Quantum Dynamics for Classical Systems E-Book

Fabio Bagarello

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 367

Veröffentlichungsjahr: 2012

Beschreibung

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.

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Library of Congress Cataloging-in-Publication Data:

Bagarello, Fabio, 1964- author.

Quantum dynamics for classical systems : with applications of the number

operator / Fabio Bagarello.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-37068-1 (hardback)

1. Social sciences-Mathematics. 2. Business mathematics. 3. Quantum

theory-Mathematics. I. Title.

H61.25.B34 2013

300.1′53012-dc23

2012020074

Science never solves a problem

without creating ten more.

George Bernard Shaw

The most exciting phrase to hear in science,

the one who heralds new discoveries,

is not Eureka but That's funny...

Isaac Asimov

Table of Contents

Title Page

Copyright

Quotes

Preface

Acknowledgments

Chapter 1: Why a Quantum Tool in Classical Contexts?

1.1 A First View of (Anti-)Commutation Rules

1.2 Our Point of View

1.3 Do Not Worry About Heisenberg!

1.4 Other Appearances of Quantum Mechanics in Classical Problems

1.5 Organization of the Book

Chapter 2: Some Preliminaries

2.1 The Bosonic Number Operator

2.2 The Fermionic Number Operator

2.3 Dynamics for a Quantum System

2.4 Heisenberg Uncertainty Principle

2.5 Some Perturbation Schemes in Quantum Mechanics

2.6 Few Words on States

2.7 Getting an Exponential Law from a Hamiltonian

2.8 Green's Function

Part I : Systems with Few Actors

Chapter 3: Love Affairs

3.1 Introduction and Preliminaries

3.2 The First Model

3.3 A Love Triangle

3.4 Damped Love Affairs

3.5 Comparison with Other Strategies

Chapter 4: Migration and Interaction Between Species

4.1 Introduction and Preliminaries

4.2 A First Model

4.3 A Spatial Model

4.4 The Role of a Reservoir

4.5 Competition Between Populations

4.6 Further Comments

Chapter 5: Levels of Welfare: The Role of Reservoirs

5.1 The Model

5.2 The Small λ Regime

5.3 Back to

5.4 Final Comments

Chapter 6: An Interlude: Writing the Hamiltonian

6.1 Closed Systems

6.2 Open Systems

6.3 Generalizations

Part II : Systems with Many Actors

Chapter 7: A First Look at Stock Markets

7.1 An Introductory Model

Chapter 8: All-in-One Models

8.1 The Genesis of the Model

8.2 A Two-Traders Model

8.3 Many Traders

Chapter 9: Models with an External Field

9.1 The Mixed Model

9.2 A Time-Dependent Point of View

9.3 Final Considerations

Chapter 10: Conclusions

10.1 Other Possible Number Operators

10.2 What Else?

Bibliography

Index

Preface

In 2005 or so, I started wondering whether that particular part of quantum mechanics that theoretical physicists call second quantization could be used in the analysis of some particular, somehow discrete, classical system. In particular, I started considering stock markets, or a reasonably simplified version of these, since at that time this was a very fashionable topic: econophysics was in its early years, and the general feeling was that there was still room, and need, for many other contributions from physicists and mathematicians. I got the idea that the analysis of the huge amount of information going around a real market was only part of what was interesting to do. I was much more interested in considering the viewpoint of the single trader, who is more likely interested in having some control of his own portfolio. Therefore, I constructed a model of a simplified market, just to see if this strange approach could be interesting for such a hypothetical trader, and I suddenly realized that “yes, it might make sense to carry on in this line of research, but, wow, it is hard to have such a paper accepted in a good journal.” However, after a few weeks, I also realized that this topic seemed to be interesting not only for me, but also for a large community of scientists, and that this community was increasing very fast, producing more and more contributions on the ArXiv. People started citing my first paper, and I was contacted by people interested in what I was doing and who wanted to discuss my point of view. This pushed me in the direction of considering more sophisticated models for stock markets, using my knowledge of quantum mechanics for systems with infinite degrees of freedom in this other, and apparently completely different, field. I thought that this was essentially the end of the story: quantum versus economics. Unexpectedly, a few years ago during a conference in Acireale where I gave a talk on my quantum stock markets, I had a discussion with an old friend of mine, Franco Oliveri, and he suggested using the same general strategy in a different classical context. I remember that in our first discussion, we were thinking of foxes and chicken, a topic that was not very exciting for me. After a while, we realized that what we were discussing could also have been used to describe something completely different: a love story. And that was the beginning of a long story that still continues. Since then, we have constructed several models for different classical systems, playing with our understanding of these systems and looking for some phenomenological description. It turned out that these models quite often produce nontrivial and, in my opinion, quite interesting features that are not fully explored yet. Moreover, what is also very intriguing to me is that the same general framework can be used in many different contexts, giving rise to a sort of unifying setting.

