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[资源] Introduction to Relativistic Statistical Mechanics Classical and Quantum

Contents
Preface xvii
Notations and Conventions xix
Introduction xxi
1. The One-Particle Relativistic Distribution Function 1
1.1 The One-Particle Relativistic Distribution Function . . . 1
1.1.1 The phase space “volume element” . . . . . . . . 5
1.2 The Jüttner–Synge Equilibrium Distribution . . . . . . . 6
1.2.1 Thermodynamics of the Jüttner–Synge gas . . . 9
1.2.2 Thermal velocity . . . . . . . . . . . . . . . . . . 10
1.2.3 Moments of the Jüttner–Synge function . . . . . 12
1.2.4 Orthogonal polynomials . . . . . . . . . . . . . . 13
1.2.5 Zero mass particles . . . . . . . . . . . . . . . . . 15
1.3 From the Microcanonical Distribution
to the Jüttner–Synge One . . . . . . . . . . . . . . . . . 16
1.4 Equilibrium Fluctuations . . . . . . . . . . . . . . . . . . 19
1.5 One-Particle Liouville Theorem . . . . . . . . . . . . . . 21
1.5.1 Relativistic Liouville equation from the
Hamiltonian equations of motion . . . . . . . . . 22
1.5.2 Conditions for the Jüttner–Synge functions
to be an equilibrium . . . . . . . . . . . . . . . . 24
1.6 The Relativistic Rotating Gas . . . . . . . . . . . . . . . 24
2. Relativistic Kinetic Theory and the BGK Equation 27
2.1 Relativistic Hydrodynamics . . . . . . . . . . . . . . . . 29
2.1.1 Sound velocity . . . . . . . . . . . . . . . . . . . 31
2.1.2 The Eckart approach . . . . . . . . . . . . . . . . 32
2.1.3 The Landau–Lifschitz approach . . . . . . . . . . 34
vii
viii Introduction to Relativistic Statistical Mechanics: Classical and Quantum
2.2 The Relaxation Time Approximation . . . . . . . . . . . 35
2.3 The Relativistic Kinetic Theory Approach
to Hydrodynamics . . . . . . . . . . . . . . . . . . . . . 36
2.4 The Static Conductivity Tensor . . . . . . . . . . . . . . 40
2.5 Approximation Methods for the Relativistic Boltzmann
Equation and Other Kinetic Equations . . . . . . . . . . 41
2.5.1 A simple Chapman–Enskog approximation . . . 42
2.6 Transport Coefficients for a System Embedded
in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . 43
3. Relativistic Plasmas 47
3.1 Electromagnetic Quantities in Covariant Form . . . . . . 47
3.2 The Static Conductivity Tensor . . . . . . . . . . . . . . 50
3.3 Debye–Hückel Law . . . . . . . . . . . . . . . . . . . . . 51
3.4 Derivation of the Plasma Modes . . . . . . . . . . . . . . 52
3.4.1 Evaluation of the various integrals . . . . . . . . 55
3.4.2 Collective modes in extreme cases . . . . . . . . 56
3.5 Brief Discussion of the Plasma Modes . . . . . . . . . . . 57
3.6 The Conductivity Tensor . . . . . . . . . . . . . . . . . . 62
3.7 Plasma–Beam Instability . . . . . . . . . . . . . . . . . . 63
3.7.1 Perturbed dispersion relations
for the plasma–beam system . . . . . . . . . . . 63
3.7.2 Stability of the beam–plasma system . . . . . . . 64
4. Curved Space–Time and Cosmology 67
4.1 Basic Modifications . . . . . . . . . . . . . . . . . . . . . 68
4.2 Thermal Equilibrium in a Gravitational Field . . . . . . 70
4.2.1 Thermal equilibrium in a static
isotropic metric . . . . . . . . . . . . . . . . . . . 71
4.3 Einstein–Vlasov Equation . . . . . . . . . . . . . . . . . 71
4.3.1 Linearization of Einstein’s equation . . . . . . . . 72
4.3.2 The formal solution to the linearized
Einstein equation . . . . . . . . . . . . . . . . . . 74
4.3.3 The self-consistent kinetic equation
for the gravitating gas . . . . . . . . . . . . . . . 76
Contents ix
