Mathematical quantum physics : a foundational introduction / / Gabriele Nunzio Tornetta

• Record Nr.: UNISA996495163103316
• Autore: Tornetta Gabriele Nunzio
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031148125; 9783031148118
• Descrizione fisica: 1 online resource (187 pages)
• Disciplina: 016.61483
• Soggetti: Quantum theory - Mathematics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: Intro -- Preface -- References -- Contents -- 1 The Basic Postulates -- 1.1 The Measuring Process -- 1.2 The Mathematical Framework of Quantum Mechanics -- 1.2.1 Projections -- 1.2.2 State Transitions -- 1.2.3 Interference -- 1.3 The Relation with Wave Mechanics -- 1.4 Symmetries -- 1.4.1 Dynamics -- 1.4.2 Regular Representations -- 1.4.3 Superselection Rules à la Wick-Wightman-Wigner -- References -- 2 Relativistic Invariance -- 2.1 The Lorentz Group -- 2.2 The Poincaré Group -- 2.3 Massive Particles -- 2.4 The Generalised Dirac Equation -- 2.4.1 Massless Particles -- 2.4.2 The Dirac Equation -- 2.4.3 Minimal Coupling -- References -- 3 Quantum Mechanics -- 3.1 The Weyl Relations -- 3.1.1 Irreducibility of the Schrödinger Representation -- 3.2 The Heisenberg Group -- 3.2.1 Uniqueness of the Schrödinger Representation -- 3.3 Quantum Harmonic Oscillator -- 3.4 Non-commutative Functional Calculus -- References -- 4 Quantum Field Theory -- 4.1 Infinite Degrees of Freedom -- 4.2 The Segal Field -- 4.3 Second Quantisation -- 4.3.1 Fock Space Revisited -- 4.3.2 Infinite Tensor Products -- 4.3.3 Gauge Invariance -- 4.4 Canonical Anticommutation Relations -- 4.4.1 The Fermi Oscillator -- 4.4.2 Representations of the CAR Algebra -- 4.4.3 Gauge-invariant States -- 4.5 Quantum Fields -- 4.5.1 The Scalar Field -- 4.5.2 Covariance -- 4.5.3 The Spectral Condition -- 4.5.4 Irreducibility -- 4.5.5 Complexification -- 4.6 Field Algebras -- 4.6.1 Isotony -- 4.6.2 ; Locality -- 4.6.3 Canonical Commutation Relations -- 4.6.4 Additivity -- 4.6.5 Duality -- 4.6.6 Property B -- 4.7 Modular Structure -- 4.7.1 The KMS Condition -- 4.7.2 Araki Duality for Wedges -- 4.8 The Charged Scalar Free Field -- 4.8.1 Superselection Sectors -- 4.9 The Charged Free Spinor Field -- 4.9.1 The Generic j2 Spin Case -- 4.10 The Wightman Axioms -- 4.10.1 Wightman Fields.; 4.10.2 Vacuum Expectation Values -- 4.10.3 The Reconstruction Theorem -- 4.10.4 The Borchers-Uhlmann Algebra -- 4.10.5 Further Properties of Wightman Fields -- 4.10.6 Constructive Quantum Field Theory -- References -- 5 Further Topics -- 5.1 Scattering Theory -- 5.2 The DHR Theory -- 5.2.1 Superselection Sectors -- 5.2.2 Statistics -- 5.3 The DFR Models for Quantum Space-time -- 5.3.1 The C*-algebra of the Basic Model -- 5.3.2 Field Algebras on the DFR Quantum Space-time -- 5.3.3 The Scale-covariant Model -- References -- Appendix A Elements of Group Theory -- A.1 Topological Groups -- A.2 Group Cohomology -- A.2.1 Lifting of Projective Representations -- A.3 Representation Theory -- A.3.1 Induced Representations -- A.3.2 The Little Group Method -- A.3.3 The GNS Construction -- Appendix B Infinite Tensor Products -- B.1 For C*-algebras -- B.2 For Hilbert Spaces -- B.3 For Von Neumann Algebras -- B.4 Equivalences -- References -- Index.