
Ubiquitous Quantum Structure
From Psychology to Finance
Hardcover | 12 February 2010
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232 Pages
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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 (Vaexjoe 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.
| Quantum-like Paradigm | p. 1 |
| Applications of Mathematical Apparatus of QM Outside of Physics | p. 1 |
| Irreducible Quantum Randomness, Copenhagen Interpretation | p. 2 |
| Quantum Reductionism in Biology and Cognitive Science | p. 3 |
| Statistical (or Ensemble) Interpretation of QM | p. 4 |
| No-Go Theorems | p. 5 |
| Einstein's and Bohr's Views on Realism | p. 6 |
| Quantum and Quantum-like Models | p. 7 |
| Quantum-like Representation Algorithm – QLRA | p. 7 |
| Non-Kolmogorov Probability | p. 8 |
| Contextual Probabilistic Model – Växjö Model | p. 9 |
| Experimental Verification | p. 10 |
| Violation of Savage's Sure Thing Principle | p. 11 |
| Quantum-like Description of the Financial Market | p. 11 |
| Quantum and Quantum-like Games | p. 13 |
| Terminology: Context, Contextual Probability, Contextuality | p. 14 |
| Formula of Total Probability | p. 15 |
| Formula of Total Probability with Interference Term | p. 15 |
| Quantum-like Representation of Contexts | p. 16 |
| Classical (Kolmogorovian) and Quantum (Born) Probability | p. 19 |
| Kolmogorovian Probabilistic Model | p. 19 |
| Probability Space | p. 19 |
| Conditional Probability | p. 22 |
| Formula of Total Probability | p. 24 |
| Probabilistic Incompatibility: Bell–Boole Inequalities | p. 25 |
| Views of Boole, Kolmogorov, and Vorob'ev | p. 26 |
| Bell's and Wigner's Inequalities | p. 28 |
| Bell-type Inequalities for Conditional Probabilities | p. 28 |
| Quantum Probabilistic Model | p. 29 |
| Postulates | p. 30 |
| Quantization | p. 33 |
| Interpretations of Wave Function | p. 34 |
| Quantum Conditional Probability | p. 35 |
| Interference of Probabilities in Quantum Mechanics | p. 36 |
| Contextual Point of View of Interference | p. 38 |
| Bell's Inequality in Quantum Physics | p. 38 |
| Växjö Interpretation of Quantum Mechanics | p. 40 |
| Contextual Probabilistic Model – Växjö model | p. 41 |
| Contextual Description of Observations | p. 41 |
| Contextual Probability Space and Model | p. 41 |
| Selection Contexts; Analogy with Projection Postulate | p. 43 |
| Transition Probabilities, Reference Observables | p. 43 |
| Covariance | p. 44 |
| Interpretations of Contextual Probabilities | p. 45 |
| Formula of Total Probability with Interference Term | p. 46 |
| Quantum-like Representation Algorithm – QLRA | p. 49 |
| Inversion of Born's Rule | p. 50 |
| QLRA: Complex Representation | p. 51 |
| Visualization on Bloch's Sphere | p. 55 |
| The Case of Non-Doubly Stochastic Matrices | p. 57 |
| QLRA: Hyperbolic Representation | p. 58 |
| Hyperbolic Born's Rule | p. 58 |
| Hyperbolic Hilbert Space Representation | p. 60 |
| Bloch's Hyperboloid | p. 61 |
| The Quantum-like Brain | p. 65 |
| Quantum and Quantum-like Cognitive Models | p. 65 |
| Interference of Minds | p. 68 |
| Cognitive and Social Contexts; Observables | p. 68 |
| Quantum-like Structure of Experimental Mental Data | p. 69 |
| Contextual Redundancy | p. 70 |
| Mental Wave Function | p. 72 |
| Quantum-like Projection of Mental Reality | p. 72 |
| Social Opinion Poll | p. 72 |
| Quantum-like Functioning of Neuronal Structures | p. 73 |
| Quantum-like Consciousness | p. 75 |
| The Brain as a Quantum-like Computer | p. 76 |
| Evolution of Mental Wave Function | p. 76 |
| Structure of a Set of Mental States | p. 77 |
| Combining Neuronal Realism with Quantum-like Formalism | p. 78 |
| Experimental Tests of Quantum-like Behavior of the Mind | p. 79 |
| Theoretical Foundations of Experiment | p. 79 |
| Gestalt Perception Theory | p. 80 |
| Gestalt-like Experiment for Quantum-like Behavior of the Mind | p. 81 |
| Analysis of Cognitive Entities | p. 84 |
| Description of Experiment on Image Recognition | p. 86 |
| Preparation | p. 87 |
| First Experiment: Slight Deformations Versus Short Exposure Time | p. 87 |
| Second Experiment: Essential Deformations Versus Long Exposure Time | p. 88 |
| Interference Effect at the Financial Market? | p. 90 |
| Supplementary (“complementary”) Stocks | p. 90 |
| Experiment Design | p. 91 |
| Quantum-like Decision Making and Disjunction Effect | p. 93 |
| Sure Thing Principle, Disjunction Effect | p. 93 |
| Quantum-like Decision Making: General Discussion and Postulates | p. 96 |
| Superposition of Choices | p. 98 |
| Parallelism of Creation and Processing of Mental Wave function | p. 99 |
