Preface ...................................................... xiii
Acknowledgments ............................................... xix
About the author .............................................. xxi
I Classical optics in global vacuum ............................ 1
1 Heading for photon physics ................................... 3
2 Fundamentals of free electromagnetic fields .................. 7
2.1 Maxwell equations and wave equations .................... 7
2.2 Transverse and longitudinal vector fields ............... 8
2.3 Complex analytical signals ............................. 10
2.4 Monochromatic plane-wave expansion of the
electromagnetic field .................................. 13
2.5 Polarization of light .................................. 14
2.5.1 Transformation of base vectors .................. 14
2.5.2 Geometrical picture of polarization states ...... 15
2.6 Wave packets as field modes ............................ 18
2.7 Conservation of energy, moment of energy, momentum,
and angular momentum ................................... 21
2.8 Riemann-Silberstein formalism .......................... 22
2.9 Propagation of analytical signal ....................... 24
3 Optics in the special theory of relativity .................. 27
3.1 Lorentz transformations and proper time ................ 27
3.2 Tensors ................................................ 30
3.3 Four-vectors and -tensors .............................. 31
3.4 Manifest covariance of the free Maxwell equations ...... 33
3.5 Lorentz transformation of the (transverse) electric
and magnetic fields. Duality ........................... 35
3.6 Lorentz transformation of Riemann-Silberstein
vectors. Inner-product invariance ...................... 38
II Light rays and geodesies. Maxwell theory in general
relativity .................................................. 39
4 The light-particle and wave pictures in classical physics ... 41
5 Eikonal theory and Fermat's principle ....................... 45
5.1 Remarks on geometrical optics. Inhomogeneous vacuum .... 45
5.2 Eikonal equation. Geometrical wave surfaces and rays ... 47
5.3 Geodetic line: Fermat's principle ...................... 52
6 Geodesies in general relativity ............................. 55
6.1 Metric tensor. Four-dimensional Riemann space .......... 55
6.2 Time-like metric geodesies ............................. 56
6.3 The Newtonian limit: Motion in a weak static
gravitational field .................................... 59
6.4 Null geodesies and "light particles" ................... 61
6.5 Gravitational redshift. Photon in free fall ............ 62
7 The space-time of general relativity ........................ 67
7.1 Tensor fields .......................................... 67
7.2 Covariant derivative ................................... 69
7.3 Parallel transport ..................................... 70
7.4 Riemann curvature tensor ............................... 71
7.5 Algebraic properties of the Riemann curvature tensor ... 73
7.6 Einstein field equations in general relativity ......... 74
7.7 Metric compatibility ................................... 76
7.8 Geodesic deviation of light rays ....................... 76
8 Electromagnetic theory in curved space-time ................. 79
8.1 Vacuum Maxwell equations in general relativity ......... 79
8.2 Covariant curl and divergence in Riemann space ......... 80
8.3 A uniform formulation of electrodynamics in curved
and fiat space-time .................................... 81
8.3.1 Maxwell equations with normal derivatives ....... 81
8.3.2 Maxwell equations with E, B, D, and H fields .... 83
8.3.3 Microscopic Maxwell-Lorentz equations in
curved space-time ............................... 84
8.3.4 Constitutive relations in curved space-time ..... 85
8.3.5 Remarks on the constitutive relations in
Minkowskian space ............................... 87
8.3.6 Permittivity and permeability for static
metrics ......................................... 88
8.4 Permittivity and permeability in expanding universe .... 89
8.5 Electrodynamics in potential description. Eikonal
theory and null geodesies .............................. 91
8.6 Gauge-covariant derivative ............................. 95
III Photon wave mechanics ...................................... 97
9 The elusive light particle .................................. 99
10 Wave mechanics based on transverse vector potential ........ 105
10.1 Gauge transformation. Covariant and noncovariant
gauges ................................................ 105
10.2 Tentative wave function and wave equation for
transverse photons .................................... 107
10.3 Transverse photon as a spin-1 particle ................ 110
10.4 Neutrino wave mechanics. Massive eigenstate
neutrinos ............................................. 113
11 Longitudinal and scalar photons. Gauge and near-field
light quanta ............................................... 119
11.1 L- and S-photons. Wave equations ...................... 119
11.2 L- and S-photon neutralization in free space .......... 120
11.3 NF- and G-photons ..................................... 122
11.4 Gauge transformation within the Lorenz gauge .......... 124
