Introduction to Relativistic Statistical Mechanics: Classical and Quantum

Contents......Page 8
Preface......Page 18
Notations and Conventions......Page 20
Introduction......Page 22
1.1 The One-Particle Relativistic Distribution Function......Page 30
1.1.1 The phase space “volume element”......Page 34
1.2 The Juttner–Synge Equilibrium Distribution......Page 35
1.2.1 Thermodynamics of the Juttner–Synge gas......Page 38
1.2.2 Thermal velocity......Page 39
1.2.3 Moments of the Juttner–Synge function......Page 41
1.2.4 Orthogonal polynomials......Page 42
1.2.5 Zeromass particles......Page 44
1.3 From the Microcanonical Distribution to the Juttner–Synge One......Page 45
1.4 Equilibrium Fluctuations......Page 48
1.5 One-Particle Liouville Theorem......Page 50
1.5.1 Relativistic Liouville equation from the Hamiltonian equations of motion......Page 51
1.6 The Relativistic Rotating Gas......Page 53
2. Relativistic Kinetic Theory and the BGK Equation......Page 56
2.1 Relativistic Hydrodynamics......Page 58
2.1.1 Sound velocity......Page 60
2.1.2 The Eckart approach......Page 61
2.1.3 The Landau–Lifschitz approach......Page 63
2.2 The Relaxation Time Approximation......Page 64
2.3 The Relativistic Kinetic Theory Approach to Hydrodynamics......Page 65
2.4 The Static Conductivity Tensor......Page 69
2.5 Approximation Methods for the Relativistic Boltzmann Equation and Other Kinetic Equations......Page 70
2.5.1 A simple Chapman–Enskog approximation......Page 71
2.6 Transport Coefficients for a System Embedded in aMagnetic Field......Page 72
3.1 Electromagnetic Quantities in Covariant Form......Page 76
3.2 The Static Conductivity Tensor......Page 79
3.3 Debye–H¨uckel Law......Page 80
3.4 Derivation of the Plasma Modes......Page 81
3.4.1 Evaluation of the various integrals......Page 84
3.4.2 Collective modes in extreme cases......Page 85
3.5 Brief Discussion of the Plasma Modes......Page 86
3.6 The Conductivity Tensor......Page 91
3.7.1 Perturbed dispersion relations for the plasma–beamsystem......Page 92
3.7.2 Stability of the beam–plasma system......Page 93
4. Curved Space–Time and Cosmology......Page 96
4.1 Basic Modifications......Page 97
4.2 Thermal Equilibrium in a Gravitational Field......Page 99
4.3 Einstein–Vlasov Equation......Page 100
4.3.1 Linearization of Einstein’s equation......Page 101
4.3.2 The formal solution to the linearized Einstein equation......Page 103
4.4 An Illustration in Cosmology......Page 105
4.4.1 The two-timescale approximation......Page 107
4.4.2 Derivation of the dispersion relations (a rough outline)......Page 109
4.5 Cosmology and Relativistic Kinetic Theory......Page 110
4.5.1 Cosmology: a very brief overview......Page 111
4.5.2 Kinetic theory and cosmology......Page 114
4.5.3 Kinetic theory of the observed universe......Page 116
4.5.4 Statistical mechanics in the primeval universe......Page 117
4.5.5 Particle survival......Page 119
5.1 The Dynamical Problem......Page 123
5.2 Statement of the Main Statistical Problems......Page 125
5.2.1 The initial value problem: observations andmeasures......Page 126
5.2.2 Phase space and the Gibbs ensemble......Page 129
5.3 Many-Particle Distribution Functions......Page 131
5.3.1 Statistics of the particles’ manifolds......Page 132
5.4 The Relativistic BBGKY Hierarchy......Page 134
5.4.1 Cluster decomposition of the relativistic distribution functions......Page 136
5.5 Self-interaction and Radiation......Page 138
5.5.1 An alternative treatment of radiation reaction......Page 140
5.5.2 Remarks on irreversibility......Page 142
5.5.3 Remarks on thermal equilibrium......Page 143
5.6 Radiation Quantities......Page 145
5.7.1 Derivation of the covariant Landau equation......Page 147
5.7.2 The relativistic Vlasov equation with radiation effects......Page 150
5.7.3 Radiation effects for a relativistic plasma in amagnetic field......Page 153
5.8 Statistics of Fields and Particles......Page 154
6. Relativistic Stochastic Processes and Related Questions......Page 157
6.1 Stochastic Processes in Minkowski Space–Time......Page 158
6.1.1 Basic definitions......Page 159
6.1.3 Markovian processes in space–time......Page 160
6.2 Stochastic Processes in µ Space......Page 162
