1 Covariant Fluid Models for Magnetized Plasmas .............. 1
1.1 Covariant Description of a Magnetostatic Field ............. 1
1.1.1 Maxwell 4-Tensor .................................... 2
1.1.2 Projection Tensors gǀμν and gǀǀμν ..................... 4
1.1.3 Basis 4-Vectors ..................................... 5
1.1.4 Linear and Nonlinear Response Tensor ................ 8
1.2 Covariant Cold Plasma Model ............................... 11
1.2.1 Fluid Description of a Cold Plasma ................. 11
1.2.2 Linear Response Tensor: Cold Plasma ................ 13
1.2.3 Tensor τμν(ω) ...................................... 14
1.2.4 Cold Plasma Dielectric Tensor ...................... 16
1.3 Inclusion of Streaming Motions ............................ 18
1.3.1 Lorentz Transformation to Streaming Frame .......... 18
1.3.2 Dielectric Tensor for a Streaming Distribution ..... 21
1.3.3 Cold Counterstreaming Electrons and Positrons ...... 22
1.4 Relativistic Magnetohydrodynamics ......................... 23
1.4.1 Covariant Form of the MHD Equations ................ 24
1.4.2 Derivation from Kinetic Theory ..................... 26
1.4.3 Generalized Ohm's Law .............................. 27
1.4.4 Two-Fluid Model for a Pair Plasma .................. 28
1.4.5 MHD Wave Modes ..................................... 29
1.5 Quantum Fluid Theory ...................................... 32
1.5.1 Early QFT Theories ................................. 32
1.5.2 Generalizations of QFT ............................. 34
1.5.3 Quasi-classical Models for Spin .................... 35
1.5.4 Spin-Dependent Cold Plasma Response ................ 36
References ................................................ 38
2 Response Tensors for Magnetized Plasmas ................... 41
2.1 Orbit of a Spiraling Charge ............................... 42
2.1.1 Orbit of a Spiraling Charge ........................ 42
2.1.2 Characteristic Response Due to a Spiraling Charge .. 44
2.1.3 Expansion in Bessel Functions ...................... 45
2.1.4 Gyroresonance Condition ............................ 46
2.2 Perturbation Expansions ................................... 48
2.2.1 Perturbation Expansion of the 4-Current ............ 48
2.2.2 Small-Gyroradius Approximation ..................... 50
2.3 General Forms for the Linear Response 4-Tensor ............ 51
2.3.1 Forward-Scattering Method for a Magnetized Plasma .. 51
2.3.2 Forward-Scattering Form Summed over Gyroharmonics .. 53
2.3.3 Vlasov Method for a Magnetized Plasma .............. 55
2.3.4 Vlasov Form for the Linear Response Tensor ......... 57
2.3.5 Vlasov Form Summed over Gyroharmonics .............. 59
2.4 Response of a Relativistic Thermal Plasma ................. 60
2.4.1 Trubnikov's Response Tensor for a Magnetized
Plasma ............................................. 60
2.4.2 Forward-Scattering Form of Trubnikov's Tensor ...... 64
2.4.3 Other Forms of Пμν(k) for a Juttner Distribution ... 65
2.4.4 Relativistic Plasma Dispersion Functions (RPDFs) ... 66
2.4.5 Strictly-Perpendicular Juttner Distribution ........ 67
2.5 Weakly Relativistic Thermal Plasma ........................ 69
2.5.1 Nonrelativistic Plasma Dispersion Function ......... 69
2.5.2 Response Tensor for a Maxwellian Distribution ...... 71
2.5.3 Mildly Relativistic Limit of Trubnikov's Tensor .... 74
2.5.4 Shkarofsky and Dnestrovskii Functions .............. 76