This book might be considered as a first attempt to summarize what I have done so far in this field. My idea was to make the book reasonably simple and self-contained. This is because I expect that some not necessarily mathematical-minded readers might be intrigued by the title, and I do not want to lose these readers. However, a complex system can be made easy only up to a certain extent, and this is exactly what I have tried to do in these notes. Even the love story I will consider in Chapter 3, which from the purely dynamic point of view is surely the simplest situation, is not simple at all. This is not a big surprise, as almost every lover knows very well from personal experience. I should also clarify that it is not my main interest to discuss the psychological aspects behind a love story, a migration process, or the choices of traders in a stock market. I am not even interested in giving any abstract, or too general, description of these systems. Here I want to be quantitative! I want to deduce formulas from general ideas, and I want to see what these formulas imply for the system I have in mind, and if they have some predictive power. However, this ultimate goal implies some effort to the reader, who is required to create his own background on quantum mechanics (if needed) by reading Chapter 2. Dear reader, if you can understand Chapter 2, you can understand the rest! On the other hand, if Chapter 2 is too technical for you, do not worry: you could still try to read the book, simply jumping over this chapter. Of course, if you are not a physicist, you will lose a lot. But you can still get the feeling of what is going on. It is up to you! I really hope you enjoy reading this book!

Fabio Bagarello

Acknowledgments

It is always a pleasure to thank old friends of mine such as Camillo Trapani and Franco Oliveri for their very precious help in so many different situations that I cannot even remember myself. This could be seen as (the first) evidence of the death of my neurons. But, do not worry! I have still enough neurons left in my brain to remember that if something in functional analysis is not particularly clear, Camillo is the right one! And I also have a post-it on my desk that says “Any numerical problem? Call Franco!” What is more funny is that they still answer my questions, even when they are very, very busy! Franco is also responsible, in part, for what is discussed in these notes, and I also thank him for this scientific collaboration and for his enthusiasm.

It is also a pleasure to thank the various editors and referees whom I have met along these years, including the ones who referred this book (before it became a book!). Most of them have contributed significantly to the growth of my research, with many useful, and sometimes unexpected, suggestions. Particular thanks goes to Wiley, for the enthusiasm shown for my manuscript.

I dedicate this book to my beloved parents Giovanna and Benedetto, to my brother Vincenzo, and, dulcis in fundo, to Federico, Giovanna, and Grazyna. When I look at them I often ask myself the same question, a question for which I have no answer, yet: how can they resist with so much mathematics and physics going around?

Chapter 1: Why a Quantum Tool in Classical Contexts?

Of course, there is no other way to begin this book. In our experience, this is the first question that a referee usually raises when he receives a paper of ours. Hence this is the question that we try to answer in this chapter, to motivate our analysis.

Taking a look at the index, we see that the applications discussed here cover a large variety of problems, from love affairs to migration, from competition between species to stock markets. First of all, we have to stress that we are not claiming that, for instance, a love affair has a quantum nature! (Even though, as every lover knows very well, each love story is surely characterized by a strong stochastic ingredient that could be analyzed, for instance, using tools from probability theory. It is not surprising, then, that one could try to use quantum mechanics as well, in view of its probabilistic interpretation.) Therefore, we are not going to discuss any quantum love affair. Rather, we just claim that some quantum tools, and the number operator in particular, can be used, cum grano salis, in the analysis of several dynamical systems in which the variables are seen as operator-valued functions. The interesting fact is that the results we deduce using these tools describe very well the dynamics of the system we are considering. This is shown to be true for love affairs first, but this same conclusion apparently holds in other, completely different, contexts (migrations, stock markets, competition between species, etc.).

However, to answer in more detail the question raised in the title, we need a long introduction, and this is the main content of this chapter. We begin with a few useful facts on (anti-)commutation rules, which are used to motivate our answer. Then, we describe briefly other appearances of quantum mechanics in the description of classical systems, proposed by several authors in recent years. We conclude the chapter with the plan of the book.

