Ubiquitous Quantum Structure (eBook)

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.

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