Quantum Dynamics for Classical Systems

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 Bosonic Number Operator 11

2.2 The Fermionic Number Operator 15

2.3 Dynamics for a Quantum System 16

2.3.1 Schr¨odinger 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 ? 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