Contextual Approach to Quantum Formalism

  • Authors
  • Andrei Khrennikov

Part of the Fundamental Theories of Physics book series (FTPH, volume 160)

Table of contents

  1. Front Matter
    Pages i-xxviii
  2. Quantum and Classical Probability

    1. Front Matter
      Pages 1-2
    2. Andrei Khrennikov
      Pages 27-43
  3. Contextual Probability and Quantum-Like Models

    1. Front Matter
      Pages 45-46
    2. Andrei Khrennikov
      Pages 47-79
  4. Bell’s Inequality

    1. Front Matter
      Pages 169-170
    2. Andrei Khrennikov
      Pages 171-191
    3. Andrei Khrennikov
      Pages 193-203
    4. Andrei Khrennikov
      Pages 223-237
  5. Interrelation between Classical and Quantum Probabilities

    1. Front Matter
      Pages 239-240
    2. Andrei Khrennikov
      Pages 241-267
  6. Hyperbolic Quantum Mechanics

    1. Front Matter
      Pages 291-292

About this book

Introduction

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.

Keywords

Bell's Inequality Contextual Probabilistic Model Contextual Probability Foundation of Physics Interference of Probabilities Observable Probability distribution Probability space Quantum Formalism Quantum Physics Quantum Probability quantum mechanics statistical model

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4020-9593-1
  • Copyright Information Springer Netherlands 2009
  • Publisher Name Springer, Dordrecht
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-1-4020-9592-4
  • Online ISBN 978-1-4020-9593-1
  • About this book