| Quantum-like Rationality | p. 99 |
| Quantum-like Ethics | p. 100 |
| Rational Behavior, Prisoner's Dilemma | p. 100 |
| Contextual Analysis of Experiments with Disjunction Effect | p. 101 |
| Prisoner's Dilemma | p. 101 |
| Gambling Experiment | p. 104 |
| Exam's Result and Hawaii Experiment | p. 105 |
| Reason-Based Choice and Its Quantum-like Interpretation | p. 105 |
| Coefficients of Interference and Quantum-like Representation | p. 106 |
| Non-double Stochasticity of Matrices of Transition Probabilities in Cognitive Psychology | p. 107 |
| Decision Making | p. 108 |
| Bayesian Updating of Mental State Distribution | p. 110 |
| Mixed State Representation | p. 112 |
| Comparison with Standard Quantum Decision-Making Theory | p. 112 |
| Bayes Risk | p. 113 |
| Conclusion | p. 114 |
| Macroscopic Games and Quantum Logic | p. 115 |
| Spin-One-Half Example of a Quantum-like Game | p. 117 |
| Spin-One Quantum-like Game | p. 122 |
| Interference of Probability in Quantum-like Games | p. 127 |
| Wave Functions in Macroscopic Quantum-like Games | p. 129 |
| Spin-One-Half Game with Three Observables | p. 132 |
| Heisenberg's Uncertainty Relations | p. 134 |
| Cooperative Quantum-like Games, Entanglement | p. 135 |
| Contextual Approach to Quantum-like Macroscopic Games | p. 137 |
| Quantum Probability and Game Theory | p. 137 |
| Wine Testing Game | p. 138 |
| Extensive Form Game with Imperfect Information | p. 141 |
| Quantum-like Representation of the Wine Testing Game | p. 142 |
| Superposition of Preferences | p. 143 |
| Interpretation of Gambling Wave Function | p. 143 |
| The Role of Bayes Formula | p. 144 |
| Action at a Distance? | p. 145 |
| Wine Game with Three Players | p. 145 |
| Simulation of the Wine Game | p. 146 |
| Bell's Inequality for Averages of Payoffs | p. 147 |
| Psycho-financial Model | p. 151 |
| Deterministic and Stochastic Models of Financial Markets | p. 151 |
| Efficient Market Hypothesis | p. 151 |
| Deterministic Models for Dynamics of Prices | p. 152 |
| Behavioral Finance and Economics | p. 153 |
| Quantum-like Model for Behavioral Finance | p. 154 |
| Classical Econophysical Model of the Financial Market | p. 155 |
| Financial Phase Space | p. 155 |
| Classical Dynamics | p. 157 |
| Critique of Classical Econophysics | p. 159 |
| Quantum-like Econophysical Model of the Financial Market | p. 160 |
| Financial Pilot Waves | p. 160 |
| Dynamics of Prices Guided by Financial Pilot Wave | p. 161 |
| Application of Quantum Formalism to the Financial Market | p. 165 |
| Standard Deviation of Price | p. 166 |
| Comparison with Conventional Models of the Financial Market | p. 167 |
| Stochastic Model | p. 167 |
| Deterministic Dynamical Model | p. 169 |
| Stochastic Model and Expectations of Agents of the Financial Market | p. 170 |
| The Problem of Smoothness of Bohmian Trajectories | p. 171 |
| Existence Theorems for Nonsmooth Financial Forces | p. 171 |
| The Problem of Smoothness of Price Trajectories | p. 171 |
| Picard's Theorem and its Generalization | p. 173 |
| The Problem of Quadratic Variation | p. 176 |
| Singular Potentials and Forces | p. 177 |
| Example | p. 177 |
| Singular Quantum Potentials | p. 177 |
| Classical and Quantum Financial Randomness | p. 178 |
| Randomness of Initial Conditions | p. 179 |
| Random Financial Mass | p. 179 |
| Bohm–Vigier Stochastic Mechanics | p. 180 |
| Bohmian Model and Models with Stochastic Volatility | p. 182 |
| Classical and Quantum Contributions to Financial Randomness | p. 183 |
| Appendix | p. 185 |
| Independence | p. 185 |
| Kolmogorovian Model | p. 185 |
| Quantum Model | p. 186 |
| Växjö model | p. 187 |
| Proof of Wigner's Inequality | p. 188 |
| Projection Postulate | p. 190 |
| Contextual View of Kolmogorov and Quantum Models | p. 190 |
| Contextual Models Induced by the Classical (Kolmogorov) Model | p. 190 |
| Contextual Models Induced by the Quantum (Dirac-von neumann) Model | p. 191 |
| Generalization of Quantum Formalism | p. 191 |
| Bohmian Mechanics | p. 194 |
| References | p. 199 |
| Index | p. 213 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9783642051005
ISBN-10: 3642051006
Published: 12th February 2010
Format: Hardcover
Language: English
Number of Pages: 232
Audience: General Adult
Publisher: Springer Nature B.V.
Country of Publication: DE
Dimensions (cm): 24.13 x 16.51 x 1.27
Weight (kg): 0.45
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- Non-FictionSciencePhysicsQuantum Physics & Quantum Mechanics & Quantum Field Theory
- Non-FictionMathematicsProbability & Statistics
- Non-FictionPsychologyCognition & Cognitive Psychology
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