Khrennikov Ubiquitous Quantum Structure

From Psychology to Finance
1. Auflage 2010
ISBN: 978-3-642-05101-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 226 Seiten

Quantum-like structure is present practically everywhere. Quantum-like (QL) models, i.e. models based on the mathematical formalism of quantum mechanics and its generalizations can be successfully applied to cognitive science, psychology, genetics, economics, finances, and game theory. This book is not about quantum mechanics as a physical theory. The short review of quantum postulates is therefore mainly of historical value: quantum mechanics is just the first example of the successful application of non-Kolmogorov probabilities, the first step towards a contextual probabilistic description of natural, biological, psychological, social, economical or financial phenomena. A general contextual probabilistic model (Växjö model) is presented. It can be used for describing probabilities in both quantum and classical (statistical) mechanics as well as in the above mentioned phenomena. This model can be represented in a quantum-like way, namely, in complex and more general Hilbert spaces. In this way quantum probability is totally demystified: Born's representation of quantum probabilities by complex probability amplitudes, wave functions, is simply a special representation of this type.

Professor Khrennikov works actively in quantum foundations concentrating his research on such fundamental problems as inter-relation of quantum and classical probability, quantum nonlocality, Bell's inequality, interference of probabilities. He was one of the first in the world who started to apply quantum mathematics outside physics - in psychology, cognitive science, genetics, economy and finances. He published about 300 papers in the most prestigious journals in physics, mathematics, biology, psychology, finances, cognitive science. He is the author of 11 monographs on foundations of probability, quantum physics, p-adic and non-Archimedean analysis and their applications. Since 2001, Prof. Khrennikov is the director of The International Center for Mathematical Modeling in Physics and Cognitive Science, University of Vaxjo Sweden. This center has already organized 10 conferences on foundations of probability and quantum physics, workshops on quantum psychology and quantum finances.

Autoren/Hrsg.

Khrennikov, Andrei Y.

