Quantum Tools for Macroscopic Systems 3031302796, 9783031302794

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Synthesis Lectures on Mathematics & Statistics

Fabio Bagarello · Francesco Gargano · Francesco Oliveri

Quantum Tools for Macroscopic Systems

Synthesis Lectures on Mathematics & Statistics Series Editor Steven G. Krantz, Department of Mathematics, Washington University, Saint Louis, MO, USA

This series includes titles in applied mathematics and statistics for cross-disciplinary STEM professionals, educators, researchers, and students. The series focuses on new and traditional techniques to develop mathematical knowledge and skills, an understanding of core mathematical reasoning, and the ability to utilize data in specific applications.

Fabio Bagarello · Francesco Gargano · Francesco Oliveri

Quantum Tools for Macroscopic Systems

Fabio Bagarello University of Palermo Palermo, Italy

Francesco Gargano University of Palermo Palermo, Italy

Francesco Oliveri University of Messina Messina, Italy

ISSN 1938-1743 ISSN 1938-1751 (electronic) Synthesis Lectures on Mathematics & Statistics ISBN 978-3-031-30279-4 ISBN 978-3-031-30280-0 (eBook) https://doi.org/10.1007/978-3-031-30280-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

We have been working together since many years on several projects, with different strategies and attitudes, bringing new ideas in the team and proposing new problems to solve. All of this was always done with pleasure, in a nice atmosphere, drinking coffee (avoiding F. G.’s coffee that tastes like tungsten); we often discussed at lunch with typical Sicilian street food and drinks, and this helped us to work better. We like what we do, and we like when we have the possibility to work together. This book gave us the possibility to stop for a while, and try to summarize some of our joint results on mathematical modeling. We are confident that the reader will appreciate our approach, and will be curious enough to carry on her/his own research, to contribute to this quantum-like line of research, together with us and many other scientists localized everywhere in the world. It is really a pleasure to thank all the people who significantly contributed to our research in this field along the years, and those who were interested with what we have been doing: Belal Baaquie, Irina Basieva, Jerome Busemeyer, Rosa Di Salvo, Matteo Gorgone, Emmanuel Haven, Andrei Khrennikov, Emmanuel Pothos, Federico Roccati are just few of them! As always, F. B. dedicates this book to his beloved parents, Giovanna and Benedetto, and to Federico, Giovanna and Grazyna, always on his side! F. G. dedicates this book to his beloved parents, Giuseppe and Elena, to his “sisters” …and to the little Vittoria! F. O. dedicates this book to his wife, Mimma, and daughters Maria Lucia and Elena Sofia, who say often that without mathematics he would be a lost soul swimming in a fish bowl. Palermo, Italy Palermo, Italy Messina, Italy March 2023

Fabio Bagarello Francesco Gargano Francesco Oliveri

v

Introduction

This book is mainly constructed on the possibility of using quantum ideas in contexts which are not necessarily those of the usual quantum world. In particular, we are going to show that tools like the Schrödinger or the Heisenberg equations of motion, the ladder operators, or still the number or the Hamiltonian operators, can be efficiently used in the description of some macroscopic systems, with few or many agents, in various fields of research. This approach, which was somehow strange just few years ago, is now well accepted by many researchers. A lot of people, all over the world, started to be interested, and then to use with success, quantum ideas in biology, finance, social science, decision making, and in many other fields. And this because of the mathematical power of Hilbert spaces and operators theory, but also because some quantum ideas are very deeply connected with real problems. Just to mention an example, decision under uncertainty is one of such problems. It is well known in fact that non commuting operators are intrinsically connected to the description of decisions in presence of some alea. This is, indeed, the Heisenberg uncertainty principle, restated for macroscopic variables. But this is not the only case where operators or vectors in some Hilbert space are clearly relevant in macroscopic systems. The role of superpositions (think of the famous Schrödinger’s cat) is useful also, for instance, in finance or in biology. Among all possible operators, in this book a very special role is played by the lowering and raising operators related to the canonical commutation or anticommutation relations (CCR or CAR, respectively) which can be used in the description of processes where some relevant quantities change discontinuously, like, just to quote few examples, stock markets, migration processes, or some biological systems like cells proliferation. Moreover, other operators have been used in the past, and in this book: Pauli matrices, or truncated bosonic operators are some of the other possible tools, but not the only ones, and they are used both in the description of the system, and in the construction of the Hamiltonian operator which, in our approach, is the main object driving the dynamics of the system. Indeed, as we will see, sometimes a Hamiltonian is not sufficient, and in this book the so-called (H, ρ)-induced dynamics will play a special role. Here, H is the Hamiltonian of the system, while ρ is some (external or internal) operation acting (mostly periodically) on the system, and modifying it as a consequence of its action. This is an efficient way to vii

viii

Introduction

implement in the description of the system some effects which cannot easily be described in any purely Hamiltonian approach. The reader should be aware of the fact that, along the book, more than mathematical rigor, we will be interested in the possibility of getting concrete (exact, possibly, or at least approximate) results. This is useful to check that the expected behavior of a given system is, indeed, recovered by our operatorial approach and, even more, if we can reproduce also the details of some experimental data. However, we will do our best to clarify to which extent, within the context of the model we are considering, our results can be trusted or should be refined further.

Organization of the Book The book, based on a series of papers of ours (mainly [5–7, 9–13]; see also references therein), is divided in two main parts. In the first part, which includes Chaps. 1 and 2, we focus on some general ideas used in the rest of the book. More in details, to keep the book self-contained, we start giving some definitions on ladder operators of different nature (bosonic or fermionic) and introducing the role of the Hamiltonian of a system S in the analysis of its own dynamics. Then, we extend this purely Hamiltonian approach adding a rule, which could modify some details of S as a consequence of some check on S itself. In view of its relevance in concrete applications, the role of the related (H, ρ)induced dynamics in the determination of some equilibrium for S is discussed in detail. Chapter 1 ends with two simple prototypical examples, useful to show the main ideas used everywhere in this book. The second part, starting with Chap. 3, is all focused on applications. In particular, in Chap. 3 we describe a recent model of epidemics, with an explicit application to COVID 19. We will see that the model is efficient enough to capture the asymptotic behavior of the first wave, producing densities of the various compartments of the populations (dead, recovered, infected) which are in good agreement with the real data. In Chap. 4, we adopt our ladder operators in the description of a network where some information is diffused. In particular, we consider the possibility of having two different kind of news, the good and the fake news. The difference between a purely Hamiltonian description and the use of the (H, ρ)-induced dynamics is discussed in detail. Chapter 5 describes an application to a real, large scale, problem. After some preliminary considerations, intended to simplify the numerical difficulties that an operatorial approach presents, we describe a problem in population dynamics. Here, the populations are drifters in the Mediterranean sea, and our final aim is to describe their densities as a function of time. In Chap. 6, we show that ladder operators can be also used efficiently in the description of a political system, and in particular in the analysis of politician who, during a Legislature, change political group going, e.g., from a left-side party to a right-oriented

Introduction

ix

group or vice versa. We also compare the model with some real data coming from the XVII Legislature in Italy. Chapter 7 is slightly different. Here, the role of the dynamics is not central, but it is useful only in connection with a particular tool in quantum statistical mechanics, the so-called KMS–condition. This will be used to check if and when a group of people can undergo to a sort of phase transition so that this group comes out with a common decision. This is a very natural problem in decision making. We end our book with a chapter of conclusions and plans for the future. The variety of topics is evident. For us, this is a nice indication that the framework we are using in this book is really promising, and deserves more study. Besides the books [3, 4] by one of us on these topics, several other authors (see, e.g., [1, 2, 8, 14– 16]) used quantum techniques for describing phenomena outside the microscopic world; this enforces our belief that the approach we are considering and refining since some years, can be useful in several contexts, and can help to get some insight in situations and systems where the ordinary approaches, based on partial differential equations or stochastic models, do not work well. But now it is time to start!

References 1. B. E. Baaquie. Quantum Finance. Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press, New York, 2004. 2. B. E. Baaquie. Interest Rates and Coupon Bonds in Quantum Finance. Cambridge University Press, New York, 2009. 3. F. Bagarello. Quantum dynamics for classical systems: with applications of the Number operator. John Wiley & Sons, New York, 2012. 4. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 5. F. Bagarello, R. Di Salvo, F. Gargano, and F. Oliveri. (H, ρ)-induced dynamics and large time behaviors. Physica A: Statistical Mechanics and its Applications, 505:355–373, 2018. 6. F. Bagarello, F. Gargano, and F. Oliveri. Spreading of competing information in a network. Entropy, 22:1169, 2020. 7. F. Bagarello, F. Gargano, and F. Roccati. Modeling epidemics through ladder operators. Chaos, Solitons & Fractals, 140:110193, 2020. 8. J. R. Busemeyer and P. D. Bruza. Quantum Models of Cognition and Decision. Cambridge University Press, Cambridge, 2013. 9. R. Di Salvo, M. Gorgone, and F. Oliveri. (H, ρ)-induced political dynamics: facets of the disloyal attitudes into the public opinion. International Journal of Theoretical Physics, 56:3912– 3922, 2017. 10. R. Di Salvo, M. Gorgone, and F. Oliveri. Political dynamics affected by turncoats. International Journal of Theoretical Physics, 56:3604–3614, 2017. 11. R. Di Salvo, M. Gorgone, and F. Oliveri. Generalized hamiltonian for a two-mode fermionic model and asymptotic equilibria. Physica A: Statistical Mechanics and its Applications, 540:12032, 2020.

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12. R. Di Salvo and F. Oliveri. An operatorial model for complex political system dynamics. Mathematical Methods in the Applied Sciences, 40:5668–5682, 2017. 13. F. Gargano. Population dynamics based on ladder bosonic operators. Applied Mathematical Modelling, 96:39–52, 2021. 14. E. Haven and A. Khrennikov. Quantum social science. Cambridge University Press, Cambridge, 2013. 15. A. Khrennikov. Ubiquitous Quantum Structure: from Psychology to Finance. Springer-Verlag, Berlin, 2012. 16. A. Khrennikov. Social laser. Jenny Stanford Publishing, Singapore, 2020.

Contents

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Bosonic Number Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Fermionic Number Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Dynamics for a Quantum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Schrödinger Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Heisenberg Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 From Operators to Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 A Comment on Large Time Behavior and Equilibria . . . . . . . . . . . . . . . . . . 1.6 The (H , ρ)-Induced Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 The Rule ρ as a Map from H to H . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 The Rule ρ as a Map in the Space of the Parameters of H . . . . . . 1.7 A Short Comment on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Tutorial 1: A Two-Mode System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Tutorial 2: A First View to Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 6 7 9 10 10 11 13 15 17 19

2 Dynamics with Asymptotic Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Two-Mode Fermionic System and (H , ρ)-Induced Dynamics . . . . . . . . . . 2.3 A Generalized Hamiltonian Leading to Asymptotic Equilibria . . . . . . . . . 2.4 Asymptotic Steady States and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 23 26 32 35 36

3 Epidemics: Some Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Model and Its Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Lockdown Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 40 46 48 50 51 xi

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4 Spreading of Information in a Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Model and Its Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Hamiltonian and Its Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Effect of the Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 A Simple Network with 3 Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 A Network with 7 Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 55 57 58 59 61 63 64

5 Population Dynamics in Large Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Transport Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Application to Large Domain: The 2D Model . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 First Application: Prey-Predator Dynamics . . . . . . . . . . . . . . . . . . . . 5.4.2 Second Application: Marine Dynamics . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 67 69 72 73 75 81 82

6 Political Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Different Behaviors Depending on the Cross Interactions . . . . . . . 6.3 The Advanced Model: Rule Induced Dynamics . . . . . . . . . . . . . . . . . . . . . . 6.4 A Case Study: The Dynamics of Turncoats in the Italian XVII Legislature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 85 89 91 95 100 102

7 Phase Transitions and Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Introducing the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 104 106 110 111

8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 114

1

Preliminaries

This chapter is meant to give a very short review of some basic facts about quantum mechanics, together with few new concepts and results that will be used later in some of the applications considered hereafter. Most of what we are going to write can be found in any monograph about Quantum Mechanics, as in [8, 9], just to cite two. In particular, we sketch briefly some basic elements of what is often called second quantization, which will be one of the essential tools throughout these notes. For us, second quantization is nothing more than the functional settings associated to the canonical commutation relations (CCR) and to the canonical anticommutation relations (CAR). Then, we discuss some definitions and results on quantum dynamics mainly for closed systems, focusing in particular on both the Schrödinger and the Heisenberg representations, together with their dynamical contents. Then, we recall few facts about the states of a quantum system. We also introduce what has been called (H , ρ)-induced dynamics, [4, 5], which, roughly speaking, can be thought of as a modified version of the Heisenberg dynamics in presence of what is called a rule, accounting for effects that cannot be easily described in terms of any Hamiltonian operator.

1.1

The Bosonic Number Operator

Let H be a Hilbert space, and B(H) the set of all the bounded operators on H. Let S be our physical system and A the set of all the operators useful for a complete description of S , which includes the observables of S , i.e., those quantities which need to be measured in a concrete experiment. Here H is the Hilbert space where S is defined. A could coincide or not with B(H). In particular, if H is infinite-dimensional, A could contain some unbounded operators, like it happens when CCR are used. This aspect will be clarified soon.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_1

1

2

1 Preliminaries

A special role in our analysis is played by the CCR: we say that a set of operators {a j , a †j , j = 1, 2, . . . , L}, acting on the Hilbert space H, satisfy the CCR if the relations [a j , ak† ] = δ jk 11,

[a j , ak ] = [a †j , ak† ] = 0,

j, k = 1, . . . , L,

(1.1)

11 being the identity operator on H, hold true. From these operators we can construct the self adjoint operators nˆ j = a †j a j and Nˆ = Lj=1 nˆ j , that is nˆ j = nˆ †j and Nˆ = Nˆ † . In particular, nˆ j is the number operator for the jth mode, while Nˆ is the number operator for the system

S described by our operators. An orthonormal (o.n.) basis of H can be constructed as follows: we introduce first the vacuum of the theory, that is a vector ϕ0 which is annihilated by all the operators a j : a j ϕ0 = 0 for all j = 1, 2, . . . , L. Then we act on ϕ0 with the operators a †j and their powers, 1 (a † )n 1 (a2† )n 2 · · · (a L† )n L ϕ0 , ϕn 1 ,n 2 ,...,n L := √ n1! n2! . . . n L ! 1

(1.2)

n j = 0, 1, 2, . . ., for all j; as a result, the vectors so obtained are normalized. The set of the ϕn 1 ,n 2 ,...,n L ’s in (1.2) forms a complete and o.n. set in H, and they are eigenstates of both nˆ j and Nˆ : nˆ j ϕn 1 ,n 2 ,...,n L = n j ϕn 1 ,n 2 ,...,n L and L

Nˆ ϕn 1 ,n 2 ,...,n L = N ϕn 1 ,n 2 ,...,n L ,

where N = j=1 n j . Hence, n j and N are eigenvalues of nˆ j and Nˆ , respectively. Moreover, using the CCR we deduce that   nˆ j a j ϕn 1 ,n 2 ,...,n L = (n j − 1)(a j ϕn 1 ,n 2 ,...,n L ), for n j ≥ 1 while, if n j = 0, a j annihilates the vector, and   nˆ j a †j ϕn 1 ,n 2 ,...,n L = (n j + 1)(a †j ϕn 1 ,n 2 ,...,n L ), for all j and for all n j . For these reasons the following interpretation is given in the literature: if the L different modes of bosons of S are described by the vector ϕn 1 ,n 2 ,...,n L , this means that n 1 bosons are in the first mode, n 2 in the second mode, and so on. The operator nˆ j acts on ϕn 1 ,n 2 ,...,n L and returns n j , which is exactly the number of bosons in the jth mode. The operator Nˆ counts the total number of bosons. Moreover, the operator a j destroys a boson in the jth mode, if there is at least one. Otherwise a j simply destroys the state. Its adjoint, a †j , creates a boson in the same mode. This is why in the physical literature a j and a †j are often called the annihilation and the creation operators.

1.2 The Fermionic Number Operator

3

Remark 1.1 It is useful to observe that a j , a †j , nˆ j and Nˆ are all unbounded [7]. Hence, when CCR are involved, A is usually different from B(H). The vector ϕn 1 ,n 2 ,...,n L in (1.2) defines a vector (or number) state over the set A as ωn 1 ,n 2 ,...,n L (X ) = ϕn 1 ,n 2 ,...,n L , X ϕn 1 ,n 2 ,...,n L ,

(1.3)

where  ,  is the scalar product in the Hilbert space H, and X ∈ A. These states will be used to project from quantum to classical dynamics and to fix the initial conditions of the system under consideration in a way which will be used several times in the chapters of this book. Something more concerning states of this and of different kind will be discussed later in this chapter.