1.1 A First View of (Anti-)Commutation Rules

In Chapter 2, we show, among other things, that the eigenvalues of are 0, 1, 2, … , whereas those of are simply 0, 1. This is related to the fact that the fermionic and the bosonic Hilbert spaces differ as the first one is finite dimensional, whereas the second is infinite dimensional. Needless to say, this produces severe differences from a technical point of view. In particular, operators acting on a (finite modes) fermionic Hilbert space are automatically bounded, whereas those acting on a bosonic Hilbert space are quite often unbounded.

1.2 Our Point of View

Analogously, if we consider what in these notes is called asimplified stock market (SSM), that is, a group of people (the traders) with some money and a certain number of shares of different kind, which are exchanged between the traders who pay some cash for that, it is again clear that natural numbers play a crucial role: in our SSM, a trader may have only a natural number of shares (30, 5000, or 106, but not 0.75 shares), and a natural number of units of cash (there is nothing < 1 cent of euro, for instance). Hence, if two traders buy or sell a share, the number of shares in their portfolios increases or decreases by one unit, and the amount of money they possess also changes by an integer multiple of the unit of cash.

In the first part of these notes, we also consider some quantities that change continuously but that can also still be measured, quite naturally, using discrete values: this is the case, for instance, of the love affair between Alice and Bob described in Chapter 3: in some old papers, see Strogatz (1988) and Rinaldi (1998a,b) for instance, the mutual affection between the two actors of the love affair is described by means of two continuous functions. However, it is not hard to imagine how a similar description could be given in terms of discrete quantities: this is what we have done, for instance, in Bagarello and Oliveri (2010, 2011): Bob's affection for Alice is measured by a discrete index, nB, which, when it increases during a time interval [ti, tf], from, say, a value 1 to the value 2, describes the fact that Bob's love for Alice increases during that particular time interval. Analogously, Alice's affection for Bob can be naturally measured by a second discrete index, nA, which, when its value decreases from, say, 1 to 0, describes the fact that Alice's love for Bob simply disappears.

These are just a few examples, all described in detail in these notes, showing how the use of discrete quantities is natural and can be used in the description of several systems, in very different situations. Of course, at first sight, this may look as a simple discretization of a continuous problem, for which several procedures have been proposed along the years. However, this is not our point of view. We adopt here a rather different philosophy, which can be summarized as follows: the discrete quantities used in the description of the system under analysis are closely related to the eigenvalues of some self-adjoint operator. Moreover, these operators can be quite often approximated with effective, finite dimensional, self-adjoint matrices, whose dimensions are somehow fixed by the initial conditions; see, for instance, Chapter 3. Then the natural question is the following: how can we deduce the dynamical behavior of ? This is, of course, the hard part of the job! Along all our work, we have chosen to use a Heisenberg-like dynamics, or its Schrödinger counterpart, which we believe is a good choice for the following reasons:

This list shows that we have two technical and one a posteriori reason to use an energy-like operator H to compute the dynamics of . This is not, of course, the end of the story, but, in our opinion, it is already a very good starting point.

1.3 Do Not Worry About Heisenberg!

People with a quantum mechanical background know very well that, whenever incompatible (i.e., not commuting) observables appear in the description of a given physical system , some uncertainty results follow. Hence, one may wonder how our quantum-like description could be compatible with the classical nature of , whose observable quantities are not expected to be affected by any error, except, at most, by the error due to the experimental settings. This problem, actually, does not exist at all in the applications considered in these notes as all the observables we are interested in form a commuting subset of a larger nonabelian algebra. Therefore, they can be diagonalized simultaneously and a common orthonormal (o.n.) basis of the Hilbert space used in the description of , made of eigenstates of these observables, can be indeed obtained, as we see several times in Chapters 3–9. This means that, in the complete description of , all the results that are deduced using our approach are not affected by any uncertainty because all the relevant self-adjoint operators whose eigenvalues are relevant to us are compatible, that is, mutually commuting.

