This monograph presents recent developments in quantum field theory at finite temperature. By using Lie groups, ideas from thermal theory are considered with concepts of symmetry, allowing for applications not only to quantum field theory but also to transport theory, quantum optics and statistical mechanics. This includes an analysis of geometrical and topological aspects of spatially confined systems with applications to the Casimir effect, superconductivity and phase transitions. Finally, some developments in open systems are also considered. The book provides a unified picture of the fundamental aspects in thermal quantum field theory and their applications, and is important to the field as a result, since it combines several diverse ideas that lead to a better understanding of different areas of physics.
Contents: General Principles: Elements of Thermodynamics; Elements of Statistical Mechanics; Partition Function and Path Integral; Zero Temperature Interacting Fields; Thermal Fields: Thermofield Dynamics: Kinematical Symmetry Algebraic Basis; Thermal Oscillators: Bosons and Fermions; Thermal PoincarГѓВ© and Galilei Groups; Thermal Propagator; Scattering Process at Finite Temperature; Topics on Renormalization Theory; Ward-Takahashi Relations and Gauge Symmetry; Applications to Quantum Optics: Thermalized States of a Field Mode; Nonclassical Properties of Thermal Quantum States; SU(2) and SU(1,1) Systems: Entanglement; Compactified Fields: Compactified Fields; Casimir Effect for the Electromagnetic Field; Casimir Effect for Fermions; Compactified 4 Theory; Phase Transitions in Confined Systems: Application to Superconducting Films; Second-Order Phase Transition in Wires and Grains; First-Order Phase Transitions in Confined Systems; Applications to Open Systems: Thermo-Algebras in Phase Space: Quantum and Classical Systems; Real-Time Method for Nonequilibrium Quantum Mechanics; Dressed and Bare State Approaches to the Thermalization Process.
Author(s): Faqir C. Khanna, Adolfo P C Malbouisson, Jorge M C Malbouisson, Ademir E Santana
Publisher: World Scientific Publishing Company
Year: 2009
Language: English
Pages: 482
Contents......Page 14
Preface......Page 8
Part I – General Principles......Page 22
1. Elements of Thermodynamics......Page 24
1.1 Kinematical aspects of thermal physics......Page 25
1.2 Dynamical aspects of thermal physics......Page 27
1.3 Equations of state......Page 28
1.4 The meaning of intensive variables......Page 30
1.5 Thermodynamical potentials......Page 33
1.6 Gibbs-Duhem relation......Page 36
1.7 Second derivatives......Page 37
1.8.1 State equation for an ideal gas......Page 40
1.8.2 The van der Waals equation......Page 41
1.9 Stability conditions and phase transitions......Page 42
2.1 Macro- and micro-physics......Page 48
2.2 Liouville-von Neumann equation......Page 50
2.3.1 Micro-canonical ensemble......Page 52
2.3.2 Canonical ensemble......Page 53
2.3.3 Grand-canonical ensemble......Page 54
2.3.4 Equivalence among the ensembles......Page 55
2.4 Wigner function formalism......Page 56
3.1 Partition function and the propagator......Page 60
3.2 Path integral in quantum mechanics......Page 63
3.3 Classical fields......Page 64
3.4 Canonical quantization of scalar fields......Page 67
3.5 Path integral for a scalar field......Page 69
3.6 Canonical quantization of the Dirac field......Page 72
3.7 Path integral for the Dirac field......Page 76
4.1 Generating functional for bosons......Page 80
4.1.1 Feynman rules in momentum space......Page 83
4.2 The effective action......Page 84
4.3 Gauge fields......Page 87
4.4 Generating functional for gauge fields......Page 91
4.5 U(1) gauge theory......Page 94
4.6 SU(3) gauge theory......Page 95
4.7 Scattering amplitudes......Page 98
4.8 S-matrix in the canonical approach......Page 106
Part II – Thermal Fields......Page 108
5. Thermo eld Dynamics: Kinematical Symmetry Algebraic Basis......Page 110
5.1 Thermal Hilbert space......Page 112
5.2.1 Generators of symmetry and observables......Page 114
5.2.2 Doubled Lie algebra......Page 115
5.2.3 Tilde conjugation rules......Page 117
