Quantum Social Science

Contents......Page 9
Foreword......Page 15
Preface......Page 19
Acknowledgements......Page 21
List of symbols......Page 23
I Physics concepts in social science? A discussion......Page 27
1.1 Newtonian mechanics......Page 29
1.2 References......Page 32
1.3 The Hamiltonian formalism......Page 33
1.4 Statistical mechanics and the Liouville equation......Page 35
1.5 The classical kingdom . . .......Page 37
1.7 Classical fields......Page 38
1.9 The Born–Sommerfeld quantization......Page 40
1.11 Theory of quantum waves......Page 43
1.13 Heisenberg’s symbolic calculus......Page 44
1.14 Heisenbergian symbolism in physics: a version of symbolism in art......Page 49
1.16 Completeness of quantum mechanics and a possibility to apply quantum mechanics outside of physics......Page 50
1.18 Brownian motion......Page 54
1.20 The Schr¨odinger equation......Page 57
1.22 Do not be afraid of no-go theorems!......Page 61
1.23 References......Page 62
2.1 Science and social science: econophysics?......Page 63
2.3 The physics-based “Fokker–Planck” PDE in economics......Page 66
2.5 Potential and kinetic energy in social science......Page 67
2.6 References......Page 68
2.7 The backward Kolmogorov PDE in finance......Page 69
2.8 References......Page 74
2.9 What a weird world! Martingales and fake probabilities......Page 75
2.11 Whetting the quantum appetite......Page 78
2.12 References......Page 79
3.1 What is quantum social science?......Page 80
3.2 Key findings from quantum social science......Page 88
3.3 References......Page 90
II Mathematics and physics preliminaries......Page 95
4.1 Linear spaces......Page 97
4.3 State space: Hilbert space......Page 98
4.5 Operators......Page 99
4.7 Dirac brakets and bras and kets......Page 101
4.8 References......Page 102
4.11 Hermiticity......Page 103
4.13 Projection operators......Page 104
4.16 ODEs and PDEs......Page 105
4.18 Basics of stochastic mathematics, Brownian motion, non-arbitrage condition, It\Zo’s Lemma......Page 106
4.19 References......Page 108
5.1 Mathematical formalism of quantum mechanics: brief introduction......Page 110
5.2 References......Page 115
5.3 Double slit experiment: rationale for the existence of probability waves......Page 116
5.5 Quantum mechanical postulates......Page 118
5.6 References......Page 120
5.8 References......Page 121
5.11 Heisenberg’s uncertainty principle......Page 122
5.13 The time-dependent and time-independent Schr¨odinger PDE......Page 123
5.15 Classical limit ideas: Ehrenfest’s approach and the correspondence principle......Page 124
5.16 References......Page 126
6.1 Short introduction to Bohmian mechanics......Page 128
6.3 Mathematical formalism......Page 129
6.4 References......Page 131
6.5 Non-locality......Page 132
6.7 Criticisms of Bohmian mechanics......Page 133
6.8 References......Page 134
III Quantum probabilistic effects in psychology: basic questions and answers......Page 137
7.1 Decision making in social science: general overview......Page 139
7.2 References......Page 142
7.3 Modeling risk: some basic approaches......Page 143
7.5 Possible remedies to the paradox: a brief discussion......Page 146
7.7 The role of the law of total probability (LTP): a brief overview......Page 148
7.8 Reference......Page 149
8.1 Classical decision making and the Bayesian approach......Page 150
8.3 Non-classical decision making: violation of the LTP (law of total probability) and the quantum Bayesian approach......Page 151
8.4 Contextual probabilistic formalization......Page 154
8.6 Savage sure-thing principle......Page 158
8.7 Behavioral games: Prisoner’s Dilemma......Page 160
8.8 Violation of rationality in the experiments of Shafir and Tversky......Page 161
8.9 Prisoner’s dilemma-type experiment: Shafir and Tversky......Page 162
8.10 Violation of double stochasticity for matrices of transition probabilities......Page 163
8.11 Prisoner’s dilemma-type experiment: Croson......Page 164
8.12 Gambling experiment – 1: Tversky and Shafir......Page 165
8.14 The Hawaii vacation experiment......Page 167
8.15 Non-classicality of statistical data: non-zero coefficients of interference......Page 168
8.16 The constructive wave function approach and fit to data from the experiments of Shafir and Tversky......Page 170
8.17 Other experiments......Page 171
8.18 References......Page 178
9.2 Two-player game and rational behavior......Page 181
9.3 Construction of a mental state......Page 182
9.4 A process of decision making......Page 184
9.5 Example: decision making in PD......Page 187
Appendix 1: Channels and liftings......Page 188
Appendix 2: Quantum Markov chain description of data from experiments in cognitive psychology......Page 189
9.6 References......Page 196
IV Other quantum probabilistic effects in economics, finance, and brain sciences......Page 197
10.1 Relevance of the concepts of efficiency and non-arbitrage: a brief discussion......Page 199
10.2 References......Page 202
10.3 George Soros’ interpretation of the crisis and the use of classical quantum physics in finance......Page 203
10.4 References......Page 207
10.5 The need for an information modeling device in economics and finance......Page 208
10.6 Reference......Page 209
11.1 The pilot wave function and its uses outside of quantum mechanics......Page 210
11.2 References......Page 211
12.1 The Bohm–Vigier model in finance......Page 212
12.3 The Newton–Bohm equation: path simulation......Page 213
12.4 Reference......Page 217
13.1 The (non-)Hermiticity of finance-based operators?......Page 218
13.2 References......Page 222
13.3 Implications of the non-Hermiticity of a Black–Scholes Hamiltonian operator on the use of the classical limit arguments......Page 223
13.6 Interpretations of the wave function: a brief discussion......Page 224
13.7 The wave function and non-observed state prices......Page 226
13.8 Price and superposition of values......Page 230
13.10 Arbitrage and negative probabilities......Page 233
13.11 References......Page 236
13.12 The Li–Zhang and WKB approach......Page 237
13.14 The wave function as a Radon–Nikodym derivative......Page 240
13.16 Universal Brownian motion: definition and discussion......Page 246
13.18 Universal Brownian motion and option pricing......Page 248
13.20 Wave functions in drift-dependent option pricing......Page 252
13.22 Generalizations of It stochastics: path integration and other tools......Page 253
13.24 q-calculus and finance......Page 254
13.25 References......Page 261
14.1 Introduction......Page 263
14.2 Why could the brain use the quantum-like representation of information which is based on classical electromagnetic waves?......Page 265
14.3 Prequantum classical statistical field theory: non-composite systems......Page 268
14.4 Cognitive model: two regimes of brain’s functioning......Page 272
14.5 Classical regime: time representation......Page 276
14.6 Classical signal processing of mental images......Page 278
14.7 Quantum-like processing of mental images......Page 281
14.8 Composite systems......Page 285
14.9 References......Page 287
15 Conclusion......Page 289
Glossary of mathematics, physics, and economics/finance terms......Page 291
Index......Page 300