Contextual Approach to Quantum Formalism (eBook)

Prof. Andrei Khrennikov is the director of International center for mathematical modeling in physics, engineering and cognitive science, University of Växjö, Sweden, which was created 8 years ago to perform interdisciplinary research.Two series of conferences on quantum foundations (especially probabilistic aspects) were established on the basis of this center: "Foundations of Probability and Physics" and "Quantum Theory: Reconsideration of Foundations". These series became well known in the quantum community (including quantum information groups). Hundreds of theoreticians (physicists and mathematicians), experimenters and even philosophers participated in these conferences presenting a huge diversity of views to quantum foundations. Contacts with these people played the crucial role in creation of the present book.Prof. Andrei Khrennikov published about 300 papers in internationally recognized journals in mathematics, physics and biology and 9 monographs – in p-adic and non-Archimedean analysis with applications to mathematical physics and cognitive sciences as well as foundations of probabilityu theory.

Dedication 6
Preface 7
Acknowledgements 16
Contents 19
Part I Quantum and Classical Probability 27
Quantum Mechanics: Postulates and Interpretations 29
Quantum Mechanics 29
Mathematical Basis 29
Postulates 31
Projection Postulate, Collapse of Wave Function, Schrödinger's Cat 37
Von Neumann's Projection Postulate 38
Collapse of Wave Function 38
Schrödinger's Cat 39
Lüders Projection Postulate 40
Statistical Mixtures 41
Von Neumann's and Lüders' Postulates for Mixed States 43
Conditional Probability 47
Derivation of Interference of Probabilities 49
Classical Probability Theories 53
Kolmogorov Measure-Theoretic Model 54
Formalism 54
Discussion 57
Von Mises Frequency Model 59
Collective (Random Sequence) 59
Difficulties with Definition of Randomness 60
S-sequences 62
Operations for Collectives 62
Combining and Independence of Collectives 67
Part II Contextual Probability and Quantum-Like Models 70
Contextual Probability and Interference 72
Växjö Model: Contextual Probability 74
Contexts 74
Observables 74
Contextual Probability Space and Model 75
Växjö Models Induced by the Kolmogorov Model 78
Växjö Models Induced by the Quantum Model 80
Växjö Models Induced by the von Mises Model 81
Contextual Probabilistic Description of Double Slit Experiment 82
Formula of Total Probability and Measures of Supplementarity 84
Supplementary Observables 87
Principle of Supplementarity 88
Supplementarity and Kolmogorovness 89
Double Stochasticity as the Law of Probabilistic Balance 91
Probabilistically Balanced Observables 91
Symmetrically Conditioned Observables 94
Incompatibility, Supplementarity and Existence of Joint Probability Distribution 95
Joint Probability Distribution 96
Compatibility and Probabilistic Compatibility 97
Interpretational Questions 99
Contextuality 99
Realism 100
Historical Remark: Comparing with Mackey's Model 101
Subjective and Contextual Probabilities in Quantum Theory 103
Quantum-Like Representation of Contextual Probabilistic Model 105
Trigonometric, Hyperbolic, and Hyper Trigonometric Contexts 106
Quantum-Like Representation Algorithm-QLRA 108
Probabilistic Data about Context 108
Construction of Complex Probabilistic Amplitudes 109
Hilbert Space Representation of b-Observable 112
Born's Rule 112
Fundamental Physical Observable: Views of De Broglie and Bohm 113
b-Observable as Multiplication Operator 113
Interference 114
Hilbert Space Representation of a-Observable 115
Conventional Quantum and Quantum-Like Representations 115
a-Basis from Interference 116
Necessary and Sufficient Conditions for Born's Rule 117
Choice of Probabilistic Phases 119
Contextual Dependence of a-Basis 120
Existence of Quantum-Like Representation with Born's Rule for Both Reference Observables 121
``Pathologies'' 123
Properties of Mapping of Trigonometric Contexts into Complex Amplitudes 124
Classical-Like Contexts 124
Non-Injectivity of Representation Map 125
Non-Double Stochastic Matrix: Quantum-Like Representations 125
Noncommutativity of Operators Representing Observables 128
Symmetrically Conditioned Observables 129
b-Selections Are Trigonometric Contexts 130
Extension of Representation Map 133
Formalization of the Notion of Quantum-Like Representation 133
Domain of Application of Quantum-Like Representation Algorithm 137
Ensemble Representation of Contextual Statistical Model 139
Systems and Contexts 139
Interference of Probabilities: Ensemble Derivation 141
Classical and Nonclassical Probabilistic Behaviors 146
Classical Probabilistic Behavior 148