5.3 Tilde and non-tilde operators......Page 118
5.4 Liouville-von Neumann equation......Page 121
5.5 Physical implications of thermo-algebras......Page 123
6.1 Boson oscillators......Page 126
6.1.1 Thermal vacuum......Page 127
6.1.2 Bogoliubov transformation......Page 128
6.1.3 Thermal operators......Page 129
6.1.4 Matrix notation......Page 132
6.2.1 Thermal vacuum......Page 133
6.2.2 Bogoliubov transformation......Page 134
6.2.3 Thermal operators......Page 135
6.2.4 Matrix notation......Page 136
6.3 TFD and spin 1/2 lattices......Page 137
6.3.1 Boson representation for the SU(2) algebra......Page 138
6.3.2 Thermo-SU(2) algebra......Page 139
7.1 The Poincar e group......Page 142
7.2 Relativistic density matrices......Page 143
7.2.1 Bosons......Page 145
7.2.2 Fermions......Page 146
7.3 The Galilei group......Page 147
7.4 Galilean density matrices......Page 149
7.5 Lagrangians......Page 152
8.1 Thermal Klein-Gordon field......Page 154
8.2 Thermal Dirac field......Page 160
8.3 Doubled notation for bosons......Page 163
8.4 Generating functional for bosons......Page 164
8.5 Generating functional for fermions......Page 166
8.6 Thermal gauge fields......Page 167
9. Scattering Process at Finite Temperature......Page 170
9.1 Scattering matrix in TFD......Page 171
9.2 Reaction rates......Page 173
9.3 Decay of particles and generalized Cutkosky rules......Page 174
9.4 Decay of Higgs meson......Page 177
9.5 The detailed balance......Page 178
9.6.2 Fermion-fermion scattering......Page 179
9.7 Fermion-boson scattering......Page 180
10.1 Ultraviolet divergences......Page 182
10.2 Regularization......Page 185
10.3 Renormalization......Page 186
10.3.1 Renormalization parts in the 4 theory......Page 192
10.3.2 The Callan-Zimanzik equation......Page 193
10.4 Bogoliubov recurrence......Page 194
10.4.1 Dimensional renormalization......Page 196
10.4.2 Other renormalization procedures......Page 198
10.5 Temperature effects......Page 200
11. Ward-Takahashi Relations and Gauge Symmetry......Page 204
11.2 Ward-Takahashi relations......Page 205
11.3.1 W-T relations for the case of n-body current amplitudes......Page 207
11.3.2 Ward-Takahashi relations at nite temperature......Page 209
11.4 Transverse Ward-Takahashi relations......Page 212
11.5 Transverse W-T relation in momentum space......Page 215
11.5.1 Full vertex for the fermion-gauge boson vertex......Page 216
11.6 W-T relations and spontaneous symmetry breaking......Page 217
Part III – Applications to Quantum Optics......Page 220
12.1 Thermalized states......Page 222
12.1.2 Thermal coherent states......Page 223
12.1.3 Thermal displaced number states......Page 225
12.1.4 Thermal squeezed states......Page 226
12.2 Physical interpretation......Page 228
12.3.1 Thermal tilde states......Page 234
12.3.2 Physical meaning of the thermal tilde states......Page 236
12.3.3 General states of HT......Page 238
13.1 Photon statistics......Page 242
13.1.1 Thermal states......Page 243
13.1.2 Thermal tilde states......Page 248
13.2 Quadrature squeezing......Page 249
13.3 Atomic population inversion......Page 252
13.4 Phase space representation......Page 258
13.4.1 Q-function of the thermal number state......Page 259
13.4.2 Wigner function of the thermal number state......Page 260
13.4.3 R-representation and nonclassical depth of the thermal number state......Page 261
13.4.4 Phase space representations of the thermal tilde number state......Page 262
14.1 Maximum entanglement......Page 266
14.2 Maximally entangled states and SU(1; 1) symmetry......Page 268
14.3 Maximally entangled states and SU(2) symmetry......Page 270
14.4 Entanglement of a system with fixed spin......Page 271
14.5 Entanglement of two-boson squeezed states......Page 273
14.6 Coherent fermion states and density matrix operators......Page 276
14.7 Entanglement of two-mode squeezed fermion states......Page 279
Part IV – Compactified Fields......Page 282
15. Compactified Fields......Page 284
15.1.1 Compactification of one space dimension......Page 285
15.1.2 Compactification of time dimension......Page 289
15.1.3 Compactification of space and time......Page 290
