Foundations of Classical Electrodynamics, Draft Version (Progress in Mathematical Physics)

FOUNDATIONS OF CLASSICAL ELECTRODYNAMICS......Page 1
Contents......Page 3
Description of the book......Page 11
Preface......Page 15
Five plus one axioms......Page 18
Topological approach......Page 20
Electromagnetic spacetime relation as fifth axiom......Page 21
A reminder: Electrodynamics in 3-dimensional Euclidean vector calculus......Page 23
On the literature......Page 26
References......Page 29
Part A: Mathematics: Some exterior calculus......Page 36
Why exterior differential forms?......Page 37
A.1.1 A real vector space and its dual......Page 43
A.1.2 Tensors of type [p q]......Page 47
A.1.3 A generalization of tensors: geometric quantities......Page 48
A.1.4 Almost complex structure......Page 50
A.1.5 Exterior p-forms......Page 51
A.1.6 Exterior multiplication......Page 53
A.1.7 Interior multiplication of a vector with a form......Page 56
A.1.8 Volume elements on a vector space, densities, orientation......Page 57
A.1.9 Levi–Civita symbols and generalized Kronecker deltas......Page 61
A.1.10 The space M^6 of two-forms in four dimensions......Page 65
A.1.11 Almost complex structure on M^6......Page 70
A.1.12 Computer algebra......Page 73
A.2 Exterior calculus......Page 87
A.2.1 Differentiable manifolds......Page 88
A.2.2 Vector fields......Page 92
A.2.3 One-form fields, differential p-forms......Page 93
A.2.4 Images of vectors and one-forms......Page 94
A.2.5 Volume forms and orientability......Page 97
A.2.6 Twisted forms......Page 98
A.2.7 Exterior derivative......Page 100
A.2.8 Frame and coframe......Page 103
A.2.9 Maps of manifolds: push-forward and pull-back......Page 105
A.2.10 Lie derivative......Page 107
A.2.11 Excalc, a Reduce package......Page 114
A.2.12 Closed and exact forms, de Rham cohomology groups......Page 118
A.3.1 Integration of 0-forms and orientability of a manifold......Page 123
A.3.2 Integration of n-forms......Page 124
A.3.3 Integration of p-forms with 0 < p < n......Page 126
A.3.4 Stokes's theorem......Page 131
A.3.5 De Rham's theorems......Page 134
References......Page 141
Part B: Axioms of classical electrodynamics......Page 146
B.1.1 Counting charges. Absolute and physical dimension......Page 149
B.1.2 Spacetime and the first axiom......Page 155
B.1.3 Electromagnetic excitation......Page 157
B.1.4 Time-space decomposition of the inhomogeneous Maxwell equation......Page 158
B.2.1 Electromagnetic field strength......Page 163
B.2.2 Second axiom relating mechanics and electrodynamics......Page 165
B.2.3 The first three invariants of the electromagnetic field......Page 167
B.3.1 Third axiom......Page 171
B.3.2 Electromagnetic potential......Page 176
B.3.3 Abelian Chern–Simons and Kiehn 3-forms......Page 177
B.3.4 Measuring the excitation......Page 179
B.4.1 Integral version and Maxwell's equations......Page 187
B.4.2 Jump conditions for electromagnetic excitation and field strength......Page 193
B.4.3 Arbitrary local non-inertial frame: Maxwell's equations in components......Page 194
B.4.4 Electrodynamics in flatland: 2-dimensional electron gas and quantum Hall effect......Page 196
B.5.1 Fourth axiom: localization of energy-momentum......Page 209
B.5.2 Properties of the energy-momentum current, electric-magnetic reciprocity......Page 212
B.5.3 Time-space decomposition of the energy-momentum current......Page 222
B.5.4 Action......Page 224
B.5.5 Coupling of the energy-momentum current to the coframe......Page 228
B.5.6 Maxwell's equations and the energy–momentum current in Excalc......Page 232
References......Page 237
Part C: More mathematics......Page 242
C.1 Linear connection......Page 243
C.1.1 Covariant differentiation of tensor fields......Page 244
C.1.2 Linear connection 1-forms......Page 246
C.1.3 Covariant differentiation of a general geometric quantity......Page 249
C.1.4 Parallel transport......Page 250
C.1.5 Torsion and curvature......Page 251
C.1.6 Cartan's geometric interpretation of torsion and curvature......Page 256
