Foundations of Quantum Mechanics I: (Theoretical and Mathematical Physics)

This book is the first volume of a two-volume work on the Foundations of Quantum Mechanics, and is intended as a new edition of the author's book Die Grundlagen der Quantenmechanik [37] which was published in 1954. In this two-volume work we will seek to obtain an improved formulation of the interpretation of quantum mechanics based on experiments. The second volume will appear shortly. Since the publication of [37] there have been several attempts to develop a basis for quantum mechanics which is, in the large part, based upon the work of J. von Neumann [38]. In particular, we mention the books ofG. W. Mackey [39], J. Jauch [40], C. Piron [41], M. Drieschner [9], and the original work ofS. P. Gudder [42], D. J. Foulis and C. H. Randall [43], and N. Zierler [44]. Here we do not seek to compare these different formulations of the foundations of quantum mechanics. We refer interested readers to [45] for such comparisons.

Table of Contents:
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.- 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.- 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.- 2 Structures in the Class of 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.- 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.- 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.- 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.- 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.