Contextual Approach to Quantum Formalism [electronic resource] / by Andrei Khrennikov.

Quantum and Classical Probability -- Quantum Mechanics: Postulates andInterpretations -- Classical Probability Theories -- Contextual Probability and Quantum-Like Models -- Contextual Probability and Interference -- Quantum-Like Representation of Contextual Probabilistic Model -- Ensemble Representation of Contextual Statistical Model -- Latent Quantum-Like Structure intheKolmogorov Model -- Interference of Probabilities from Law of Large Numbers -- Bells Inequality -- Probabilistic Analysis of Bells Argument -- Bells Inequality for Conditional Probabilities -- Frequency Probabilistic Analysis of Bell-Type Considerations -- Original EPR-Experiment: Local Realistic Model -- Interrelation between Classical and Quantum Probabilities -- Discrete Time Dynamics -- Noncommutative Probability in Classical Disordered Systems -- Derivation of Schrȵdingers Equation intheContextual Probabilistic Framework -- Hyperbolic Quantum Mechanics -- Representation of Contextual Statistical Model by Hyperbolic Amplitudes -- Hyperbolic Quantum Mechanics as Deformation of Conventional Classical Mechanics.

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 Bells 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 Bells 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.

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