Ubiquitous Quantum Structure: From Psychology to Finance / Edition 1

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Overview

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, 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 space. 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.

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Product Details

  • ISBN-13: 9783642051005
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 2/1/2010
  • Edition description: 2009
  • Edition number: 1
  • Pages: 216
  • Product dimensions: 6.20 (w) x 9.20 (h) x 0.80 (d)

Meet the Author

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.

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Table of Contents

1 Quantum-like Paradigm 1

1.1 Applications of Mathematical Apparatus of QM Outside of Physics 1

1.2 Irreducible Quantum Randomness, Copenhagen Interpretation 2

1.3 Quantum Reductionism in Biology and Cognitive Science 3

1.4 Statistical (or Ensemble) Interpretation of QM 4

1.5 No-Go Theorems 5

1.6 Einstein's and Bohr's Views on Realism 6

1.7 Quantum and Quantum-like Models 7

1.8 Quantum-like Representation Algorithm – QLRA 7

1.9 Non-Kolmogorov Probability 8

1.10 Contextual Probabilistic Model – Växjö Model 9

1.11 Experimental Verification 10

1.12 Violation of Savage's Sure Thing Principle 11

1.13 Quantum-like Description of the Financial Market 11

1.14 Quantum and Quantum-like Games 13

1.15 Terminology: Context, Contextual Probability, Contextuality 14

1.16 Formula of Total Probability 15

1.17 Formula of Total Probability with Interference Term 15

1.18 Quantum-like Representation of Contexts 16

2 Classical (Kolmogorovian) and Quantum (Born) Probability 19

2.1 Kolmogorovian Probabilistic Model 19

2.1.1 Probability Space 19

2.1.2 Conditional Probability 22

2.1.3 Formula of Total Probability 24

2.2 Probabilistic Incompatibility: Bell–Boole Inequalities 25

2.2.1 Views of Boole, Kolmogorov, and Vorob'ev 26

2.2.2 Bell's and Wigner's Inequalities 28

2.2.3 Bell-type Inequalities for Conditional Probabilities 28

2.3 Quantum Probabilistic Model 29

2.3.1 Postulates 30

2.3.2 Quantization 33

2.3.3 Interpretations of Wave Function 34

2.4 Quantum Conditional Probability 35

2.5 Interference of Probabilities in Quantum Mechanics 36

2.6 Contextual Point of View of Interference 38

2.7 Bell's Inequality in Quantum Physics 38

2.8 Växjö Interpretation of Quantum Mechanics 40

3 Contextual Probabilistic Model – Växjö model 41

3.1 Contextual Description of Observations 41

3.1.1 Contextual Probability Space and Model 41

3.1.2 Selection Contexts; Analogy with Projection Postulate 43

3.1.3 Transition Probabilities, Reference Observables 43

3.1.4 Covariance 44

3.1.5 Interpretations of Contextual Probabilities 45

3.2 Formula of Total Probability with Interference Term 46

4 Quantum-like Representation Algorithm – QLRA 49

4.1 Inversion of Born's Rule 50

4.2 QLRA: Complex Representation 51

4.3 Visualization on Bloch's Sphere 55

4.4 The Case of Non-Doubly Stochastic Matrices 57

4.5 QLRA: Hyperbolic Representation 58

4.5.1 Hyperbolic Born's Rule 58

4.5.2 Hyperbolic Hilbert Space Representation 60

4.6 Bloch's Hyperboloid 61

5 The Quantum-like Brain 65

5.1 Quantum and Quantum-like Cognitive Models 65

5.2 Interference of Minds 68

5.2.1 Cognitive and Social Contexts; Observables 68

5.2.2 Quantum-like Structure of Experimental Mental Data 69

5.2.3 Contextual Redundancy 70

5.2.4 Mental Wave Function 72

5.3 Quantum-like Projection of Mental Reality 72

5.3.1 Social Opinion Poll 72

5.3.2 Quantum-like Functioning of Neuronal Structures 73

5.4 Quantum-like Consciousness 75

5.5 The Brain as a Quantum-like Computer 76

5.6 Evolution of Mental Wave Function 76

5.6.1 Structure of a Set of Mental States 77

5.6.2 Combining Neuronal Realism with Quantum-like Formalism 78

6 Experimental Tests of Quantum-like Behavior of the Mind 79

6.1 Theoretical Foundations of Experiment 79

6.2 Gestalt Perception Theory 80

6.3 Gestalt-like Experiment for Quantum-like Behavior of the Mind 81

6.4 Analysis of Cognitive Entities 84

6.5 Description of Experiment on Image Recognition 86

6.5.1 Preparation 87

6.5.2 First Experiment: Slight Deformations Versus Short Exposure Time 87

6.5.3 Second Experiment: Essential Deformations Versus Long Exposure Time 88