1.2

The Fermionic Number Operator

Other classes of annihilation and creation operators, not related to CCR, can also be considered. Given a set of operators {b j , b†j , j = 1, 2, . . . , L} acting on a certain Hilbert space H F , we say that they satisfy the CAR if the conditions {b j , bk† } = δ j,k 11,

{b j , bk } = {b†j , bk† } = 0,

j, k = 1, . . . , L,

(1.4)

hold true. Here, {x, y} := x y + yx is the anticommutator of x and y and 11 is now the identity operator on H F . These operators are considered in many textbooks on quantum mechanics (see, for instance, [10]), and are used to describe L different modes of fermions. As for bosons, from these operators we can construct the self-adjoint operators nˆ j = b†j b j  and Nˆ = Lj=1 nˆ j . In particular, nˆ j is the number operator for the jth mode, while Nˆ is the

global number operator for S . Compared with bosonic operators, the operators introduced here satisfy a very important feature: if we try to square them, we simply get zero: for instance, (1.4) implies that b2j = 0. This is of course related to the fact that fermions satisfy the Pauli exclusion principle [10] while bosons do not. The Hilbert space of our system is constructed as for bosons: we introduce the vacuum of the theory, that is a vector 0 which is annihilated by all the operators b j : b j 0 = 0 for all j = 1, 2, . . . , L. Then we act on 0 with the operators (b†j )n j : n 1 ,n 2 ,...,n L := (b1† )n 1 (b2† )n 2 · · · (b†L )n L 0 ,

(1.5)

n j = 0, 1, for all j. Of course, we do not consider higher powers of the b†j ’s since these powers would simply destroy the vector. This is the reason why no nontrivial normalization appears, differently from what happens in (1.2). These vectors form an o.n. set spanning H F , and they are eigenstates of both nˆ j and Nˆ , similarly to what we have seen for CCR:

4

1 Preliminaries

nˆ j n 1 ,n 2 ,...,n L = n j n 1 ,n 2 ,...,n L and where N =

Nˆ n 1 ,n 2 ,...,n L = N n 1 ,n 2 ,...,n L ,

L

j=1 n j ,

with n j = 0, 1. Moreover, using the CAR, we deduce that

  nˆ j b j n 1 ,n 2 ,...,n L = and nˆ j



b†j n 1 ,n 2 ,...,n L





(n j − 1)(b j n 1 ,n 2 ,...,n L ), if n j = 1 0, if n j = 0,

 =

(n j + 1)(b†j n 1 ,n 2 ,...,n L ), if n j = 0 0, if n j = 1,

for all j. The interpretation of these formulas does not differ much from that for bosons, and this explains why b j and b†j are again called the annihilation and the creation operators. However, in some sense, b†j is also an annihilation operator since, acting on a state with n j = 1, it destroys that state: we are trying to put together two identical fermions, and this operation is forbidden by the Pauli exclusion principle. Of course, H F has a finite dimension. In particular, for just one mode of fermions, dim(H F ) = 2, while dim(H F ) = 4 if L = 2. Then all the fermionic operators are bounded and can be represented by finite-dimensional matrices. As for bosons, the vector n 1 ,n 2 ,...,n L in (1.5) defines a vector (or number) state over the algebra A of the operators over H F as ωn 1 ,n 2 ,...,n L (X ) = n 1 ,n 2 ,...,n L , X n 1 ,n 2 ,...,n L ,

(1.6)

where  ,  is the scalar product in H F , and X ∈ A. Again, these states will be used to project from quantum to classical dynamics and to fix the initial conditions of the considered system. This aspect will be clarified later in concrete applications. Here we just want to stress that CAR can be naturally used to describe densities, while CCR are more useful to described quantities which, in principle, can grow without an upper bound. The operators a j and b j considered in these sections are just two examples of ladder operators. Other (intermediate) possibilities also exist, and are used in the literature (refer to [2, 3] for some recents results on these operators): in particular, a finite dimensional version of the bosonic operators will be considered in Chap. 5.

1.3

Dynamics for a Quantum System

In what follows we will be mainly interested to the description of the time evolution of some (physical, biological, ecological, …) system. For this, and considering that our dynamical variables are operators, it is convenient to introduce some definitions and some results on

1.3

Dynamics for a Quantum System

5

the dynamics in the quantum context since this is the key situation in which the dynamical variables are, indeed, operators. Our review here will be very concise. We refer to [2, 8, 9] for more extended reviews, and for many relevant references. Let S be a closed quantum system. This means that S does not interact with any external reservoir (hence, it is closed) and that its size is comparable with that of, say, the hydrogen atom: S is a microscopic system (so, it is a quantum system). Incidentally, it may be useful to observe that, if S also interacts with another system SE , of the same size, then we can always consider the union S f ull = S ∪ SE as a larger, but still closed, system. If the size of SE is much larger than that of S , then we will say that S is an open quantum system. We will briefly discuss the dynamics for systems like these in Sect. 1.9, in a very particular situation. A general treatment of open systems in our context can be found in [1, 2].

1.3.1

Schrödinger Representation

First we observe that, in a closed (conservative) system, the total probability should be preserved: suppose, to be concrete, that S is a particle. It is described by a wave function (r , t) whose evolution is driven by some unitary operator: (r , t) = U (t, t0 )(r , t0 ),

(1.7)

where U −1 (t, t0 ) = U † (t, t0 ). In this way, in fact, the L2 (R3 )-norm for  is preserved in time, and therefore  maintains its probabilistic interpretation. Due to the fact that the spatial dependence of  is not relevant for us here, from now on we will simplify the notation by using (t) rather than (r , t). The operator U (t, t0 ) is such that U (t0 , t0 ) = 11 and U (t2 , t1 )U (t1 , t0 ) = U (t2 , t0 ), for all t0 ≤ t1 ≤ t2 . Then, if we take in particular t0 = t2 , we deduce that U −1 (t1 , t0 ) = U (t0 , t1 ), for all t0 and t1 . It is now easy to deduce the differential equation for (t). This is the well known Schrödinger equation ∂(t) = H (t)(t), (1.8) i ∂t where H (t) is a self-adjoint operator, the Hamiltonian of S , H (t) = H † (t), which in general can be explicitly dependent on time. To show how (1.8) follows from (1.7) we compute first

∂U (t, t0 ) −1 ∂U (t, t0 ) † ∂U (t, t0 ) ∂(t) = (t0 ) = U (t, t0 )(t) = U (t, t0 ) (t), ∂t ∂t ∂t ∂t which can be written as in (1.8) defining H (t) = i

∂U (t, t0 ) † U (t, t0 ). ∂t

We refer to [1, 2] for the proof that H (t) is self-adjoint and independent of t0 .

(1.9)

6

1 Preliminaries

We have seen that (t) and H (t) can both be derived, in principle, from the operators U (t, t0 ) as shown in (1.7) and (1.9). Reversing the procedure, U (t, t0 ) can be deduced from H (t). The explicit result, however, is deeply linked to the time dependence of H , as we show now. First we rewrite (1.8) as i

∂U (t, t0 ) (t0 ) = H (t) U (t, t0 )(t0 ), ∂t

which should be satisfied for all possible initial states (t0 ). This means that the equation for U (t, t0 ) is ∂U (t, t0 ) (1.10) i = H (t) U (t, t0 ). ∂t This equation can be solved easily if H (t) = H , i.e., if the Hamiltonian of S does not depend explicitly on time. In this case, in fact, the formal solution of (1.10) is U (t, t0 ) = exp(−i H (t − t0 )),

(1.11)

so that (1.7) becomes (t) = exp(−i H (t − t0 ))(t0 ). U (t, t0 ) is often called the propagator, since it propagates the wave function from t0 to t, preserving its probabilistic interpretation. The situation is more complicated, in general, if H depends on time and if, as it often happens, [H (t1 ), H (t2 )] = 0. This situation is not so useful in this book, and will not be considered further. We refer to [2] for further results, and for more relevant references.

1.3.2

Heisenberg Representation

The possibility of changing representation in quantum mechanics is based on the fact that what is usually relevant for us is not really the time evolution of the state of S , or the time evolution of its observables, but only the mean values of the observables in the state of S . This is because in experiments people usually measures these mean values, and not the operators themselves. In what follows it is convenient to introduce a suffix to distinguish between the Schrödinger and the Heisenberg representations1 : in particular, we use respectively  S and X S , or  H and X H , to indicate the state  and the observable X in the Schrödinger, or in the Heisenberg, representation. The link between the two representations is provided by the relation  S (t), A S  S (t) =  H , A H (t) H  , (1.12)

1 This suffix will not be used in the rest of these notes to avoid useless complications in the notation.

1.4

From Operators to Functions

7

where it is explicitly shown that the wave function depends on time in the Schrödinger but not in the Heisenberg representation, and that, vice versa, the observables depend on time in the Heisenberg but not in the Schrödinger representation. From (1.12) and (1.7), where  is identified with  S , it follows that (putting for simplicity t0 = 0), A H (t) = U † (t, 0) A S U (t, 0).

(1.13)

It is clear that, in particular, A H (0) = A S and that 11 H = 11 S . Moreover, A H (t) = A S if A S commutes with U (t, 0). In particular, if H does not depend on time explicitly, then A H (t) = A S if A S commutes with H . Indeed, we have A H (t) = exp(i H t)A S exp(−i H t), ⇒

A˙ H (t) = i exp(i H t)[H , A S ] exp(−i H t),

which is zero if [H , A S ] = 0. Hence, A H (t) = A H (0) = A S . Also, if A and B are two observables, calling C S = A S B S their product in the Schrödinger representation, then C H (t) = A H (t)B H (t). This implies that, if D S = [A S , B S ], then D H (t) = [A H (t), B H (t)]. The next step consists in finding the differential equation for A H (t). For that we observe that, taking the adjoint of (1.10), we get −i

∂U † (t, t0 ) = U † (t, t0 ) HS (t), ∂t

so that, after few simple computations, i A˙ H (t) = [A H (t), H H (t)] + i where



∂ A S (t) ∂t



∂ A S (t) ∂t

,

(1.14)

H





∂ A S (t) = U (t, t0 ) U (t, t0 ). ∂t †

H

Of course, this term disappears if A S (t) does not depend explicitly on time, i.e., if A S (t) = A S . In this case, in particular, if A S commutes with HS , [A S , HS ] = 0, then, for what we have already seen, [A H (t), H H (t)] = 0 as well, and Eq. (1.14) implies that A H (t) = A H (0) = A S . When this happens, the operator A S is called a constant or integral of motion. The existence of integrals of motion for a given system S can be useful in the analysis of S , and of its dynamics.

1.4

From Operators to Functions

In our approach, states play an essential role, since they are used to transfer the time evolution of the system S we are dealing with from an operatorial to a classical level, while fixing the initial conditions. The strategy is the following: we use operators to describe certain

8

1 Preliminaries

macroscopic systems. Let S be one such system, and X be an observable relevant in the analysis of S . Its time evolution X (t) is deduced from the Hamiltonian of S , as discussed before, adopting, for instance, the Heisenberg representation. The Hamiltonian H is written by considering all the interactions existing between the various agents (or compartments) of S , following few minimal (and natural) rules proposed in [1] and briefly reviewed later, in Sect. 1.7. Hence, if  is the vector describing S at time zero (and therefore also at time t, see Sect. 1.3), x(t) = , X (t) is interpreted as the time evolution of the classical dynamical variable represented, in our approach, by the operator X , when S is described, at t = 0, by . In this sense, x(t) is the projection of the quantum-like dynamics of X , X (t), to its original classical level. We recall that through (1.12) we also have x(t) = (t), X (t), where here it is the vector  that evolves in time, X being X (0). In this book, we shall give particular relevance to the case where X is a number operator of the kind introduced in Sects. 1.1 and 1.2:

 n(t) = , n(t) ˆ .

(1.15)

A state of this kind, ϕ, Y ϕ, for some normalized vector ϕ ∈ H and some operator Y on H is called a vector state. More in general, a state ω over a certain *-algebra A is a positive linear functional which is normalized. In other words, for each element A ∈ A, ω satisfies the following: ω(A† A) ≥ 0, and ω(11) = 1, where 11 is the identity of A. It is known that ω(A† ) = ω(A). These properties are clearly satisfied by any vector state. In fact, independently of the choice of normalized ϕ, we have:



 ϕ, 11ϕ = ϕ2 = 1. Moreover, ϕ, A† Aϕ = Aϕ2 ≥ 0. Finally, ϕ, A† ϕ = Aϕ, ϕ = ϕ, Aϕ. Other types of states also exist and are relevant in many physical applications. For instance, the so called KMS-states, i.e., the equilibrium states for systems with infinite degrees of freedom, can be used to check the existence of phase transitions. Without going into the mathematical rigorous definition, see [7], a KMS-state ω with inverse temperature β satisfies the following equality, known as the KMS condition: ω(A B(iβ)) = ω(B A),

(1.16)

where A and B are general elements of A, and B(iβ) is the time evolution of the operator B computed at the complex value iβ of the time.2 It is well known, [11], that, when restricted to a finite size system, a KMS-state is nothing more than a Gibbs state which can be expressed in terms of a trace with a suitable weight involving the Hamiltonian of the system:   tr e−β H A  .  ω(A) = tr e−β H 2 More precisely, B(iβ) is an analytic extension of B(t) to the complex domain [7].

1.5

A Comment on Large Time Behavior and Equilibria

9

In general, a KMS state is used to describe a thermal bath interacting with a physical system S , if the bath has a non-zero temperature.

1.5

A Comment on Large Time Behavior and Equilibria

Suppose we are interested in determining whether the system we are studying reaches some dynamical equilibrium, that is if some mean value like (1.15) attains an asymptotic value for large time. For instance, this could be useful in a procedure of decision making, where we call d(t) the function (a mean value of a suitable operator) which describes the time evolution of the decision process of an agent, say, Alice. Of course, it is quite natural to expect that, for time large enough, d(t) converges to some value, d∞ . This is what should be understood as Alice’s final decision. Of course, it is reasonable to ask what is the acceptable time needed to take a decision. However, there is no general answer to this question: in some situations the decision should be extremely fast, while in other cases there is no real need to hurry up. Some authors just fix the decision time td from the very beginning. Hence, even if d(t) is oscillating, when t = td , Alice just takes d(td ) as her final decision. Of course, this approach implies that fixing a different value for td Alice takes a (possibly) completely different decision. Hence, the result, and the whole procedure, is extremely sensitive to this choice. For this reason, we prefer to adopt here other approaches, for instance the one in which the decision is driven by the interaction between Alice and her environment (a group of people, a set of information, a group of neurons, …), or some other self-adjusting mechanism which produces a d(t) converging to some finite value. This convergence cannot be achieved for any finite dimensional physical system, i.e., for any system living in a finite-dimensional Hilbert space, if the dynamics is driven by a self-adjoint Hamiltonian. Suppose indeed that the Hamiltonian of S , h = h † , is a n × n matrix. Given a certain observable D of S we have D(t) = exp(i ht)D exp(−i ht). But, since h = h † , an unitary matrix U exists such that U hU −1 = h d , with diagonal h d . Of course, the diagonal elements of h d are the eigenvalues of h. Then D(t) = U −1 exp(i h d t)U DU −1 exp(−i h d t)U , where, of course, exp(±i h d t) are diagonal matrices. Hence, the mean value D(t) of D(t) on any vector can only be periodic or quasi-periodic, depending on the relation between the eigenvalues of h: no stable asymptotic value is possible for D(t), which keeps on oscillating for all t. Therefore, if we imagine that D(t) describes how Alice is processing her decision, the only natural way to stop the procedure, and concretely decide something, is to introduce, from outside, an ad hoc decision time td . A possible way to obtain an asymptotic value, i.e., to take some decision, consists in letting Alice to communicate with the outer world, which is huge, when compared with Alice herself (see the tutorial in Sect. 1.9, and Chap. 3 for a more concrete application).

10

1 Preliminaries

Another possibility is described in the following section, and refers to the self-adjusting dynamics mentioned before: we shall also apply this approach in several chapters of this book, namely in Chaps. 2, 3, 4 and 6.

1.6

The (H, ρ)-Induced Dynamics

It is not always possible to encode all what is going on a system S with a simple Hamiltonian operator. It might happen that, during its time evolution, something happens which modifies suddenly some aspect of S : some poll before elections, some earthquake, some financial crash, and so on. These effects can be modelled using the so-called (H , ρ)-induced dynamics in which, besides the Hamiltonian H , we have to introduce a rule ρ, describing some effect which cannot naturally (and easily) be described by any term in H . This rule can act on S in different ways. In particular, it can modify the state of the system, i.e., its wave function, or some of the details of the dynamics, say some aspects of H . These situations will be described separately below.

