Contextual Approach to Quantum Formalism

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.

List of contents

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.

About the author

Prof. Andrei Khrennikov is the director of International center for mathematical modeling in physics, engineering and cognitive science, University of Växjö, Sweden, which was created 8 years ago to perform interdisciplinary research.
Two series of conferences on quantum foundations (especially probabilistic aspects) were established on the basis of this center: "Foundations of Probability and Physics" and "Quantum Theory: Reconsideration of Foundations". These series became well known in the quantum community (including quantum information groups). Hundreds of theoreticians (physicists and mathematicians), experimenters and even philosophers participated in these conferences presenting a huge diversity of views to quantum foundations. Contacts with these people played the crucial role in creation of the present book.

Prof. Andrei Khrennikov published about 300 papers in internationally recognized journals in mathematics, physics and biology and 9 monographs – in p-adic and non-Archimedean analysis with applications to mathematical physics and cognitive sciences as well as foundations of probabilityu theory.

Summary

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.

Additional text

From the reviews:
“In this book the author presents an unorthodox account of quantum probability theory according to which all the latter’s key features can be reproduced and brought under a common formalism (known as the ‘Växjö model’) with classical statistical mechanics by using a formalism based on contextual probability … . the ideas contained in this book are of potentially very high importance.” (Dean Rickles, Mathematical Reviews, Issue 2011 i)

Report

From the reviews:
"In this book the author presents an unorthodox account of quantum probability theory according to which all the latter's key features can be reproduced and brought under a common formalism (known as the 'Växjö model') with classical statistical mechanics by using a formalism based on contextual probability ... . the ideas contained in this book are of potentially very high importance." (Dean Rickles, Mathematical Reviews, Issue 2011 i)

Product details

• Authors: Andrei Khrennikov, Andrei Y Khrennikov, Andrei Y. Khrennikov
• Publisher: Springer Netherlands
• Languages: English
• Product format: Hardback
• Released: 01.02.2011
• EAN: 9781402095924
• ISBN: 978-1-4020-9592-4
• No. of pages: 354
• Dimensions: 159 mm x 27 mm x 239 mm
• Weight: 724 g
• Illustrations: XXVIII, 354 p.
• Series: Fundamental Theories of Physics; Fundamental Theories of Physics; Fundamental Theories of Physic
• Subject: Natural sciences, medicine, IT, technology > Physics, astronomy > Theoretical physics