6.2.1 An overview.......Page 163
6.2.2 Markovian processes......Page 164
6.2.3 An alternative approach......Page 166
6.2.4 Markovian processes......Page 168
6.2.5 A simple illustration......Page 169
6.3 Relativistic Brownian Motion......Page 171
6.4 Random Gravitational Fields: An Open Problem......Page 173
6.4.1 A simple example......Page 177
6.4.2 The case of thermal equilibrium......Page 178
6.4.3 Matter-induced fluctuations......Page 179
6.4.4 Random Einstein equations......Page 180
7. The Density Operator......Page 181
7.1 The Density Operator for Thermal Equilibrium......Page 182
7.1.1 Thermodynamic properties......Page 183
7.1.2 The partition function of the relativistic ideal gas......Page 185
7.1.3 The average occupation number......Page 187
7.2 Relativistic Bosons in Thermal Equilibrium......Page 188
7.2.1 The complex scalar field......Page 190
7.2.3 A few remarks on the calculation of various integrals......Page 193
7.2.4 Bose–Einstein condensation......Page 194
7.2.5 Interactions......Page 196
7.3 Free Fermions in Thermal Equilibrium......Page 200
7.4 Thermodynamic Properties of the Relativistic Ideal Fermi–DiracGas......Page 203
7.4.2 The degenerate Fermi gas......Page 204
7.4.3 Thermal corrections: Sommerfeld expansion......Page 206
7.4.4 Corrections for various thermodynamic quantities......Page 208
7.4.5 High temperature expansion (nondegenerate)......Page 209
7.5 White Dwarfs: The Degenerate Electron Gas......Page 210
7.5.1 Cooling of white dwarfs......Page 214
7.6 Functional Representation of the Partition Function......Page 216
7.6.1 The partition function for gauge particles (photons)......Page 217
7.6.2 The photons’ partition function......Page 218
7.6.3 Illustration in the case of the Lorentz gauge......Page 220
8. The Covariant Wigner Function......Page 223
8.1 The Covariant Wigner Function for Spin 1/2 Particles......Page 224
8.1.1 Basic equations......Page 226
8.1.2 The equilibrium Wigner function for free fermions......Page 229
8.1.3 Polarized media......Page 230
8.2 Equilibrium Fluctuations of Fermions......Page 233
8.3 A Simple Example......Page 236
8.4 The BBGKY Relativistic Quantum Hierarchy......Page 237
8.5 Perturbation Expansion of the Wigner Function......Page 240
8.6 The Wigner Function for Bosons......Page 242
8.6.1 The example of the λ.4 theory......Page 245
8.6.2 Four-current .uctuations of the complex scalar field......Page 246
8.7.1 Gauge-invariant Wigner functions......Page 247
8.7.2 A few remarks......Page 251
8.7.3 Gauge-invariant Wigner functions for the photon field......Page 252
8.7.4 Another gauge-invariant Wigner function......Page 254
8.7.5 Gauge invariance and approximations......Page 255
9. Fermions Interacting via a Scalar Field: A Simple Example......Page 257
9.1 Thermal Equilibrium......Page 258
9.2 Collective Modes......Page 262
9.3 Two-Body Correlations......Page 263
9.3.1 A brief discussion......Page 266
9.3.2 Exchange correlations......Page 267
9.4 Renormalization — An Illustration of the Procedure......Page 269
9.4.1 Regularization of the gap equation......Page 270
9.4.2 Regularization of the energy–momentum tensor......Page 273
9.4.3 Determination of the constants (AF,BF, CF,DF)......Page 274
9.5 Qualitative Discussion of the Effects of Renormalization......Page 275
9.6 Thermodynamics of the System......Page 278
9.6.1 The gap equation as a minimum of the free energy......Page 279
9.6.2 Thermodynamics......Page 280
9.7 Renormalization of the Excitation Spectrum......Page 282
9.7.1 Comparison with the semiclassical case......Page 286
9.8 A Short Digression on Bosons......Page 287
10. Covariant Kinetic Equations in the Quantum Domain......Page 291
10.1 General Form of the Kinetic Equation......Page 293
10.2 An Introductory Example......Page 294
10.3 A General Relaxation Time Approximation......Page 298
10.3.1 Properties of the kinetic system......Page 299
10.3.2 The collision term......Page 301
10.3.3 General form of F(1)......Page 303
11. Application to Nuclear Matter......Page 306
11.1 Thermodynamic Properties at Finite Temperature......Page 308
11.1.1 Thermodynamics in some important cases......Page 311
11.2 Remarks on the Oscillation Spectra of Mesons......Page 314