2.5.5 RPDFs Involving Hypergeometric Functions ........... 80
2.6 Pulsar Plasma ............................................. 82
2.6.1 Pulsars ............................................ 82
2.6.2 Response Tensor for a ID Pair Plasma ............... 85
2.6.3 Specific Distributions for a ID Electron Gas ....... 87
2.7 Response Tensor for a Synchrotron-Emitting Gas ............ 90
2.7.1 Synchrotron Approximation .......................... 90
2.7.2 Expansion About a Point of Stationary Phase ........ 93
2.7.3 Transverse Components of the Response Tensor ....... 94
2.7.4 Airy Integral Approximation ........................ 96
2.7.5 Power-Law and Juttner Distributions ................ 97
2.8 Nonlinear Response Tensors ................................ 99
2.8.1 Quadratic Response Tensor for a Cold Plasma ....... 100
2.8.2 Higher Order Currents ............................. 101
2.8.3 Quadratic Response Tensor for Arbitrary
Distribution ...................................... 102
References ............................................... 103
3 Waves in Magnetized Plasmas .............................. 105
3.1 Wave Dispersion .......................................... 106
3.1.1 Invariant Dispersion Equation ..................... 106
3.1.2 Polarization 3-Vector ............................. 109
3.1.3 Ratio of Electric to Total Energy ................. 110
3.1.4 Absorption Coefficient ............................ 111
3.2 Waves in Cool Electron-Ion Plasmas ....................... 112
3.2.1 Cold Plasma Dispersion Equation ................... 112
3.2.2 Parallel and Perpendicular Propagation ............ 114
3.2.3 Cutoffs and Resonances ............................ 115
3.2.4 Hybrid Waves ...................................... 116
3.2.5 Low-Frequency Cold-Plasma Waves ................... 118
3.2.6 Inertial and Kinetic Alfven Waves .................
3.2.7 MHD-Like Waves .................................... 120
3.3 Waves in Cold Electronic Plasmas ......................... 122
3.3.1 Magnetoionic Waves ................................ 122
3.3.2 Four Branches of Magnetoionic Modes ............... 124
3.3.3 QL and QT Limits .................................. 125
3.3.4 High-Frequency Limit .............................. 127
3.3.5 Effect of an Admixture of Positrons ............... 128
3.3.6 Lorentz Transformation of Magnetoionic Waves ...... 130
3.3.7 Transformation of the Polarization Vector ......... 132
3.4 Waves in Weakly Relativistic Thermal Plasmas ............. 132
3.4.1 Cyclotron-Harmonic Modes .......................... 133
3.4.2 Inclusion of Weakly Relativistic Effects .......... 137
3.5 Waves in Pulsar Plasma
3.5.1 Cold-Plasma Model ................................. 140
3.5.2 Effect of the Cyclotron Resonance ................. 141
3.5.3 Effect of a Spread in Lorentz Factors ............. 143
3.5.4 Wave Modes of a Counter-Streaming Pair Plasma ..... 146
3.5.5 Instabilities in a Pulsar Plasma .................. 148
3.5.6 Counter-Streaming Instabilities ................... 150
3.6 Weak-Anisotropy Approximation ............................ 151
3.6.1 Projection onto the Transverse Plane .............. 152
3.6.2 Stokes Parameters ................................. 153
3.6.3 High-Frequency Waves .............................. 155
3.6.4 Mode Coupling ..................................... 158
References ...............................................
