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 ...

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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.

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Editorial Reviews

From the Publisher
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)

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Product Details

  • ISBN-13: 9789048181643
  • Publisher: Springer Netherlands
  • Publication date: 12/14/2010
  • Series: Fundamental Theories of Physics Series, #160
  • Edition description: Softcover reprint of hardcover 1st ed. 2009
  • Edition number: 1
  • Pages: 354
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.79 (d)

Meet 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.

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Table of Contents

Part I: Quantum and Classical Probability

Chapter 1: Quantum Mechanics: Postulates and Interpretations
1.1 Quantum Mechanics
1.1.1 Mathematical Basis
1.1.2 Postulates
1.2 Projection Postulate, Collapse of Wave Function, Schr\"odinger's Cat
1.2.1 Von Neumann's Projection Postulate
1.2.2 Collapse of Wave Function
1.2.3 Schr\"odinger's Cat
1.2.4 L\"uders Projection Postulate
1.3 Statistical Mixtures
1.4 Von Neumann's and L\"uders' Postulates for Mixed States
1.5 Conditional Probability
1.6 Derivation of Interference of Probabilities

Chapter 2: Classical Probability Theories
2.1 Kolmogorov Measure-Theoretic Model
2.1.1 Formalism
2.1.2 Discussion
2.2 Von Mises Frequency Model
2.2.1 Collective (Random Sequence)
2.2.2 Difficulties with Definition of Randomness
2.2.3 $S$-sequences
2.2.4 Operations for Collectives
2.3 Combining and Independence of Collectives

Part I I: Contextual Probability and\\ Quantum-Like Models

Chapter 1: Contextual Probability and Interference
1.1 V\"axj\"o model: Contextual Probability
1.1.1 Contexts
1.1.2 Observables
1.1.3 Contextual Probability Space and Model
1.1.4 V\"axj\"o Models Induced by the Kolmogorov Model
1.1.5 V\"axj\"o Models Induced by QM
1.1.6 V\"axj\"o Models Induced by the von Mises Model
1.2 Contextual Probabilistic Description of Double Slit Experiment
1.3 Formula of Total Probability and Measures of Supplementarity
1.4 Supplementary Observables
1.5 Principle of Supplementarity
1.6 Supplementarity and Kolmogorovness
1.6.1 Double Shasticity as the Law of Probabilistic Balance
1.6.2 Probabilistically Balanced Observables
1.6.3 Symmetrically Conditioned Observables
1.7 Incompatibility, Supplementarity and Existence of Joint Probability Distribution
1.7.1 Joint Probability Distribution
1.7.2 Incompatible and Supplementary Observables
1.7.3 Compatibility and Probabilistic Compatibility
1.8 Interpretational Questions
1.8.1 Contextuality
1.8.2 Realism
1.9 Historical Remark: Comparing with Mackey's Model
1.10 Subjective and Contextual Probabilities in Quantum Theory

Chapter 2: Quantum-Like Representation of Contextual Probabilistic Model
2.1 Trigonometric, Hyperbolic, and Hyper-Trigonometric Contexts
2.2 Quantum-Like Representation Algorithm — QLRA
2.2.1 Probabilistic Data about Context
2.2.2 Construction of Complex Probabilistic Amplitudes
2.3 Hilbert Space Representation of $b$-Observable
2.3.1 Born's Rule
2.3.2 Fundamental Physical Observable: Views of De Broglie and Bohm
2.3.3 $b$-Observable as Multiplication Operator
2.3.4 Interference
2.4 Hilbert Space Representation of $a$-Observable
2.4.1 Conventional Quantum and Quantum-Like Representations
2.4.2 $a$-Basis from Interference
2.4.3 Necessary and Sufficient Conditions for Born's Rule
2.4.4 Choice of Probabilistic Phases
2.4.5 Contextual Dependence of $a$-Basis
2.4.6 Existence of Quantum-Like Representation with Born's Rule for Both Reference Observables
2.4.7 &'grave;Pathologies''
2.5 Properties of Mapping of Trigonometric Contexts into Complex Amplitudes
2.5.1 Classical-Like Contexts
2.5.2 Non Injectivity of Representation Map
2.6 Non-Double Shastic Matrix: Quantum-Like Representations
2.7 Noncommutativity of Operators Representing Observables
2.8 Symmetrically Conditioned Observables
2.8.1 $b$-Selections are Trigonometric Contexts
2.8.2 Extension of Representation Map
2.9 Formalization of the Notion of Quantum-Like Representation
2.10 Domain of Application of Quantum-Like Representation Algorithm

Chapter 3: Ensemble Representation of Contextual Statistical Model
3.1 Systems and Contexts
3.2 Interference of Probabilities: Ensemble Derivation
3.3 Classical and Nonclassical Probabilistic Behaviors
3.3.1 Classical Probabilistic Behavior
3.3.2 Quantum Probabilistic Behavior
3.3.3 Neither Classical nor Quantum Probabilistic Behavior
3.4 Hyperbolic probabilistic behavior

