Foundations of Quantum Theory : : From Classical Concepts to Operator Algebras.

• Superior document: Fundamental Theories of Physics Series ; v.188
• Author: Landsman, Klaas.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2017.; ©2017.
• Year of Publication: 2017
• Edition: 1st ed.
• Language: English
• Series: Fundamental Theories of Physics Series
• Physical Description: 1 online resource (881 pages)

Table of Contents

Intro -- Preface -- Contents -- Introduction -- Part I C0(X) and B(H) -- 1 Classical physics on a finite phase space -- 1.1 Basic constructions of probability theory -- 1.2 Classical observables and states -- 1.3 Pure states and transition probabilities -- 1.4 The logic of classical mechanics -- 1.5 The GNS-construction for C(X) -- Notes -- 2 Quantum mechanics on a finite-dimensional Hilbert space -- 2.1 Quantum probability theory and the Born rule -- 2.2 Quantum observables and states -- 2.3 Pure states in quantum mechanics -- 2.4 The GNS-construction for matrices -- 2.5 The Born rule from Bohrification -- 2.6 The Kadison-Singer Problem -- 2.7 Gleason's Theorem -- 2.8 Proof of Gleason's Theorem -- 2.9 Effects and Busch's Theorem -- 2.10 The quantum logic of Birkhoff and von Neumann -- Notes -- 3 Classical physics on a general phase space -- 3.1 Vector fields and their flows -- 3.2 Poisson brackets and Hamiltonian vector fields -- 3.3 Symmetries of Poisson manifolds -- 3.4 The momentum map -- Notes -- 4 Quantum physics on a general Hilbert space -- 4.1 The Born rule from Bohrification (II) -- 4.2 Density operators and normal states -- 4.3 The Kadison-Singer Conjecture -- 4.4 Gleason's Theorem in arbitrary dimension -- Notes -- 5 Symmetry in quantum mechanics -- 5.1 Six basic mathematical structures of quantum mechanics -- 5.2 The case -- 5.3 Equivalence between the six symmetry theorems -- 5.4 Proof of Jordan's Theorem -- 5.5 Proof of Wigner's Theorem -- 5.6 Some abstract representation theory -- 5.7 Representations of Lie groups and Lie algebras -- 5.8 Irreducible representations of -- 5.9 Irreducible representations of compact Lie groups -- 5.10 Symmetry groups and projective representations -- 5.11 Position, momentum, and free Hamiltonian -- 5.12 Stone's Theorem -- Notes -- Part II Between C0(X) and B(H).

6 Classical models of quantum mechanics -- 6.1 From von Neumann to Kochen-Specker -- 6.2 The Free Will Theorem -- 6.3 Philosophical intermezzo: Free will in the Free Will Theorem -- 6.4 Technical intermezzo: The GHZ-Theorem -- 6.5 Bell's theorems -- 6.6 The Colbeck-Renner Theorem -- Notes -- 7 Limits: Small h̄ -- 7.1 Deformation quantization -- 7.2 Quantization and internal symmetry -- 7.3 Quantization and external symmetry -- 7.4 Intermezzo: The Big Picture -- 7.5 Induced representations and the imprimitivity theorem -- 7.6 Representations of semi-direct products -- 7.7 Quantization and permutation symmetry -- Notes -- 8 Limits: large N -- 8.1 Large quantum numbers -- 8.2 Large systems -- 8.3 Quantum de Finetti Theorem -- 8.4 Frequency interpretation of probability and Born rule -- 8.5 Quantum spin systems: Quasi-local C*-algebras -- 8.6 Quantum spin systems: Bundles of C*-algebras -- Notes -- 9 Symmetry in algebraic quantum theory -- 9.1 Symmetries of C*-algebras and Hamhalter's Theorem -- 9.2 Unitary implementability of symmetries -- 9.3 Motion in space and in time -- 9.4 Ground states of quantum systems -- 9.5 Ground states and equilibrium states of classical spin systems -- 9.6 Equilibrium (KMS) states of quantum systems -- Notes -- 10 Spontaneous Symmetry Breaking -- 10.1 Spontaneous symmetry breaking: The double well -- 10.2 Spontaneous symmetry breaking: The flea -- 10.3 Spontaneous symmetry breaking in quantum spin systems -- 10.4 Spontaneous symmetry breaking for short-range forces -- 10.5 Ground state(s) of the quantum Ising chain -- 10.6 Exact solution of the quantum Ising chain: -- 10.7 Exact solution of the quantum Ising chain: -- 10.8 Spontaneous symmetry breaking in mean-field theories -- 10.9 The Goldstone Theorem -- 10.10 The Higgs mechanism -- Notes -- 11 The measurement problem -- 11.1 The rise of orthodoxy.