4 Gyromagnetic Processes ................................... 161
4.1 Gyromagnetic Emission .................................... 161
4.1.1 Probability of Emission for Periodic Motion ....... 161
4.1.2 Gyroresonance Condition ........................... 163
4.1.3 Quantum Recoil .................................... 164
4.1.4 Resonance Ellipses ................................ 165
4.1.5 Differential Changes .............................. 167
4.1.6 Quasilinear Equations ............................. 168
4.2 Gyromagnetic Emission in Vacuo ........................... 169
4.2.1 Gyromagnetic Emission of Transverse Waves ......... 169
4.2.2 Radiation Reaction to Gyromagnetic Emission ....... 172
4.3 Cyclotron Emission ....................................... 175
4.3.1 Emissivity in a Magnetoionic Mode ................. 176
4.3.2 Gyromagnetic Emission by Thermal Particles ........ 177
4.3.3 Semirelativistic Approximation .................... 179
4.3.4 Electron Cyclotron Maser Emission ................. 181
4.4 Synchrotron Emission ..................................... 182
4.4.1 Synchrotron Emissivity ............................ 183
4.4.2 Synchrotron Absorption ............................ 185
4.4.3 Synchrotron Absorption: Thermal ................... 188
4.4.4 Razin Suppression ................................. 190
4.5 Thomson Scattering in a Magnetic Field ................... 191
4.5.1 Probability for Thomson Scattering ................ 192
4.5.2 Quasilinear Equations for Scattering .............. 194
4.5.3 Scattering Cross Section .......................... 195
4.5.4 Scattering of Magnetoionic Waves .................. 196
4.5.5 Resonant Thomson Scattering ....................... 198
References ............................................... 200
5 Magnetized Dirac Electron ................................ 201
5.1 Dirac Wavefunctions in a Magnetostatic Field ............. 202
5.1.1 Review of the Dirac Equation for В = 0 ............ 202
5.1.2 The Dirac Equation in a Magnetostatic Field ....... 203
5.1.3 Construction of the Wavefunctions ................. 204
5.1.4 Johnson-Lippmann Wavefunctions .................... 208
5.1.5 Orthogonality and Completeness Relations .......... 209
5.2 Spin Operators and Eigenfunctions ........................ 210
5.2.1 Helicity Eigenstates in a Magnetic Field .......... 210
5.2.2 Magnetic-Moment Eigenstates ....................... 212
5.2.3 Eigenstates in the Cylindrical Gauge .............. 214
5.2.4 Average over the Position of the Gyrocenter ....... 215
5.3 Electron Propagator in a Magnetostatic Field ............. 217
5.3.1 Statistically Averaged Electron Propagator ........ 217
5.3.2 Electron Propagator as Green Function ............. 220
5.3.3 Spin Projection Operators ......................... 222
5.4 Vertex Function in a Magnetic Field ...................... 223
5.4.1 Definition of the Vertex Function ................. 223
5.4.2 Gauge-Dependent Factor Along an Electron Line ..... 225
5.4.3 Vertex Function for Arbitrary Spin States ......... 226
5.4.4 Sum over Initial and Final Spin States ............ 228
5.5 Ritus Method and the Vertex Formalism .................... 230
5.5.1 Factorization of the Dirac Equation ............... 231
5.5.2 Reduced Wavefunctions ............................. 233
5.5.3 Reduced Propagator in the Ritus Method ............ 234
5.5.4 Propagator in the Vertex Formalism ................ 235
5.5.5 Vertex Matrix in the Ritus Method ................. 236
5.5.6 Calculation of Traces Using the Ritus Method ...... 238
5.6 Feynman Rules for QPD in a Magnetized Plasma ............. 239
5.6.1 Rules for an Unmagnetized System .................. 239
5.6.2 Modified Rules for Magnetized Systems ............. 243
5.6.3 Probability of Transition ......................... 245
5.6.4 Probabilities for Second-Order Processes .......... 248
References ............................................... 249
6 Quantum Theory of Gyromagnetic Processes ................. 251
6.1 Gyromagnetic Emission and Pair Creation .................. 251
6.1.1 Probability of Gyromagnetic Transition ............ 251
6.1.2 Resonant Momenta and Energies ..................... 255
6.1.3 Kinetic Equations for Gyromagnetic Processes ...... 258
6.1.4 Anharmonicity and Quantum Oscillations ............ 261
6.2 Quantum Theory of Cyclotron Emission ..................... 263
6.2.1 Cyclotron Approximation ........................... 264
