Page 411
42 Deflection of Light by the Sun and Comparisonwith General Relativity......Page 418
43 Fully Linear Gravity: Parallel Plane (pp)Wavesand Gravitational ShockWave Solutions......Page 427
44 Fully Linear Gravity: Parallel Plane (pp)Wavesand Gravitational ShockWave Solutions......Page 440
45 Time-Dependent Gravity:The Friedmann-Lemaître-Robertson-Walker(FLRW) Cosmological Solution......Page 453
46 Vielbein-Spin Connection Formulation ofGeneral Relativity and Gravitational Instantons......Page 461
References......Page 475
Index......Page 478