Foundations of Quantum Mechanics I

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.