1.6.1

The Rule ρ as a Map from H to H

We first consider the case where the rule ρ maps the Hilbert space H to H. In other words, ρ changes the wave function of S . It is useful to notice that there exists a one-to-one correspondence between a vector ϕn , see (1.2), and its label n: once we know n, ϕn is clearly identified, and vice versa. Suppose now that, at time t = 0, the system S , made by L modes, is in a state n0 = (n 01 , n 02 , . . . , n 0L ) (or, which is equivalent, S is described by the vector ϕn0 ). Then, once we have fixed a positive value of τ , this vector evolves in the time interval [0, τ [ according to the Schrödinger recipe: exp(−i H t)ϕn0 . Here H = H † is the (time independent) Hamiltonian of S . Let us set (τ − ) = lim exp(−i H t)ϕn0 , t→τ −

(1.17)

where t converges to τ from below.3 Now, at time t = τ , ρ is applied to (τ − ), and the output of this action is a new vector which we assume here to be again an eigenstate of each operator nˆ j , but with possibly different eigenvalues, ϕn1 . In other words, ρ looks at the explicit expression of (τ − ) and, according to its form, returns a new vector n1 = (n 11 , n 12 , . . . , n 1L ) and, as a consequence, a new vector ϕn1 of H: ϕn1 = ρ((τ − )): hence, the state of the system is changed (in general) discontinuously by ρ. Now, the procedure is iterated: we take ϕn1 as the initial vector, and we consider its evolution for another time interval of length τ , whence we compute 3 We use here τ − , 2τ − , . . ., as argument of  to emphasize that, for instance, before τ the time evolution is only due to H . In fact, as we will see now, ρ really acts at t = τ , 2τ , . . ..

1.6 The (H , ρ)-Induced Dynamics

11

(2τ − ) = lim exp(−i H t)ϕn1 , t→τ −

(1.18)

and the new vector ϕn2 is deduced as the result of the action of rule ρ on (2τ − ): ϕn2 = ρ((2τ − )). In general, for all k ≥ 1, we have (kτ − ) = lim exp(−i H t)ϕnk−1 , t→τ −

and then

  ϕnk = ρ (kτ − ) .

(1.19)

Now, let X be a generic operator on H, bounded or unbounded. In this latter case, we need to check that the various ϕnk belong to the domain of X (t) = exp(i H t)X exp(−i H t) for all t ∈ [0, τ ]. Definition 1.1 The sequence of functions 

xk+1 (t) := ϕnk , X (t)ϕnk ,

(1.20)

for t ∈ [0, τ ] and k ∈ N0 , is called the (H , ρ)-induced dynamics of X . Now, from X (t) = (x1 (t), x2 (t), . . .) it is possible to define a new function of time, which can be understood as the evolution of the observable X of S under the action of H and ρ, in the following way: ⎧ x1 (t) t ∈ [0, τ [, ⎪ ⎪ ⎨ (t − τ ) t ∈ [τ , 2τ [, x 2 (1.21) X˜ (t) = ⎪ (t − 2τ ) t ∈ [2τ , 3τ [, x 3 ⎪ ⎩ ... It is clear that X˜ (t) may have discontinuities in kτ , for positive integers k. The meaning of X˜ (t) is the following: for t ∈ [0, τ [ X˜ (t) is just the mean value of the operator X (t) in the state defined by ϕn0 . For larger values of t, and in particular if t ∈ [τ , 2τ [, then 

X˜ (t) = ϕn1 , X (t − τ )ϕn1 , which is again the mean value of X (t) shifted back in time, but in a state labeled by ϕn1 , and so on. In other words, X˜ (t) is constructed by considering a set of mean values of X (s) with s ∈ [0, τ [, but on different states, ϕn0 , ϕn1 , ϕn2 and so on, which are the vectors identified by ρ via its successive actions. This is because, in our interpretation of the rule, we assume that the time evolution of any observable starts again and again any time the rule acts on the system.

1.6.2

The Rule ρ as a Map in the Space of the Parameters of H

Formula (1.19) shows that the action of the rule considered in the previous section produces a change in the state of the system, from an input to an output vector. And, to be more explicit, a very special change: from an eigenstate of the nˆ j ’s to another eigenstate of the same compatible operators. The other ingredients of S , and in particular its Hamiltonian

12

1 Preliminaries

H , are not modified by the rule and stay unchanged. As we have already mentioned, this is not the only possibility. We now describe a rule that acts on S changing some details of its Hamiltonian, and in particular adjusting the values of some of the parameters of H . The formalization of the (H , ρ)-induced approach can be summarized as follows. Let us start considering a self-adjoint quadratic Hamiltonian operator H (1) ; the corresponding evolution of a certain observable X reads X (t) = exp(i H (1) t)X exp(−i H (1) t),

(1.22)

and its mean value computed on an eigenstate ϕn is x(t) = ϕn , X (t)ϕn ,

(1.23)

for t in a time interval of length τ > 0. Then, let us modify some of the parameters involved in H (1) , on the basis of the variations of the x(t) computed according to (1.23) after the time τ has elapsed. In this way, we get a new Hamiltonian operator H (2) , having the same functional form as H (1) , but (in general) with different values of (some of) its parameters. Actually, we do not restart the evolution of the system from a new initial condition, but we simply continue to follow the evolution with the only difference that for t ∈]τ , 2τ ] the new Hamiltonian H (2) rules the process. And so on. The rule has to be thought of as a map acting on the space of the parameters involved in the Hamiltonian. Therefore, the global evolution is governed by a sequence of similar Hamiltonian operators, and the parameters entering the model can be considered stepwise (in time) constant. To be more precise, let us consider a time interval [0, T ], and split it in n = T /τ subintervals of length τ . Assume n to be integer. In the kth subinterval [(k − 1)τ , kτ [, consider a Hamiltonian H (k) ruling the dynamics. The global dynamics arises from the sequence of Hamiltonians τ τ τ τ (1.24) H (1) −→ H (2) −→ H (3) −→ . . . −→ H (n) , the complete evolution being obtained by glueing the local evolutions, i.e., ⎛ X (t) = exp(iH (k+1) (t − kτ )) ⎝

k  j=1

⎞

⎛

exp(iH (k− j+1) τ )⎠ X ⎝

k 

⎞ exp(−iH ( j) τ )⎠ exp(−iH (k+1) (t − kτ )),

j=1

 where t ∈ [kτ , (k + 1)τ ], k = 0, . . . , n − 1, and in the product nk=1 ak the order of the operators ak ’s is in general crucial. This rule-induced stepwise dynamics clearly may generate discontinuities in the first order derivatives of the operators, but prevents the occurrence of jumps in their evolutions and, consequently, in the mean values of the number operators. By adopting this rule, we are implicitly considering the possibility of having a time dependent Hamiltonian (for a detailed comparison between the two approaches see [5]). A sort of time dependence is somehow hidden: in each subinterval [(k − 1)τ , kτ [ the Hamiltonian does not depend on time, but at kτ some changes may occur, according to the system evolution.

1.7

A Short Comment on H

13

This means that, using this formulation, the evolution of the system is strongly influenced both by the rule and by the choice of the value of τ . In the kth subinterval [(k − 1)τ , kτ [, the dynamics will be driven by the Hamiltonian (k) H . Hence, the global dynamics arises from the sequence of Hamiltonians as in (1.24), and we have a system like ˙ = U (k) A(t), A(t)

t ∈ [(k − 1)τ , kτ [,

(1.25)

k ≥ 1, at least if each H ( j) is quadratic, and where U (k) is constructed out of the parameters of H (k) , by rewriting conveniently the Heisenberg equations of motion. The analytical solution of this system is not difficult, at least if the various equations are linear. The complete evolution is obtained by glueing the local evolutions.

1.7

A Short Comment on H

In our analysis, the dynamics of a system is always deduced out of some Hamiltonian H , with or without the addition of a rule ρ. In most of the cases considered so far we have used H = H † and H time independent. Hence, it is clear that the way in which H should be constructed becomes essential. This is not a surprise: it is true already for simple classical systems, where the Hamiltonian is usually (i.e., for conservative systems) the energy of the system under study, and it is also true for quantum systems, from the simplest to the most complicated ones. We refer to [1, 2] for a detailed description of few rules to follow in the construction of the Hamiltonian. Here we only want to observe that we usually consider some differences between closed and open systems, mainly due to the fact that, while closed systems usually refer to few agents, open systems are often made by many (or even very many) agents. However, except that for this difference, the general ideas producing H are the same in both cases. In particular, H is divided in two terms, H = H0 + H I , which are the free and the interaction Hamiltonians, respectively. Then we have the following guiding (natural) rules. Rule 1.1 In absence of interactions between the various agents of S , all its observables stay constant in time. In terms of the Hamiltonian of S , this means that H0 should commute with all the observables of the system, and that only the interactions cause a significant change in the status of the system. Rule 1.2 In presence of interactions, if some global observable of the system must stay constant, then the operator which represent this observable must commute with H . Just to clarify the meaning of this, suppose that S is a closed market: the money cannot leave or enter the market, and the shares cannot be created or destroyed. Hence, the number

14

1 Preliminaries

operators Cˆ j and Sˆ j , measuring respectively the units of cash and the number of shares to the trader τ j in the market may change with time, since τ j may trade with τk . However,   ˆ ˆ j C j and j S j , i.e., the total amount of cash and the full set of shares in S , must be   constant. Hence [H , j Cˆ j ] = [H , j Sˆ j ] = 0. The details of the third rule are more complicated to explain. It is sufficient here to give the idea which is behind this rule. More details are given in [1, 2], and many more will be given in the concrete applications discussed later in this book. Roughly speaking we may express it as follows. Rule 1.3 The interaction Hamiltonian Hint responsible for exchanges between the various agents of S should contains powers of lowering and of raising operators. In particular, lowering operators describe the decrease of some variable of the agents, while raising operators describe their increase. For open system, as already stated, the situation does not change much, except that for the huge degrees of freedom necessarily involved in the description of the full system. More explicitly, an open system for us is simply a system S interacting with a reservoir R. From a dynamical point of view, the Hamiltonian HS˜ of this larger system, S˜ := S ∪ R, appears to be the Hamiltonian of an interacting system. This means that the general expression of HS˜ is the sum of three contributions: HS˜ = HS + H0,R + HS ,R .

(1.26)

Here, HS is the Hamiltonian for S , whose explicit expression should be deduced adopting Rules 1.1, 1.2 and 1.3. H0,R is the free Hamiltonian of the reservoir, again written using (in particular) Rule 1.1 above, while HS ,R contains the interactions between the system and the reservoir, for which an extended version of Rule 1.3 is usually adopted. In particular, in all the models discussed in this book, we adopt the following rule. Rule 1.4 In the interaction Hamiltonian HS ,R each annihilation operator of the system is linearly coupled to a corresponding creation operator of the reservoir, and vice versa. This last condition allows us to deduce analytical solutions for most of the models we have proposed along the years, but it is clearly quite strong since we do not expect all the interactions between a system and its reservoir should be so simple. But in these notes simplicity is often preferred to a more detailed description of the system, because it guarantees a way to find analytical results which could not be found when adopting some more elaborated version of H .

1.8 Tutorial 1: A Two-Mode System

1.8

15

Tutorial 1: A Two-Mode System

We conclude this chapter with two explicit applications of what we have discussed so far, in the attempt of clarifying the main ideas behind our construction. We begin with a simple application to a sort of prey-predator model, described as a simple closed system. In the next section, we also consider an open system. We anticipate that, as discussed before, we will find an always oscillating solution for the finite dimensional case discussed in this section, while an asymptotic limit will be deduced in the open system discussed in Sect. 1.9. Let us consider a system S , having two (fermionic) degrees of freedom, and described by the Hamiltonian H = H0 + λH I ,

H0 = ω1 a1† a1 + ω2 a2† a2 ,

H I = a1† a2 + a2† a1 ,

(1.27)

where ω j and λ are real (and positive) quantities in order to ensure that H is self-adjoint. The operators a j and a †j are assumed to satisfy the following CAR: {ai , a †j } = δi, j 11,

{ai , a j } = {ai† , a †j } = 0,

(1.28)

i, j = 1, 2, where, as usual, 11 is the identity operator. Of course, when λ = 0, the two agents of S are not interacting. As already mentioned, S can be seen as a sort of predator-pray system. We will return on this interpretation later in this section. The eigenstates of the number operators nˆ j = a †j a j are easily obtained: if ϕ0,0 is the ground vector of S , a1 ϕ0,0 = a2 ϕ0,0 = 0, an o.n. basis of the four-dimensional Hilbert space H of S is given by the following vectors: ϕ0,0 ,

ϕ1,0 := a1† ϕ0,0 ,

ϕ0,1 := a2† ϕ0,0 ,

ϕ1,1 := a1† a2† ϕ0,0 ,

(1.29)

as in (1.5). We have nˆ 1 ϕn 1 ,n 2 = n 1 ϕn 1 ,n 2 ,

nˆ 2 ϕn 1 ,n 2 = n 2 ϕn 1 ,n 2 .

(1.30)

Since the possible eigenvalues are only 0 and 1, we interpret 0 has very low density and 1 as very high density of one of the two species of S . According to Heisenberg’s view, the equations of motion for the annihilation operators a j (t) are a straightforward consequence of the CAR in (1.28): a˙ 1 (t) = −iω1 a1 (t) − iλa2 (t),

a˙ 2 (t) = −iω2 a2 (t) − iλa1 (t),

(1.31)

For instance, the first equation in (1.31) can be deduced following what we have discussed in Sect. 1.3.2: a˙ 1 (t) = i exp(i H t)[H , a1 ] exp(i H t) = i exp(i H t) ([H0 , a1 ] + λ[H I , a1 ]) exp(−i H t).

16

1 Preliminaries

But, since [a2† a2 , a1 ] = a2† a2 a1 − a1 a2† a2 = a2† a2 a1 − (−1)2 a2† a2 a1 = 0,   [H0 , a1 ] = ω1 a1† a1 a1 − ω1 a1 a1† a1 = −ω1 {a1 , a1† } − a1† a1 a1 = −ω1 a1 , and     [H I , a1 ] = a1† a2 + a2† a1 a1 − a1 a1† a2 + a2† a1 = a1† a2 a1 − a1 a1† a2 = −{a1† , a1 }a2 = −a2 .

Then we have a˙ 1 (t) = i exp(i H t) (−ω1 a1 − λa2 ) exp(−i H t), which is the first equation in (1.31). Going back to (1.31), these can be solved recalling that a1 (0) = a1 and a2 (0) = a2 , and the solution is 1 (a1 ((ω1 − ω2 )− (t) + δ+ (t)) + 2λa2 − (t)) , 2δ 1 a2 (t) = (a2 (−(ω1 − ω2 )− (t) + δ+ (t)) + 2λa1 − (t)) , 2δ

a1 (t) =

where

δ=



(ω1 − ω2 )2 + 4λ2 ,   δt it(ω1 + ω2 ) cos , + (t) = 2 exp − 2 2   δt it(ω1 + ω2 ) − (t) = −2i exp − sin . 2 2 

Then, the functions n j (t) := ϕn 1 ,n 2 , nˆ j (t)ϕn 1 ,n 2 are    δt δt + n 2 sin2 , n 1 cos2 2 2    n 2 (ω1 − ω2 )2 4λ2 2 δt 2 δt n 2 (t) = + 2 n 2 cos + n 1 sin , δ2 δ 2 2 n 1 (t) =

(1.32)

n 1 (ω1 − ω2 )2 4λ2 + 2 2 δ δ

(1.33)

(1.34)

which oscillate in time. These functions could be interpreted, in agreement with other similar applications, as the densities of two species, S1 and S2 , interacting as in (1.27) in a given (small) region.4 The interaction Hamiltonian H I in (1.27) describes a sort of prey-predator mechanism, and this is reflected by the solutions in (1.34), which show how the two densities, because of the interaction between S1 and S2 , oscillate in the interval [0, 1]. Otherwise, if λ = 0, n j (t) = n j : the densities stay constant, and nothing interesting happens in S . We observe

4 Other interpretations are also possible. For instance, they can play the role of the decision functions

of two interacting agents, trying to decide on some binary question.

1.9 Tutorial 2: A First View to Open Systems

17

that the formulas in (1.34) automatically imply that n 1 (t) + n 2 (t) = n 1 + n 2 , independently of t and λ: the oscillations are such that they sum up to zero. The first trivial remark is that the functions n 1 (t) and n 2 (t) in (1.34) do not admit any asymptotic limit, except when n 1 = n 2 (or when λ = 0, which is not so interesting and, as such, is excluded here). In this case, clearly, n 1 (t) = n 2 (t) = n 1 = n 2 . On the other hand, if n 1 = n 2 , then both n 1 (t) and n 2 (t) always oscillate in time. This is not surprising since we have already proved that, if S is a system living in a finite dimensional Hilbert space, and if its dynamics is driven by a time independent, self-adjoint, Hamiltonian H˜ , then its evolution is necessarily periodic or quasi-periodic. Then the conclusion is that, if we are interested in getting (non trivial) asymptotic values, we need to modify the way in which the time evolution is taken, at least for large t.

