Foundations of Quantum Mechanics I

Inhaltsverzeichnis

• I The Problem: An Axiomatic Basis for Quantum Mechanics.
• 1 The Axiomatic Formulation of a Physical Theory.
• 2 The Fundamental Domain for Quantum Mechanics.
• 3 The Measurement Problem.
• II Microsystems, Preparation, and Registration Procedures.
• 1 The Concept of a Physical Object.
• 2 Selection Procedures.
• 3 Statistical Selection Procedures.
• 4 Physical Systems.
• 4.1 Preparation Procedures.
• 4.2 Registration Procedures.
• 4.3 The Dependence of Registration upon Preparation.
• 4.4 The Concept of a Physical System.
• 4.5 The Structure of Probability Fields for Physical Systems.
• III Ensembles and Effects.
• 1 Combinations of Preparation and Registration Methods.
• 2 Mixtures and Decompositions of Ensembles and Effects.
• 3 General Laws: Preparation and Registration of Microsystems.
• 4 Properties and Pseudoproperties.
• 4.1 Properties and Physical Objects.
• 4.2 Pseudoproperties.
• 5 Ensembles and Effects in Quantum Mechanics.
• 6 Decision Effects and Faces of K.
• IV Coexistent Effects and Coexistent Decompositions.
• 1 Coexistent Effects and Observables.
• 1.1 Coexistent Registrations.
• 1.2 Coexistent Effects.
• 1.3 Commensurable Decision Effects.
• 1.4 Observables.
• 2 Structures in the Class of Observables.
• 2.1 The Spaces ?(?) and ?’(?).
• 2.2 Mixture Morphisms Corresponding to an Observable.
• 2.3 The Kernel of an Observable; Mixture of Effects for an Observable.
• 2.4 Mixtures and Decompositions of Observables.
• 2.5 Measurement Scales for Observables.
• 3 Coexistent and Complementary Observables.
• 4 Realizations of Observables.
• 5 Coexistent Decompositions of Ensembles.
• 6 Complementary Decompositions of Ensembles.
• 7 Realizations of Decompositions.
• 8 Objective Properties and Pseudoproperties of Microsystems.
• 8.1 Objective Properties of Microsystems and Superselection Rules.
• 8.2 Pseudoproperties of Microsystems.
• 8.3 Logic of Decision Effects?.
• V Transformations of Registration and Preparation Procedures. Transformations of Effects and Ensembles.
• 1 Morphisms for Selection Procedures.
• 2 Morphisms of Statistical Selection Procedures.
• 3 Morphisms of Preparation and Registration Procedures.
• 4 Morphisms of Ensembles and Effects.
• 4.1 Morphisms of Ensembles.
• 4.2 Morphisms of Effects.
• 4.3 Coexistent Operations and Coexistent Effects Morphisms.
• 5 Isomorphisms and Automorphisms of Ensembles and Effects.
• VI Representation of Groups by Means of Effect Automorphisms and Mixture Automorphisms.
• 1 Homomorphic Maps of a Group
  𝒢 in the Group 𝓐 of ?-continuous Effect Automorphisms.
• 1.1 Generation of a Representation of
  𝒢 in 𝓐 by Means of a Representation of 𝒢 by r-Automorphisms.
• 1.2 Some General Properties of a Representation of 𝒢 in 𝓐.
• 1.3 Topologies on the Group 𝓐.
• 1.4 The Representation of 𝒢 in Phase Space ?.
• 2 The 𝒢-invariant Structure Corresponding to a Group Representation.
• 3 Properties of Representations of 𝒢 which are Dependent on the Special Structure of 𝓐(?) in Quantum Mechanics.
• 3.1 The Topological Structure of the Group 𝓐(?).
• 3.2 The Topological Properties of a Representation of 𝒢.
• 3.3 Unitary and Anti-unitary Representations Up to a Factor.
• VII The Galileo Group.
• 1 The Galileo Group as a Set of Transformations of Registration Procedures Relative to Preparation Procedures.
• 2 Irreducible Representations of the Galileo Group and Their Physical Meaning.
• 3 Irreducible Representations of the Rotation Group.
• 4 Position and Momentum Observables.
• 5 Energy and Angular Momentum Observables.
• 6 Time Observable?.
• 7 Spatial Reflections (Parity Transformations).
• 8 The Problem of the Space 𝓓 for Elementary Systems.
• 9 The Problem of Differentiability.
• VIII Composite Systems.
• 1 Registrations and Effects of the Inner Structure.
• 2 Composite Systems Consisting of Two Different Elementary Systems.
• 3 Composite Systems Consisting of Two Identical Elementary Systems.
• 4 Composite Systems Consisting of Electrons and Atomic Nuclei.
• 5 The Hamiltonian Operator.
• 6 Microsystems in External Fields.
• 7 Criticism of the Description of Interaction in Quantum Mechanics and the Problem of the Space 𝓓.
• Appendix I.
• Summary of Lattice Theory.
• 1 Definition of a Lattice.
• 2 Orthomodularity.
• 3 Boolean Rings.
• 4 Set Lattices.
• Appendix II.
• Remarks about Topological and Uniform Structures.
• 1 Topological Spaces.
• 2 Uniform Spaces.
• 3 Baire Spaces.
• 4 Connectedness.
• Appendix III.
• Banach Spaces.
• 1 Linear Vector Spaces.
• 2 Normed Vector Spaces and Banach Spaces.
• 3 The Dual Space for a Banach Space.
• 4 Weak Topologies.
• 5 Linear Maps of Banach Spaces.
• 6 Ordered Vector Spaces.
• Appendix IV.
• Operators in Hubert Space.
• 1 The Hubert Space Structure Type.
• 2 Orthogonal Systems and Closed Subspaces.
• 3 The Banach Space of Bounded Operators.
• 4 Bounded Linear Forms.
• 6 Projection Operators.
• 7 Isometric and Unitary Operators.
• 8 Spectral Representation of Self-adjoint and Unitary Operators.
• 9 The Spectrum of Compact Self-adjoint Operators.
• 10 Spectral Representation of Unbounded Self-adjoint Operators.
• 11 The Trace as a Bilinear Form.
• 12 Gleason’s Theorem.
• 13 Isomorphisms and Anti-isomorphisms.
• 14 Products of Hubert Spaces.
• References.
• List of Frequently Used Symbols.
• List of Axioms.