Introduces number operators with a focus on the relationship between quantum mechanics and social science Mathematics is increasingly applied to classical problems in finance, biology, economics, and elsewhere. Quantum Dynamics for Classical Systems describes how quantum tools the number operator in particular can be used to create dynamical systems in which the variables are operator-valued functions and whose results explain the presented model. The book presents mathematical results and their applications to concrete systems and discusses the methods used, results obtained, and techniques developed for the proofs of the results. The central ideas of number operators are illuminated while avoiding excessive technicalities that are unnecessary for understanding and learning the various mathematical applications. The presented dynamical systems address a variety of contexts and offer clear analyses and explanations of concluded results. Additional features in Quantum Dynamics for Classical Systems include: Applications across diverse fields including stock markets and population migration as well as a unique quantum perspective on these classes of models Illustrations of the use of creation and annihilation operators for classical problems Examples of the recent increase in research and literature on the many applications of quantum tools in applied mathematics Clarification on numerous misunderstandings and misnomers while shedding light on new approaches in the field Quantum Dynamics for Classical Systems is an ideal reference for researchers, professionals, and academics in applied mathematics, economics, physics, biology, and sociology. The book is also excellent for courses in dynamical systems, quantum mechanics, and mathematical models.

Introduces number operators with a focus on the relationshipbetween quantum mechanics and social scienceMathematics is increasingly applied to classical problems infinance, biology, economics, and elsewhere. Quantum Dynamics forClassical Systems describes how quantum tools--the numberoperator in particular--can be used to create dynamicalsystems in which the variables are operator-valued functions andwhose results explain the presented model. The book presentsmathematical results and their applications to concrete systems anddiscusses the methods used, results obtained, and techniquesdeveloped for the proofs of the results.The central ideas of number operators are illuminated whileavoiding excessive technicalities that are unnecessary forunderstanding and learning the various mathematical applications.The presented dynamical systems address a variety of contexts andoffer clear analyses and explanations of concluded results.Additional features in Quantum Dynamics for ClassicalSystems include:* Applications across diverse fields including stock markets andpopulation migration as well as a unique quantum perspective onthese classes of models* Illustrations of the use of creation and annihilation operatorsfor classical problems* Examples of the recent increase in research and literature onthe many applications of quantum tools in applied mathematics* Clarification on numerous misunderstandings and misnomers whileshedding light on new approaches in the fieldQuantum Dynamics for Classical Systems is an idealreference for researchers, professionals, and academics in appliedmathematics, economics, physics, biology, and sociology. The bookis also excellent for courses in dynamical systems, quantummechanics, and mathematical models.

PREFACE xiACKNOWLEDGMENTS xv1 WHY A QUANTUM TOOL IN CLASSICAL CONTEXTS? 11.1 A First View of (Anti-)Commutation Rules 21.2 Our Point of View 41.3 Do Not Worry About Heisenberg! 61.4 Other Appearances of Quantum Mechanics in Classical Problems71.5 Organization of the Book 82 SOME PRELIMINARIES 112.1 The Bosonic Number Operator 112.2 The Fermionic Number Operator 152.3 Dynamics for a Quantum System 162.3.1 Schr¨odinger Representation 172.3.2 Heisenberg Representation 202.3.3 Interaction Representation 212.4 Heisenberg Uncertainty Principle 262.5 Some Perturbation Schemes in Quantum Mechanics 272.5.1 A Time-Dependent Point of View 282.5.2 Feynman Graphs 312.5.3 Dyson's Perturbation Theory 332.5.4 The Stochastic Limit 352.6 Few Words on States 382.7 Getting an Exponential Law from a Hamiltonian 392.7.1 Non-Self-Adjoint Hamiltonians for Damping 422.8 Green's Function 44I SYSTEMS WITH FEW ACTORS 473 LOVE AFFAIRS 493.1 Introduction and Preliminaries 493.2 The First Model 503.2.1 Numerical Results for M >1 543.3 A Love Triangle 613.3.1 Another Generalization 663.4 Damped Love Affairs 713.4.1 Some Plots 763.5 Comparison with Other Strategies 804 MIGRATION AND INTERACTION BETWEEN SPECIES 814.1 Introduction and Preliminaries 824.2 A First Model 844.3 A Spatial Model 884.3.1 A Simple Case: Equal Coefficients 904.3.2 Back to the General Case: Migration 954.4 The Role of a Reservoir 1004.5 Competition Between Populations 1034.6 Further Comments 1055 LEVELS OF WELFARE: THE ROLE OF RESERVOIRS 1095.1 The Model 1105.2 The Small lambda Regime 1165.2.1 The Sub-Closed System 1175.2.2 And Now, the Reservoirs! 1195.3 Back to S 1215.3.1 What If M = 2? 1235.4 Final Comments 1256 AN INTERLUDE: WRITING THE HAMILTONIAN 1296.1 Closed Systems 1296.2 Open Systems 1336.3 Generalizations 136II SYSTEMS WITH MANY ACTORS 1397 A FIRST LOOK AT STOCK MARKETS 1417.1 An Introductory Model 1428 ALL-IN-ONE MODELS 1518.1 The Genesis of the Model 1518.1.1 The Effective Hamiltonian 1558.2 A Two-Traders Model 1628.2.1 An Interlude: the Definition of cP^ 1638.2.2 Back to the Model 1648.3 Many Traders 1698.3.1 The Stochastic Limit of the Model 1728.3.2 The FPL Approximation 1779 MODELS WITH AN EXTERNAL FIELD 1879.1 The Mixed Model 1889.1.1 Interpretation of the Parameters 1949.2 A Time-Dependent Point of View 1969.2.1 First-Order Corrections 2009.2.2 Second-Order Corrections 2039.2.3 Feynman Graphs 2049.3 Final Considerations 20610 CONCLUSIONS 21110.1 Other Possible Number Operators 21110.1.1 Pauli Matrices 21210.1.2 Pseudobosons 21310.1.3 Nonlinear Pseudobosons 21310.1.4 Algebra for an M + 1 Level System 21510.2 What Else? 217BIBLIOGRAPHY 219INDEX 225