11.2 The rise of modernity: Swiss approach and Decoherence -- 11.3 Insolubility theorems -- 11.4 The Flea on Schrödinger's Cat -- Notes -- 12 Topos theory and quantum logic -- 12.1 C*-algebras in a topos -- 12.2 The Gelfand spectrum in constructive mathematics -- 12.3 Internal Gelfand spectrum and intuitionistic quantum logic -- 12.4 Internal Gelfand spectrum for arbitrary C*-algebras -- 12.5 "Daseinisation" and Kochen-Specker Theorem -- Notes -- Appendix A Finite-dimensional Hilbert spaces -- A.1 Basic definitions -- A.2 Functionals and the adjoint -- A.3 Projections -- A.4 Spectral theory -- A.5 Positive operators and the trace -- Notes -- Appendix B Basic functional analysis -- B.1 Completeness -- B.2 lp spaces -- B.3 Banach spaces of continuous functions -- B.4 Basic measure theory -- B.5 Measure theory on locally compact Hausdorff spaces -- B.6 Lp spaces -- B.7 Morphisms and isomorphisms of Banach spaces -- B.8 The Hahn-Banach Theorem -- B.9 Duality -- B.10 The Krein-Milman Theorem -- B.11 Choquet's Theorem -- B.12 A précis of infinite-dimensional Hilbert space -- B.13 Operators on infinite-dimensional Hilbert space -- B.14 Basic spectral theory -- B.15 The spectral theorem -- B.16 Abelian ∗-algebras in B(H) -- B.17 Classification of maximal abelian ∗-algebras in B(H) -- B.18 Compact operators -- B.19 Spectral theory for self-adjoint compact operators -- B.20 The trace -- B.21 Spectral theory for unbounded self-adjoint operators -- Notes -- Appendix C Operator algebras -- C.1 Basic definitions and examples -- C.2 Gelfand isomorphism -- C.3 Gelfand duality -- C.4 Gelfand isomorphism and spectral theory -- C.5 C*-algebras without unit: general theory -- C.6 C*-algebras without unit: commutative case -- C.7 Positivity in C*-algebras -- C.8 Ideals in Banach algebras -- C.9 Ideals in C*-algebras -- C.10 Hilbert C*-modules and multiplier algebras.

C.11 Gelfand topology as a frame -- C.12 The structure of C*-algebras -- C.13 Tensor products of Hilbert spaces and C*-algebras -- C.14 Inductive limits and infinite tensor products of C*-algebras -- C.15 Gelfand isomorphism and Fourier theory -- C.16 Intermezzo: Lie groupoids -- C.17 C*-algebras associated to Lie groupoids -- C.18 Group C*-algebras and crossed product algebras -- C.19 Continuous bundles of C*-algebras -- C.20 von Neumann algebras and the σ-weak topology -- C.21 Projections in von Neumann algebras -- C.22 The Murray-von Neumann classification of factors -- C.23 Classification of hyperfinite factors -- C.24 Other special classes of C*-algebras -- C.25 Jordan algebras and (pure) state spaces of C*-algebras -- Notes -- Appendix D Lattices and logic -- D.1 Order theory and lattices -- D.2 Propositional logic -- D.3 Intuitionistic propositional logic -- D.4 First-order (predicate) logic -- D.5 Arithmetic and set theory -- Notes -- Appendix E Category theory and topos theory -- E.1 Basic definitions -- E.2 Toposes and functor categories -- E.3 Subobjects and Heyting algebras in a topos -- E.4 Internal frames and locales in sheaf toposes -- E.5 Internal language of a topos -- Notes -- References -- Index.