Contents Quantum-like Paradigm Classical (Kolmogorovian) and Quantum (Born) Probability

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1 1 Quantum-like Paradigm Applications of Mathematical Apparatus of QM Outside ofphysics Irreducible Quantum Randomness, Copenhagen Interpretation Quantum Reductionism in Biology and Cognitive Science Statistical(orEnsemble)InterpretationofQM No-Go Theorems Einstein sandbohr sviewsonrealism Quantum and Quantum-like Models Quantum-like Representation Algorithm QLRA Non-Kolmogorov Probability Contextual Probabilistic Model Växjö Model Experimental Verification Violation of Savage s Sure Thing Principle Quantum-like Description of the Financial Market Quantum and Quantum-like Games Terminology: Context, Contextual Probability, Contextuality Formula of Total Probability Formula of Total Probability with Interference Term Quantum-like Representation of Contexts Classical (Kolmogorovian) and Quantum (Born) Probability Kolmogorovian Probabilistic Model Probability Space Conditional Probability Formula of Total Probability Probabilistic Incompatibility: Bell Boole Inequalities Views of Boole, Kolmogorov, and Vorob ev Bell s and Wigner s Inequalities Bell-type Inequalities for Conditional Probabilities Quantum Probabilistic Model Postulates ix

2 x Contents Quantization Interpretations of Wave Function Quantum Conditional Probability Interference of Probabilities in Quantum Mechanics Contextual Point of View of Interference Bell s Inequality in Quantum Physics Växjö Interpretation of Quantum Mechanics Contextual Probabilistic Model Växjö Model ContextualDescriptionofObservations Contextual Probability Space and Model Selection Contexts; Analogy with Projection Postulate Transition Probabilities, Reference Observables Covariance Interpretations of Contextual Probabilities Formula of Total Probability with Interference Term Quantum-like Representation Algorithm QLRA InversionofBorn srule QLRA:ComplexRepresentation Visualization on Bloch s Sphere The Case of Non-Doubly Stochastic Matrices QLRA: Hyperbolic Representation Hyperbolic Born s Rule Hyperbolic Hilbert Space Representation Bloch s Hyperboloid The Quantum-like Brain Quantum and Quantum-like Cognitive Models Interference of Minds Cognitive and Social Contexts; Observables Quantum-like Structure of Experimental Mental Data Contextual Redundancy Mental Wave Function Quantum-like Projection of Mental Reality Social Opinion Poll Quantum-like Functioning of Neuronal Structures Quantum-like Consciousness The Brain as a Quantum-like Computer Evolution of Mental Wave Function StructureofaSetofMentalStates Combining Neuronal Realism with Quantum-like Formalism... 78

3 xi 6 Experimental Tests of Quantum-like Behavior of the Mind Theoretical Foundations of Experiment Gestalt Perception Theory Gestalt-like Experiment for Quantum-like Behavior of the Mind Analysis of Cognitive Entities Description of Experiment on Image Recognition Preparation First Experiment: Slight Deformations Versus Short Exposure Time Second Experiment: Essential Deformations Versus Long Exposure Time Interference Effect at the Financial Market? Supplementary ( complementary ) Stocks Experiment Design Quantum-like Decision Making and Disjunction Effect SureThingPrinciple,DisjunctionEffect Quantum-like Decision Making: General Discussion andpostulates Superposition of Choices Parallelism of Creation and Processing of Mental Wavefunction Quantum-like Rationality Quantum-like Ethics Rational Behavior, Prisoner s Dilemma Contextual Analysis of Experiments with Disjunction Effect Prisoner s Dilemma Gambling Experiment Exam s Result and Hawaii Experiment Reason-Based Choice and Its Quantum-like Interpretation Coefficients of Interference and Quantum-like Representation Non-double Stochasticity of Matrices of Transition Probabilities in Cognitive Psychology DecisionMaking Bayesian Updating of Mental State Distribution MixedStateRepresentation Comparison with Standard Quantum Decision-Making Theory Bayes Risk Conclusion Macroscopic Games and Quantum Logic Spin-One-Half Example of a Quantum-like Game Spin-One Quantum-like Game Interference of Probability in Quantum-like Games

4 xii Contents 8.4 Wave Functions in Macroscopic Quantum-like Games Spin-One-HalfGamewithThreeObservables Heisenberg s Uncertainty Relations Cooperative Quantum-like Games, Entanglement Contextual Approach to Quantum-like Macroscopic Games Quantum Probability and Game Theory WineTestingGame ExtensiveFormGamewithImperfectInformation Quantum-like Representation of the Wine Testing Game Superposition of Preferences Interpretation of Gambling Wave Function The Role of Bayes Formula Action at a Distance? Wine Game with Three Players SimulationoftheWineGame Bell s Inequality for Averages of Payoffs Psycho-financial Model Deterministic and Stochastic Models of Financial Markets Efficient Market Hypothesis Deterministic Models for Dynamics of Prices Behavioral Finance and Economics Quantum-like Model for Behavioral Finance Classical Econophysical Model of the Financial Market Financial Phase Space Classical Dynamics Critique of Classical Econophysics Quantum-like Econophysical Model of the Financial Market Financial Pilot Waves Dynamics of Prices Guided by Financial Pilot Wave Application of Quantum Formalism to the Financial Market Standard Deviation of Price Comparison with Conventional Models of the Financial Market Stochastic Model Deterministic Dynamical Model Stochastic Model and Expectations of Agents of the Financial Market The Problem of Smoothness of Bohmian Trajectories Existence Theorems for Nonsmooth Financial Forces The Problem of Smoothness of Price Trajectories Picard s Theorem and its Generalization The Problem of Quadratic Variation

5 xiii 11.3 Singular Potentials and Forces Example Singular Quantum Potentials Classical and Quantum Financial Randomness Randomness of Initial Conditions Random Financial Mass Bohm Vigier Stochastic Mechanics Bohmian Model and Models with Stochastic Volatility Classical and Quantum Contributions to Financial Randomness Appendix Independence Kolmogorovian Model Quantum Model Växjö Model Proof of Wigner s Inequality ProjectionPostulate Contextual View of Kolmogorov and Quantum Models Contextual Models Induced by the Classical (Kolmogorov) Model Contextual Models Induced by the Quantum (Dirac von neumann) Model Generalization of Quantum Formalism Bohmian Mechanics References Index...213