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An Axiomatic Basis for Quantum Mechanics
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Table of Contents

I The Problem of Formulating an Axiomatics for Quantum Mechanics.- 1 Is There an Axiomatic Basis for Quantum Mechanics?.- 2 Concepts Unsuitable in a Basis for Quantum Mechanics.- 3 Experimental Situations Describable Solely by Pretheories.- 4 Mathematical Problems.- 5 Progress to More Comprehensive Theories.- II Pretheories for Quantum Mechanics.- 1 State Space and Trajectory Space.- 2 Preparation and Registration Procedures.- 2.1 Statistical Selection Procedures.- 2.2 Preparation Procedures.- 2.3 Registration Procedures.- 2.4 Dependence of Registration on Preparation.- 3 Trajectory Preparation and Registration Procedures.- 3.1 Trajectory Effects.- 3.2 Trajectory Ensembles.- 3.3 The Dynamic Laws and the Objectivating Manner of Description.- 3.4 Dynamically Continuous Systems.- 4 Transformations of Preparation and Registration Procedures.- 4.1 Time Translations of the Trajectory Registration Procedures.- 4.2 Time Translations of the Preparation Procedures.- 4.3 Further Transformations of Preparation and Registration Procedures.- 5 The Macrosystems as Physical Objects.- III Base Sets and Fundamental Structure Terms for a Theory of Microsystems.- 1 Composite Macrosystems.- 2 Preparation and Registration Procedures for Composite Macrosystems.- 3 Directed Interactions.- 4 Action Carriers.- 5 Ensembles and Effects.- 5.1 The Problem of Combining Preparation and Registration Procedures.- 5.2 Physical Systems.- 5.3 Mixing and De-mixing of Ensembles and Effects.- 5.4 Re-elimination of the Action Carrier.- 6 Objectivating Method of Describing Experiments.- 6.1 The Method of Describing Composite Macrosystems in the Trajectory Space.- 6.2 Trajectory Effects of the Composite Systems.- 6.3 Trajectory Ensembles of the Composite Systems.- 6.4 The Structure of the Trajectory Measures for Directed Action.- 6.5 Complete Description by Trajectories.- 6.6 Use of the Interaction for the Registration of Macrosystems.- 6.7 The Relation Between the Two Forms of an Axiomatic Basis.- 7 Transport of Systems Relative to Each Other.- IV Embedding of Ensembles and Effect Sets in Topological Vector Spaces.- 1 Embedding of K, L in a Dual Pair of Vector Spaces.- 2 Uniform Structures of the Physical Imprecision on K and L.- 3 Embedding of K and L in Topologically Complete Vector Spaces.- 4 ?, ?', D, D' Considered as Ordered Vector Spaces.- 5 The Faces of K and L.- 6 Some Convergence Theorems.- V Observables and Preparators.- 1 Coexistent Effects and Observables.- 1.1 Coexistent Registrations.- 1.2 Coexistent Effects.- 1.3 Observables.- 2 Mixture Morphisms.- 3 Structures in the Class of Observables.- 3.1 The Spaces ? (?) and ?' (?) Assigned to a Boolean Ring ?.- 3.2 Mixture Morphism Corresponding to an Observable.- 3.3 The Kernel of an Observable.- 3.4 De-mixing of Observables.- 3.5 Measurement Scales of Observables and Totally Ordered Subsets of L.- 4 Coexistent and Complementary Observables.- 5 Realization of Observables.- 6 Coexistent De-mixing of Ensembles.- 7 Complementary De-mixings of Ensembles.- 8 Realizations of De-mixings.- 9 Preparators and Faces of K.- 10 Physical Objects as Action Carriers.- 11 Operations and Transpreparators.- VI Main Laws of Preparation and Registration.- 1 Main Laws for the Increase in Sensitivity of Registrations.- 1.1 Increase in Sensitivity Relative to Two Effect Procedures.- 1.2 Some Experimental and Intuitive Indications for the Law of Increase in Sensitivity.- 1.3 Decision Effects.- 1.4 The Increase in Sensitivity of an Effect.- 2 Relations Between Preparation and Registration Procedures.- 2.1 Main Law for the De-mixing of Ensembles and Related Possibilities of Registering.- 2.2 Some Consequences of Axiom AV2.- 3 The Lattice G.- 4 Commensurable Decision Effects.- 5 The Orthomodularity of G.- 6 The Main Law for Not Coexistent Registrations.- 6.1 Experimental Hints for Formulating the Main Law for Not Coexistent Registrations.- 6.2 Some Important Equivalenees.- 6.3 Formulation of the Main Law and Some Consequences.- 7 The Main Law of Quantization.- 7.1 Intuitive Indications for Formulating the Main Law of Quantization.- 7.2 Simple Consequences of the Main Law of Quantization.- VII Decision Observables and the Center.- 1 The Commutator of a Set of Decision Effects.- 2 Decision Observables.- 3 Structures in That Class of Observables Whose Range also Contains Elements of G.- 4 Commensurable Decision Observables.- 5 Decomposition of ? and ?' Relative to the Center Z.- 5.1 Reduction of the Elements of ?' by the Elements of G.- 5.2 Reduction by Center Elements.- 5.3 Classical Systems.- 5.4 Decomposition into Irreducible Parts.- 6 System Types and Super Selection Rules.- VIII Representation of ?, ?' by Banach Spaces of Operators in a Hilbert Space.- 1 The Finite Elements of G.- 2 The General Representation Theorem for Irreducible G.- 3 Some Topological Properties of G.- 4 The Representation Theorem for K, L.- 4.1 The Representation Theorem for G.- 4.2 The Ensembles and Effects.- 4.3 Coexistence, Commensurability, Uncertainty Relations, and Commutability of Operators.- 5 Some Theorems for Finite-dimensional and Irreducible ?.- A II Banach Lattices.- A III The Axiom AVid and the Minimal Decomposition Property.- A IV The Bishop-Phelps Theorem and the Ellis Theorem.- List of Frequently Used Symbols.- List of Axioms.

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