Contents. Appendix A Strong limit and weak limit 35. Appendix B Glauber coherent states 37. Appendix C Generalized coherent states 41

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1 Contents Preface 1. The structure of the space of the physical states Introduction The space of the states of physical particles The Weyl Heisenberg algebra and the Fock space Irreducible representations of the canonical commutation relations Unitarily equivalent representations The Stone von Neumann theorem Unitarily inequivalent representations The deformation of Weyl Heisenberg algebra Self-similarity, fractals and the Fock Bargmann representation The physical particle energy and momentum operator The physical Fock space and the physical fields Appendix A Strong limit and weak limit 35 vii Appendix B Glauber coherent states 37 Appendix C Generalized coherent states 41 Appendix D q-wh algebra, coherent states and theta functions 49 xiii

2 xiv Quantum Field Theory and its Macroscopic Manifestations 2. Inequivalent representations of the canonical commutation relations Introduction Heisenberg fields, physical fields and the dynamical map Examples of inequivalent representations The Haag theorem and non-perturbative physics The momentum operator Time evolution and asymptotic limits Inequivalent representations in flavor mixing Appendix E Computation of 0 ψ i (x) α n α m 85 Appendix F Computation of 0(θ) 87 Appendix G Orthogonality of flavor vacua at different times 89 Appendix H Entanglement in neutrino oscillations Spontaneous breakdown of symmetry and the Goldstone theorem Introduction Invariance and symmetry Irreducible representations of the symmetry group Symmetry and the vacuum manifold Boson transformation and inequivalent representations Spontaneous symmetry breaking and functional integrals The Goldstone theorem U(1) symmetry SU(2) symmetry Spontaneous symmetry breaking in local gauge theories The U(1) local gauge model The chiral gauge model Finite volume effects Space-time dimensionality Appendix I The order parameter space 133 Appendix J The Mermin Wagner Coleman theorem 135

3 Contents xv 4. Dynamical rearrangement of symmetry and macroscopic manifestations of QFT Introduction Dynamical rearrangement of symmetry SU(2) symmetry Global U(1) symmetry Local U(1) symmetry and the emergence of classical Maxwell equations The boson transformation theorem and the nonhomogeneous boson condensation Topological singularities, gapless modes and macroscopic observables Defect formation in the process of symmetry breaking phase transitions Group contraction and spontaneous symmetry breaking The infrared effect Group contraction, boson condensation and macroscopic quantum systems The collective behavior of quantum components and group contraction Quantum fluctuations and macroscopic stability Quantum mechanical decoherence and stability of macroscopic quantum systems Appendix K Group contraction and Virasoro algebra 179 Appendix L Phase locking in the N atom system Thermal field theory and trajectories in the space of the representations Introduction Doubling the degrees of freedom The two-slit experiment Thermo Field Dynamics: A brief introduction The propagator structure in TFD Non-hermitian representation of TFD TFD for fields with continuous mass spectrum.. 202

4 xvi Quantum Field Theory and its Macroscopic Manifestations 5.4 The q-deformed Hopf algebra and the doubling of the field degrees of freedom Free energy, entropy and the arrow of time. Intrinsic thermal nature of QFT Entropy and system-environment entanglement Thermal field theory and the gauge field Boson condensation at finite temperature Free energy and classical energy Trajectories in the space of representations Selected topics in thermal field theory Introduction The Gell-Mann Low formula and the closed time-path formalism The functional integral approach Generating functionals for Green s functions The Feynman Matthews Salam formula More on generating functionals The effective action and the Schwinger Dyson equations Imaginary-time formalism Geometric background for thermal field theories The η-ξ spacetime Fields in η-ξ spacetime Appendix M Thermal Wick theorem 271 Appendix N Coherent state functional integrals 275 N.1 Glauber coherent states N.2 Generalized coherent states Appendix O Imaginary-time formalism and phase transitions 287 O.1 Landau Ginzburg treatment Appendix P Proof of Bogoliubov inequality Topological defects as non-homogeneous condensates. I Introduction Quantum field dynamics and classical soliton solutions. 298

5 Contents xvii The dynamical map and the boson transformation The quantum coordinate The λφ 4 kink solution The kink solution and temperature effects The kink solution: closed time-path approach The sine-gordon solution The quantum image of the Bäcklund transformations Soliton solutions of the non-linear Schrödinger equation The ferromagnetic chain Non-linear Schrödinger equation with Toda lattice back-reaction potential Ring solitons in the Scheibe aggregates Fermions in topologically non-trivial background fields Superfluid vortices Topological defects as non-homogeneous condensates. II Introduction Vortices in U(1) local gauge theory Topological solitons in gauge theories Homogeneous boson condensation The vortex of scalar electrodynamics The t Hooft Polyakov monopole The sphaleron The SU(2) instanton Dissipation and quantization Introduction The exact action for damped motion Quantum Brownian motion Quantum dissipation and unitarily inequivalent representations in QFT The arrow of time and squeezed coherent states Dissipative non-commutative plane The dissipative quantum phase interference Gauge structure and thermal features in particle mixing Dissipation and the many-body model of the brain Quantization and dissipation

6 xviii Quantum Field Theory and its Macroscopic Manifestations Appendix Q Entropy and geometrical phases in neutrino mixing 413 Appendix R Trajectories in the memory space Elements of soliton theory and related concepts Introduction The Korteweg de Vries soliton Topological solitons in (1 + 1)-d relativistic field theories The sine-gordon soliton The λφ 4 kink Topological solitons in gauge theories The Nielsen Olesen vortex The t Hooft Polyakov monopole Topological defect classification and the Kibble Zurek mechanism for defect formation Exact homotopy sequences Topological defects in theories with SSB Derrick theorem and defect stability Bogomol nyi bounds Non-topological solitons Instantons and their manifestations Collective coordinates and fermionic zero modes. 480 Appendix S Bäcklund transformation for the sine-gordon system 483 Appendix T Elements of homotopy theory 487 Bibliography 497 Index 519