1.9

Tutorial 2: A First View to Open Systems

We suppose here that the system S , described in terms of bosonic operators a, a † and nˆ a = a † a, interacts with an (infinitely extended) reservoir R, whose particles are described by an ˆ = b† (k)b(k), k ∈ R. In agreement with infinite set of bosonic operators b(k), b† (k) and n(k) what discussed in Sect. 1.7, we assume here the following Hamiltonian for S f ull = S ∪ R:    †  H = H0 + λH I , H0 = ω nˆ a + ab (k) + a † b(k) f (k)dk, ω(k)n(k) ˆ dk, H I = R

R

(1.35) where [a, a † ] = 11, [b(k), b† (q)] = 11δ(k − q), while all the other commutators are zero. All the constant appearing in (1.35), as well as the regularizing function f (k), are real, so that H = H † . In our simple system, H models Alice and her interactions with some environment, which can be thought of as the set of her friends, parents and relatives, for instance. Remark 1.2 It is useful to observe that the Hamiltonian in (1.35) extends, in a natural way, the one in (1.27). This is, of course, due to the change of statistics (bosonic, here, rather than fermionic), but mainly because the second species is now made of infinite agents. This explains the appearance of the index k, together with the integrals in H0 and H I in (1.35). Remark 1.3 From a more mathematical side, saying that H in (1.35) is self-adjoint should be understood only as a formal claim, not really mathematically motivated. Indeed, to turn this claim into a rigorous result, we should first compute the domain of H , the domain of H † , and check that they coincide. But these mathematical details are not so relevant in the analysis carried out in this book, and for this reason sometimes we will be satisfied with formal, rather than rigorous, results.

18

1 Preliminaries

 Notice that an integral of motion exists also for S f ull , nˆ a + R n(k) ˆ dk, which extends the one for S ∪ S˜ , nˆ a + nˆ b . With this choice of H , the Heisenberg equations of motion are   a(t) ˙ = i[H , a(t)] = −iωa(t) − iλ R f (k) b(k, t) dk, (1.36) ˙ t) = i[H , b(k, t)] = −iω(k)b(k, t) − iλ f (k) a(t), b(k, which are supplemented by the initial conditions a(0) = a and b(k, 0) = b(k). Remark 1.4 We stress once more the similarity between these equations and those in (1.31), the main difference being the presence of integrals in k. The last equation in (1.36) can be rewritten in integral form as  t b(k, t) = b(k) exp(−iω(k)t) − iλ f (k) a(t1 ) exp(−iω(k)(t − t1 )) dt1 . 0

We fix now, as it is often done in the literature (and also to maintain the similarity with the fermionic case), f (k) = 1, ω(k) = k, and replace b(k, t) in the first equation in (1.36).  Then, with a change of the order of integration, and recalling that R exp(−ik(t − t1 )) dk = t 2πδ(t − t1 ) and 0 g(t1 )δ(t − t1 ) dt1 = 21 g(t) for any test function g(t), we conclude that  2 a(t) ˙ = −(iω + πλ )a(t) − iλ b(k) exp(−ikt) dk. (1.37) R

This equation can be solved, and the solution is   a(t) = a − iλ dkη(k, t)b(k) exp(−(iω + πλ2 )t), R

(1.38)

1 where η(k, t) = ρ(k) (exp(ρ(k)t) − 1) and ρ(k) = i(ω − k) + πλ2 . Using complex contour integration it is possible to check that [a(t), a † (t)] = 11 for all t: this means that the apparent decay of a(t), described in (1.38), is balanced by a contribution of the reservoir. 

Let us now consider a state over S f ull ,  X S ⊗ X R  = ϕn a , X S ϕn a  X R R , in which ϕn a is the eigenstate of the number operator nˆ a and < >R is a state of the reservoir, see Sect. 1.4. Here X S ⊗ X R is the tensor product of an operator of the system, X S , and an operator of the reservoir, X R . Stated differently, .R is a normalized, positive linear functional 

over the algebra of the operators of R, while ϕn a , . ϕn a is a vector state over the algebra of S . The state .R is assumed to satisfy some properties which extend to R similar properties

 of the vector states over S . In particular, we require that b† (k)b(q) R = n b (k)δ(k − q), and that  b(k)b(q)R = 0. These are standard choices in the context of open quantum systems and of quantum optics in particular [6]. Then, if for simplicity we take the function n b (k) to be constant in k, we get

References

19

  n a (t) :=< nˆ a (t) >=< a † (t)a(t) >= n a exp(−2λ2 πt) + n b 1 − exp(−2λ2 πt) , (1.39) which goes to n b as t → ∞. Hence, if 0 ≤ n b < n a , the value of n a (t) decreases with time. If, on the other hand, n b > n a , then the value of n a (t) increases for large t. In both cases, n a (t) admits an asymptotic value which, identifying n a (t) with the decision function D(t) introduced in Sect. 1.5, can be interpreted as Alice’s final decision. We refer to [2] for different choices of f (k), ω(k) and n b (k), and for their interpretations. Remark 1.5 In the Hamiltonian (1.35) the operators a and b(k) are assumed to satisfy the CCR. If we rather use fermionic operators, so that {a, a † } = 11, with {a, a} = 0, and {b(k), b† (q)} = δ(k − q) 11,

{b(k), b(q)} = 0,

with {a  , b (k)} = 0, where x  indicates either x or x † , then, assuming the same analytic expression for the Hamiltonian (1.35), after some computations we deduce the same function n a (t) as in (1.39). In other words, the analytic expression for n a (t) does not depend on our choice of ladder operators, bosonic or fermionic. However, while the only initial conditions for the CAR are zero and one, for CCR we have plenty other choices. The different use of bosonic and fermionic operators is connected, for us, to the nature of S we have to describe: bosons are better for systems with many levels. Fermions are a natural choice to describe densities or decision functions.

References 1. F. Bagarello. Quantum dynamics for classical systems: with applications of the Number operator. John Wiley & Sons, New York, 2012. 2. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 3. F. Bagarello. Pseudo-Bosons and their coherent states. Mathematical Physics Studies. Springer, New York, 2022. 4. F. Bagarello, R. Di Salvo, F. Gargano, and F. Oliveri. (H , ρ)-induced dynamics and the quantum game of life. Applied Mathematical Modelling, 43:15–32, 2017. 5. F. Bagarello, R. Di Salvo, F. Gargano, and F. Oliveri. (H , ρ)-induced dynamics and large time behaviors. Physica A: Statistical Mechanics and its Applications, 505:355–373, 2018. 6. S. M. Barnett and P. M. Radmore. Methods in theoretical quantum optics. Clarendon Press, Oxford, 1997. 7. O. Bratteli and D.W. Robinson. Operator algebras and quantum statistical mechanics 1. Springer-Verlag, New York, 2002. 8. E. Merzbacher. Quantum mechanics. John Wiley & Sons, New York, 1970. 9. A. Messiah. Quantum mechanics, volume 2. North Holland Publishing Company, Amsterdam, 1962. 10. P. Roman. Advanced quantum mechanics. Addison-Wesley, New York, 1965. 11. G. L. Sewell. Quantum Theory of Collective Phenomena. Oxford University Press, Oxford, 1989.

2

Dynamics with Asymptotic Equilibria

2.1

Introduction

The description of the dynamics of operatorial models merely ruled by a time independent self-adjoint Hamiltonian operator has some limitations; in fact, as already observed in Sect. 1.5, for systems with a finite number of degrees of freedom, we can only obtain periodic or quasiperiodic evolutions. Then, if we expect the system S we wish to model to have some asymptotic final state, it is evident that such a description does not work, and the modeling framework needs to be modified. This is not new: in quantum optics, or for two or three levels atoms, if we need to describe a transition from one level to another, in some cases an effective finite dimensional non-Hermitian Hamiltonian can be considered [5], and decays are well described phenomenologically. A recent application outside the realm of Physics for describing a dynamical system of tumor cells proliferation can be found in [4]. Another way consists in opening the system allowing for the interaction of the “atoms” with some infinite reservoir, but in such a case the full system (S f ull , i.e., S plus a suitable reservoir) is not finite dimensional. This is exactly the approach considered in Sect. 1.9, see in particular formula (1.39). In [2], where a quantum version of game of life is considered, an extended version of the Heisenberg dynamics has been proposed. This was useful to take into account effects which may occur during the time evolution of the system, and which can not apparently be included in a purely Hamiltonian description. This idea was sketched in Sect. 1.6, and will be used later on in this chapter. In particular, during the evolution of the system, driven by a time independent Hermitian Hamiltonian H , at fixed times some checks on the system are performed, and these periodical measures are used to change the state of the system itself according to an explicit prescription. Alternatively, the periodical checks on the state of the system can be used to change the values of some of the parameters entering the Hamiltonian operator, without modifying the functional form of the Hamiltonian itself.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_2

21

22

2 Dynamics with Asymptotic Equilibria

This latter approach proved to be quite efficient in operatorial models of stressed bacterial populations [9], as well as in models of political systems affected by turncoat-like behaviors of part of their members, i.e., systems characterized by internal fluxes between different political parties [6, 7, 10]. In some sense, such an approach allows us to describe a sort of discrete self-adaptation of the model depending on the evolution of the state of the system, without the need of opening the system itself to any external reservoir, or considering more complex time dependent formulations. This (H , ρ)-induced dynamics approach has been analyzed from a general viewpoint in [3], where it has been also considered a comparison with the well established frameworks where the Hamiltonian operator is time dependent or where the system is open and interacting with a reservoir. The strategy of the (H , ρ)-induced dynamics may give interesting results if the rule ρ is not considered as a purely mathematical expedient, but is somehow physically justified. For instance, in [9], the rule describes the modifications in the metabolic activity of bacteria due to lack of nutrients and/or to the presence of waste material, whereas, in [6, 7, 10], the rule modifies the attitudes of the members of a political party with regard to their tendency to shift allegiance from one loyalty or ideal to another one. In this chapter, we describe the possibility of having a dynamics approaching an equilibrium state with a generalized approach modeling a system that continuously adjusts itself during its evolution. The procedure is illustrated by analyzing a rather simple model involving two fermionic operators, whose dynamics is ruled by a time independent, Hermitian quadratic Hamiltonian. In such a situation, without the rule, the resulting dynamics is necessarily periodic. This is exactly what we have seen in Sect. 1.8. If we consider the strategy of the (H , ρ)-induced dynamics (according to which the rule at fixed instants changes some of the parameters entering the Hamiltonian on the basis of the variations of the state of the system), the effect is that the system approaches asymptotically an equilibrium state. Then, we introduce a natural generalization of the rule by means of a suitable limit; in such a way, the parameters entering the Hamiltonian may gain a sort of dependence on the observables of the system, so that the corresponding values change continuously according to the instantaneous evolution of the system. This leads to the introduction of what we call a generalized Hamiltonian somewhat embedding in its formulation the state of the system. Stated differently, the use of the (H , ρ)-induced dynamics can efficiently replace either an explicit time dependence in H , or some non-linear effect in the differential equations of motion. This is very useful to make the analytical and numerical analysis of the resulting system of differential equations affordable. Remarkably, by considering different sets of initial parameters, we are able to find a mathematical relation linking the values of the parameters of the Hamiltonian with the value of the asymptotic equilibrium state. This will be clarified later on.

2.2 Two-Mode Fermionic System and (H , ρ)-Induced Dynamics

2.2

23

Two-Mode Fermionic System and (H, ρ)-Induced Dynamics

Consider the two-mode system S described in Sect. 1.8, ruled by the Hamiltonian H = H0 + λH I ,

H0 = ω1 a1† a1 + ω2 a2† a2 ,

H I = a1 a2† + a2 a1† ,

(2.1)

whose time evolution, computed analytically following Heisenberg view, provides the mean values of number operators      n 1 (ω1 − ω2 )2 4λ2 2 δt 2 δt + n 2 sin , + 2 n 1 cos n 1 (t) = δ2 δ 2 2 (2.2)      n 2 (ω1 − ω2 )2 4λ2 2 δt 2 δt n 2 (t) = + n 1 sin , + 2 n 2 cos δ2 δ 2 2  with δ = (ω1 − ω2 )2 + 4λ2 , exhibiting a periodic trend (see Fig. 2.1). In the following, we exploit the possibility of getting some limiting values for n 1 (t) and n 2 (t) for large values of t, when λ  = 0. The functions n 1 (t) and n 2 (t) given in (2.2) do not admit any asymptotic limit, except when n 1 = n 2 (or when λ = 0). If n 1  = n 2 , and λ  = 0, then both n 1 (t) and n 2 (t) always oscillate in time in opposition of phase. The description of the dynamics can be enriched by introducing a rule able to include in the model some effects that can not be embedded in the definition of H . As already stated, a possible rule is a law that modifies at fixed instants some of the values of the parameters involved in the Hamiltonian according to the variation of the current state of the system. This means that the model is able to adjust itself during the time evolution.

Fig. 2.1 Plot of the time evolution of the mean values of the number operators of the system ruled by the Hamiltonian (2.1) with ω1 = 0.5, ω2 = 0.7, λ = 0.1; initial conditions: ϕ1,0 (a), 0.5 ϕ1,0 + 0.3 ϕ0,1 (b)

24

2 Dynamics with Asymptotic Equilibria

Considering the Hamiltonian operator (2.1), let us illustrate the (H , ρ)-induced approach according to which the rule ρ at fixed times kτ (k = 1, 2, . . .), once we chose a positive value of τ , modifies the parameters ω1 and ω2 in (2.1) according to the variations of the mean values n 1 (t) and n 2 (t) (computed on some assigned vector, representing some given initial condition) in the time interval [(k − 1)τ , kτ ] . Let us fix the initial values of the parameters, say ω1 , ω2 , λ, the initial conditions n 1 and n 2 (or, equivalently, ϕn 1 ,n 2 ), and assume the rule ρ defined by ρ(ω1 ) = ω1 (1 + δ1 (k)),

δ1 (k) = n 1 (kτ ) − n 1 ((k − 1)τ ),

ρ(ω2 ) = ω2 (1 + δ2 (k)),

δ2 (k) = n 2 (kτ ) − n 2 ((k − 1)τ ).

(2.3)

Here n j (t) = ϕn 1 ,n 2 , exp(i H t)a †j a j exp(−i H t)ϕn 1 ,n 2 , j = 1, 2. As we can see in Figs. 2.2 and 2.3, the system may reach some asymptotic equilibrium states in correspondence of different choices of τ . We observe that the transient behavior of the time evolution changes with τ , and, at least for τ ≤ 4, an equilibrium is obtained. For larger values of τ , the time

(a) τ = 1

(c) τ = 4

(b) τ = 2

(d) τ = 8

Fig. 2.2 Time evolution with rule (2.3) for different choices of τ ; ω1 = 0.5, ω2 = 0.7, λ = 0.1; ϕ1,0 as initial condition

2.2 Two-Mode Fermionic System and (H , ρ)-Induced Dynamics

25

(a) τ = 1

(b) τ = 2

(c) τ = 4

(d) τ = 8

Fig. 2.3 Time evolution with rule (2.3) for different choices of τ ; ω1 = 0.5, ω2 = 0.7, λ = 0.1; 0.5 ϕ1,0 + 0.3 ϕ0,1 as initial condition

needed to reach the equilibrium is increasing, or is not reached at all. This is because a large τ is not really much different from having no rule at all! The rule (2.3) makes ω1 to increase (decrease, respectively) if δ1 (k) > 0 (δ1 (k) < 0, respectively). Moreover, since n 1 (t) + n 2 (t) is a constant, δ1 (k) + δ2 (k) ≡ 0, so that the variations of the inertia parameters ω1 and ω2 are opposite. Other choices are of course possible, depending on the effects we want to include in the model. We stress once more that, for some sets of the initial parameters and different rules, it may happen that the system does not reach any equilibrium or takes a very long time to reach it. In each subinterval [(k − 1)τ , kτ [ the Hamiltonian does not depend on time, but at kτ some changes may occur, according to the system evolution. Therefore, it is not a surprise that, using this formulation, the evolution of the system is strongly influenced both by the rule and by the choice of the value of τ . We will consider an explicit application of this rule in Chap. 6, in connection with politics.