It should also be mentioned that, in some specific applications, the impossibility of observing simultaneously two (apparently) classical quantities has been taken as a strong indication of the relevance of a quantumlike structure in the description of that process, showing, in particular, the importance of noncommuting operators. This is what was proposed, for instance, in Segal and Segal (1998), which is based on the natural assumption that a trader in a market cannot know, at the same time, the price of a certain share and its forward time derivative. The reason is clear: if the trader has access to both these information with absolute precision, then he is surely able to earn as much as he wants! For this reason, Segal and Segal proposed to use two noncommuting operators to describe the price and its time derivative. Going back to the title of this section, although in this book we are happy to not deal with the uncertainty principle, in other approaches this is actually seen as the main motivation to use a quantum or noncommutative approach for a macroscopic system. For this reason, also in view of possible future applications, we describe in Section 2.4 a possible mathematical derivation of a rather general inequality for noncommuting operators, which, fixing the operators in a suitable way, gives back the Heisenberg uncertainty relation.

1.4 Other Appearances of Quantum Mechanics in Classical Problems

Going back to the crucial aspect of this book, which is surely the mixture of quantum and classical words, we want to stress again that this is surely not the first place in which such a mixture is extensively adopted. On the contrary, in the past few years, a growing interest in classical applications of quantum ideas appeared in the literature, showing that more and more people believe that there is not a really big difference between these two worlds or that, at least, some mathematical tool originally introduced in quantum mechanics may also play a significant role in the analysis of classical systems. These kinds of mixtures can be found in very different fields such as economics (Aerts et al., 2012; Segal and Segal, 1998; Schaden, 2002; Baaquie, 2004; Accardi and Boukas, 2006; Al, 2007; Choustova, 2007; Ishio and Haven, 2009; Khrennikov, 2010; Romero et al., 2011; Pedram, 2012), biology (Engel et al., 2007; Arndt et al., 2009; Pusuluk and Deliduman; Martin-Delgado; Panitchayangkoon et al., 2011; Ritz et al. 2004), sociology, and psychology (Shi, 2005; Jimenez and Moya, 2005; Busemeyer et al., 2006; Khrennikov, 2006; Aerts et al., 2009, 2010; Yukalov and Sornette, 2009a,b; Aerts, 2010; Mensky, 2010; Makowski and Piotrowski, 2011; Yukalov and Sornette), and also in more general contexts (Abbott et al., 2008; Khrennikov, 2010), just to cite a few. The number of scientific contributions having classical applications of quantum mechanics as their main subject is fast increasing. To have an idea of what is going on, it is enough to follow the arXiv at xxx.lanl.gov, where almost everyday new papers are submitted. This, of course, provides encouragement to pursue our analysis and to check how far we can go with our techniques and how our results can be used to explain some aspects of the real macroscopic world.

1.5 Organization of the Book

This book is essentially organized in three parts. In the first part, Chapter 2, we review some important aspects of quantum mechanics, which are used in the rest of the book. In particular, we discuss the dynamical problem in ordinary quantum mechanics using several representations and describing the relations between them. We also discuss in great detail CCR, CAR, and some perturbative approaches, which are used sometimes in the book, as well as other tools and aspects related to quantum mechanics of certain interest for us, such as the Green's functions, the states over the algebra of bounded operators, and the Heisenberg uncertainty principle.

In the second part, Chapters 3–5, we show how the CCR and the CAR can be used for classical systems with few degrees of freedom. In particular, we discuss our point of view on love relations, describing also the role of the environment surrounding the people involved in the love affair. Later, in Chapter 4, we show how the same general framework can be used in the description of competitions between species and for other biological systems. For instance, we describe a migration process involving two populations, one living in a rich area and the second one in a poor region of a two-dimensional lattice.

Chapter 5 is dedicated to the description of the dynamical behavior of a biological-like system (e.g., some kind of bacteria) coupled to two reservoirs, one describing the food needed by the system to survive, and the other mimicking the garbage that is produced by the system itself.

Chapter 6 is a sort of an interlude, useful to fix the ideas on the role of the Hamiltonian of the system we are describing, and on how this Hamiltonian should be constructed. More explicitly, we identify three main steps in the analysis of : the first step consists in understanding the main mechanisms taking place in , with a particular interest to the interactions between its constituents. Second, we deduce the Hamiltonian for , H, following a set of rather general rules, which is listed and explained in detail. The final step in our analysis of is the deduction, from H, of its dynamics. This is usually the hardest part of the job.

The last part of the book, Chapters 7, 8, and 9, is concerned with systems with many degrees of freedom and in particular, with our closed SSM. We propose several models for an SSM, from very simple to more complicated ones, and we consider some of the related dynamical features. In particular, most of the times, we will be interested in the deduction of the time evolution of the portfolio of each single trader, but in Chapter 9, we also compute a transition probability between different states of the SSM.

