Contents -- Preface -- Part I - General Principles -- 1. Elements of Thermodynamics -- 1.1 Kinematical aspects of thermal physics -- 1.2 Dynamical aspects of thermal physics -- 1.3 Equations of state -- 1.4 The meaning of intensive variables -- 1.5 Thermodynamical potentials -- 1.6 Gibbs-Duhem relation -- 1.7 Second derivatives -- 1.8 Example: ideal gas and generalizations -- 1.8.1 State equation for an ideal gas -- 1.8.2 The van der Waals equation -- 1.9 Stability conditions and phase transitions -- 2. Elements of Statistical Mechanics -- 2.1 Macro- and micro-physics -- 2.2 Liouville-von Neumann equation -- 2.3 Gibbs ensembles -- 2.3.1 Micro-canonical ensemble -- 2.3.2 Canonical ensemble -- 2.3.3 Grand-canonical ensemble -- 2.3.4 Equivalence among the ensembles -- 2.4 Wigner function formalism -- 3. Partition Function and Path Integral -- 3.1 Partition function and the propagator -- 3.2 Path integral in quantum mechanics -- 3.3 Classical fields -- 3.4 Canonical quantization of scalar fields -- 3.5 Path integral for a scalar field -- 3.6 Canonical quantization of the Dirac field -- 3.7 Path integral for the Dirac field -- 4. Zero Temperature Interacting Fields -- 4.1 Generating functional for bosons -- 4.1.1 Feynman rules in momentum space -- 4.2 The effective action -- 4.3 Gauge fields -- 4.4 Generating functional for gauge fields -- 4.5 U(1) gauge theory -- 4.6 SU(3) gauge theory -- 4.7 Scattering amplitudes -- 4.8 S-matrix in the canonical approach -- Part II - Thermal Fields -- 5. Thermo eld Dynamics: Kinematical Symmetry Algebraic Basis -- 5.1 Thermal Hilbert space -- 5.2 The meaning of the doubling: thermo-algebras -- 5.2.1 Generators of symmetry and observables -- 5.2.2 Doubled Lie algebra -- 5.2.3 Tilde conjugation rules -- 5.3 Tilde and non-tilde operators -- 5.4 Liouville-von Neumann equation.

5.5 Physical implications of thermo-algebras -- 6. Thermal Oscillators: Bosons and Fermions -- 6.1 Boson oscillators -- 6.1.1 Thermal vacuum -- 6.1.2 Bogoliubov transformation -- 6.1.3 Thermal operators -- 6.1.4 Matrix notation -- 6.2 Fermion oscillators -- 6.2.1 Thermal vacuum -- 6.2.2 Bogoliubov transformation -- 6.2.3 Thermal operators -- 6.2.4 Matrix notation -- 6.3 TFD and spin 1/2 lattices -- 6.3.1 Boson representation for the SU(2) algebra -- 6.3.2 Thermo-SU(2) algebra -- 7. Thermal Poincar e and Galilei Groups -- 7.1 The Poincar e group -- 7.2 Relativistic density matrices -- 7.2.1 Bosons -- 7.2.2 Fermions -- 7.3 The Galilei group -- 7.4 Galilean density matrices -- 7.5 Lagrangians -- 8. Thermal Propagator -- 8.1 Thermal Klein-Gordon field -- 8.2 Thermal Dirac field -- 8.3 Doubled notation for bosons -- 8.4 Generating functional for bosons -- 8.5 Generating functional for fermions -- 8.6 Thermal gauge fields -- 9. Scattering Process at Finite Temperature -- 9.1 Scattering matrix in TFD -- 9.2 Reaction rates -- 9.3 Decay of particles and generalized Cutkosky rules -- 9.4 Decay of Higgs meson -- 9.5 The detailed balance -- 9.6 Scattering cross-section of 1 + 2 ! 10 + 20 -- 9.6.1 Boson-boson scattering -- 9.6.2 Fermion-fermion scattering -- 9.7 Fermion-boson scattering -- 10. Topics on Renormalization Theory -- 10.1 Ultraviolet divergences -- 10.2 Regularization -- 10.3 Renormalization -- 10.3.1 Renormalization parts in the 4 theory -- 10.3.2 The Callan-Zimanzik equation -- 10.4 Bogoliubov recurrence -- 10.4.1 Dimensional renormalization -- 10.4.2 Other renormalization procedures -- 10.4.3 Borel summability -- 10.5 Temperature effects -- 11. Ward-Takahashi Relations and Gauge Symmetry -- 11.1 Ward relation -- 11.2 Ward-Takahashi relations -- 11.3 Applications of generalized Ward-Takahashi relations.