26

2 Dynamics with Asymptotic Equilibria

In the next section, with the aim of making the evolution of S independent of τ , we can define a generalized Hamiltonian, and still obtain a dynamics approaching an equilibrium state

2.3

A Generalized Hamiltonian Leading to Asymptotic Equilibria

In this section, we introduce a suitable limit of the (H , ρ)-induced dynamics whereupon the definition of a new Hamiltonian operator, giving rise to a time evolution asymptotically approaching an equilibrium state, is naturally suggested [8]. Moreover, as it will be shown in the next section, a mathematical relation, linking the values of the parameters appearing in the Hamiltonian with the value of the equilibrium state can be devised. The rule (2.3) requires the choice of the length τ of the subinterval where we compute the variation of the state of the system which is needed to update the values of ω1 and ω2 . Different choices of τ imply different transient evolutions before the asymptotic equilibrium state, if any, is reached; moreover, the choice of τ is somehow arbitrary, if not physically motivated, for instance if it is related to some periodic check on the system; in this case, τ is exactly the time interval between a check and the next one. In order to have an evolution that is independent of τ , once the initial condition ϕn 1 ,n 2 is chosen, we consider in a small interval [t, t + τ ] the ratio n j (t + τ ) − n j (t) , τ

j = 1, 2,

(2.4)

i.e., the mean rate of change in the subinterval, and then take the limit for τ going to zero, so obtaining the instantaneous rate of change of n j (t). By using (1.31), the Heisenberg equations of motion for the single mode number operators nˆ j read as (2.5) n˙ˆ 1 = −n˙ˆ 2 = iλ(a1† a2 − a2† a1 ). Therefore, after computing the mean values n˙ i of n˙ˆ i (i = 1, 2) on the initial state ϕn 1 ,n 2 , we may assume that at each instant of time t the values of ω1 and ω2 are given by ω1 = ω10 (1 + α1 n˙ 1 ), ω2 = ω20 (1 + α2 n˙ 2 ),

(2.6)

where ω10 and ω20 are the initial inertia parameters, whereas α1 and α2 are some constants whose meaning is explained below. Therefore, the dynamics of our system is ruled by the following set of nonlinear operatorvalued differential equations, which extend those in (1.31):

2.3

A Generalized Hamiltonian Leading to Asymptotic Equilibria

27

  a˙ 1 = i λa2 − ω10 (1 + iα1 λϕn 1 ,n 2 , (a1† a2 − a2† a1 )ϕn 1 ,n 2 )a1 ,   a˙ 2 = i λa1 − ω20 (1 + iα2 λϕn 1 ,n 2 , (a2† a1 − a1† a2 )ϕn 1 ,n 2 )a2 , to be solved with the initial condition ⎛ ⎞ 0100 ⎜0 0 0 0⎟ ⎟ a1 (0) = ⎜ ⎝0 0 0 1⎠, 0000

⎛

0 ⎜0 a2 (0) = ⎜ ⎝0 0

0 0 0 0

1 0 0 0

⎞ 0 −1 ⎟ ⎟, 0 ⎠ 0

(2.7)

(2.8)

once the vector ϕn 1 ,n 2 has been fixed. Of course, we will be interested to the mean values of the occupation numbers on the initial condition given by ϕn 1 ,n 2 . Equation (2.7) can be written formally as

, a j ], a˙ j = i[ H

j = 1, 2,

(2.9)

along with the generalized Hamiltonian

= ω10 (1 + iα1 λϕn 1 ,n 2 , (a † a2 − a † a1 )ϕn 1 ,n 2 )a1 a † H 1 2 1 + ω20 (1 − iα2 λϕn 1 ,n 2 , (a1† a2 − a2† a1 )ϕn 1 ,n 2 )a2 a2† + λ(a1 a2†

(2.10)

+ a2 a1† ),

whose definition includes the term ϕn 1 ,n 2 , (a1† a2 − a2† a1 )ϕn 1 ,n 2 , proportional to the time derivative of the mean values of the occupation numbers on the assigned initial condition. It is trivial to observe that ϕn 1 ,n 2 , (a1 (0)† a2 (0) − a2 (0)† a1 (0))ϕn 1 ,n 2  = 0, but, during the evolution ruled by (2.7), ϕn 1 ,n 2 , (a1 (t)† a2 (t) − a2 (t)† a1 (t))ϕn 1 ,n 2   = 0, approaching zero in a neighborhood of the equilibrium state. Thus, to obtain the time evolution of a j (t) ( j = 1, 2), we have to integrate Eq. (2.7), but we can not compute

t).

t)a j (0) exp(−i H exp(i H Notice also that, from Eq. (2.7) it can be easily derived that n 1 (t) + n 2 (t) is a conserved quantity. The constants α1 and α2 are related to the effects we want to model. We can choose their values in order to take into account into the Hamiltonian only one between n˙ 1 and n˙ 2 (α1 α2 = 0, α1 + α2  = 0), or both (α1 α2  = 0); if α1 α2 > 0 the coefficients in the free

28

2 Dynamics with Asymptotic Equilibria

in (2.10) change in the opposite way, since n˙ 1 (t) = −n˙ 2 (t); on the contrary, they part of H change in the same way if α1 α2 < 0. Of course, if α1 = α2 = 0, we reduce to the standard case where no rule works. To keep the situation simple, let us take, without loss of generality, α1 , α2 ∈ {−1, 0, 1}. In fact, the numerical solutions show that the values of the asymptotic states are not changed by choosing non zero values of α1 and α2 different from {−1, 1}. The numerical integration of (2.7) for various choices of the values of ω10 , ω20 and λ, as well as the parameters α1 , α2 , shows that the system, after a transient, excluding very few particular situations whose aspects will be discussed below, always approaches an asymptotic equilibrium state. Of course, for values of t large enough, when the equilibrium state is reached, the contribution ϕn 1 ,n 2 , (a1 (t)† a2 (t) − a2 (t)† a1 (t))ϕn 1 ,n 2  is vanishing. In the simulations shown hereafter, we used the initial condition ϕ1,0 , i.e., the initial mean values n 1 = 1, n 2 = 0. This is not limiting because we consider all the possible cases: ω10 < ω20 , ω20 < ω10 , and ω10 = ω20 . Figures 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 2.10 clearly show that our model reaches (with a different speed) its equilibrium state apparently for all values of α1 and α2 . This is indeed true if α1 α2 = 0 and when α1 α2 = 1 while, as we will show later, it is not true if α1 α2 = −1 and eq ω10 = ω20 (see Fig. 2.10d). As already remarked, the asymptotic equilibrium n 1 does not change when the non zero values of α1 and α2 do not belong to the set {−1, 1}: what changes eq is only the transient evolution. As shown in Table 2.1, the asymptotic equilibrium n 1 for the parameters (α1 , α2 ) is the complement to 1 of the one for the parameters (−α1 , −α2 ), i.e., it depends on sign(α1 + α2 ) if the latter is different from zero. In the cases where α1 α2 = −1 (this means that the coefficients of the free part of H both increase or decrease during the evolution), we observe a long transient period before the model reaches the equilibrium state (see Fig. 2.6).

(a) α = (0,

1)

(b) α = ( 1, 0)

Fig. 2.4 Plots of the solutions of Eq. (2.7) with λ = 0.1, and the starting values of the inertia parameters ω10 = 0.5, ω20 = 0.7, when α1 + α2 = −1, α1 α2 = 0; ϕ1,0 as initial condition

2.3

A Generalized Hamiltonian Leading to Asymptotic Equilibria

(a) α = (0, 1)

29

(b) α = (1, 0)

Fig. 2.5 Plots of the solutions of Eq. (2.7) with λ = 0.1, and the starting values of the inertia parameters ω10 = 0.5, ω20 = 0.7, when α1 + α2 = 1, α1 α2 = 0; ϕ1,0 as initial condition

(a) α = (−1, −1)

(c) α = ( 1, 1)

(b) α = (1, 1)

(d) α = (1,

1)

Fig. 2.6 Plots of the solutions of Eq. (2.7) with λ = 0.1, and the starting values of the inertia parameters ω10 = 0.5, ω20 = 0.7, when α1 α2 = ±1; ϕ1,0 as initial condition

30

2 Dynamics with Asymptotic Equilibria

(a) α = (0, −1)

(b) α = (−1, 0)

Fig. 2.7 Plots of the solutions of Eq. (2.7) with λ = 0.3, and the starting values of the inertia parameters ω10 = 0.5, ω20 = 0.7, when α1 + α2 = −1, α1 α2 = 0; ϕ1,0 as initial condition

(a) α = (0, 1)

(b) α = (1, 0)

Fig. 2.8 Plots of the solutions of Eq. (2.7) with λ = 0.3, and the starting values of the inertia parameters ω10 = 0.5, ω20 = 0.7, when α1 + α2 = +1, α1 α2 = 0; ϕ1,0 as initial condition

Choosing ω10 = 0.7, ω20 = 0.5, the time evolutions we get for α1 α2  = −1, as one would expect, are those depicted in Figs. 2.4, 2.5 and 2.6 provided that we change the signs of α1 and α2 ; for instance, if ω10 > ω20 , the evolution in the case α = (0, −1) is that obtained for ω10 < ω20 , in the case α = (0, 1), and so on. On the contrary, if α1 α2 = −1, we get the same time evolution when we exchange the initial values of ω10 and ω20 . The same qualitative considerations apply if we change the value of λ. What can be observed is that higher values of λ determine a shorter time to reach the equilibrium state (see Figs. 2.7, 2.8 and 2.9); this is reasonable, since the value of λ measures the strength

2.3

A Generalized Hamiltonian Leading to Asymptotic Equilibria

(a) α = ( 1,

1)

(c) α = ( 1, 1)

31

(b) α = (1, 1)

(d) α = (1,

1)

Fig. 2.9 Plots of the solutions of Eq. (2.7) with λ = 0.3, and the starting values of the inertia parameters ω10 = 0.5, ω20 = 0.7, when α1 α2 = ±1; ϕ1,0 as initial condition

of the interaction and the term ϕn 1 ,n 2 , (a1† a2 − a2† a1 )ϕn 1 ,n 2  in the dynamical Eq. (2.7) is multiplied by λ; moreover, also the value of the asymptotic equilibrium changes with λ (see Table 2.1). The case where ω10 = ω20 is much simpler. In fact, the time evolution for α = (α1 , α2 ) is the same as that obtained for α = (−α1 , −α2 ). The cases where α1 α2 = −1 produce the particular situations we mentioned before in which the time evolution remains periodic (n 1 (t) and n 2 (t) oscillate in the whole interval [0, 1]), and no asymptotic equilibrium is reached by the system. This is not surprising, since we start with equal inertia parameters and their variation during the evolution is always in the same direction. We also observe that eq the asymptotic equilibrium state is always n 1 = 0.5; when the solution is periodic, 0.5 is the integral mean of the solution itself. A final comment is in order. When the evolution of the system is very close to the equilibrium state, the Hamiltonian (2.10), since n˙ 1 = n˙ 2 ≈ 0, returns that in (1.27). The latter, due to its structure, is such that n 1 (t) and n 2 (t) should oscillate periodically in opposition of phase; nevertheless, n 1 (t) and n 2 (t) do not. Therefore, we can say that a synchroniza-

32

2 Dynamics with Asymptotic Equilibria

(a) α = (0,

(c) α = ( 1,

1), α = ( 1, 0)

1), α = (1, 1)

(b) α = (1, 0), α = (1, 0)

(d) α = ( 1, 1), α = (1,

1)

Fig. 2.10 Plots of the solutions of Eq. (2.7) with λ = 0.1, and the starting values of the inertia parameters ω10 = ω20 = 0.5 according to various choices of α1 and α2 ; ϕ1,0 as initial condition

tion of the oscillations of n 1 (t) and n 2 (t) has emerged, whereupon, since n 1 (t) + n 2 (t) is a conserved quantity, an equilibrium state arises.

2.4

Asymptotic Steady States and Parameters

In this section, we show that, independently of the choice of (α1 , α2 ), a mathematical relation can be found giving the value of the asymptotic equilibrium state for n 1 (t) (which of course determines the asymptotic equilibrium value of n 2 (t) because of the constraint n 1 (t) + n 2 (t) = constant) as a function of ω10 , ω20 , λ, α1 and α2 . eq Let us fix ω10 = 1 and compute the value n 1 varying ω20 between 0.1 and 1.9, for λ = 0.1. We start considering the case where α = (0, −1). The obtained data suggest to look for a nonlinear fit with the function

2.4

Asymptotic Steady States and Parameters

33

Table 2.1 Asymptotic equilibrium state for n 1 (t); ω10 = 0.5, ω20 = 0.7, λ = 0.1, λ = 0.2, and λ = 0.3 eq

eq

eq

α1

α2

n 1 (λ = 0.1)

n 1 (λ = 0.2)

n 1 (λ = 0.3)

0

−1

0.1468

0.2771

0.3432

−1

0

0.1466

0.2768

0.3424

0

1

0.8535

0.7236

0.6581

1

0

0.8535

0.7236

0.6581

−1

0.1466

0.2769

0.3425

1

0.8535

0.7236

0.6581

1

0.8536

0.7236

0.6581

−1

0.1468

0.2771

0.3431

−1 1 −1 1

(a) α = (0,

(b) α = (0, 1)

1)

Fig. 2.11 Equilibrium values of n 1 versus ω10 − ω20 for λ = 0.1, fitted by means of function (2.11)

f (x) = a + b tanh(cx),

(2.11)

where x = ω10 − ω20 , and a, b and c are constant. The data are well fitted by using (2.11) along with the parameters a = 0.5013, b = 0.4793, c = 4.7420.

(2.12)

If we consider the case α = (0, 1), function (2.11) also well reproduces the equilibrium eq values n 1 by using the parameters a = 0.5001, b = −0.4782, c = 4.7458.

(2.13)

Both situations are depicted in Fig. 2.11. Moreover, if we look at the values of a, b and c obtained in the case α = (0, −1) in correspondence to the different values of λ (reported in Table 2.2), we observe that the

34

2 Dynamics with Asymptotic Equilibria

Table 2.2 Parameters entering the fitting (2.11) as a function of λ λ

a

b

c

λ

a

b

c

0.05

0.5012

0.4903

8.9566

0.55

0.5021

0.4231

1.0733

0.1

0.5013

0.4793

4.7420

0.6

0.5020

0.4229

0.9832

0.15

0.5014

0.4688

3.3296

0.65

0.5022

0.4222

0.9092

0.2

0.5015

0.4595

2.6037

0.7

0.5024

0.4167

0.8582

0.25

0.5015

0.4515

2.1494

0.75

0.5024

0.4147

0.8062

0.3

0.5016

0.4457

1.8292

0.8

0.5018

0.4154

0.7516

0.35

0.5018

0.4395

1.6031

0.85

0.5021

0.4140

0.7116

0.4

0.5017

0.4354

1.4215

0.9

0.5022

0.4047

0.6904

0.45

0.5017

0.4310

1.2816

0.95

0.5023

0.4053

0.6534

0.5

0.5019

0.4251

1.1761

1.0

0.5024

0.4168

0.6013

parameter a does not seem to depend on λ, whereas b depends weakly upon λ, and c is strongly affected by the value of λ. Since the equilibrium value in the case α = (0, 1) corresponds to the complement to 1 of the equilibrium value in the case α = (0, −1), we should have exactly a = 1/2. Moreover, neglecting the unavoidable numerical errors in the nonlinear fitting, it is reasonable to assume b = 1/2 in (2.11). Along with these assumptions, looking for a fit relating c to λ, we obtain the following relation: 0.5107 1 c= ≈ . λ 2λ Therefore, we assume in the case α = (0, −1) that the law    1 ω10 − ω20 eq 1 + tanh (2.14) n1 = 2 2λ provides the relation between the constant parameters entering the Hamiltonian (2.10) and eq n 1 . By setting ω10 − ω20 μ= , 2λ we also notice that the quantity δ in (2.2) (related to the periodicity of the evolution when α = (0, 0)) can be written as  δ = 2λ μ2 + 1. It can be verified that the relation (2.14) holds also when α = (−1, 0) and α = (−1, −1). On the contrary, in the cases α = (0, 1), α = (1, 0) and α = (1, 1) the relation giving the value of the asymptotic equilibrium state reads    ω10 − ω20 1 eq 1 − tanh . (2.15) n1 = 2 2λ

2.5

Conclusions and Perspectives

(a) α = (−1, 1)

35

(b) α = (1, −1)

Fig. 2.12 Equilibrium values of n 1 versus ω10 − ω20 for λ = 0.1, fitted by means of function (2.17) eq

Therefore, in all the cases where α1 α2  = −1, the relation giving the value of n 1 in terms of the initial parameters entering the Hamiltonian can be written in the unified way    1 ω10 − ω20 eq 1 − sign(α1 + α2 )tanh . (2.16) n1 = 2 2λ The situation is a little bit different when α1 α2 = −1, where the formula    1 ω10 − ω20 1 − sign(α1 )sign(ω10 − ω20 )tanh , n 1 eq = 2 2λ

(2.17)

giving the equilibrium state for n 1 (t) (see Fig. 2.12), can be derived. As already pointed out, when α1 α2 = −1, and ω10 = ω20 , the evolution remains periodic, with n 1 (t) and n 2 (t) oscillating between 0 and 1 without approaching any equilibrium state. Nevertheless, formula (2.17) predicts an equilibrium value equal to 21 : the formula can be considered coherent also in this case since the mean value of n 1 (t) (oscillating between 0 and 1) is exactly 21 . At the present, we do not have a rigorous argument in order to justify mathematically the relation between the parameters of the model and the value of the equilibrium; further extensions to models with higher number of fermionic modes are possible in order to consider more complicated and realistic systems.