12 Massive photon field ....................................... 127
12.1 Proca equation ........................................ 127
12.2 Dynamical equations for E and A ....................... 129
12.3 Diamagnetic interaction: Transverse photon mass ....... 130
12.4 Massive vector boson (photon) field ................... 132
12.5 Massive photon propagator ............................. 136
13 Photon-energy wave function formalism ...................... 143
13.1 The Oppenheimer light quantum theory .................. 143
13.2 Interlude: From spherical to Cartesian
representation ........................................ 146
13.3 Photons and antiphotons: Bispinor wave functions ...... 150
13.4 Four-momentum and spin of photon wave packet .......... 153
13.5 Relativistic scalar product. Lorentz-invariant
integration on the energy shell ....................... 155
IV Single-photon quantum optics in Minkowskian space ......... 159
14 The photon of the quantized electromagnetic field .......... 161
15 Polychromatic photons ...................................... 165
15.1 Canonical quantization of the transverse
electromagnetic field ................................. 165
15.2 Energy, momentum, and spin operators of the
transverse field ...................................... 168
15.3 Monochromatic plane-wave photons. Fock states ......... 171
15.4 Single-photon wave packets ............................ 173
15.5 New T-photon "mean" position state .................... 177
15.6 T-photon wave function and related dynamical
equation .............................................. 179
15.7 The non-orthogonality of T-photon position states ..... 181
16 Single-photon wave packet correlations ..................... 183
16.1 Wave-packet basis for one-photon states ............... 183
16.2 Wave-packet photons related to a given i-matrix ....... 184
16.3 Integral equation for the time evolution operator in
the interaction picture ............................... 186
16.4 Atomic and field correlation matrices ................. 189
16.5 Single-photon correlation matrix: The wave function
fingerprint ........................................... 194
17 Interference phenomena with single-photon states ........... 197
17.1 Wave-packet mode interference ......................... 197
17.2 Young-type double-source interference ................. 198
17.3 Interference between transition amplitudes ............ 201
17.4 Field correlations in photon mean position state ...... 201
17.4.1 Correlation supermatrix ........................ 202
17.4.2 Relation between the correlation supermatrix
and the transverse photon propagator ........... 203
18 Free-field operators: Time evolution and commutation
relations .................................................. 205
18.1 Maxwell operator equations. Quasi-classical states .... 205
18.2 Generalized Landau-Peierls-Sudarshan equations ........ 207
18.3 Commutation relations ................................. 208
18.3.1 Commutation relations at different times
(τ≠0) .......................................... 209
18.3.2 Equal-time commutation relations ............... 210
V Photon embryo states ....................................... 213
19 Attached photons in rim zones .............................. 215
20 Evanescent photon fields ................................... 221
20.1 Four-potential description in the Lorenz gauge ........ 221
20.2 Sheet current density: T-, L-, and S-parts ............ 223
20.3 Evanescent T-, L-, and S-potentials ................... 225
20.4 Four-potential photon wave mechanics .................. 229
20.5 Field-quantized approach .............................. 231
20.6 Near-field photon: Heisenberg equation of motion and
coherent state ........................................ 234
21 Photon tunneling ........................................... 237
21.1 Near-field interaction. The photon measurement
problem ............................................... 237
21.2 Scattering of a wave-packet band from a single
current-density sheet ................................. 238
21.3 Incident fields generating evanescent tunneling
potentials ............................................ 243
21.4 Interlude: Scalar propagator in various domains ....... 246
21.5 Incident polychromatic single-photon state ............ 247
21.6 Photon tunneling-coupled sheets ....................... 250
22 Near-field photon emission in 3D ........................... 255
22.1 T-, L-, and S-potentials of a classical point-
particle .............................................. 255
22.1.1 General considerations on source fields ........ 255
22.1.2 Point-particle potentials ...................... 257
22.2 Cerenkov shock wave ................................... 260
22.2.1 Four-potential of point-particle in uniform
motion in vacuum ............................... 260
22.2.2 Transverse and longitudinal response theory
in matter ...................................... 263
22.2.3 The transverse Cerenkov phenomenon ............. 266
22.2.4 Momenta associated to the transverse and
longitudinal parts of the Cerenkov field ....... 269
22.2.5 Screened canonical particle momentum ........... 272
VI Photon source domain and propagators ....................... 275