Inhaltsverzeichnis

1;Preface;6
2;Acknowledgements;8
3;Contents;10
4;1 Quantum-like Paradigm;16
4.1;1.1 Applications of Mathematical Apparatus of QM Outsideof Physics;16
4.2;1.2 Irreducible Quantum Randomness, Copenhagen Interpretation;17
4.3;1.3 Quantum Reductionism in Biology and Cognitive Science;18
4.4;1.4 Statistical (or Ensemble) Interpretation of QM;19
4.5;1.5 No-Go Theorems;20
4.6;1.6 Einstein's and Bohr's Views on Realism;21
4.7;1.7 Quantum and Quantum-like Models;22
4.8;1.8 Quantum-like Representation Algorithm -- QLRA;22
4.9;1.9 Non-Kolmogorov Probability;23
4.10;1.10 Contextual Probabilistic Model -- Växjö Model;24
4.11;1.11 Experimental Verification;25
4.12;1.12 Violation of Savage's Sure Thing Principle;26
4.13;1.13 Quantum-like Description of the Financial Market;26
4.14;1.14 Quantum and Quantum-like Games;28
4.15;1.15 Terminology: Context, Contextual Probability, Contextuality;29
4.16;1.16 Formula of Total Probability;30
4.17;1.17 Formula of Total Probability with Interference Term;30
4.18;1.18 Quantum-like Representation of Contexts;31
5;2 Classical (Kolmogorovian) and Quantum (Born) Probability;33
5.1;2.1 Kolmogorovian Probabilistic Model;33
5.1.1;2.1.1 Probability Space;33
5.1.2;2.1.2 Conditional Probability;36
5.1.3;2.1.3 Formula of Total Probability;38
5.2;2.2 Probabilistic Incompatibility: Bell--Boole Inequalities;39
5.2.1;2.2.1 Views of Boole, Kolmogorov, and Vorob'ev;40
5.2.2;2.2.2 Bell's and Wigner's Inequalities;42
5.2.3;2.2.3 Bell-type Inequalities for Conditional Probabilities;42
5.3;2.3 Quantum Probabilistic Model;43
5.3.1;2.3.1 Postulates;44
5.3.2;2.3.2 Quantization;47
5.3.3;2.3.3 Interpretations of Wave Function;48
5.4;2.4 Quantum Conditional Probability;49
5.5;2.5 Interference of Probabilities in Quantum Mechanics;50
5.6;2.6 Contextual Point of View of Interference;52
5.7;2.7 Bell's Inequality in Quantum Physics;52
5.8;2.8 Växjö Interpretation of Quantum Mechanics;54
6;3 Contextual Probabilistic Model -- Växjö Model;55
6.1;3.1 Contextual Description of Observations;55
6.1.1;3.1.1 Contextual Probability Space and Model;55
6.1.2;3.1.2 Selection Contexts; Analogy with Projection Postulate;57
6.1.3;3.1.3 Transition Probabilities, Reference Observables;57
6.1.4;3.1.4 Covariance;58
6.1.5;3.1.5 Interpretations of Contextual Probabilities;59
6.2;3.2 Formula of Total Probability with Interference Term;60
7;4 Quantum-like Representation Algorithm -- QLRA;63
7.1;4.1 Inversion of Born's Rule;64
7.2;4.2 QLRA: Complex Representation;65
7.3;4.3 Visualization on Bloch's Sphere;69
7.4;4.4 The Case of Non-Doubly Stochastic Matrices;71
7.5;4.5 QLRA: Hyperbolic Representation;72
7.5.1;4.5.1 Hyperbolic Born's Rule;72
7.5.2;4.5.2 Hyperbolic Hilbert Space Representation;74
7.6;4.6 Bloch's Hyperboloid;75
8;5 The Quantum-like Brain;79
8.1;5.1 Quantum and Quantum-like Cognitive Models;79
8.2;5.2 Interference of Minds;82
8.2.1;5.2.1 Cognitive and Social Contexts; Observables;82
8.2.2;5.2.2 Quantum-like Structure of Experimental Mental Data;83
8.2.3;5.2.3 Contextual Redundancy;84
8.2.4;5.2.4 Mental Wave Function;86
8.3;5.3 Quantum-like Projection of Mental Reality;86
8.3.1;5.3.1 Social Opinion Poll;86
8.3.2;5.3.2 Quantum-like Functioning of Neuronal Structures;87
8.4;5.4 Quantum-like Consciousness;89
8.5;5.5 The Brain as a Quantum-like Computer;90
8.6;5.6 Evolution of Mental Wave Function;90
8.6.1;5.6.1 Structure of a Set of Mental States;91
8.6.2;5.6.2 Combining Neuronal Realism with Quantum-like Formalism;92
9;6 Experimental Tests of Quantum-like Behavior of the Mind;93
9.1;6.1 Theoretical Foundations of Experiment;93
9.2;6.2 Gestalt Perception Theory;94
9.3;6.3 Gestalt-like Experiment for Quantum-like Behavior of the Mind;95
9.4;6.4 Analysis of Cognitive Entities;98
9.5;6.5 Description of Experiment on Image Recognition;100
9.5.1;6.5.1 Preparation;101
9.5.2;6.5.2 First Experiment: Slight Deformations Versus ShortExposure Time;101
9.5.3;6.5.3 Second Experiment: Essential Deformations VersusLong Exposure Time;102
9.6;6.6 Interference Effect at the Financial Market?;104
9.6.1;6.6.1 Supplementary (``complementary'') Stocks;104
9.6.2;6.6.2 Experiment Design;105
10;7 Quantum-like Decision Making and Disjunction Effect;107
10.1;7.1 Sure Thing Principle, Disjunction Effect;107
10.2;7.2 Quantum-like Decision Making: General Discussion and Postulates;110