6.2.2 Spontaneous Gyromagnetic Emission in Vacuo ........ 268
6.2.3 Cyclotron Emission and Absorption Coefficients .... 271
6.3 Quantum Theory of Synchrotron Emission ................... 272
6.3.1 Quantum Synchrotron Parameter ..................... 273
6.3.2 Synchrotron Approximation ......................... 274
6.3.3 Transition Rate for Synchrotron Emission .......... 279
6.3.4 Change in the Spin During Synchrotron Emission .... 282
6.3.5 Gyromagnetic Emission in Supercritical Fields ..... 283
6.4 One-Photon Pair Creation ................................. 285
6.4.1 Probability of Pair Creation and Decay ............ 285
6.4.2 Rate of Pair Production Near Threshold ............ 288
6.4.3 Creation of Relativistic Pairs .................... 289
6.4.4 Spin- and Polarization-Dependent Decay Rates ...... 293
6.4.5 Energy Distribution of the Pairs .................. 296
6.4.6 One-Photon Pair Annihilation ...................... 298
6.5 Positronium in a Superstrong Magnetic Field .............. 300
6.5.1 Qualitative Description of Positronium ............ 300
6.5.2 Approximate Form of Schrodinger's Equation ........ 302
6.5.3 Bound States of Positronium ....................... 303
6.5.4 Evolution of Photons into Bound Pairs ............. 304
6.5.5 Tunneling Across the Intersection Point ........... 305
References ............................................... 306
7 Second Order Gyromagnetic Processes ...................... 309
7.1 General Properties of Compton Scattering ................. 309
7.1.1 Probability of Compton Scattering ................. 310
7.1.2 Kinetic Equations for Compton Scattering .......... 312
7.1.3 Sum over Intermediate States: Vertex Formalism .... 313
7.1.4 Sum over Intermediate States: Ritus Method ........ 314
7.2 Compton Scattering by an Electron with и = 0 ............. 317
7.2.1 Scattering Probability for n = 0 .................. 318
7.2.2 Transitions n = 0 → n' ≥ 0 ....................... 319
7.2.3 Resonant Compton Scattering ....................... 321
7.3 Scattering in the Cyclotron Approximation ............... 323
7.3.1 Cyclotron-Like Approximation ...................... 324
7.3.2 Scattering in the Birefringent Vacuum ............. 32ч
7.3.3 Inverse Compton Scattering ........................ 321
7.3.4 Special Case n = 0, x' = 0 ........................ 326
7.4 Two-Photon Processes ..................................... 328
7.4.1 Kinetic Equations for Two-Photon Processes ........ 32$
7.4.2 Double Cyclotron Emission ......................... 330
7.4.3 Two-Photon Pair Creation and Annihilation ......... 332
7.5 Electron-Ion and Electron-Electron Scattering ............ 333
7.5.1 Collisional Excitation by a Classical Ion ......... 334
7.5.2 Electron-Electron (Møller) Scattering ............. 335
References ............................................... 337
8 Magnetized Vacuum ........................................ 339
8.1 Linear Response of the Magnetized Vacuum ................. 339
8.1.1 Vacuum Polarization Tensor ........................ 340
8.1.2 Unregularized Tensor: Geheniau Form ............... 342
8.1.3 Regularization of the Vacuum Polarization Tensor .. 345
8.1.4 Vacuum Polarization Tensor: Vertex Formalism ...... 347
8.1.5 Antihermitian Part of the Vacuum Polarization
Tensor ............................................ 350
8.1.6 Vacuum Polarization: Limiting Cases ............... 352
8.1.7 Wave Modes of the Magnetized Vacuum ............... 353
8.2 Schwinger's Proper-Time Method ........................... 355
8.2.1 Proper-Time Method ................................ 355
8.2.2 Propagator in an Electromagnetic Field ............ 358
8.2.3 Weisskopf's Lagrangian ............................ 360
8.2.4 Generalization of Heisenberg-Euler Lagrangian ..... 362
8.3 Vacuum in an Electromagnetic Wrench ...................... 363
8.3.1 Response Tensor for an Electromagnetic Wrench ..... 364
8.3.2 Response Tensors for ω << m ....................... 365
8.3.3 Nonlinear Response Tensors for ω << m ............. 369
8.3.4 Spontaneous Pair Creation ......................... 370
8.4 Waves in Strongly Magnetized Vacuum ...................... 372
8.4.1 Weak-Field, Weak-Dispersion Limit ................. 373
8.4.2 Vacuum Wave Modes: General Case ................... 375
8.4.3 Vacuum Plus Cold Electron Gas ..................... 378
8.4.4 Vacuum Resonance .................................. 380
