Contextual Approach to Quantum Formalism [electronic resource] /
Khrennikov, Andrei Y.author.
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.
Description based upon print version of record.
Includes bibliographical references (p.325-347) and index.
Quantum and Classical Probability -- Quantum Mechanics: Postulates and Interpretations -- 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 in the Kolmogorov Model -- Interference of Probabilities from Law of Large Numbers -- Bell’s Inequality -- Probabilistic Analysis of Bell’s Argument -- Bell’s 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 Schrödinger’s Equation in the Contextual 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 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.
Quantum physics.
Elementary particles (Physics).
Quantum field theory.
Probabilities.
Quantum computers.
Spintronics.
Statistical physics.
Dynamical systems.
Mathematical physics.
Quantum Physics.
Elementary Particles, Quantum Field Theory.
Probability Theory and Stochastic Processes.
Quantum Information Technology, Spintronics.
Complex Systems.
Theoretical, Mathematical and Computational Physics.