4.4 An Illustration in Cosmology . . . . . . . . . . . . . . . 76
4.4.1 The two-timescale approximation . . . . . . . . . 78
4.4.2 Derivation of the dispersion relations
(a rough outline) . . . . . . . . . . . . . . . . . . 80
4.5 Cosmology and Relativistic Kinetic Theory . . . . . . . 81
4.5.1 Cosmology: a very brief overview . . . . . . . . . 82
4.5.2 Kinetic theory and cosmology . . . . . . . . . . . 85
4.5.3 Kinetic theory of the observed universe . . . . . 87
4.5.4 Statistical mechanics in the primeval
universe . . . . . . . . . . . . . . . . . . . . . . . 88
4.5.5 Particle survival . . . . . . . . . . . . . . . . . . 90
5. Relativistic Statistical Mechanics 94
5.1 The Dynamical Problem . . . . . . . . . . . . . . . . . . 94
5.2 Statement of the Main Statistical Problems . . . . . . . 96
5.2.1 The initial value problem: observations
and measures . . . . . . . . . . . . . . . . . . . . 97
5.2.2 Phase space and the Gibbs ensemble . . . . . . . 100
5.3 Many-Particle Distribution Functions . . . . . . . . . . . 102
5.3.1 Statistics of the particles’ manifolds . . . . . . . 103
5.4 The Relativistic BBGKY Hierarchy . . . . . . . . . . . . 105
5.4.1 Cluster decomposition of the relativistic
distribution functions . . . . . . . . . . . . . . . 107
5.5 Self-interaction and Radiation . . . . . . . . . . . . . . . 109
5.5.1 An alternative treatment of radiation
reaction . . . . . . . . . . . . . . . . . . . . . . . 111
5.5.2 Remarks on irreversibility . . . . . . . . . . . . 113
5.5.3 Remarks on thermal equilibrium . . . . . . . . . 114
5.6 Radiation Quantities . . . . . . . . . . . . . . . . . . . . 116
5.7 A Few Relativistic Kinetic Equations . . . . . . . . . . . 118
5.7.1 Derivation of the covariant Landau
equation . . . . . . . . . . . . . . . . . . . . . . . 118
5.7.2 The relativistic Vlasov equation
with radiation effects . . . . . . . . . . . . . . . . 121
5.7.3 Radiation effects for a relativistic plasma
in a magnetic field . . . . . . . . . . . . . . . . . 124
5.8 Statistics of Fields and Particles . . . . . . . . . . . . . . 125
x Introduction to Relativistic Statistical Mechanics: Classical and Quantum
6. Relativistic Stochastic Processes and Related Questions 128
6.1 Stochastic Processes in Minkowski Space–Time . . . . . 129
6.1.1 Basic definitions . . . . . . . . . . . . . . . . . . 130
6.1.2 Conditional currents . . . . . . . . . . . . . . . . 131
6.1.3 Markovian processes in space–time . . . . . . . . 131
6.2 Stochastic Processes in μ Space . . . . . . . . . . . . . . 133
6.2.1 An overview . . . . . . . . . . . . . . . . . . . . . 134
6.2.2 Markovian processes . . . . . . . . . . . . . . . . 135
6.2.3 An alternative approach . . . . . . . . . . . . . . 137
6.2.4 Markovian processes . . . . . . . . . . . . . . . . 139
6.2.5 A simple illustration . . . . . . . . . . . . . . . . 140
6.3 Relativistic Brownian Motion . . . . . . . . . . . . . . . 142
6.4 Random Gravitational Fields: An Open Problem . . . . 144
6.4.1 A simple example . . . . . . . . . . . . . . . . . 148
6.4.2 The case of thermal equilibrium . . . . . . . . . 149
6.4.3 Matter-induced fluctuations . . . . . . . . . . . . 150
6.4.4 Random Einstein equations . . . . . . . . . . . . 151
7. The Density Operator 152
7.1 The Density Operator for Thermal Equilibrium . . . . . 153
7.1.1 Thermodynamic properties . . . . . . . . . . . . 154
7.1.2 The partition function of the relativistic
ideal gas . . . . . . . . . . . . . . . . . . . . . . . 156
7.1.3 The average occupation number . . . . . . . . . 158
7.2 Relativistic Bosons in Thermal Equilibrium . . . . . . . 159