8.5 Photon Splitting ......................................... 382
8.5.1 Photon Splitting as a Three-Wave Interaction ...... 382
8.5.2 Three-Wave Interactions in the Vacuum ............. 385
8.5.3 Decay Rates in the Weak-Field Approximation ....... 386
8.5.4 5-Matrix Approach ................................. 388
References ............................................... 390
9 Response of a Magnetized Electron Gas .................... 393
9.1 Response of a Magnetized Electron Gas .................... 393
9.1.1 Calculation of the Response Tensor ................ 394
9.1.2 Vertex Form of Пμν(k) ............................. 394
9.1.3 Summed Form of Пμν(k) ............................. 397
9.1.4 Nongyrotropic and Gyrotropic Parts of Пμν (k) ..... 399
9.1.5 Response Tensor: Ritus Method ..................... 401
9.1.6 Neglect of Quantum Effects ........................ 402
9.1.7 Nonquantum Limit of Пμν(k) ........................ 404
9.2 Relativistic Plasma Dispersion Functions ................. 406
9.2.1 Dispersion-Integral Method ........................ 406
9.2.2 Evaluation of Dispersion Integrals ................ 409
9.2.3 Nondispersive Part ................................ 411
9.2.4 Plasma Dispersion Function Znє (t0) ............... 412
9.2.5 RPDF Form of Пμν(k) ............................... 414
9.2.6 Imaginary Parts of RPDFs .......................... 416
9.3 Magnetized Thermal Distributions ......................... 418
9.3.1 Fermi-Dirac Distribution for Magnetized
Electrons ......................................... 418
9.3.2 Completely Degenerate and Nondegenerate Limits .... 419
9.3.3 RPDFs: Completely Degenerate Limit ................ 421
9.3.4 Dissipation in a Completely Degenerate Electron
Gas ............................................... 423
9.3.5 RPDFs: Nondegenerate Limit ........................ 424
9.3.6 Nonrelativistic Distributions ..................... 427
9.4 Special and Limiting Cases of the Response Tensor ........ 430
9.4.1 Parallel Propagation .............................. 430
9.4.2 Small-x Approximation ............................. 433
9.4.3 One-Dimensional (ID) Electron Gas ................. 434
9.4.4 5-Function Distribution Function .................. 436
9.5 Wave Dispersion: Parallel, Degenerate Case ............... 438
9.5.1 Static Response ................................... 439
9.5.2 Dispersion Equation for Parallel Propagation ...... 440
9.5.3 Longitudinal Modes ................................ 441
9.5.4 Transverse Modes .................................. 444
9.5.5 Discussion of GA and PC Modes ..................... 449
9.6 Response of a Spin-Polarized Electron Gas ................ 450
9.6.1 Spin-Dependent Occupation Number .................. 450
9.6.2 General Forms for Пmμν(k) ......................... 451
9.6.3 Small-x Approximation to Пmμν(k) .................. 454
9.6.4 Reduction to the Cold-Plasma Limit ................ 455
9.7 Nonlinear Response Tensors ............................... 456
9.7.1 Closed Loop Diagrams .............................. 457
9.7.2 Quadratic Response Tensor for the Vacuum .......... 457
9.7.3 Nonlinear Responses: The Vertex Formalism ......... 459
9.7.4 Quadratic Response Tensor ......................... 461
9.7.5 Cubic Response Tensor ............................. 462
References ............................................... 463
A Special Functions ........................................ 465
A.l Bessel Functions and J-Functions ......................... 465
A.1.1 Ordinary Bessel Functions ......................... 465
A.1.2 Modified Bessel Functions Iν(z) ................... 466
A.1.3 Macdonald Functions Kν(z) ......................... 466
A.1.4 Airy Functions .................................... 468
A.1.5 J-Functions ....................................... 468
A.2 Relativistic Plasma Dispersion Functions ................. 473
A.2.1 Relativistic Thermal Function T(z, p) ............. 473
A.2.2 Trubnikov Functions ............................... 473
A.2.3 Shkarofsky and Dnestrovskii Functions ............. 474
A.3 Dirac Algebra ............................................ 477
A.3.1 Definitions and the Standard Representation ....... 477
A.3.2 Basic Set of Dirac Matrices ....................... 478
References ............................................... 479
Index ......................................................... 481
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