Chapter 4: Latent Quantum-Like Structure in the Kolmogorov Model
4.1 Contextual Model with &'grave;Continuous Observables''
4.1.1 Measure-Theoretic Representation of V\"axj\"o Model
4.2 Measure-Theoretic Derivation of Interference
4.3 Quantum-Like Representation of Kolmogorov Model
4.4 Example of Quantum-Like Representation of Contextual Kolmogorovian Model
4.4.1 Contextual Kolmogorovian Probability Model
4.4.2 Quantum-Like Representation
4.5 Features of Quantum-Like Representation of Contextual Kolmogorovian Model
4.6 Dispersion-Free States
4.7 Complex Amplitudes of Probabilities: Multi-Valued Variables
Chapter 5: Interference of Probabilities from Law of Large Numbers
5.1 Kolmogorovian Description of Quantum Measurements
5.2 Limit Theorems and Formula of Total Probability with Interference Term

Part III: Bell's Inequality

Chapter 1: Probabilistic Analysis of Bell's Argument
1.1 Measure-Theoretic Derivations of Bell-Type Inequalities
1.1.1 Bell's Inequality
1.1.2 Wigner's Inequality
1.1.3 Clauser-Horne-Shimony-Holt's Inequality
1.2 Correspondence between Classical and Quantum Statistical Models
1.3 Von Neumann Postulates on Classical$\to $ Quantum Correspondence
1.4 Bell-Type No-Go Theorems
1.5 Range of Values (&'grave;Spectral'') Postulate
1.6 Contextuality
1.6.1 Non-Injectivity of Classical $\to $ Quantum Correspondence
1.6.2 Bell's Inequality and Experiment
1.7 Bell-Contextuality and Action at Distance

Chapter 2 : Bell Inequality for Conditional Probabilities
2.1 Measure-Theoretic Probability Models
2.1.1 Conventional Probability Model and Classical Statistical Mechanics
2.1.2 Bell's Probability Model and Classical Statistical Mechanics
2.1.3 Confronting Bell's Classical Probabilistic Model and Quantum Mechanics
2.2 Wigner-Type Inequality for Conditional Probabilities
2.3 Impossibility of Classical Probabilistic description of Spin Projections of Single Electron

Chapter 3: Frequency Probabilistic Analysis of Bell-type Considerations
3.1 Frequency Probabilistic Description of Models with Hidden Variables for EPR-Bohm Experiment
3.2 Bell Locality (Bell-Clauser-Horne Factoribility) Condition
3.3 Chaos of Hidden Variables does not Contradict to Stabilization of Frequencies for Macro-Observables
3.4 Fluctuating Distributions of Hidden Variables
3.5 Generalized Bell's Inequality
3.6 Transmission of Information with the Aid of Dependent Collectives

Chapter 4: Local Realistic Representation for Correlations in the EPR-experiment for Position and Momentum
4.1 Space and Arguments of Einstein, Podolsky, Rosen, and Bell
4.1.1 Bell's Local Realism
4.1.2 Einstein's Local Realism
4.1.3 Local Realist Representation for Quantum Spin Correlations
4.1.4 EPR versus Bohm and Bell
4.2 Bell's Theorem and Ranges of Values of Observables
4.3 Correlation Functions in EPR Model
4.4 Space-Time Dependence of Correlation Functions and Disentanglement
4.4.1 Modified Bell's Equation
4.4.2 Disentanglement
4.5 Role of Space-Time in EPR Argument

Part IV: Interrelation between Classical and Quantum Probabilities

Chapter 1: Discrete Time Dynamics 1.1 Discrete Time in Newton's Equations
1.2 Diffraction Pattern in a Single Slit Scattering
1.3 Interference in the Two Slit Experiment for Deterministic Particles
1.4 Physical Interpretation of Results of Computer Simulation
1.5 Discrete Time Dynamics
1.6 Motion in Central Potential
1.7 Energy Levels of Hydrogen Atom
1.8 Spectrum of Harmonic Oscillator
1.9 General Case of Arbitrary Spectrum
1.10 Energy Spectrum in Various Potentials
1.11 Discussion and Conclusion

Chapter 2: Noncommutative Probability in Classical Disordered Systems
2.1 Noncommutative Probability and Time Averaging
2.2 Noncommutative Probability and Disordered Systems

Chapter 3: Derivation of Schr\"odinger's Equation in the Contextual Probabilistic Framework
3.1 Representation of Contextual Probabilistic Dynamics in the Complex Hilbert Space 3.2 Characterization of Schr\"odinger Dynamics through Dynamics of the Coefficients of Supplementarity

Part V Hyperbolic Quantum Mechanics 297

Chapter 1: Representation of Contextual Statistical Model by Hyperbolic Amplitudes 1.1 Hyperbolic Algebra
1.2 Hyperbolic Version of Quantum-like Representation Algorithm
1.2.1 Hyperbolic Probability Amplitude, Hyperbolic Born's Rule
1.2.2 Hyperbolic Hilbert Space
1.2.3 Hyperbolic Hilbert Space Representation
1.3 Hyperbolic Quantization
1.4 Experimental Verification of Hyperbolic Quantum Mechanics

Chapter 2: Hyperbolic Quantum Mechanics as Deformation of Conventional Classical Mechanics
2.1 On the Classical Limit of Hyperbolic Quantum Mechanics
2.2 Ultra-distributions and Pseudo-differential Operators over the Hyperbolic Algebra 2.3 The Classical Limit of the Hyperbolic Quantum Field Theory
2.4 Hyperbolic Fermions and Hyperbolic Supersymmetry


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