7.2.1 The complex scalar field . . . . . . . . . . . . . . 161
7.2.2 Charge fluctuations . . . . . . . . . . . . . . . . 164
7.2.3 A few remarks on the calculation
of various integrals . . . . . . . . . . . . . . . . . 164
7.2.4 Bose–Einstein condensation . . . . . . . . . . . . 165
7.2.5 Interactions . . . . . . . . . . . . . . . . . . . . . 167
7.3 Free Fermions in Thermal Equilibrium . . . . . . . . . . 171
7.4 Thermodynamic Properties of the Relativistic
Ideal Fermi–Dirac Gas . . . . . . . . . . . . . . . . . . . 174
7.4.1 Remarks on the numerical calculations
of various physical quantities . . . . . . . . . . . 175
7.4.2 The degenerate Fermi gas . . . . . . . . . . . . . 175
7.4.3 Thermal corrections: Sommerfeld expansion . . . 177
Contents xi
7.4.4 Corrections for various thermodynamic
quantities . . . . . . . . . . . . . . . . . . . . . . 179
7.4.5 High temperature expansion (nondegenerate) . . 180
7.5 White Dwarfs: The Degenerate Electron Gas . . . . . . . 181
7.5.1 Cooling of white dwarfs . . . . . . . . . . . . . . 185
7.5.2 Pycnonuclear reactions . . . . . . . . . . . . . . . 187
7.6 Functional Representation of the Partition Function . . 187
7.6.1 The partition function for gauge
particles (photons) . . . . . . . . . . . . . . . . . 188
7.6.2 The photons’ partition function . . . . . . . . . . 189
7.6.3 Illustration in the case of the Lorentz gauge . . . 191
8. The Covariant Wigner Function 194
8.1 The Covariant Wigner Function for Spin 1/2
Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.1.1 Basic equations . . . . . . . . . . . . . . . . . . . 197
8.1.2 The equilibrium Wigner function
for free fermions . . . . . . . . . . . . . . . . . . 200
8.1.3 Polarized media . . . . . . . . . . . . . . . . . . . 201
8.2 Equilibrium Fluctuations of Fermions . . . . . . . . . . . 204
8.3 A Simple Example . . . . . . . . . . . . . . . . . . . . . 207
8.4 The BBGKY Relativistic Quantum Hierarchy . . . . . . 208
8.5 Perturbation Expansion of the Wigner Function . . . . . 211
8.6 The Wigner Function for Bosons . . . . . . . . . . . . . 213
8.6.1 The example of the λϕ 4 theory . . . . . . . . . . 216
8.6.2 Four-current fluctuations of the complex
scalar field . . . . . . . . . . . . . . . . . . . . . 217
8.7 Gauge Properties of the Wigner Function . . . . . . . . 218
8.7.1 Gauge-invariant Wigner functions . . . . . . . . 218
8.7.2 A few remarks . . . . . . . . . . . . . . . . . . . 222
8.7.3 Gauge-invariant Wigner functions
for the photon field . . . . . . . . . . . . . . . . . 223
8.7.4 Another gauge-invariant Wigner function . . . . 225
8.7.5 Gauge invariance and approximations . . . . . . 226
9. Fermions Interacting via a Scalar Field: A Simple Example 228
9.1 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . 229
9.2 Collective Modes . . . . . . . . . . . . . . . . . . . . . . 233
xii Introduction to Relativistic Statistical Mechanics: Classical and Quantum
9.3 Two-Body Correlations . . . . . . . . . . . . . . . . . . . 234
9.3.1 A brief discussion . . . . . . . . . . . . . . . . . . 237
9.3.2 Exchange correlations . . . . . . . . . . . . . . . 238
9.4 Renormalization — An Illustration of the Procedure . . 240
9.4.1 Regularization of the gap equation . . . . . . . . 241
9.4.2 Regularization of the energy–momentum
tensor . . . . . . . . . . . . . . . . . . . . . . . . 244
9.4.3 Determination of the constants
(A F ,B F ,C F ,D F ) . . . . . . . . . . . . . . . . . 245
9.5 Qualitative Discussion of the Effects
of Renormalization . . . . . . . . . . . . . . . . . . . . . 246
