Contents -- Preface -- 1. The structure of the space of the physical states -- 1.1 Introduction -- 1.2 The space of the states of physical particles -- 1.3 The Weyl-Heisenberg algebra and the Fock space -- 1.4 Irreducible representations of the canonical commutation relations -- 1.5 Unitarily equivalent representations -- 1.6 The Stone-von Neumann theorem -- 1.7 Unitarily inequivalent representations -- 1.8 The deformation of Weyl-Heisenberg algebra -- 1.8.1 Self-similarity, fractals and the Fock-Bargmann representation -- 1.9 The physical particle energy and momentum operator -- 1.10 The physical Fock space and the physical fields -- Appendix A Strong limit and weak limit -- Appendix B Glauber coherent states -- Appendix C Generalized coherent states -- Appendix D q-WH algebra, coherent states and theta functions -- 2. Inequivalent representations of the canonical commutation relations -- 2.1 Introduction -- 2.2 Heisenberg fields, physical fields and the dynamical map -- 2.3 Examples of inequivalent representations -- 2.4 The Haag theorem and non-perturbative physics -- 2.5 The momentum operator -- 2.6 Time evolution and asymptotic limits -- 2.7 Inequivalent representations in flavor mixing -- Appendix E Computation of h0

3.8 Spontaneous symmetry breaking in local gauge theories -- 3.8.1 The U(1) local gauge model -- 3.8.2 The chiral gauge model -- 3.9 Finite volume effects -- 3.10 Space-time dimensionality -- Appendix I The order parameter space -- Appendix J The Mermin-Wagner-Coleman theorem -- 4. Dynamical rearrangement of symmetry and macroscopic manifestations of QFT -- 4.1 Introduction -- 4.2 Dynamical rearrangement of symmetry -- 4.2.1 SU(2) symmetry -- 4.2.2 Global U(1) symmetry -- 4.2.3 Local U(1) symmetry and the emergence of classical Maxwell equations -- 4.3 The boson transformation theorem and the nonhomogeneous boson condensation -- 4.3.1 Topological singularities, gapless modes and macroscopic observables -- 4.3.2 Defect formation in the process of symmetry breaking phase transitions -- 4.4 Group contraction and spontaneous symmetry breaking -- 4.4.1 The infrared effect -- 4.4.2 Group contraction, boson condensation and macroscopic quantum systems -- 4.4.3 The collective behavior of quantum components and group contraction -- 4.5 Quantum fluctuations and macroscopic stability -- 4.5.1 Quantum mechanical decoherence and stability of macroscopic quantum systems -- Appendix K Group contraction and Virasoro algebra -- Appendix L Phase locking in the N atom system -- 5. Thermal field theory and trajectories in the space of the representations -- 5.1 Introduction -- 5.2 Doubling the degrees of freedom -- 5.2.1 The two-slit experiment -- 5.3 Thermo Field Dynamics: A brief introduction -- 5.3.1 The propagator structure in TFD -- 5.3.2 Non-hermitian representation of TFD -- 5.3.3 TFD for fields with continuous mass spectrum -- 5.4 The q-deformed Hopf algebra and the doubling of the field degrees of freedom -- 5.5 Free energy, entropy and the arrow of time. Intrinsic thermal nature of QFT -- 5.5.1 Entropy and system-environment entanglement.

5.6 Thermal field theory and the gauge field -- 5.7 Boson condensation at finite temperature -- 5.7.1 Free energy and classical energy -- 5.8 Trajectories in the space of representations -- 6. Selected topics in thermal field theory -- 6.1 Introduction -- 6.2 The Gell-Mann-Low formula and the closed time-path formalism -- 6.3 The functional integral approach -- 6.3.1 Generating functionals for Green's functions -- 6.3.2 The Feynman-Matthews-Salam formula -- 6.3.3 More on generating functionals -- 6.4 The effective action and the Schwinger-Dyson equations -- 6.5 Imaginary-time formalism -- 6.6 Geometric background for thermal field theories -- 6.6.1 The - spacetime -- 6.6.2 Fields in - spacetime -- Appendix M Thermal Wick theorem -- Appendix N Coherent state functional integrals -- N.1 Glauber coherent states -- N.2 Generalized coherent states -- Appendix O Imaginary-time formalism and phase transitions -- O.1 Landau-Ginzburg treatment -- Appendix P Proof of Bogoliubov inequality -- 7. Topological defects as non-homogeneous condensates. I -- 7.1 Introduction -- 7.2 Quantum field dynamics and classical soliton solutions -- 7.2.1 The dynamical map and the boson transformation -- 7.2.2 The quantum coordinate -- 7.3 The 4 kink solution -- 7.3.1 The kink solution and temperature effects -- 7.3.2 The kink solution: closed time-path approach -- 7.4 The sine-Gordon solution -- 7.4.1 The quantum image of the Bäcklund transformations -- 7.5 Soliton solutions of the non-linear Schrödinger equation -- 7.5.1 The ferromagnetic chain -- 7.5.2 Non-linear Schrödinger equation with Toda lattice back-reaction potential -- 7.5.3 Ring solitons in the Scheibe aggregates -- 7.6 Fermions in topologically non-trivial background fields -- 7.7 Superfluid vortices -- 8. Topological defects as non-homogeneous condensates. II -- 8.1 Introduction.

8.2 Vortices in U(1) local gauge theory -- 8.3 Topological solitons in gauge theories -- 8.3.1 Homogeneous boson condensation -- 8.3.2 The vortex of scalar electrodynamics -- 8.3.3 The 't Hooft-Polyakov monopole -- 8.3.4 The sphaleron -- 8.4 The SU(2) instanton -- 9. Dissipation and quantization -- 9.1 Introduction -- 9.2 The exact action for damped motion -- 9.2.1 Quantum Brownian motion -- 9.3 Quantum dissipation and unitarily inequivalent representations in QFT -- 9.3.1 The arrow of time and squeezed coherent states -- 9.4 Dissipative non-commutative plane -- 9.4.1 The dissipative quantum phase interference -- 9.5 Gauge structure and thermal features in particle mixing -- 9.6 Dissipation and the many-body model of the brain -- 9.7 Quantization and dissipation -- Appendix Q Entropy and geometrical phases in neutrino mixing -- Appendix R Trajectories in the memory space -- 10. Elements of soliton theory and related concepts -- 10.1 Introduction -- 10.2 The Korteweg-de Vries soliton -- 10.3 Topological solitons in (1 + 1)-d relativistic field theories -- 10.3.1 The sine-Gordon soliton -- 10.3.2 The 4 kink -- 10.4 Topological solitons in gauge theories -- 10.4.1 The Nielsen-Olesen vortex -- 10.4.2 The 't Hooft-Polyakov monopole -- 10.5 Topological defect classification and the Kibble-Zurek mechanism for defect formation -- 10.5.1 Exact homotopy sequences -- 10.5.2 Topological defects in theories with SSB -- 10.6 Derrick theorem and defect stability -- 10.7 Bogomol'nyi bounds -- 10.8 Non-topological solitons -- 10.9 Instantons and their manifestations -- 10.9.1 Collective coordinates and fermionic zero modes -- Appendix S Backlund transformation for the sine-Gordon system -- Appendix T Elements of homotopy theory -- Bibliography -- Index.