Contextual Approach to Quantum Formalism

Part I: Quantum and Classical Probability Chapter 1: Quantum Mechanics: Postulates and Interpretations 1.1 Quantum Mechanics 1.1.1 Mathematical Basis 1.1.2 Postulates 1.2 Projection Postulate, Collapse of Wave Function, Schrodingers Cat 1.2.1 Von Neumanns Projection Postulate 1.2.2 Collapse of Wave Function 1.2.3 Schrodingers Cat 1.2.4 Luders Projection Postulate 1.3 Statistical Mixtures 1.4 Von Neumanns and Luders Postulates for Mixed States 1.5 Conditional Probability 1.6 Derivation of Interference of Probabilities Chapter 2: Classical Probability Theories 2.1 Kolmogorov Measure-Theoretic Model 2.1.1 Formalism 2.1.2 Discussion 2.2 Von Mises Frequency Model 2.2.1 Collective (Random Sequence) 2.2.2 Difficulties with Definition of Randomness 2.2.3 $S$-sequences 2.2.4 Operations for Collectives 2.3 Combining and Independence of Collectives Part I I: Contextual Probability and\ Quantum-Like Models Chapter 1: Contextual Probability and Interference 1.1 Vaxjo model: Contextual Probability 1.1.1 Contexts 1.1.2 Observables 1.1.3 Contextual Probability Space and Model 1.1.4 Vaxjo Models Induced by the Kolmogorov Model 1.1.5 Vaxjo Models Induced by QM 1.1.6 Vaxjo Models Induced by the von Mises Model 1.2 Contextual Probabilistic Description of Double Slit Experiment 1.3 Formula of Total Probability and Measures of Supplementarity 1.4 Supplementary Observables 1.5 Principle of Supplementarity 1.6 Supplementarity and Kolmogorovness 1.6.1 Double Stochasticity as the Law of Probabilistic Balance 1.6.2 Probabilistically Balanced Observables 1.6.3 Symmetrically Conditioned Observables 1.7 Incompatibility, Supplementarity and Existence of Joint Probability Distribution 1.7.1 Joint Probability Distribution 1.7.2 Incompatible and Supplementary Observables 1.7.3 Compatibility and Probabilistic Compatibility 1.8 Interpretational Questions 1.8.1 Contextuality 1.8.2 Realism 1.9 Historical Remark: Comparing with Mackeys Model 1.10 Subjective and Contextual Probabilities in Quantum Theory Chapter 2: Quantum-Like Representation of Contextual Probabilistic Model 2.1 Trigonometric, Hyperbolic, and Hyper-Trigonometric Contexts 2.2 Quantum-Like Representation Algorithm -- QLRA 2.2.1 Probabilistic Data about Context 2.2.2 Construction of Complex Probabilistic Amplitudes 2.3 Hilbert Space Representation of $b$-Observable 2.3.1 Borns Rule 2.3.2 Fundamental Physical Observable: Views of De Broglie and Bohm 2.3.3 $b$-Observable as Multiplication Operator 2.3.4 Interference 2.4 Hilbert Space Representation of $a$-Observable 2.4.1 Conventional Quantum and Quantum-Like Representations 2.4.2 $a$-Basis from Interference 2.4.3 Necessary and Sufficient Conditions for Borns Rule 2.4.4 Choice of Probabilistic Phases 2.4.5 Contextual Dependence of $a$-Basis 2.4.6 Existence of Quantum-Like Representation with Borns Rule for Both Reference Observables 2.4.7 Pathologies 2.5 Properties of Mapping of Trigonometric Contexts into Complex Amplitudes 2.5.1 Classical-Like Contexts 2.5.2 Non Injectivity of Representation Map 2.6 Non-Double Stochastic Matrix: Quantum-Like Representations 2.7 Noncommutativity of Operators Representing Observables 2.8 Symmetrically Conditioned Observables 2.8.1 $b$-Selections are Trigonometric Contexts 2.8.2 Extension of Representation Map 2.9 Formalization of the Notion of Quantum-Like Representation 2.10 Domain of Application of Quantum-Like Representation Algorithm Chapter 3: Ensemble Representation o