10.2.1;7.2.1 Superposition of Choices;112
10.2.2;7.2.2 Parallelism of Creation and Processing of Mental Wave function;113
10.2.3;7.2.3 Quantum-like Rationality;113
10.2.4;7.2.4 Quantum-like Ethics;114
10.3;7.3 Rational Behavior, Prisoner's Dilemma;114
10.4;7.4 Contextual Analysis of Experiments with Disjunction Effect;115
10.4.1;7.4.1 Prisoner's Dilemma;115
10.4.2;7.4.2 Gambling Experiment;118
10.4.3;7.4.3 Exam's Result and Hawaii Experiment;119
10.5;7.5 Reason-Based Choice and Its Quantum-like Interpretation;119
10.6;7.6 Coefficients of Interference and Quantum-like Representation;120
10.7;7.7 Non-double Stochasticity of Matrices of Transition Probabilities in Cognitive Psychology;121
10.8;7.8 Decision Making;122
10.9;7.9 Bayesian Updating of Mental State Distribution;124
10.10;7.10 Mixed State Representation;126
10.11;7.11 Comparison with Standard Quantum Decision-Making Theory;126
10.12;7.12 Bayes Risk;127
10.13;7.13 Conclusion;128
11;8 Macroscopic Games and Quantum Logic;129
11.1;8.1 Spin-One-Half Example of a Quantum-like Game;131
11.2;8.2 Spin-One Quantum-like Game;136
11.3;8.3 Interference of Probability in Quantum-like Games;141
11.4;8.4 Wave Functions in Macroscopic Quantum-like Games;143
11.5;8.5 Spin-One-Half Game with Three Observables;146
11.6;8.6 Heisenberg's Uncertainty Relations;148
11.7;8.7 Cooperative Quantum-like Games, Entanglement;149
12;9 Contextual Approach to Quantum-like Macroscopic Games;150
12.1;9.1 Quantum Probability and Game Theory;150
12.2;9.2 Wine Testing Game;151
12.3;9.3 Extensive Form Game with Imperfect Information;154
12.3.1;9.3.1 Quantum-like Representation of the Wine Testing Game;155
12.3.2;9.3.2 Superposition of Preferences;156
12.3.3;9.3.3 Interpretation of Gambling Wave Function;156
12.3.4;9.3.4 The Role of Bayes Formula;157
12.3.5;9.3.5 Action at a Distance?;158
12.4;9.4 Wine Game with Three Players;158
12.5;9.5 Simulation of the Wine Game;159
12.6;9.6 Bell's Inequality for Averages of Payoffs;160
13;10 Psycho-financial Model;163
13.1;10.1 Deterministic and Stochastic Models of Financial Markets;163
13.1.1;10.1.1 Efficient Market Hypothesis;163
13.1.2;10.1.2 Deterministic Models for Dynamics of Prices;164
13.1.3;10.1.3 Behavioral Finance and Economics;165
13.1.4;10.1.4 Quantum-like Model for Behavioral Finance;166
13.2;10.2 Classical Econophysical Model of the Financial Market;167
13.2.1;10.2.1 Financial Phase Space;167
13.2.2;10.2.2 Classical Dynamics;169
13.2.3;10.2.3 Critique of Classical Econophysics;171
13.3;10.3 Quantum-like Econophysical Model of the Financial Market;172
13.3.1;10.3.1 Financial Pilot Waves;172
13.3.2;10.3.2 Dynamics of Prices Guided by Financial Pilot Wave;173
13.4;10.4 Application of Quantum Formalism to the Financial Market;177
13.5;10.5 Standard Deviation of Price;178
13.6;10.6 Comparison with Conventional Models of the Financial Market;179
13.6.1;10.6.1 Stochastic Model;179
13.6.2;10.6.2 Deterministic Dynamical Model;181
13.6.3;10.6.3 Stochastic Model and Expectations of Agents of the Financial Market;182
14;11 The Problem of Smoothness of Bohmian Trajectories;183
14.1;11.1 Existence Theorems for Nonsmooth Financial Forces;183
14.1.1;11.1.1 The Problem of Smoothness of Price Trajectories;183
14.1.2;11.1.2 Picard's Theorem and its Generalization;185
14.2;11.2 The Problem of Quadratic Variation;188
14.3;11.3 Singular Potentials and Forces;189
14.3.1;11.3.1 Example;189
14.3.2;11.3.2 Singular Quantum Potentials;189
14.4;11.4 Classical and Quantum Financial Randomness;190
14.4.1;11.4.1 Randomness of Initial Conditions;191
14.4.2;11.4.2 Random Financial Mass;191
14.5;11.5 Bohm--Vigier Stochastic Mechanics;192
14.6;11.6 Bohmian Model and Models with Stochastic Volatility;194
14.7;11.7 Classical and Quantum Contributions to Financial Randomness;195
15;12 Appendix;196
15.1;12.1 Independence;196
15.1.1;12.1.1 Kolmogorovian Model;196
15.1.2;12.1.2 Quantum Model;197
15.1.3;12.1.3 Växjö Model;198
15.2;12.2 Proof of Wigner's Inequality;199
15.3;12.3 Projection Postulate;201
15.4;12.4 Contextual View of Kolmogorov and Quantum Models;201
15.4.1;12.4.1 Contextual Models Induced by the Classical (Kolmogorov) Model;201
15.4.2;12.4.2 Contextual Models Induced by the Quantum (Dirac--von neumann) Model;202
15.5;12.5 Generalization of Quantum Formalism;202
15.6;12.6 Bohmian Mechanics;205
16;References;209
17;Index;223