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The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By taking into account the dependence of (classical) probabilities on contexts (i.e. complexes of physical conditions), one can reproduce all distinct features of quantum probabilities such as the interference of probabilities and the violation of Bell's inequality. Moreover, by starting with a formula for the interference of probabilities (which generalizes the well known classical formula of total probability), one can construct the representation of contextual probabilities by complex probability amplitudes or, in the abstract formalism, by normalized vectors of the complex Hilbert space or its hyperbolic generalization. Thus the Hilbert space representation of probabilities can be naturally derived from classical probabilistic assumptions. An important chapter of the book critically reviews known no-go theorems: the impossibility to establish a finer description of micro-phenomena than provided by quantum mechanics; and, in particular, the commonly accepted consequences of Bell's theorem (including quantum non-locality). Also, possible applications of the contextual probabilistic model and its quantum-like representation in complex Hilbert spaces in other fields (e.g. in cognitive science and psychology) are discussed.

Contextual Approach to Quantum Formalism

Part I: Quantum and Classical ProbabilityChapter 1: Quantum Mechanics: Postulates and Interpretations1.1 Quantum Mechanics1.1.1 Mathematical Basis1.1.2 Postulates1.2 Projection Postulate, Collapse of Wave Function, Schr'odinger's Cat1.2.1 Von Neumann's Projection Postulate1.2.2 Collapse of Wave Function1.2.3 Schr'odinger's Cat1.2.4 L'uders Projection Postulate1.3 Statistical Mixtures1.4 Von Neumann's and L'uders' Postulates for Mixed States1.5 Conditional Probability1.6 Derivation of Interference of ProbabilitiesChapter 2: Classical Probability Theories2.1 Kolmogorov Measure-Theoretic Model2.1.1 Formalism2.1.2 Discussion2.2 Von Mises Frequency Model2.2.1 Collective (Random Sequence)2.2.2 Difficulties with Definition of Randomness2.2.3 $S$-sequences2.2.4 Operations for Collectives2.3 Combining and Independence of CollectivesPart I I: Contextual Probability and\ Quantum-Like ModelsChapter 1: Contextual Probability and Interference1.1 V'axj'o model: Contextual Probability1.1.1 Contexts1.1.2 Observables1.1.3 Contextual Probability Space and Model1.1.4 V'axj'o Models Induced by the Kolmogorov Model1.1.5 V'axj'o Models Induced by QM1.1.6 V'axj'o Models Induced by the von Mises Model1.2 Contextual Probabilistic Description of Double Slit Experiment1.3 Formula of Total Probability and Measures of Supplementarity1.4 Supplementary Observables1.5 Principle of Supplementarity1.6 Supplementarity and Kolmogorovness1.6.1 Double Stochasticity as the Law of Probabilistic Balance1.6.2 Probabilistically Balanced Observables1.6.3 Symmetrically Conditioned Observables1.7 Incompatibility, Supplementarity and Existence of Joint Probability Distribution1.7.1 Joint Probability Distribution1.7.2 Incompatible and Supplementary Observables1.7.3 Compatibility and Probabilistic Compatibility1.8 Interpretational Questions1.8.1 Contextuality1.8.2 Realism1.9 Historical Remark: Comparing with Mackey's Model1.10 Subjective and Contextual Probabilities in Quantum TheoryChapter 2: Quantum-Like Representation of Contextual Probabilistic Model2.1 Trigonometric, Hyperbolic, and Hyper-Trigonometric Contexts2.2 Quantum-Like Representation Algorithm -- QLRA2.2.1 Probabilistic Data about Context2.2.2 Construction of Complex Probabilistic Amplitudes2.3 Hilbert Space Representation of $b$-Observable2.3.1 Born's Rule2.3.2 Fundamental Physical Observable: Views of De Broglie and Bohm2.3.3 $b$-Observable as Multiplication Operator2.3.4 Interference2.4 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 Phases2.4.5 Contextual Dependence of $a$-Basis2.4.6 Existence of Quantum-Like Representation with Born's Rule for Both Reference Observables2.4.7 ``Pathologies''2.5 Properties of Mapping of Trigonometric Contexts into Complex Amplitudes2.5.1 Classical-Like Contexts2.5.2 Non Injectivity of Representation Map2.6 Non-Double Stochastic Matrix: Quantum-Like Representations2.7 Noncommutativity of Operators Representing Observables2.8 Symmetrically Conditioned Observables2.8.1 $b$-Selections are Trigonometric Contexts2.8.2 Extension of Representation Map2.9 Formalization of the Notion of Quantum-Like Representation2.10 Domain of Application of Quantum-Like Representation AlgorithmChapter 3: Ensemble Representation o