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

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

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Bibliographic Details
Superior document:Fundamental Theories of Physics Series ; v.188
:
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
Online Access:
Physical Description:1 online resource (881 pages)
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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.

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