Foundations of Quantum Theory From Classical Concepts to Operator Algebras
This text studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created...
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| Hlavní autor: | |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Cham
Springer Nature
2017
Springer Open Springer International Publishing AG SpringerOpen |
| Vydání: | 1 |
| Edice: | Fundamental Theories of Physics |
| Témata: | |
| ISBN: | 9783319517773, 3319517775, 3319517767, 9783319517766 |
| On-line přístup: | Získat plný text |
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- 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)
- 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
- 11.2 The rise of modernity: Swiss approach and Decoherence -- 11.3 Insolubility theorems -- 11.4 The Flea on Schr¨odinger'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 pr´ecis 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
- 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 ¯h -- 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