11.3.1 W-T relations for the case of n-body current amplitudes -- 11.3.2 Ward-Takahashi relations at nite temperature -- 11.4 Transverse Ward-Takahashi relations -- 11.5 Transverse W-T relation in momentum space -- 11.5.1 Full vertex for the fermion-gauge boson vertex -- 11.5.2 Tranverse W-T relation for axial current -- 11.5.3 Transverse W-T relation at nite temperature -- 11.6 W-T relations and spontaneous symmetry breaking -- Part III - Applications to Quantum Optics -- 12. Thermalized States of a Field Mode -- 12.1 Thermalized states -- 12.1.1 Thermal number states -- 12.1.2 Thermal coherent states -- 12.1.3 Thermal displaced number states -- 12.1.4 Thermal squeezed states -- 12.2 Physical interpretation -- 12.3 Other possibilities of thermalized states -- 12.3.1 Thermal tilde states -- 12.3.2 Physical meaning of the thermal tilde states -- 12.3.3 General states of HT -- 13. Nonclassical Properties of Thermal Quantum States -- 13.1 Photon statistics -- 13.1.1 Thermal states -- 13.1.2 Thermal tilde states -- 13.2 Quadrature squeezing -- 13.3 Atomic population inversion -- 13.4 Phase space representation -- 13.4.1 Q-function of the thermal number state -- 13.4.2 Wigner function of the thermal number state -- 13.4.3 R-representation and nonclassical depth of the thermal number state -- 13.4.4 Phase space representations of the thermal tilde number state -- 14. SU(2) and SU(1 -- 1) Systems: Entanglement -- 14.1 Maximum entanglement -- 14.2 Maximally entangled states and SU(1 -- 1) symmetry -- 14.3 Maximally entangled states and SU(2) symmetry -- 14.4 Entanglement of a system with fixed spin -- 14.5 Entanglement of two-boson squeezed states -- 14.6 Coherent fermion states and density matrix operators -- 14.7 Entanglement of two-mode squeezed fermion states -- Part IV - Compactified Fields -- 15. Compactified Fields.

15.1 Compactification and topology -- 15.1.1 Compactification of one space dimension -- 15.1.2 Compactification of time dimension -- 15.1.3 Compactification of space and time -- 15.1.4 Compactification in d-dimensions -- 15.2 Generalized Bogoliubov transformation -- 15.3 Field theory -- 15.4 Feynman rules -- 16. Casimir Effect for the Electromagnetic Field -- 16.1 The vacuum state of the electromagnetic field -- 16.2 The Casimir effect -- 16.2.1 Casimir effect at zero temperature -- 16.2.2 Casimir effect at non-zero temperature -- 16.3 Casimir-Boyer model -- 17. Casimir Effect for Fermions -- 17.1 Casimir effect in -- 17.2 Compactification in higher dimensions -- 17.3 Casimir effect for two plates -- 17.4 Casimir effect for a waveguide -- 17.5 Casimir effect for a box -- 17.6 Casimir effect for a non-interacting massless QCD -- 18. Compactified '4 Theory -- 18.1 Compactification of a d-dimensional subspace -- 18.2 Subtraction scheme -- 18.3 The zero-temperature compacti ed model -- 18.3.1 Wick-ordered model -- 18.3.2 The model without Wick-ordering -- 18.4 The compactified model at finite temperature: spontaneous symmetry breaking -- 18.4.1 Mass behavior and critical curve -- 19. Phase Transitions in Con ned Systems: Application to Superconducting Films -- 19.1 Overview -- 19.2 Second-order phase transition in superconducting films -- 19.2.1 The effective potential for the Ginzburg-Landau model with one compactified dimension -- 19.3 Mass renormalization and transition temperature -- 19.3.1 Effect of the coupling-constant correction on Tc(L) -- 19.4 Critical behavior of type-II superconducting films in a magnetic field -- 19.4.1 Coupling-constant correction in the presence of an external magnetic field -- 19.4.2 The gap equation and the critical curve -- 20. Second-Order Phase Transition in Wires and Grains.

20.1 Compactification of a d-dimensional subspace -- 20.2 Critical behavior for wires -- 20.3 Critical behavior for grains -- 20.4 Boundary effects on the coupling constant -- 20.5 Effects of the boundary-corrected coupling constant on the critical behavior -- 20.5.1 Effects of the boundary-corrected coupling constant on the phase transition for wires -- 20.5.2 Effects of the boundary-corrected coupling constant on the phase transition for grains -- 20.6 Universal behavior of size-effects in second-order phase transitions -- 21. First-Order Phase Transitions in Confined Systems -- 21.1 Effective potential with compactification of a d-dimensional subspace -- 21.2 The film, the wire and the grain -- Part V - Applications to Open Systems -- 22. Thermo-Algebras in Phase Space: Quantum and Classical Systems -- 22.1 Wigner function for the Schr odinger field -- 22.2 Wigner function for the Klein-Gordon field -- 22.3 Wigner function for the Dirac field -- 22.4 Representations for classical systems -- 22.4.1 Thermo-Lie groups for classical systems -- 22.4.2 SU(1 -- 1) and the thermal classical oscillator -- 22.5 Classical unitary representations -- 22.6 Liouville equation for the oscillator -- 22.7 Non-relativistic symmetries in the Sch onberg-Fock space -- 22.8 Classical relativistic representation -- 22.9 Boltzmann equation and non-relativistic limit -- 23. Real-Time Method for Nonequilibrium Quantum Mechanics -- 23.1 Schr odinger, Heisenberg and Liouville pictures -- 23.2 Linear model for phase transition -- 23.3 Nonlinear model for phase transition -- 23.3.1 Correlation functions in coherent state -- 23.3.2 Correlation functions in thermal state -- 23.4 Beyond the Hartree approximation for nonlinear model -- 23.4.1 Beyond the Hartree approximation -- 23.4.2 Stability of the Liouville-von Neumann method -- 23.5 TFD for time-dependent boson system.

24. Dressed and Bare State Approaches to the Thermalization Process.