9.6 Thermodynamics of the System . . . . . . . . . . . . . . 249
9.6.1 The gap equation as a minimum
of the free energy . . . . . . . . . . . . . . . . . . 250
9.6.2 Thermodynamics . . . . . . . . . . . . . . . . . . 251
9.7 Renormalization of the Excitation Spectrum . . . . . . . 253
9.7.1 Comparison with the semiclassical case . . . . . 257
9.8 A Short Digression on Bosons . . . . . . . . . . . . . . . 258
10. Covariant Kinetic Equations in the Quantum Domain 262
10.1 General Form of the Kinetic Equation . . . . . . . . . . 264
10.2 An Introductory Example . . . . . . . . . . . . . . . . . 265
10.3 A General Relaxation Time Approximation . . . . . . . 269
10.3.1 Properties of the kinetic system . . . . . . . . . . 270
10.3.2 The collision term . . . . . . . . . . . . . . . . . 272
10.3.3 General form of F (1) . . . . . . . . . . . . . . . . 274
11. Application to Nuclear Matter 277
11.1 Thermodynamic Properties at Finite Temperature . . . 279
11.1.1 Thermodynamics in some important cases . . . . 282
11.2 Remarks on the Oscillation Spectra of Mesons . . . . . . 285
11.3 Transport Coefficients of Nuclear Matter . . . . . . . . . 286
11.3.1 Chapman–Enskog expansion . . . . . . . . . . . 288
11.3.2 Transport coefficients: Eckart versus
Landau–Lifschitz representations . . . . . . . . . 290
11.3.3 Entropy production . . . . . . . . . . . . . . . . 293
11.3.4 A brief comparison: BGK versus BUU . . . . . . 297
11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Contents xiii
11.5 Dense Nuclear Matter: Neutron Stars . . . . . . . . . . . 302
11.5.1 The static equilibrium of a neutron star . . . . . 303
11.5.2 The composition of matter in a neutron star . . 304
11.5.3 Beyond the drip point . . . . . . . . . . . . . . . 307
12. Strong Magnetic Fields 309
12.1 Relations Obeyed by the Magnetic Field . . . . . . . . . 312
12.2 The Partition Function . . . . . . . . . . . . . . . . . . . 314
12.2.1 Magnetization of an electron gas . . . . . . . . . 317
12.3 Relativistic Quantum Liouville Equation . . . . . . . . . 319
12.3.1 Solution of the inhomogeneous equation . . . . . 321
12.3.2 The initial value problem . . . . . . . . . . . . . 323
12.4 The Equilibrium Wigner Function for Noninteracting
Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
12.4.1 Thermodynamic quantities . . . . . . . . . . . . 325
12.5 The Wigner Function of the Ideal Magnetized
Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . 326
12.5.1 The nonmagnetic field limit . . . . . . . . . . . . 328
12.5.2 Equations of state . . . . . . . . . . . . . . . . . 329
12.5.3 Is the pressure isotropic? . . . . . . . . . . . . . 330
12.5.4 The completely degenerate case . . . . . . . . . . 331
12.5.5 Magnetization . . . . . . . . . . . . . . . . . . . 333
12.5.6 Landau orbital ferromagnetism:
LOFER states . . . . . . . . . . . . . . . . . . . 335
12.6 The Magnetized Vacuum . . . . . . . . . . . . . . . . . . 336
12.6.1 The general structure of the vacuum
Wigner function . . . . . . . . . . . . . . . . . . 336
12.6.2 The Wigner function of the magnetized
vacuum . . . . . . . . . . . . . . . . . . . . . . . 338
12.6.3 Renormalization of the vacuum
Wigner function . . . . . . . . . . . . . . . . . . 339
12.7 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 340
12.7.1 Fluctuations of the four-current . . . . . . . . . . 341
12.8 Polarization Tensors of the Magnetized Electron Gas
and of the Magnetized Vacuum . . . . . . . . . . . . . . 348
12.8.1 The vacuum polarization tensor . . . . . . . . . . 349