11.3 Transport Coe.cients of Nuclear Matter......Page 315
11.3.1 Chapman–Enskog expansion......Page 317
11.3.2 Transport coefficients: Eckart versus Landau–Lifschitz representations......Page 319
11.3.3 Entropy production......Page 322
11.3.4 A brief comparison: BGK versus BUU......Page 326
11.4 Discussion......Page 328
11.5 Dense Nuclear Matter: Neutron Stars......Page 331
11.5.1 The static equilibrium of a neutron star......Page 332
11.5.2 The composition of matter in a neutron star......Page 333
11.5.3 Beyond the drip point......Page 336
12. Strong Magnetic Fields......Page 338
12.1 Relations Obeyed by the Magnetic Field......Page 341
12.2 The Partition Function......Page 343
12.2.1 Magnetization of an electron gas......Page 346
12.3 Relativistic Quantum Liouville Equation......Page 348
12.3.1 Solution of the inhomogeneous equation......Page 350
12.3.2 The initial value problem......Page 352
12.4 The Equilibrium Wigner Function for Noninteracting Electrons......Page 353
12.4.1 Thermodynamic quantities......Page 354
12.5 The Wigner Function of the Ideal Magnetized ElectronGas......Page 355
12.5.1 The nonmagnetic field limit......Page 357
12.5.2 Equations of state......Page 358
12.5.3 Is the pressure isotropic?......Page 359
12.5.4 The completely degenerate case......Page 360
12.5.5 Magnetization......Page 362
12.5.6 Landau orbital ferromagnetism: LOFER states......Page 364
12.6.1 The general structure of the vacuum Wigner function......Page 365
12.6.2 The Wigner function of the magnetized vacuum......Page 367
12.6.3 Renormalization of the vacuum Wigner function......Page 368
12.7 Fluctuations......Page 369
12.7.1 Fluctuations of the four-current......Page 370
12.8 Polarization Tensors of the Magnetized Electron Gas and of theMagnetized Vacuum......Page 377
12.8.1 The vacuum polarization tensor......Page 378
12.9 Remarks on the Transport Coe.cients of the Magnetized Electron Gas......Page 379
12.10 Astrophysical Aspects......Page 382
13. Statistical Mechanics of Relativistic Quasiparticles......Page 385
13.1 Classical Fields......Page 388
13.1.1 Internal symmetries and conserved currents......Page 389
13.1.2 Space–time symmetries......Page 392
13.1.3 A general remark......Page 396
13.2 Quantum Quasiparticles......Page 399
13.2.1 Formal quantization......Page 400
13.3.1 A first example......Page 403
13.3.2 Another example the QED plasma......Page 405
13.3.3 Migdal’s approach......Page 406
13.4 The Covariant Wigner Function......Page 408
13.5 Equilibrium Properties......Page 411
13.6 A Simple Example: The λφ4 Model......Page 414
13.7 Remarks on the Thermodynamics of Quasiparticles......Page 417
13.8 Equilibrium Fluctuations......Page 420
13.9 Remarks on the Negative Energy Modes......Page 423
13.10 InteractingQuasibosons......Page 424
13.10.1 The long wavelength and low frequency limit......Page 427
14.1 Independent Quasifermions......Page 429
14.1.1 Quantization and observables......Page 431
14.1.2 Statistical expressions......Page 434
14.1.3 Thermal equilibrium......Page 435
14.2 InteractingQuasifermions......Page 436
14.2.1 The long wavelength and low frequency limit......Page 438
14.3 Kinetic Equation for Quasiparticles......Page 439
14.4 Remarks on the Relativistic Landau Theory......Page 441
15.1 Basic Equations......Page 451
15.2 Plasma CollectiveModes......Page 452
15.3 The Fluctuation–Dissipation Theorem and Its Inverse......Page 457
15.4 Four-Current Fluctuations and the Polarization Tensor......Page 458
15.5 The Polarization Tensor at Order e2......Page 462
15.6.1 Quasiphotons in thermal equilibrium......Page 465
15.6.2 Gauge properties......Page 469
15.6.3 Quasielectron modes in thermal equilibrium......Page 471
A.1 Kelvin’s Functions......Page 475
A.2 Associated Laguerre Polynomials......Page 476
Appendix B: γ Matrices......Page 477
Appendix C: Outline of Functional Methods......Page 480
C.1 Functional Differentiation......Page 481
C.2 Functional Integration......Page 482
D.1 Ordinary Units......Page 486
D.2 Other Units of Interest......Page 487
E.1 Useful Formulae for Bosons......Page 489
E.2 Useful Formulae for Fermions......Page 491
Bibliography......Page 494
Index......Page 558