Foundations of Quantum Theory : From Classical Concepts to Operator Algebras.
| Main Author: | |
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| Format: | eBook |
| Language: | English |
| Published: |
Cham :
Springer International Publishing AG,
2017.
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| Edition: | 1st ed. |
| Series: | Fundamental Theories of Physics Series
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| Subjects: | |
| Online Access: | Click to View |
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.