15.1.4 Compactification in d-dimensions......Page 292
15.2 Generalized Bogoliubov transformation......Page 293
15.3 Field theory......Page 295
15.4 Feynman rules......Page 298
16. Casimir Effect for the Electromagnetic Field......Page 300
16.1 The vacuum state of the electromagnetic field......Page 301
16.2.1 Casimir effect at zero temperature......Page 308
16.2.2 Casimir effect at non-zero temperature......Page 309
16.3 Casimir-Boyer model......Page 311
17. Casimir Effect for Fermions......Page 314
17.1 Casimir effect in......Page 315
17.2 Compactification in higher dimensions......Page 317
17.3 Casimir effect for two plates......Page 319
17.4 Casimir effect for a waveguide......Page 320
17.5 Casimir effect for a box......Page 324
17.6 Casimir effect for a non-interacting massless QCD......Page 327
18.1 Compactification of a d-dimensional subspace......Page 332
18.2 Subtraction scheme......Page 335
18.3.2 The model without Wick-ordering......Page 338
18.4 The compactified model at finite temperature: spontaneous symmetry breaking......Page 343
18.4.1 Mass behavior and critical curve......Page 344
19.1 Overview......Page 348
19.2 Second-order phase transition in superconducting films......Page 350
19.2.1 The effective potential for the Ginzburg-Landau model with one compactified dimension......Page 351
19.3 Mass renormalization and transition temperature......Page 354
19.3.1 Effect of the coupling-constant correction on Tc(L)......Page 355
19.4.1 Coupling-constant correction in the presence of an external magnetic field......Page 358
19.4.2 The gap equation and the critical curve......Page 360
20.1 Compactification of a d-dimensional subspace......Page 364
20.2 Critical behavior for wires......Page 367
20.3 Critical behavior for grains......Page 370
20.4 Boundary effects on the coupling constant......Page 373
20.5 Effects of the boundary-corrected coupling constant on the critical behavior......Page 374
20.5.1 Effects of the boundary-corrected coupling constant on the phase transition for wires......Page 377
20.5.2 Effects of the boundary-corrected coupling constant on the phase transition for grains......Page 379
20.6 Universal behavior of size-effects in second-order phase transitions......Page 382
21.1 Effective potential with compactification of a d-dimensional subspace......Page 386
21.2 The film, the wire and the grain......Page 392
Part V – Applications to Open Systems......Page 396
22.1 Wigner function for the Schr odinger field......Page 398
22.2 Wigner function for the Klein-Gordon field......Page 402
22.3 Wigner function for the Dirac field......Page 404
22.4.1 Thermo-Lie groups for classical systems......Page 406
22.4.2 SU(1; 1) and the thermal classical oscillator......Page 408
22.5 Classical unitary representations......Page 410
22.6 Liouville equation for the oscillator......Page 415
22.7 Non-relativistic symmetries in the Sch onberg-Fock space......Page 417
22.8 Classical relativistic representation......Page 419
22.9 Boltzmann equation and non-relativistic limit......Page 422
23. Real-Time Method for Nonequilibrium Quantum Mechanics......Page 424
23.1 Schr odinger, Heisenberg and Liouville pictures......Page 425
23.2 Linear model for phase transition......Page 427
23.3 Nonlinear model for phase transition......Page 429
23.3.1 Correlation functions in coherent state......Page 430
23.3.2 Correlation functions in thermal state......Page 432
23.4 Beyond the Hartree approximation for nonlinear model......Page 433
23.4.1 Beyond the Hartree approximation......Page 434
23.4.2 Stability of the Liouville-von Neumann method......Page 436
23.5 TFD for time-dependent boson system......Page 437
24. Dressed and Bare State Approaches to the Thermalization Process......Page 442
24.1 The model......Page 443
24.2 The thermalization process in bare coordinates......Page 446
24.3 Dressed coordinates and dressed states......Page 452
24.4 Thermal behavior for a cavity of arbitrary size with dressed coordinates......Page 454
24.5 The limit of arbitrarily large cavity: unbounded space......Page 456
Epilogue......Page 462
Bibliography......Page 466
Index......Page 478