We devote Chapter 10 to some final considerations and to possible generalizations and applications. In particular, we discuss several possible extensions of our main tools, the CCR and the CAR. This could be useful to describe a nonunitary time evolution, describing some decay, as well as systems with a finite, and larger than 2, number of energy levels.

Notes

1 Along this book, we add some remarks concerning the unboundedness of some operators used in the description of the system under investigation.

Chapter 2: Some Preliminaries

In this chapter, we briefly review some basic facts in quantum mechanics. In particular, we focus on what physicists usually call second quantization, which is used in the rest of the book. This chapter is essentially meant to keep these notes self-contained and to fix the notation. Nevertheless, the reader with a background in quantum mechanics is surely in a better position to fully comprehend the material discussed in this book. Of course, people with such a background could safely skip this chapter. It might also be worth stressing that, sometimes, we discuss something more that what is really used, just to give a reasonably closed form to the arguments presented in this chapter.

2.1 The Bosonic Number Operator

2.1

or, equivalently, by the solution of the differential equation

2.2

As already discussed briefly in Chapter 1, a special role in our analysis is played by the CCR: we say that a set of operators , acting on the Hilbert space , satisfy the CCR if

2.3

2.4

and

where . Hence, nl and N are eigenvalues of and , respectively. Moreover, using the CCR we deduce that

for all l and for all nl. For these reasons, the following interpretation is given in the literature: if the L different modes of bosons of are described by the vector , this means that n1 bosons are in the first mode, n2 in the second mode, and so on.2 The operator acts on and returns nl, which is exactly the number of bosons in the lth mode. The operator counts the total number of bosons. Moreover, the operator al destroys a boson in the lth mode, whereas creates a boson in the same mode. This is why in the physical literature al and are usually called the annihilation and the creation operators, respectively.

The vector in Equation 2.4 defines a vector (or number) state over the set as

2.5

where 〈, 〉 is the scalar product in the Hilbert space . These states are used to project from quantum to classical dynamics and to fix the initial conditions of the system under consideration, in a way that is clarified later. Something more concerning states is discussed later in this chapter.

For the sake of completeness, it is interesting to now check explicitly that the operators introduced so far, al, , , and , are all unbounded. This can be easily understood as, for instance,

and it is clearly related to the fact that is infinite dimensional. It is well known that unbounded operators have severe domain problems, as they cannot be defined in all of , (Reed and Simon, 1980), but only on a dense subset of . However, this is not a major problem for us here for two reasons: first, each vector belongs to the domains of all the operators that are relevant to us, even when raised to some power or combined among them. Second, at least in the numerical calculations performed in Chapter 3, is replaced by an effective Hilbert space, eff, which becomes dynamically finite dimensional because of the existence of some conserved quantity and because of the initial conditions, which impose some constraints on the levels accessible to the members of the system. This aspect is discussed in more detail later.

2.2 The Fermionic Number Operator

Given a set of operators acting on a certain Hilbert space F, we say that they satisfy the CAR if the conditions

2.6

and

where . A major difference with respect to what happens for bosons is that the eigenvalues of are simply 0 and 1, and consequently N can take any integer value larger or equal to 0 (as for bosons). Moreover, using the CAR, we deduce that

and

As for bosons, the vector in Equation 2.7 defines a vector (or number) state over the set as

2.8

where 〈, 〉 is the scalar product in F. Again, these states are used to project from quantum to classical dynamics and to fix the initial conditions of the considered system.

Remark:

Fermions and bosons are not the only possibilities. Other elementary excitations, satisfying different, and surely less friendly, commutation rules, are briefly considered in Chapter 10.

2.3 Dynamics for a Quantum System

Let be a closed quantum system. This means that does not interact with any external reservoir and that its size is comparable with that of, say, the hydrogen atom: is a microscopic system. In this section, we describe how to find the time evolution of . More precisely, we discuss in detail three possible equivalent strategies that produce, given a measurable quantity, its dependence on time.

2.3.1 Schrödinger Representation

Following Roman (1965), it is now natural to assume that an operator T(t, t0) exists, depending on t0 and on t, which describes the time evolution of the wave function:

2.9

It is clear that . Moreover, as

must coincide with < ΦS(t0), ΦS(t0) > , and as this has to be true for all possible choices of the initial wave function ΦS(t0), it follows that . Hence T(t, t0) is an isometric operator. As a matter of fact, T is more than this: T is unitary as it satisfies the equality

2.10

2.11