C.1.7 Covariant exterior derivative......Page 258
C.1.8 The p-forms o(a), conn1(a,b), torsion2(a), curv2(a,b)......Page 260
C.2 Metric......Page 263
C.2.1 Metric vector spaces......Page 264
C.2.2 Orthonormal, half-null, and null frames, the coframe statement......Page 266
C.2.3 Metric volume 4-form......Page 270
C.2.4 Duality operator for 2-forms as a symmetric almost complex structure on M^6......Page 272
C.2.5 From the duality operator to a triplet of complex 2-forms......Page 274
C.2.6 From the triplet of complex 2-forms to a duality operator......Page 276
C.2.7 From a triplet of complex 2-forms to the metric: Schönberg–Urbantke formulas......Page 279
C.2.8 Hodge star and Excalc's #......Page 281
C.2.9 Manifold with a metric, Levi–Civita connection......Page 285
C.2.10 Codifferential and wave operator, also in Excalc......Page 287
C.2.11 Nonmetricity......Page 289
C.2.12 Post-Riemannian pieces of the connection......Page 291
C.2.13 Excalc again......Page 295
Problems......Page 298
References......Page 301
Part D: The Maxwell–Lorentz spacetime relation......Page 303
D.1.1 Linearity......Page 305
D.1.2 Extracting the Abelian axion......Page 308
D.1.3 Fresnel equation......Page 310
D.1.4 Analysis of the Fresnel equation......Page 315
D.2.1 Reciprocity implies closure......Page 321
D.2.2 Almost complex structure......Page 323
D.2.3 Algebraic solution of the closure relation......Page 324
D.3.1 Lagrangian and symmetry......Page 327
D.3.2 Duality operator and metric......Page 329
D.3.3 Algebraic solution of the closure and symmetry relations......Page 330
D.3.4 From a quartic wave surface to the lightcone......Page 336
D.4 Extracting the conformally invariant part of the metric by an alternative method......Page 343
D.4.1 Triplet of self-dual 2-forms and metric......Page 344
D.4.2 Maxwell–Lorentz spacetime relation and Minkowski spacetime......Page 347
D.4.3 Hodge star operator and isotropy......Page 348
D.4.4 Covariance properties......Page 350
D.5 Fifth axiom......Page 355
References......Page 357
Part E: Electrodynamics in vacuum and in matter......Page 362
E.1.1 Maxwell–Lorentz equations, impedance of the vacuum......Page 365
E.1.2 Action......Page 367
E.1.3 Foliation of a spacetime with a metric. Effective permeabilities......Page 368
E.1.4 Symmetry of the energy–momentum current......Page 370
E.2.1 Keeping the first four axioms fixed......Page 373
E.2.2 Mashhoon......Page 374
E.2.3 Heisenberg–Euler......Page 375
E.2.4 Born–Infeld......Page 376
E.2.5 Plebański......Page 377
E.3.1 Splitting of the current: Sixth axiom......Page 379
E.3.2 Maxwell's equations in matter......Page 381
E.3.3 Linear constitutive law......Page 382
E.3.4 Energy–momentum currents in matter......Page 383
E.3.5 Experiment of Walker & Walker......Page 389
E.4.1 Laboratory and material foliation......Page 393
E.4.2 Electromagnetic field in laboratory and material frames......Page 397
E.4.3 Optical metric from the constitutive law......Page 401
E.4.4 Electromagnetic field generated in moving continua......Page 402
E.4.5 The experiments of Röntgen and Wilson & Wilson......Page 406
E.4.6 Non-inertial "rotating coordinates"......Page 411
E.4.7 Rotating observer......Page 413
E.4.8 Accelerating observer......Page 415
E.4.9 The proper reference frame of the noninertial observer ("noninertial frame")......Page 418
E.4.10 Universality of the Maxwell–Lorentz spacetime relation......Page 420
References......Page 423
Part F: (Preliminary sketch version of) Validity of classical electrodynamics, interaction with gravity, outlook......Page 426
F.1.1 Gravitational field......Page 429
F.1.2 Classical (1st quantized) Dirac field......Page 440
F.1.3 Topology and electrodynamics......Page 442
F.1.4 Remark on possible violations of Poincaré invariance......Page 445
F.2.1 QED......Page 447
F.2.2 Electro-weak unification......Page 448
F.2.3 Quantum Chern–Simons and the QHE......Page 450
References......Page 451
Index......Page 454