2.5

Conclusions and Perspectives

In this chapter, by considering a suitable limit in the (H , ρ)-induced approach, we showed how the rule can be configured in such a way it continuously acts on the parameters embodied in the Hamiltonian; in some sense, it is possible to define a generalized Hamiltonian operator giving rise to a time evolution approaching specific equilibria. This generalized operator is

36

2 Dynamics with Asymptotic Equilibria

time independent and formally self-adjoint, but in the parameters entering its free part a dependence on the time derivative of the mean values on a given eigenstate of the number operators is embedded. We remark that the sequence of the Hamiltonian operators ruling the dynamics according to the (H , ρ) approach can be viewed as a discrete version of this latter generalized Hamiltonian. Within this generalized framework we are able to describe inside the usual Heisenberg view asymptotic behaviors, even in the case of a two-mode fermionic system living in a finite-dimensional Hilbert space. Moreover, a relation linking the value of the equilibria to the set of the parameters of the model is established. This allows us to tune the parameters of the model in order to obtain a given asymptotic limit. This can be quite useful in concrete applications, when a sort of reverse engineering approach is needed to find out the right Hamiltonian giving rise to a given dynamics. We close this chapter with a natural interpretation of the model with two fermionic modes and this generalized approach in terms of a decision making problem. Suppose there is an argument and two possible opposite decisions to make. Let us interpret the mean values n 1 and n 2 as measures of the propensity to choose between the two different decisions, i.e., n 1 (t) and n 2 (t) are two decision functions in the sense of [1], that is, we make the decision that at the end of the process of interaction between the two opposite answers possesses the larger mean value. Initially, we take the decision associated to n 1 as the candidate to be accepted. The two opinions interact according to the choice of the parameter λ and the inertia parameters. The values of the asymptotic equilibrium state determine the decision to be made.

References 1. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 2. F. Bagarello, R. Di Salvo, F. Gargano, and F. Oliveri. (H , ρ)-induced dynamics and the quantum game of life. Applied Mathematical Modelling, 43:15–32, 2017. 3. F. Bagarello, R. Di Salvo, F. Gargano, and F. Oliveri. (H , ρ)-induced dynamics and large time behaviors. Physica A: Statistical Mechanics and its Applications, 505:355–373, 2018. 4. F. Bagarello and F. Gargano. Non-Hermitian operator modelling of basic cancer cell dynamics. Entropy, 20:270, 2018. 5. S. M. Barnett and P. M. Radmore. Methods in theoretical quantum optics. Clarendon Press, Oxford, 1997. 6. R. Di Salvo, M. Gorgone, and F. Oliveri. (H , ρ)-induced political dynamics: facets of the disloyal attitudes into the public opinion. International Journal of Theoretical Physics, 56:3912–3922, 2017. 7. R. Di Salvo, M. Gorgone, and F. Oliveri. Political dynamics affected by turncoats. International Journal of Theoretical Physics, 56:3604–3614, 2017. 8. R. Di Salvo, M. Gorgone, and F. Oliveri. Generalized hamiltonian for a two-mode fermionic model and asymptotic equilibria. Physica A: Statistical Mechanics and its Applications, 540:12032, 2020.

References

37

9. R. Di Salvo and F. Oliveri. On fermionic models of a closed ecosystem with application to bacterial populations. Atti Accademia Peloritana dei Pericolanti, 94:A5, 2016. 10. R. Di Salvo and F. Oliveri. An operatorial model for complex political system dynamics. Mathematical Methods in the Applied Sciences, 40:5668–5682, 2017.

3

Epidemics: Some Preliminary Results

3.1

Introduction

Since many decades a relevant application of Mathematics is the analysis of epidemics, [1, 6, 12]. The possibility of analysing, and predicting, the evolution of a pandemics was quite relevant in the past, and it was even more relevant recently, when the SARS-CoV-2 changed drastically our lives. Many people started to be attracted by the possibility of constructing mathematical models, of very different kind and nature, able to capture some aspects of what was going on. In this respect, a mathematical (and statistical) analysis and its connection with biological aspects plays a crucial role. This analysis can include the evaluation of the up-to-date relevant quantities such as the various growth rates of the numbers of the infected, dead and recovered individuals through statistical arguments and best fit procedures with well known phenomenological models like the logistic one. Also, more sophisticated stochastic models have been considered [8, 9, 15], as well as Monte Carlo simulations [13], deep learning and fuzzy rule [2], compartments/SIR-like epidemic models [10], transmission network models, [11], only to quote a few. It is not surprising that the unpredictable intrinsic variability of any infection leads very often to the failure of the adopted model in predicting the (especially) long time evolution of the infection process. Most models, especially the very basic ones, could work only to reproduce the evolution of the infection up to the moment the model is applied, and the best parameters used to fit real data can dramatically change when the data are updated. Also, the effect of several waves is not easy to consider and in fact in many models, including the one we are going to consider in this chapter, was not considered at all. We refer to [16, 17] for some other reviews on the mathematical modelling concerning possible descriptions of epidemics. Most of the aforementioned models are based on a compartmental description which considers the population divided into distinct classes, according to their epidemiological status: for instance, the classical S I R formalism divides the population in Susceptible to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_3

39

40

3 Epidemics: Some Preliminary Results

the disease (S), currently Infectious (I ), and Recovered (R). Some enriched versions of this formalism also include other classes (for instance Exposed (E) or Dead (D)). In this chapter, following the general approach described in this book, we adopt an operatorial method based on ladder operators to describe the long time dynamics of an infected population, assuming that the main classes of the model are the infected, recovered and dead individuals. As we shall see later, we construct our system by attaching three different fermionic modes to these individuals, whereas the group of susceptible/healthy individuals (being very large in comparison with the other groups) play the role of a reservoir for the infected people. The dynamics will be deduced out of an Hamiltonian H which contains the various interactions occurring in the system. We will show that our approach is able to provide a reliable long time dynamics capable to capture the final stage (of the first wave) of the infection when the various densities of the classes have reached some sort of equilibrium values. For concreteness, we shall apply our model to the recent COVID-19 outbreaks and to the previous SARS in 2003. More details and discussion on the topic of this chapter, as well as other references to the mathematical modelling devoted to the epidemics dynamics, can be found in [5]. The chapter is organized as follows. In Sect. 3.2, we present the model and the mathematical settings, and we define the main operators used to construct the Hamiltonian of the system. We then derive the dynamics. In Sect. 3.3, we apply our model to capture the long time behavior of the recent Coronavirus pandemics in China and in the Italian region of Umbria, and that of the SARS epidemics in 2003 in China. We further show how imposing lockdown measures after 10 days from the beginning of the spreading of the epidemics is beneficial for the long time value of infected, recovered and dead individuals. Our conclusions are given in Sect. 3.4.

3.2

The Model and Its Dynamics

We introduce here our model: its components, the possible interactions among them, and the relevant Hamiltonian of the system. Then we deduce the differential equations of motion and their solutions, and we focus, in particular, on their long time behavior. Our system S consists of four different compartments: healthy, infected, recovered and dead individuals. Notice that the set of healthy people is very large, when compared to the other three. This is because, in most epidemics experienced so far in mankind’s history, only a small percentage of the whole population is infected. For this reason, using a well settled approach, we consider the healthy people as a sort of big reservoir, R, for a smaller (in size) system, SP , made by the other three compartments, so that S = R ∪ SP . These compartments interact as shown in Fig. 3.1. Healthy people can be infected; those who have been infected, either die or recover. The latter fill up again the set of healthy people and can, in principle, be infected again. Notice that this (oversimplified) analysis does not take in consideration the biological reasons why the contagion can spread all along the population. Our main interest here is just to show that the long time behavior of the number of the

3.2 The Model and Its Dynamics

41

Infected Recovered

Dead

SP

S Healthy

R

Fig. 3.1 The whole system S made by the subsystem SP and its reservoir R

three relevant populations here can be deduced, and is in good agreement in some concrete applications of our general model. As always in our operatorial approach, we start the analysis of the model by attaching some ladder operators to each compartment of the system, and we assume these operators obey suitable commutation relations, whose choice is connected to the features of the system we are interested to. Here, we will assume that the relevant rules are the canonical anticommutation relations (CAR introduced in Sect. 1.2), for reasons which will be clarified later. We use the ladder operators p1 and p1† , p2 and p2† and p3 and p3† , respectively for the infected, the recovered, and the dead individuals. To the healthy people a family of ladder operators, B(k) and B † (k) (labelled by a continuous index k ∈ R), is attached. According to our discussion in Sect. 1.9, the following rules are assumed: † { p n , pm } = δn,m 11, {B(k), B † (q)} = δ(k − q) 11,

(3.1)

for all n, m = 1, 2, 3 and k, q ∈ R. All the other anti-commutators are taken to be zero, say { pn , pm } = {B(k), B(q)} = { pn , B(k)} = { pn , B † (k)} = · · · = 0.

(3.2)

These operators are used to construct an operator, the Hamiltonian H of S , which is the generator of the dynamics of the system. In details, we will assume the following expression for H :

42

3 Epidemics: Some Preliminary Results

⎧ 0 + H1 + H2 , ⎪ ⎪ H = H  ⎪ ⎪ ⎨ H0 = 3j=1 ω j p †j p j + R (k)B † (k)B(k) dk,     † (k) + B(k) p † dk + σ † (k) + B(k) p † dk, p p H = σ B B 1 B,1 1 B,2 2 ⎪ 1 2 R ⎪ R ⎪ ⎪ † † † † ⎩H =σ p + σ p . p + p p p + p p 2 1,2 1,3 1 2 2 1 1 3 3 1

(3.3)

Here ω j and the σ’s are all real quantities, and (k) is a real-valued function, so that H appears as a (formal) Hermitian operator. The meaning of the various contributions in H , and their relations with the scheme in Fig. 3.1, is the following. The first term in H1 models the vertical line in Fig. 3.1. In particular, B(k) p1† describes the fact that the number of healthy people decreases (because of B(k), which is a lowering operator), while the density of the infected increases because of p1† , which is a raising operator. Notice that H1 also contains the adjoint term p1 B † (k), which can be seen as a recovery term: the density of infected decreases (as the effect of p1 ), while that of healthy people increases, because of B † (k). Our model also admits a two-steps recovery path: first the infected slightly recover, and then they recover completely. This double process can be thought as going from the intensive care to otherdepartments in the hospital, and then being released. These two steps are described by σ1,2 p1† p2 + p2† p1 in H2 , and by the second term in H1 . As before, we have a contribution p2 B † (k) describing full recovery, but we also have a term B(k) p2† which describes the possibility that someone, negative to a first swab, becomes positive later. This term could also be relevant in describing people who fall again infected, after a first recovery. A similar interpretation we can imagine for the term σ1,2 p1† p2 : part of those who apparently are recovering, fall again infected. As for H0 , this is a standard free term, not affecting all the populations in absence of interactions. In other words, if H = H0 , the densities of the members of each population do not change with time. We notice that the Hamilonian contains a term of the kind p1† p3 which, in principle, is not physical (dead individuals cannot become infected). However, due to the initial conditions we shall choose and the parameters of the Hamiltonian, the effective contribution of this term will be negligible as compared to others, and in particular to that deriving from the term p3† p1 . This will be explained at the end of this section. It is clear that the operator H considered here is just a (simple, but not so simple) possibility. More effects could be added in H , or some explicit time dependence might be considered. In addition, we could also consider the effect of some external rule leading to the (H , ρ)-induced dynamics described in Sect. 1.6 and in Chap. 2. In particular, this latter possibility will be considered later, and we will show that it produces interesting results. Using now (3.3) and the Heisenberg equations of motion X˙ (t) = i[H , X (t)], we get, because of the CAR (3.1) and (3.2):

3.2 The Model and Its Dynamics

⎧  p˙ 1 (t) = −iω1 p1 (t) + iσ B,1 R B(q, t) dq − iσ1,2 p2 (t) − iσ1,3 p3 (t), ⎪ ⎪  ⎪ ⎪ ⎨ p˙ 2 (t) = −iω2 p2 (t) + iσ B,2 R B(q, t) dq − iσ1,2 p1 (t), p˙ 3 (t) = −iω3 p3 (t) + iσ1,3 p1 (t), ⎪ ⎪ ⎪ ⎪ ⎩ B(q, ˙ t) = −i(q)B(q, t) + iσ B,1 p1 (t) + iσ B,2 p2 (t).

43

(3.4)

This is a system of four ordinary, operator valued, linear differential equations in four unknowns which describes the time evolution of the relevant dynamical variables of the system, the lowering operators p j (t), j = 1, 2, 3 and B(q, t). Then the system can be explicitly solved and, from this solution, we can deduce the time evolution of the relevant operators used to measure the densities of the various populations. In fact, as proposed in Chap. 1, we consider the evolution of the number operators related to the p j (t) above, Pˆ j (t) = p †j (t) p j (t), j = 1, 2, 3, and then to their mean values on suitable states describing the initial conditions of S , see below. These are for us, the initial numbers (or, more properly, densities) of the infected, recovered and dead people. Since the mean values of the number operators related to the CAR are always quantities ranging in [0, 1], this shows why we look at these quantities as densities of the populations of SP . To deduce quantities which can be compared with the data, we need to introduce suitable states over the system. Then, we start introducing the vacuum of the p j , i.e., the non zero vector ϕ0,0,0 in HP = C8 , the Hilbert space of SP , satisfying the condition p j ϕ0,0,0 = 0, j = 1, 2, 3. The other vectors of the o.n. basis Fϕ = {ϕn 1 ,n 2 ,n 3 , n 1 , n 2 , n 3 = 0, 1} for HP can be constructed as follows: ϕ1,0,0 = p1† ϕ0,0,0 , ϕ0,1,0 = p2† ϕ0,0,0 , ϕ1,1,0 = p1† p2† ϕ0,0,0 , ϕ1,1,1 = p1† p2† p3† ϕ0,0,0 , and so on. Now, the reason why Pˆ j (t) is called “number operator” is because, if we compute its action on ϕn 1 ,n 2 ,n 3 , we get Pˆ j (0)ϕn 1 ,n 2 ,n 3 = n j ϕn 1 ,n 2 ,n 3 , j = 1, 2, 3. This means that the elements of Fϕ are eigenstates of Pˆ j = Pˆ j (0), property which is lost for t > 0: in general, Pˆ j (t)ϕn 1 ,n 2 ,n 3 is only a linear combination of the ϕn 1 ,n 2 ,n 3 ’s, with time dependent coefficients. In view of what we have discussed so far, the subsystem SP is described, at t = 0, by the vector ϕ0,0,0 . This is because the eigenvalues of the Pˆ j (t)’s, or better (see below), their mean values, are proportional to the densities of the various compartments. Therefore, since at t = 0 there are no elements of P1 , P2 and P3 , SP must be described by ϕ0,0,0 . Anyway, since SP is just a part of S , we still have to introduce a state over R, as pointed out in Sect. 1.9. This is a positive linear functional [7, 14] on the algebra of the reservoir operators satisfying the following (standard) requirements: ωR (11R ) = 1, ωR (B(k)) = ωR (B † (k)) = 0, ωR (B † (k)B(q)) = N (k) δ(k − q), (3.5) for some suitable function N (k), as well as ωR (B(k)B(q)) = 0, k, q ∈ R. In analogy with what happens for SP , N (k) is also set to be constant in k. From now on, we set N (k) = N , and this value is a kind of measure of the density of people exposed to the infection: N = 0

44

3 Epidemics: Some Preliminary Results

actually means that there are no people exposed, whereas higher values of N mean that more people are exposed. This will be clear later when we will deduce a specific asymptotic solution corresponding to a particular choice of the parameters, and we will show how changing the value of N can drastically modify the whole dynamics. The state over the full system S , ., is defined, for all operators of the form X S ⊗ YR , X S being an operator of SP and YR an operator of the reservoir, as follows: 

X S ⊗ YR  := ϕn 1 ,n 2 ,n 3 , X S ϕn 1 ,n 2 ,n 3 ωR (YR ).

(3.6)

For the purposes of our model, we now introduce the mean values of the number operators, which are what we can call the density functions for the system



j = 1, 2, 3. (3.7) P j (t) := Pˆ j (t) = p †j (t) p j (t) , Once we have the equations of motion for the ladder operators and the form of the states over S , we can find the densities of the various classes of individuals. We refer to [3, 4] for further details about this computation for a different (and slightly larger) system. Here, we limit ourselves to write the final analytical expression for P j (t) in (3.7). For that, we need to introduce first some useful quantities. We start defining the symmetric block matrix U , ⎛

ωˆ 1 ⎜ γ1,2 ⎜ ⎜γ ⎜ 1,3 U =⎜ ⎜ 0 ⎜ ⎝ 0 0

γ1,2 ωˆ 2 0 0 0 0

γ1,3 0 ωˆ 3 0 0 0

0 0 0 ωˆ 1 γ1,2 γ1,3

0 0 0

0 0 0

⎞

⎟ ⎟ ⎟ ⎟ ⎟, γ1,2 γ1,3 ⎟ ⎟ ωˆ 2 0 ⎠ 0 ωˆ 3

where ωˆ 1 := iω1 + π

σ 2B,1 

, ωˆ 2 := iω2 + π

σ 2B,2 

, ωˆ 3 := iω3 , γ1,2 := iσ1,2 +

π σ B,1 σ B,2 , γ1,3 := iσ1,3 . 

