PREFACE ........................................................ xi
ACKNOWLEDGMENTS ................................................ xv
1 WHY A QUANTUM TOOL IN CLASSICAL CONTEXTS? .................... 1
1.1 A First View of (Anti-)Commutation Rules ................ 2
1.2 Our Point of View ....................................... 4
1.3 Do Not Worry About Heisenberg! .......................... 6
1.4 Other Appearances of Quantum Mechanics in Classical
Problems ................................................ 7
1.5 Organization of the Book ................................ 8
2 SOME PRELIMINARIES .......................................... 11
2.1 The Вosonic Number Operator ............................ 11
2.2 The Fermionic Number Operator .......................... 15
2.3 Dynamics for a Quantum System .......................... 16
2.3.1 Schrцdinger Representation ...................... 17
2.3.2 Heisenberg Representation ....................... 20
2.3.3 Interaction Representation ...................... 21
2.4 Heisenberg Uncertainty Principle ....................... 26
2.5 Some Perturbation Schemes in Quantum Mechanics ......... 27
2.5.1 A Time-Dependent Point of View .................. 28
2.5.2 Feynman Graphs .................................. 31
2.5.3 Dyson's Perturbation Theory ..................... 33
2.5.4 The Stochastic Limit ............................ 35
2.6 Few Words on States .................................... 38
2.7 Getting an Exponential Law from a Hamiltonian .......... 39
2.7.1 Non-Self-Adjoint Hamiltonians for Damping ....... 42
2.8 Green's Function ....................................... 44
I SYSTEMS WITH FEW ACTORS ..................................... 47
3 LOVE AFFAIRS ................................................ 49
3.1 Introduction and Preliminaries ......................... 49
3.2 The First Model 50
3.2.1 Numerical Results for M > 1 ..................... 54
3.3 A Love Triangle ........................................ 61
3.3.1 Another Generalization .......................... 66
3.4 Damped Love Affairs .................................... 71
3.4.1 Some Plots ...................................... 76
3.5 Comparison with Other Strategies ....................... 80
4 MIGRATION AND INTERACTION BETWEEN SPECIES ................... 81
4.1 Introduction and Preliminaries ......................... 82
4.2 A First Model .......................................... 84
4.3 A Spatial Model ........................................ 88
4.3.1 A Simple Case: Equal Coefficients ............... 90
4.3.2 Back to the General Case: Migration ............. 95
4.4 The Role of a Reservoir ............................... 100
4.5 Competition Between Populations ....................... 103
4.6 Further Comments ...................................... 105
5 LEVELS OF WELFARE: THE ROLE OF RESERVOIRS .................. 109
5.1 The Model ............................................. 110
5.2 The Small λ Regime .................................... 116
5.2.1 The Sub-Closed System .......................... 117
5.2.2 And Now, the Reservoirs! ....................... 119
5.3 Back to S ............................................. 121
5.3.1 What If M = 2? ................................. 123
5.4 Final Comments ........................................ 125
6 AN INTERLUDE: WRITING THE HAMILTONIAN ...................... 129
6.1 Closed Systems ........................................ 129
6.2 Open Systems .......................................... 133
6.3 Generalizations ....................................... 136
II SYSTEMS WITH MANY ACTORS .................................. 139
7 A FIRST LOOK AT STOCK MARKETS .............................. 141
7.1 An Introductory Model ................................. 142
8 ALL-IN-ONE MODELS .......................................... 151
8.1 The Genesis of the Model .............................. 151
8.1.1 The Effective Hamiltonian ...................... 155
8.2 A Two-Traders Model ................................... 162
8.2.1 An Interlude: the Definition of cp ............. 163
8.2.2 Back to the Model .............................. 164
8.3 Many Traders .......................................... 169
8.3.1 The Stochastic Limit of the Model .............. 172
8.3.2 The FPL Approximation .......................... 177
9 MODELS WITH AN EXTERNAL FIELD .............................. 187
9.1 The Mixed Model ....................................... 188
9.1.1 Interpretation of the Parameters ............... 194
9.2 A Time-Dependent Point of View ........................ 196
9.2.1 First-Order Corrections ........................ 200
9.2.2 Second-Order Corrections ....................... 203
9.2.3 Feynman Graphs ................................. 204
9.3 Final Considerations .................................. 206
10 CONCLUSIONS ................................................ 211
10.1 Other Possible Number Operators ...................... 211
10.1.1 Pauli Matrices ................................. 212
10.1.2 Pseudobosons ................................... 213
10.1.3 Nonlinear Pseudobosons ......................... 213
10.1.4 Algebra for an M + 1 Level System .............. 215
10.2 What Else? ............................................ 217
BIBLIOGRAPHY .................................................. 219
INDEX ......................................................... 225
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