12.9 Remarks on the Transport Coefficients
of the Magnetized Electron Gas . . . . . . . . . . . . . . 350
12.10 Astrophysical Aspects . . . . . . . . . . . . . . . . . . . 353
xiv Introduction to Relativistic Statistical Mechanics: Classical and Quantum
13. Statistical Mechanics of Relativistic Quasiparticles 356
13.1 Classical Fields . . . . . . . . . . . . . . . . . . . . . . . 359
13.1.1 Internal symmetries and conserved currents . . . 360
13.1.2 Space–time symmetries . . . . . . . . . . . . . . 363
13.1.3 A general remark . . . . . . . . . . . . . . . . . . 367
13.2 Quantum Quasiparticles . . . . . . . . . . . . . . . . . . 370
13.2.1 Formal quantization . . . . . . . . . . . . . . . . 371
13.3 Problems with the Quantization of Quasiparticles . . . . 374
13.3.1 A first example . . . . . . . . . . . . . . . . . . . 374
13.3.2 Another example the QED plasma . . . . . . . . 376
13.3.3 Migdal’s approach . . . . . . . . . . . . . . . . . 377
13.4 The Covariant Wigner Function . . . . . . . . . . . . . . 379
13.5 Equilibrium Properties . . . . . . . . . . . . . . . . . . . 382
13.6 A Simple Example: The λφ 4 Model . . . . . . . . . . . . 385
13.7 Remarks on the Thermodynamics of Quasiparticles . . . 388
13.8 Equilibrium Fluctuations . . . . . . . . . . . . . . . . . . 391
13.9 Remarks on the Negative Energy Modes . . . . . . . . . 394
13.10 Interacting Quasibosons . . . . . . . . . . . . . . . . . . 395
13.10.1 The long wavelength and low frequency limit . . 398
14. The Relativistic Fermi Liquid 400
14.1 Independent Quasifermions . . . . . . . . . . . . . . . . 400
14.1.1 Quantization and observables . . . . . . . . . . . 402
14.1.2 Statistical expressions . . . . . . . . . . . . . . . 405
14.1.3 Thermal equilibrium . . . . . . . . . . . . . . . . 406
14.2 Interacting Quasifermions . . . . . . . . . . . . . . . . . 407
14.2.1 The long wavelength and low frequency limit . . 409
14.3 Kinetic Equation for Quasiparticles . . . . . . . . . . . . 410
14.4 Remarks on the Relativistic Landau Theory . . . . . . . 412
15. The QED Plasma 422
15.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . 422
15.2 Plasma Collective Modes . . . . . . . . . . . . . . . . . . 423
15.3 The Fluctuation–Dissipation Theorem and Its Inverse . 428
15.4 Four-Current Fluctuations and the Polarization
Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
15.5 The Polarization Tensor at Order e 2 . . . . . . . . . . . 433
15.6 Quasiparticles in the Relativistic Plasma . . . . . . . . . 436
Contents xv
15.6.1 Quasiphotons in thermal equilibrium . . . . . . . 436
15.6.2 Gauge properties . . . . . . . . . . . . . . . . . . 440
15.6.3 Quasielectron modes in thermal equilibrium . . . 442
Appendix A: A Few Useful Properties of Some Special Functions 446
A.1 Kelvin’s Functions . . . . . . . . . . . . . . . . . . . . . 446
A.2 Associated Laguerre Polynomials . . . . . . . . . . . . . 447
Appendix B: γ Matrices 448
Appendix C: Outline of Functional Methods 451
C.1 Functional Differentiation . . . . . . . . . . . . . . . . . 452
C.2 Functional Integration . . . . . . . . . . . . . . . . . . . 453
Appendix D: Units 457
D.1 Ordinary Units . . . . . . . . . . . . . . . . . . . . . . . 457
D.2 Other Units of Interest . . . . . . . . . . . . . . . . . . . 458
Appendix E: Some Useful Formulae for Wigner Functions 460
E.1 Useful Formulae for Bosons . . . . . . . . . . . . . . . . 460
E.2 Useful Formulae for Fermions . . . . . . . . . . . . . . . 462
Bibliography 465
Index 529

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