6.6 Interference Effect at the Financial Market? 90

6.6.1 Supplementary (“complementary”) Stocks 90

6.6.2 Experiment Design 91

7 Quantum-like Decision Making and Disjunction Effect 93

7.1 Sure Thing Principle, Disjunction Effect 93

7.2 Quantum-like Decision Making: General Discussion and Postulates 96

7.2.1 Superposition of Choices 98

7.2.2 Parallelism of Creation and Processing of Mental Wave function 99

7.2.3 Quantum-like Rationality 99

7.2.4 Quantum-like Ethics 100

7.3 Rational Behavior, Prisoner's Dilemma 100

7.4 Contextual Analysis of Experiments with Disjunction Effect 101

7.4.1 Prisoner's Dilemma 101

7.4.2 Gambling Experiment 104

7.4.3 Exam's Result and Hawaii Experiment 105

7.5 Reason-Based Choice and Its Quantum-like Interpretation 105

7.6 Coefficients of Interference and Quantum-like Representation 106

7.7 Non-double Stochasticity of Matrices of Transition Probabilities in Cognitive Psychology 107

7.8 Decision Making 108

7.9 Bayesian Updating of Mental State Distribution 110

7.10 Mixed State Representation 112

7.11 Comparison with Standard Quantum Decision-Making Theory 112

7.12 Bayes Risk 113

7.13 Conclusion 114

8 Macroscopic Games and Quantum Logic 115

8.1 Spin-One-Half Example of a Quantum-like Game 117

8.2 Spin-One Quantum-like Game 122

8.3 Interference of Probability in Quantum-like Games 127

8.4 Wave Functions in Macroscopic Quantum-like Games 129

8.5 Spin-One-Half Game with Three Observables 132

8.6 Heisenberg's Uncertainty Relations 134

8.7 Cooperative Quantum-like Games, Entanglement 135

9 Contextual Approach to Quantum-like Macroscopic Games 137

9.1 Quantum Probability and Game Theory 137

9.2 Wine Testing Game 138

9.3 Extensive Form Game with Imperfect Information 141

9.3.1 Quantum-like Representation of the Wine Testing Game 142

9.3.2 Superposition of Preferences 143

9.3.3 Interpretation of Gambling Wave Function 143

9.3.4 The Role of Bayes Formula 144

9.3.5 Action at a Distance? 145

9.4 Wine Game with Three Players 145

9.5 Simulation of the Wine Game 146

9.6 Bell's Inequality for Averages of Payoffs 147

10 Psycho-financial Model 151

10.1 Deterministic and Stochastic Models of Financial Markets 151

10.1.1 Efficient Market Hypothesis 151

10.1.2 Deterministic Models for Dynamics of Prices 152

10.1.3 Behavioral Finance and Economics 153

10.1.4 Quantum-like Model for Behavioral Finance 154

10.2 Classical Econophysical Model of the Financial Market 155

10.2.1 Financial Phase Space 155

10.2.2 Classical Dynamics 157

10.2.3 Critique of Classical Econophysics 159

10.3 Quantum-like Econophysical Model of the Financial Market 160

10.3.1 Financial Pilot Waves 160

10.3.2 Dynamics of Prices Guided by Financial Pilot Wave 161

10.4 Application of Quantum Formalism to the Financial Market 165

10.5 Standard Deviation of Price 166

10.6 Comparison with Conventional Models of the Financial Market 167

10.6.1 Stochastic Model 167

10.6.2 Deterministic Dynamical Model 169

10.6.3 Stochastic Model and Expectations of Agents of the Financial Market 170

11 The Problem of Smoothness of Bohmian Trajectories 171

11.1 Existence Theorems for Nonsmooth Financial Forces 171

11.1.1 The Problem of Smoothness of Price Trajectories 171

11.1.2 Picard's Theorem and its Generalization 173

11.2 The Problem of Quadratic Variation 176

11.3 Singular Potentials and Forces 177

11.3.1 Example 177

11.3.2 Singular Quantum Potentials 177

11.4 Classical and Quantum Financial Randomness 178

11.4.1 Randomness of Initial Conditions 179

11.4.2 Random Financial Mass 179

11.5 Bohm–Vigier Stochastic Mechanics 180

11.6 Bohmian Model and Models with Stochastic Volatility 182

11.7 Classical and Quantum Contributions to Financial Randomness 183

12 Appendix 185

12.1 Independence 185

12.1.1 Kolmogorovian Model 185

12.1.2 Quantum Model 186

12.1.3 Växjö model 187

12.2 Proof of Wigner's Inequality 188

12.3 Projection Postulate 190

12.4 Contextual View of Kolmogorov and Quantum Models 190

12.4.1 Contextual Models Induced by the Classical (Kolmogorov) Model 190

12.4.2 Contextual Models Induced by the Quantum (Dirac-von neumann) Model 191

12.5 Generalization of Quantum Formalism 191

12.6 Bohmian Mechanics 194

References 199

Index 213

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