23 Super-confined T-photon sources ............................ 277
24 Transverse current density in nonrelativistic quantum
mechanics .................................................. 283
24.1 Single-particle transition current density ............ 283
24.2 The hydrogen 1s ↔ 2pz transition ...................... 286
24.3 Breathing mode: Hydrogen 1s ↔ 2pz transition .......... 289
24.4 Two-level breathing mode dynamics ..................... 292
25 Spin-1/2 current density in relativistic quantum
mechanics .................................................. 297
25.1 Dirac matrices ........................................ 297
25.2 Covariant form of the Dirac equation. Minimal
coupling. Four-current density ........................ 299
25.3 Gordon decomposition of the Dirac four-current
density ............................................... 301
25.4 Weakly relativistic spin current density .............. 303
25.5 Continuity equations for spin and space four-current
densities ............................................. 306
26 Massless photon propagators ................................ 309
26.1 From the Huygens propagator to the transverse photon
propagator ............................................ 309
26.2 T-photon time-ordered correlation of events ........... 311
26.3 Covariant correlation matrix .......................... 313
26.4 Covariant quantization of the electromagnetic field:
A brief review ........................................ 314
26.5 The Feynman photon propagator ......................... 316
26.6 Longitudinal and scalar photon propagators ............ 318
VII Photon vacuum and quanta in Minkowskian space ............. 321
27 Photons and observers ...................................... 323
28 The inertial class of observers: Photon vacuum and quanta .. 329
28.1 Transverse photon four-current density ................ 329
28.2 Boosts ................................................ 332
28.2.1 Lorentz and Lorenz-gauge transformations of
the four-potential ............................. 332
28.2.2 Plane-mode decomposition of the covariant
potential ...................................... 333
28.2.3 Mode functions ................................. 336
28.3 Physical (T-photon) vacuum ............................ 337
29 The non-inertial class of observers: The nebulous
particle concept ........................................... 345
29.1 Bogolubov transformation. Vacuum states ............... 345
29.2 The non-unique vacuum ................................. 348
29.3 The Unruh effect ...................................... 352
29.3.1 Rindler space and observer ..................... 352
29.3.2 Rindler particles in Minkowski vacuum .......... 354
30 Photon mass and hidden gauge invariance .................... 363
30.1 Physical vacuum: Spontaneous symmetry breaking ........ 363
30.2 Goldstone bosons ...................................... 366
30.3 The U(1) Higgs model .................................. 368
30.4 Photon mass and vacuum screening current .............. 372
30.5 't Hooft gauge and propagator ......................... 373
VIII Two-photon entanglement in space-time .................... 377
31 The quantal photon gas ..................................... 379
32 Quantum measurements ....................................... 385
32.1 Tensor product space (discrete case) .................. 385
32.2 Definition of an observable (discrete case) ........... 386
32.3 Reduction of the wave packet (discrete case) .......... 387
32.4 Measurements on only one part of a two-part physical
system ................................................ 387
32.5 Entangled photon polarization states .................. 390
33 Two-photon wave mechanics and correlation matrices ......... 393
33.1 Two-photon two times wave function .................... 393
33.2 Two-photon Schrödinger equation in direct space ....... 396
33.3 Two-photon wave packet correlations ................... 397
33.3.1 First-order correlation matrix ................. 397
33.3.2 Second-order correlation matrix ................ 399
34 Spontaneous one- and two-photon emissions .................. 401
34.1 Two-level atom: Weisskopf-Wigner theory of
spontaneous emission .................................. 401
34.1.1 Atom-field Hamiltonian in the electric-dipole
approximation. RWA-model ....................... 401
34.1.2 Weisskopf-Wigner state vector .................. 406
34.2 Two-level atom: Wave function of spontaneously
emitted photon ........................................ 409
34.2.1 Photon wave function in q-space ................ 409
34.2.2 The general photon wave function in r-space .... 410
34.2.3 Genuine transverse photon wave function ........ 411
34.2.4 Spontaneous photon emission in the atomic rim
zone ........................................... 413
34.3 Three-level atom: Spontaneous cascade emission ........ 417
34.3.1 Two-photon state vector ........................ 417
34.3.2 Two-photon two-times wave function ............. 420
34.3.3 The structure of Ф2,T(r1,r2,t1,t2) ............... 422
34.3.4 Far-field part of Ф(1)2,T(r1,r2,t1,t2) ............ 425
Bibliography .................................................. 429
Index ......................................................... 441
|