Quantum Probabilistic Behavior 149
Neither Classical nor Quantum Probabilistic Behavior 151
Hyperbolic Probabilistic Behavior 152
Latent Quantum-Like Structure in the Kolmogorov Model 155
Contextual Model with ``Continuous Observables'' 157
Interrelation of the Measure-Theoretic and Växjö Models 158
Measure-Theoretic Representation of the Växjö Model 158
Contextual Kolmogorov Model 159
Measure-Theoretic Derivation of Interference 160
Quantum-Like Representation of the Kolmogorov Model 163
Example of Quantum-Like Representation of Contextual Kolmogorovian Model 167
Contextual Kolmogorovian Probability Model 167
Quantum-Like Representation 168
Features of Quantum-Like Representation of Contextual Kolmogorov Model 172
Dispersion-Free States 174
Complex Amplitudes of Probabilities: Multi-Valued Variables 175
Växjö Models with Multi-Kolmogorovian Structure 181
Interference of Probabilities from Law of Large Numbers 183
Kolmogorovian Description of Quantum Measurements 184
Limit Theorems and Formula of Total Probability with Interference Term 186
Part III Bell's Inequality 192
Probabilistic Analysis of Bell's Argument 194
Measure-Theoretic Derivations of Bell-Type Inequalities 195
Bell's Inequality 195
Wigner's Inequality 196
Clauser-Horne-Shimony-Holt's Inequality 198
Correspondence between Classical and Quantum Statistical Models 199
Von Neumann Postulates on Classical -> Quantum Correspondence
Bell-Type No-Go Theorems 202
Range of Values (``Spectral'') Postulate 207
Contextuality 208
Non-Injectivity of Classical -> Quantum Correspondence
Bell's Inequality and Experiment 213
Bell-Contextuality and Action at a Distance 213
Bell's Inequality for Conditional Probabilities 215
Measure-Theoretic Probability Models 216
Conventional Probability Model and Classical Statistical Mechanics 216
Bell's Probability Model and Classical Statistical Mechanics 217
Confronting Bell's Classical Probabilistic Model and Quantum Mechanics 217
Wigner-Type Inequality for Conditional Probabilities 218
Impossibility of Classical Description of Spin of Single Electron 219
Frequency Probabilistic Analysis of Bell-Type Considerations 226
Frequency Probabilistic Description of Hidden Variables 228
Bell's Locality (Bell-Clauser-Horne Factorability) Condition 232
Chaos of Hidden Variables and Frequencies for Macro-Observables 234
Fluctuating Distributions of Hidden Variables 236
Generalized Bell's Inequality 241
Transmission of Information with the Aid of Dependent Collectives 242
Original EPR-Experiment: Local Realistic Model 244
Space and Arguments of Einstein, Podolsky, Rosen, and Bell 245
Bell's Local Realism 245
Einstein's Local Realism 246
Local Realist Representation for Quantum Spin Correlations 247
EPR versus Bohm and Bell 248
Bell's Theorem and Ranges of Values of Observables 249
Correlation Functions in EPR Model 250
Space-Time Dependence of Correlation Functions and Disentanglement 253
Modified Bell's Equation 253
Disentanglement 254
Role of Space-Time in EPR Argument 257
Part IV Interrelation between Classical and Quantum Probabilities 259
Discrete Time Dynamics 261
Discrete Time in Newton's Equations 263
Diffraction Pattern in a Single Slit Scattering 264
Interference in the Two-Slit Experiment for Deterministic Particles 268
Physical Interpretation of Results of Computer Simulation 273
Discrete Time Dynamics 274
Motion in Central Potential 276
Energy Levels of the Hydrogen Atom 278
Spectrum of Harmonic Oscillator 281
General Case of Arbitrary Spectrum 282
Energy Spectrum in Various Potentials 284
Discussion and Conclusion 285
Noncommutative Probability in Classical Disordered Systems 288
Noncommutative Probability and Time Averaging 289
Noncommutative Probability and Disordered Systems 293
Derivation of Schrödinger's Equation in the Contextual Probabilistic Framework 297
Representation of Contextual Probabilistic Dynamics in the Complex Hilbert Space 299
Schrödinger Dynamics and Coefficients of Supplementarity 304
Part V Hyperbolic Quantum Mechanics 309
Representation of Contextual Statistical Model by Hyperbolic Amplitudes 311
Hyperbolic Algebra 312
Hyperbolic Version of Quantum-Like Representation Algorithm 315
Hyperbolic Probability Amplitude, Hyperbolic Born's Rule 315
Hyperbolic Hilbert Space 317
Hyperbolic Hilbert Space Representation 318
Hyperbolic Quantization 323
Hyperbolic Quantum Mechanics as Deformation of Conventional Classical Mechanics 329
On the Classical Limit of Hyperbolic Quantum Mechanics 329
Ultra-Distributions and Pseudo-Differential Operators over the Hyperbolic Algebra 331
The Classical Limit of the Hyperbolic Quantum Field Theory 338
Hyperbolic Fermions and Hyperbolic Supersymmetry 341
References 343
Index 366