Contextual Approach to Quantum Formalism

Part I: Quantum and Classical ProbabilityChapter 1: Quantum Mechanics: Postulates and Interpretations1.1 Quantum Mechanics 1.1.1 Mathematical Basis 1.1.2 Postulates 1.2 Projection Postulate, Collapse of Wave Function, Schr ’odinger’s Cat 1.2.1 Von Neumann’s Projection Postulate 1.2.2 Collapse of Wave Function 1.2.3 Schr ’odinger’s Cat 1.2.4 L ’uders Projection Postulate 1.3 Statistical Mixtures 1.4 Von Neumann’s and L ’uders’ Postulates for Mixed States1.5 Conditional Probability1.6 Derivation of Interference of Probabilities Chapter 2: Classical Probability Theories2.1 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 Interference1.1 V ’axj ’o model: Contextual Probability 1.1.1 Contexts 1.1.2 Observables 1.1.3 Contextual Probability Space and Model 1.1.4 V ’axj ’o Models Induced by the Kolmogorov Model 1.1.5 V ’axj ’o Models Induced by QM 1.1.6 V ’axj ’o 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 Mackey’s Model1.10 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 Born’s Rule 2.3.2 Fundamental Physical Observable: Views of De Broglie and Bohm 2.3.3 $b$-Observable as Multiplication Operator2.3.4 Interference 2.4 Hilbert Space Representation of $a$-Observable2.4.1 Conventional Quantum and Quantum-Like Representations2.4.2 $a$-Basis from Interference2.4.3 Necessary and Sufficient Conditions for Born’s Rule2.4.4 Choice of Probabilistic Phases 2.4.5 Contextual Dependence of $a$-Basis2.4.6 Existence of Quantum-Like Representation with Born’s 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