Here  is a positive constant related to (k), which we assume to be linear in k: (k) = k. Further, we call Vt := exp(−U t), (Vt ) j,k its entries, and   2  ( j) ( j) pk (t) = (Vt ) j,k  , pk,l (t) = 2 (Vt ) j,k (Vt ) j,l , where (z) stands for the real part of the complex quantity z. We write the explicit formula for the density functions assuming first that the initial state on SP coincides with one of the element of Fϕ and then extending the result to the general case in which this state is a linear combination (normalized to one) of the elements of Fϕ . If the initial state is ϕn 1 ,n 2 ,n 3 we have (a) (b) (3.8) P j (t) = P j (t) + P j (t),

3.2 The Model and Its Dynamics

where (a)

P j (t) =

45

3       (Vt ) j,k 2 n k + (Vt ) j,k+3 2 (1 − n k )

(3.9)

k=1

and (b)

P j (t) =

2π N 



t

dt1  j (t − t1 ),

(3.10)

0

j = 1, 2, 3. Here, following [3, 4], 2   j (s) = σ B,1 (Vs ) j,1 + σ B,2 (Vs ) j,2  .

(3.11)

In particular, let us see what happens if the infection starts to propagate at t = 0 from a situation in which everyone is healthy. This fixes uniquely the state on which the mean values of the Pˆ j (t) must be computed. If this is the case, then the values of the n k ’s in (3.9) are all zero. Moreover, the block form of the matrix U is translated to a similar form for Vt . This implies that (Vt ) j,k+3 = 0 for all j, k = 1, 2, 3. Hence, the conclusion is that, assuming that at t = 0 the number of infected, recovered and dead is zero, the term P j(a) (t) does not contribute to P j (t) in (3.8). It is possible in this case to determine the asymptotic limit of the density functions,  2π N t P j (∞) = lim dt1  j (t − t1 ), (3.12) t→∞  0 which are directly related to the long time behavior of the various compartments in a real epidemic. As said above, the value of N modifies the asymptotic value of the density functions; a larger N implies an increasing number of infected and, consequently, of the recovered and of the dead people, whereas if N = 0 no one is infected. For a specific choice of parameters, the limits in formula (3.12) can be analytically computed. Indeed, setting σ B,2 = 0 and ω j = 0 ( j = 1, 2, 3) we get [5] P1 (∞) = N ,

P2 (∞) =

2 N σ1,2 2 + σ2 σ1,3 1,2

,

P3 (∞) =

2 N σ1,3 2 + σ2 σ1,3 1,2

.

Let us now see what happens if the state on SP has the form   = αn 1 ,n 2 ,n 3 ϕn 1 ,n 2 ,n 3 , with |αn 1 ,n 2 ,n 3 |2 = 1. n 1 ,n 2 ,n 3 =0,1

(3.13)

(3.14)

n 1 ,n 2 ,n 3 =0,1

This extension is relevant for what we are going to show later. In this case, the densities are given for all j by (a)

(a)

(b)

P j (t) = P j (t) + δ P j (t) + P j (t), where

j = 1, 2, 3,

(3.15)

46

3 Epidemics: Some Preliminary Results ⎛ P j(a) (t) = ⎝ ⎛ +⎝



   (Vt ) j,1 2 α1,n



   (Vt ) j,3 2 αn

n 1 ,n 2 =0,1

⎛

⎛

(a) δ P j (t)

2 ,n 3

n 2 ,n 3 =0,1

= 2 ⎝(Vt ) j,1 (Vt ) j,2 ⎝ ⎛ + (Vt ) j,2 (Vt ) j,3 ⎝

1 ,n 2

⎞ 2  ,1 ⎠ ,



n 3 =0,1



⎞ ⎛  2  ⎠+⎝

   (Vt ) j,2 2 αn

1 ,1,n 3

⎞ 2  ⎠

n 1 ,n 3 =0,1

⎞

⎛

α¯ 1,0,n 3 α0,1,n 3 ⎠ + (Vt ) j,1 (Vt ) j,3 ⎝



⎞ α¯ 1,n 2 ,0 α0,n 2 ,1 ⎠

n 2 =0,1

⎞⎞

α¯ n 1 ,1,0 αn 1 ,0,1 ⎠⎠ ,

n 1 =0,1

(b)

(a)

and P j (t) is the same as in (3.10). We observe that P j (a) δ P j (t)

reduces to (3.9) if the initial state (a)

coincides with (only) one of the ϕn 1 ,n 2 ,n 3 and = 0. Otherwise, δ P j (t) = 0, and it can be considered as an interference term which appears only if the initial state of SP is a non trivial superposition of the elements of Fϕ . We will show that this may produce some not negligible oscillations of the P j (t), which are particularly evident during the first part of the time evolution, while they tend to disappear for t sufficiently large. We refer to [5] for more comments on these oscillations whose analysis is not particularly relevant here. To conclude this section, we go back once more to the role of p1† p3 in H2 . This is crucial to preserve the Hermiticity of the Hamiltonian. However, as we have already discussed, it is not realistic. Nevertheless, this is not a concrete problem, since the initial conditions for SP , which is empty at t = 0, and the fact that the reservoir R interacts essentially with the infected and the recovered, produces a growth of the densities of these two compartments. In other words, during the time evolution, SP is (mainly) described by some linear combination of the vectors ϕ1,0,0 , ϕ0,1,0 , ϕ1,1,0 . This implies that the only terms in H2 which do not annihilate the state of the system are p1† p2 , p2† p1 and p3† p1 , while p1† p3 does not act on any realistic state of S .

3.3

Applications

In this section, we show how this model is sufficiently refined to reproduce the long time dynamics in some recent epidemics. In particular, we will concentrate on SARS and Coronavirus in China, and on Coronavirus in the Italian region of Umbria. Coronavirus pandemic, as far as we know today, started in Wuhan, China, most likely at the end of December 2019. Until January 2020 the epidemic was apparently confined to China with few thousands of confirmed cases, while between February and March it spread in Europe, in the US, and in other countries.

3.3

Applications

47

Fig. 3.2 SARS 2003 in China. Real data (dotted) and density functions P j (t) (solid lines) according to Eq. (3.8). (a) cumulative infected, (b) cumulative recovered, (c) cumulative deaths. The initial state of SP is ϕ0,0,0 . Parameter values:  = 1, N = 1, ω1 = 0.6, ω2 = 0.05, ω3 = 0.05, σ B,1 = 0.15, σ B,2 = 10−4 , σ1,2 = 1, σ1,3 = 0.165

Covid is not the only recent pandemic we had to fight against. In 2002, the severe acute respiratory syndrome SARS outbreak was first identified in Foshan, Guangdong, China. Up to May 2004 it caused over 8,000 individual infections from 29 different countries, and more than 900 deaths worldwide. We now present the results concerning the long time behavior of epidemic evolution as deduced trough best-fit procedures on the parameters; in particular, we have tried to minimize the relative error with the precise values of cumulative infected, recovered and dead individuals in China for both SARS and Coronavirus and in Umbria, Italy, for Coronavirus. Due to the complexity of the formal expression of the solution, in general, our fitting procedures were based on restricting first the range of the various parameters by requiring good qualitative behaviors of the solutions (when compared with the real data), and then by performing some fine-tuning variations of the parameters trying each time to minimize the errors. In all cases we started with an initial state 0 = ϕ0,0,0 corresponding to the absence of infected, recovered and dead individuals, so that the initial relevant mechanisms is the infective process occurring on some of the members of the reservoir (the healthy individuals). Densities evolutions are shown in Fig. 3.2 relative to the period March-July 2003 for SARS epidemic, that is from the first reported infection up to plateau. Similar plots are shown in Figs. 3.3 and 3.4 for Coronavirus in China and Umbria, respectively. All the densities are scaled with the final value of total confirmed cases,1 which in our cases are 5327 for the SARS, 80928 for the Coronavirus in China, and 1400 for the Coronavirus in Umbria, Italy. We can observe that after the initial transient in which infection rapidly grows, and in which the model reproduces the real data only at a qualitatively level, we end with no increments, and the model captures very well the final stage of the outbreak for all the compartments of SP , the infected, the recovered and the dead. We refer to [5] for an analysis on the error of our results.

1 When the original paper [5] was written.

48

3 Epidemics: Some Preliminary Results

Fig. 3.3 COVID-19 in China. Real data (dotted) and density functions P j (t) (solid lines) according to Eq. (3.8). (a) cumulative infected, (b) cumulative recovered, (c) cumulative deaths. The initial state of SP is ϕ0,0,0 .Parameter values:  = 4.5, N = 1, ω1 = ω2 = ω3 = 0, σ B,1 = 0.32, σ B,2 = 10−4 , σ1,2 = 1, σ1,3 = 0.22

Fig. 3.4 COVID-19 in Umbria. Real data (dotted) and density functions P j (t) (solid lines) according to Eq. (3.8). (a) cumulative infected, (b) cumulative recovered, (c) cumulative deaths. The initial state of SP is ϕ0,0,0 . Parameter values:  = 4.3, N = 1, ω1 = ω2 = ω3 = 0, σ B,1 = 0.35, σ B,2 = 0, σ1,2 = 0.037, σ1,3 = 0.009

3.3.1

Lockdown Measures

We consider now a simple variation of the model in which we adaptively change the parameters of the Hamiltonian to correct the Heisenberg dynamics to model the introduction of some countermeasures meant to fight the spreading of the disease. In particular, following the basic ideas of the (H , ρ)-induced dynamics introduced in Sect. 1.6, we change the value of the parameters (in our case just one) of the model at some specific times, mimicking in this way the occurrence of some check in the real-life situation. This induces a sort of discrete time dependence for the system. To be more specific, we suppose that at the time T the parameter N in (3.5) is adjusted to mimic the effects of some social rules, imposed by the government, with the aim of limiting the diffusion of the disease. We recall that, as suggested by (3.12), N is a measure of the density of the exposed people, i.e., the reservoir, so that decreasing (increasing, respectively) N limited (helped, respectively) the infection process: in particular, decreasing N , we are modelling what has been called social distance. Starting with an initial state 0 = ϕ0,0,0 , we let the system evolve up to T , and we compute the densities using (3.8). Then we change the value of N , taking a lower value to mimic the beginning of a lockdown, and we start the new dynamics with a new initial state

3.3

Applications

49

new =



αn 1 ,n 2 ,n 3 ϕn 1 ,n 2 ,n 3 ,

n 1 ,n 2 ,n 3

which is constructed by requiring the continuity of the densities (not their differentiability in general). The form of this state is obtained by choosing the coefficients of new as follows:  α1,0,0 = P1 (T ) − P2 (T ) − P3 (T ),  α1,1,0 = P2 (T ), (3.16)  α1,0,1 = P3 (T ),  α0,0,0 = 1 − P1 (T ), while the other coefficients are set to 0. Notice that (3.16) ensure the continuity of the densities at T , and (3.16)4 , in particular, ensures the normalization of new to 1. The requirement that the other coefficients are 0 simply follows from our interpretation of the vectors of the basis Fϕ : it is not possible to have dead, or recovered, in absence of infected. This is the reason why we impose that α0,n 2 ,n 3 = 0, if n 2 , or n 3 , or both are different from zero. Moreover, we also put α1,1,1 = 0 since there is no possibility to have the same number of infected, recovered and dead. Of course there are other forms of the coefficients which ensure the continuity of the densities, but our choice works already rather well, and it is easy to implement. In Fig. 3.5, we present the results concerning SARS and COVID-19, using the same parameters as reported in Figs. 3.2 and 3.3, respectively, and with T = 10 and N = 0.8: this is essentially like saying that, after 10 days, the lockdown restrictions lowered the normal density of the people from 1 to 0.8. A direct comparison of the density functions with and without the applied measures is shown in Fig. 3.6. Of course, different values of T and lower values of N could be chosen, producing quantitative difference in the results, but with similar qualitative outcomes (i.e., reduction of the infection). We can observe here that transient oscillations are evident, and disappear with the exception of the dead density in the SARS case: in this case, the asymptotic behavior of the density is an oscillating function (not

Fig. 3.5 Behavior of the density functions P j (t) in Eq. (3.8) for SARS (left) and COVID-19 (right), assuming a lockdown at the 10th day from the beginning of the spreading. Parameter values: same as Figs. 3.2 and 3.3 with T = 10 and N = 0.8

50

3 Epidemics: Some Preliminary Results

Fig. 3.6 Comparison of the densities of infected (black curves) and dead (gray curves) for SARS (left) and COVID-19 (right), assuming a lockdown measure at the 10th day from the beginning of the spreading. Solid curves densities without lockdown measures, dotted curves densities with measures. In the inset of the right figure the magnification of the evolution of the dead densities. Parameter values: same as Figs. 3.2 and 3.3, with T = 10 and N = 0.8

fully shown in the figure) with period T ≈ 90 and mean value P¯3 = 0.036. It is interesting to stress that this value is much lower than the asymptotic value P3 (∞) = 0.065 obtained without any lockdown measure. Therefore, the procedure of imposing a lockdown at a certain time via an increasing social distance, is able, in the model, to reduce significantly the asymptotic number of infected, recovered and dead individuals, in agreement with what observed in real life.

3.4

Conclusions

In conclusion, this chapter proposes a model for the spreading of infection diseases based on the quantum-like ideas widely explored in this book. The application to the spreading of SARS in 2003 in China and of COVID-19 in China and in the Italian region of Umbria shows how such a model is able to capture well the asymptotic behavior of the spreading. In addition, imposing lockdown measures by reducing a model parameter at a precise instant of time, we have shown that it is possible to reduce significantly the number of infected individuals, and of the recovered and dead ones as a consequence. Of course the model could be improved under several aspects, in the attempt to make it more precise and, possibly, useful. In particular, it would be interesting to improve its short time behavior, to better reproduce what happens in real life, or to include in the model the effect of the various waves of infection or even the effect of vaccination, which (we believe) could be efficiently included by adding a second rule to simple one proposed in Sect. 3.3.1.

References

51

References 1. H. Andersson and T. Britton. Stochastic epidemic models and their statistical analysis. Springer, New York, 2000. 2. P. Arora, H. Kumar, and B. Ketan Panigrahi. Prediction and analysis of COVID-19 positive cases using deep learning models: A descriptive case study of India. Chaos, Solitons & Fractals, 139:110017, 2020. 3. F. Bagarello. An operator view on alliances in politics. SIAM Journal on Applied Mathematics, 75:564–584, 2015. 4. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 5. F. Bagarello, F. Gargano, and F. Roccati. Modeling epidemics through ladder operators. Chaos, Solitons & Fractals, 140:110193, 2020. 6. N. T. G. Bailey. The mathematical theory of infectious diseases and its applications. Griffin, London, 1975. 7. O. Bratteli and D.W. Robinson. Operator algebras and quantum statistical mechanics 1. Springer-Verlag, New York, 2002. 8. F. Calleri, G. Nastasi, and V. Romano. Continuous-time stochastic processes for the spread of COVID-19 disease simulated via a monte carlo approach and comparison with deterministic models. Journal of Mathematical Biology, 83:34, 2021. 9. B. Cazelles, M. Chavez, A. J. McMichael, and S. Hales. Non stationary influence of ei nino on the synchronous dengue epidemics in thailand. PLoS Medicine, 2:313, 2005. 10. S. Çakan. Dynamic analysis of a mathematical model with health care capacity for covid-19 pandemic. Chaos, Solitons & Fractals, 139:110033, 2019. 11. T. Chen, J. Rui, and Q. Wang. A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infectious Diseases of Poverty, 9:24, 2020. 12. O. Diekmann and J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases. Wiley, Chichester, 2000. 13. S. J. Fong, G. Li, N. Dey, R. Gonzalez Crespo, and E. Herrera-Viedma. Composite monte carlo decision making under high uncertainty of novel coronavirus epidemic using hybridized deep learning and fuzzy rule induction. Applied Soft Computing, 93:106282, 2020. 14. M. Reed and B. Simon. Methods of modern mathematical physics I: Functional analysis. Academic Press, New York, 1980. 15. J. Shaman and M. Kohn. Absolute humidity modulates influenza survival, transmission, and seasonality. Proceedings of the National Academy of Sciences, 106:3243–3248, 2009. 16. C. I. Siettos and L. Russo. Mathematical modeling of infectious disease dynamics. Virulence, 4:295–306, 2003. 17. S. Unkel, P. C. Farrington, H. Paul, PH. Garthwaite, C. Robertson, and N. Andrew. Statistical methods for the prospective detection of infectious disease outbreaks: a review. Journal of the Royal Statistical Society A, 175:49–82, 2012.

4

Spreading of Information in a Network

4.1

Introduction

In this chapter we describe an application of our operatorial approach to the description of the diffusion of news (good or fake) in a network of agents that can be considered as interacting transmitters or receivers of the news. The mathematical models for the description of the spreading of information have grown recently, due to the rise of social media, personal blogs, wiki-like sites, that completely changed the way the users relate to the diffusion of news. Most models are naturally based on concepts widely used in epidemiological models and graph analysis [1, 5–7]. In our approach, the key ingredient is the definition of a Hamiltonian operator H , constructed using suitable fermionic ladder operators, that includes all the mechanisms of the transmission and distortion of the news by various agents, as well as the definition of suitable rules in the spirit of the (H , ρ)-induced dynamics that has been used also in other problems faced in this book. The observables in our analysis are the number operators attached to each ladder operator, from which we can obtain their mean values that are phenomenologically interpreted as a measure of the reliability of the news by each agent. The Hamiltonian H contains all the operators describing the interactions occurring among the different agents of the system; in particular, the terms involved in the Hamiltonian describe the diffusion of fake and good news, as well as with the ability to modify the nature of news from good to fake and vice versa. Our main goal is to describe the typical uncertainty which accompanies the diffusion of news through non-reliable agents (social media above all). We shall apply the model to two heuristic cases adopting different kinds of behavior of the agents (i.e., different rules ρ) which, as we shall see, can drastically change the way news are perceived. The chapter is organized as follows. In Sect. 4.2, we define the operatorial model for the spreading of news in a network with the definition of the Hamiltonian and the possible rules.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_4

53

54

4 Spreading of Information in a Network

Subsequently, in Sect. 4.3 we propose two applications of the general network of agents. Section 4.4 contains our conclusions. Further details and applications can be found in the original paper [4].

4.2

The Model and Its Dynamics

Our main interest is to deduce a dynamical behavior for the news N , that can be transmitted in a neutral way (good news), i.e., it is transmitted in its original form, or it is somewhat changed (fake news), for instance due to a generic agent’s convenience. We suppose that our system is made by N agents Aα , α = 1, . . . , N , able of creating, receiving and transmitting the news N . We can see the various agents as different cells of a system/network S . We say that a couple of agents (Aα , Aβ ) are neighboring if they can interchange information without any intermediary. It is natural for this problem to work in the framework of the fermionic operators described in Sect. 1.2. The reason is that with the number operator approach we can determine a finite measure of how an agent can perceive the nature of some news, good or fake. It is natural to assign a value of this perception in the interval [0, 1], with the value 0 indicating that, for instance, the agent considers the news not good at all, whereas the value 1 indicates that the agent considers the news as good (similarly for the perception of a fake news). This will be detailed below. For each agent Aα , let us introduce two families of fermionic operators that, in view of the connection of these operators with the nature of the news, we label as f α and f α† (fake news), and gα , gα† (good news). Of course, these operators satisfy CAR rules { f α , f β† } = {gα , gβ† } = δα,β 11,

2 = { f , g } = { f † , g } = { f , g † } = { f † , g † } = 0. f α2 = gα α β α β α β α β

(4.1) † †   As usual, we attach to these families the number operators Fα = f α f α and G α = gα gα . The related local four-dimensional Hilbert space Hα is constructed in the usual way: we introduce (f) (f) (g) (g) the vacua, that surely exist, ϕα,0 and ϕα,0 satisfying the equations f α ϕα,0 = gα ϕα,0 = 0. (f)

(f)

(g)

(g)

The other relevant vectors are defined with ϕα,1 = f α† ϕα,0 , ϕα,1 = gα† ϕα,0 , and (f)

ϕα:n f ,n g = ϕα,n f ⊗ ϕ(g) α,n g ,

(4.2)

where n f , n g = 0, 1. As seen in Sect. 1.2, the set Fϕ (α) = {ϕα:n f ,n g } provides an o.n. basis of Hα . Moreover, from (1.2), we have α ϕα:n f ,n g = n f ϕα:n f ,n g , F

α ϕα:n f ,n g = n g ϕα:n f ,n g . G

(4.3)

As already discussed in the previous chapters, we associate a phenomenological meaning to each vector ϕα:n f ,n g . For instance, ϕα:0,0 describes the situation in which the news N , in any of its form, has not reached the agent Aα . The vector ϕα:1,0 expresses that a fake

4.2 The Model and Its Dynamics

55

version of N has been transmitted to Aα , whereas ϕα:0,1 expresses that Aα has received the good version. Finally, ϕα:1,1 describes a situation in which a noised version of N has reached Aα that, as a consequence, is not able to discern between a good or a fake news. We now construct the global Hilbert space H = ⊗α Hα , endowed with the scalar product   f , g =  f α , gα α , α

for each f = ⊗α f α and g = ⊗α gα . The vectors of the basis of H are easily deduced: n,m = ⊗α ϕα:n α ,m α ,

(4.4)

where n = (n 1 , n 2 , . . . , n N ), m = (m 1 , m 2 , . . . , m N ). Again we can associate a phenomenological meaning to each vector, describing, in particular, which kind of information has reached any agent Aα of the whole system. The action of any operator X α on Hα is straightforwardly extended to H by the operator Xˆ α acting as X α ⊗ (⊗β=α 11β ), where 11β is the identity operator on the local Hilbert space Hβ .

4.2.1

The Hamiltonian and Its Effect

Now we have all the elements to construct the basic Hamiltonian that drives the dynamical behavior of S . To insert all the mechanisms that are relevant (in the present context) in the transmission of the news, we assume the following form of H : ⎧ H = H0 + H I , with ⎪ ⎪ ⎪   ⎪ ⎪ α + α , ⎨ H0 = ω f ,α F ωg,α G α α ⎪  (f)  (g)  ⎪ ⎪ † † † † ⎪ = p ( f f + f f ) + p (g g + g g ) + λα ( f α gα† + gα f α† ), H ⎪ I α α β β α α α,β α,β β β ⎩ α,β

α

α,β

(4.5) are all non negative. The various terms in the Hamiltonian are again quite standard in describing the dynamics of a system [2, 3]; the term H0 , describes the free part of the Hamiltonian, and the parameters ω f ,α and ωg,α measure the inertia of the various agents, i.e., their tendency to keep, or change, the original message they have received; the first two terms in H I describe, respectively, a diffusion process for fake and news; finally, the third term in H I is an interaction term which is able to ( f ,g) change the nature of N in each cell α. The diffusion coefficients pα,β in H I are assumed ( f ,g) where the various parameters ω f ,α , ωg,α , pα,β , λα

( f ,g)

( f ,g)

( f ,g)

to be symmetric ( pα,β = pβ,α ); moreover, we take pα,α = 0. It is reasonable to assume that

56

4 Spreading of Information in a Network



(f)

pα,β >



α,β

(g)

pα,β ,

α,β

since, usually, fake news diffuse faster than good news. We now derive the equations of motion for the ladder operators using the CAR in (4.1). In particular, using the Heisenberg picture described in Sect. 1.3.2, we obtain the following linear, operator-valued, system of ordinary differential equations ⎧  (f) ⎪ f˙α (t) = −iω f ,α f α (t) + 2i pα,β f β (t) + iλα gα (t), ⎪ ⎪ ⎨ β (4.6)  (g) ⎪ ⎪ (t) = −iω g (t) + 2i p g (t) + iλ f (t), g ˙ α g,α α α α β ⎪ α,β ⎩ β

where α = 1, . . . , N . Denoting with X(t) the 2N -column vector X(t) = ( f 1 (t), f 2 (t), . . . , f N (t), g1 (t), g2 (t), . . . , g N (t))T , and the following 2N × 2N matrix V , ⎛ (f) (f) −ω f ,1 2 p1,2 2 p1,3 . . ⎜ (f) (f) ⎜ 2 p1,2 −ω f ,2 2 p2,3 . . ⎜ (f) ( f ) ⎜ 2p ⎜ 1,3 2 p3,2 −ω f ,3 . . ⎜ . . . .. ⎜ ⎜ . . .. ⎜ . ⎜ (f) (f) (f) ⎜ 2p 1,N 2 p2,N 2 p3,N . . V =⎜ ⎜ λ 0 0 .. 1 ⎜ ⎜ ⎜ 0 λ2 0 .. ⎜ ⎜ 0 0 λ3 . . ⎜ ⎜ . . .. ⎜ . ⎜ ⎝ . . . .. 0 0 . ..

(f)

2 p1,N (f) 2 p2,N (f) 2 p3,N . . −ω f ,N 0 0 0 . . λN

λ1 0 0 . . 0 −ωg,1 (g) 2 p1,2 (g) 2 p1,3 . . (g) 2 p1,N

0 λ2 0 . . 0 (g) 2 p1,2 −ωg,2 (g) 2 p3,2 . . (g) 2 p2,N

0 0 λ3 . . . (g) 2 p1,3 (g) 2 p2,3 −ωg,3 . . (g) 2 p3,N

. . . . . . . . . . . .

. . . . . . . . . . . .

⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ . ⎟ ⎟ ⎟ . ⎟ ⎟ λN ⎟ ⎟, (g) 2 p1,N ⎟ ⎟ (g) ⎟ 2 p2,N ⎟ ⎟ (g) 2 p3,N ⎟ ⎟ ⎟ . ⎟ ⎟ . ⎠ −ωg,N

the system (4.6) can be rewritten as ˙ X(t) = iV X(t),

(4.7)

X(t) = exp(iV t)X(0),

(4.8)

whose solution is clearly X(0) denoting the initial condition. We now derive the explicit expression of the relevant quantities we want to obtain, that is the mean values of the number operator and a global measure of how, on average, the news is perceived by the whole system. Let vi, j (t) = (exp(iV t))i, j , i, j = 1, 2, . . . , 2N ,

4.2 The Model and Its Dynamics

57

and E = {ϕ j , j = 1, 2, . . . , 2N } be the canonical o.n. basis in H2N ≡ C2N . Then, for α = 1, . . . , N , we have f α (t) = ϕα , X (t)2N ,

gα (t) = ϕ N +α , X (t)2N .

(4.9)

Let Fβ0 and G 0β be the mean values of Fβ and G β on the vector ϕβ:n β ,m β at t = 0, say Fβ0 = ϕβ:n β ,m β , Fβ ϕβ:n β ,m β β ,

G 0β = ϕβ:n β ,m β , G β ϕβ:n β ,m β β ;

then, setting, with the same notation as before, α (t)n,m  = n,m , f α† (t) f α (t)n,m  Fα (t) = n,m , F α (t)n,m  = n,m , gα† (t)gα (t)n,m , G α (t) = n,m , G

(4.10)

we finally obtain the mean values Fα (t) =

N    |vα,β (t)|2 Fβ0 + |vα,β+N (t)|2 G 0β , β=1

G α (t) =

N  



(4.11)

|vα+N ,β (t)|2 Fβ0 + |vα+N ,β+N (t)|2 G 0β ,

β=1

and the global mean values result defined as F(t) =

N 1  Fα (t), N α=1

G(t) =

N 1  G α (t). N

(4.12)

α=1

We observe that, due to the fermionic nature of the operators therein involved, we have Fα (t), G α (t) ∈ [0, 1], and, consequently, F(t), G(t) ∈ [0, 1] as well. It is clear that (4.11) and (4.12) have different meanings. The functions Fα (t) and G α (t) are the local measure of how the agent Aα perceives the news. In the case Fα (t)  G α (t) (Fα (t) G α (t), respectively), there is no doubt the agent Aα perceives the news as fake (good, respectively), whereas the condition Fα (t) ≈ G α (t) reflects the uncertainty about the reliability of news. On the other hand, F(t) and G(t) are global mean values measuring the intensity of how the news is perceived as fake or good in R.

4.2.2

The Effect of the Rule

Some applications of the (H , ρ)-dynamics have been presented in other parts of this book. We just recall that, as explained in Sect. 1.6, the idea is to change the dynamics of the system by inducing effects derived from some rules whose action is difficult, if not impossible, to embed in a proper Hamiltonian operator. Here we apply the same concept by considering

58

4 Spreading of Information in a Network

the following basic rules. Consider a given cell α, and let α be the set of all the cells which f ,g are connected to the cell α. Denoting with wα (γ) the weights assigned to the cells γ in α , we set the weighted measure of the perception of the news as Fα (T ) := f ,g

where Wα

=

1



f Wα γ∈α



γ∈α

wαf (γ)Fγ (T ),

G α (T ) :=

1  g w (γ)G γ (T ), g Wα γ∈ α

(4.13)

α

f ,g

wα (γ) are the sums of the weights, and α (T ) = Fα (T ) − G α (T )

(4.14)

are their difference. The meaning of α (T ) is evident: if α (T ) > 0 then, at t = T , Aα is surrounded by agents distributing more fake news rather than good news, whereas the opposite happens if α (T ) < 0 (α (T ) = 0 implies a balance in the way the news is distributed). These basic ingredients are used now to define two different forms for the rules.

Rule ρ1 : Passive

Rule ρ1 acts on the state of the system in the following way. Suppose α (T− ) > 0; then, the status of news of the agent Aα , independently of its own status, changes to ϕα:1,0 . On the contrary, if α (T− ) < 0, the state is changed to ϕα:0,1 . Here T− is used to indicate that we are approaching T from below. In the case α (T ) = 0 nothing is changed in the cell α. After the check in all the cells the vector n,m in (4.4) is replaced with a new one n(1) ,m(1) deduced by the action of the rule. In the interval [T , 2T [ the evolution is again driven by the same H , and at t = 2T the rule is applied once again, changing the state into n(2) ,m(2) , and so on. This rule is passive since Aα does not contribute to its effect.

Rule ρ2 : Active

As far as the rule ρ2 is concerned, suppose that, compared to the rule ρ1 , α contains also the cell α, and hence the status of the agent Aα depends on its status too. For this reason ρ2 is called active. As before, the agent Aα changes the status at each time interval T to ϕα:1,0 or ϕα:0,1 according the value of α (kT ), k > 0.

4.3

Applications

In this section, we present some numerical simulations of the model. Different kinds of diffusion dynamics and interactions between agents and different applications of the rules are considered.

4.3

Applications

4.3.1

59

A Simple Network with 3 Agents

The first application consists of a network with 3 agents, in which initially the agent 1 diffuse good news N . We also suppose that this agent can interact and is influenced by two other agents, 2 and 3, which, in turn, do not interact between themselves (see the scheme of these interactions in Fig. 4.1). The interaction is also based on these assumptions: the agent A2 is inclined to accept only good news, whereas A3 prefers fake news. We also suppose that A3 is inclined to change the nature of N and hence to influence the agent 1 to change the nature of N . The set-up of our 0 = 0. while the parameters system is based on the initial conditions G 01 = 1, G 02,3 = F1,2,3 f ,g

f ,g

f ,g

are fixed in the following way. The inertial terms are ω1 = ω2 = 1, ω3 = 0.1, meaning that the agents A1 and A2 tends to be more inclined than A3 to maintain their perception about f f g g N . The interaction parameters are p1,2 = 1, p1,3 = 0.1, p1,2 = 0.1, p1,3 = 1 (remember f ,g

f ,g

that pi, j = p j,i , ∀i, j = 1, 2, 3, i  = j), meaning that A1 and A2 share good news, whereas A1 and A3 share fake news. The parameters related to the change of the nature of N are λ1 = λ2 = 0.2, λ3 = 1, expressing that A3 easily changes the nature of N , while A1 and A2 are more conservative. For the application of the rules ρ1 and ρ2 we take T = 60, and the weights in (4.13) are f f g g chosen as w1 (2) = 0.1, w1 (3) = 1, w1 (2) = 1, w1 (3) = 0.1 so that A1 and A2 influence themselves above all for good news, whereas A1 and A3 for fake news. For the rule ρ2 we f ,g also take wα (α) = 1, α = 1, 2, 3, meaning that each agent attributes a great role to her own perception of N . The time evolution of the main function F(t) and G(t) for the two rules ρ1 and ρ2 are shown in Fig. 4.2, while in Fig. 4.3 we show the time evolution of the functions F1 (t) and G 1 (t). In the case of rule ρ1 , we notice that in every subinterval [(k − 1)T , kT [, the evolution is periodic except for some phases of strong oscillations, representing a sort of uncertainty of how news is perceived. The appearance of the oscillations is even more pronounced if we look at the evolution for a specific agent, say A1 (the dynamics of other agents, not shown

Fig. 4.1 Schematic representation of the interactions and the diffusion of news N among the three agents: the agent 1 interacts with agents 2 and 3, while agents 2 and 3 do not interact between them

A2 N −→

A1 A3

60

4 Spreading of Information in a Network 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

100

200

300

400

0

100

(a) Rule ρ1

200

300

400

(b) Rule ρ2

Fig. 4.2 Time evolutions of F(t) and G(t) for the 3-Agent network, and rules ρ1 (a) and ρ2 (b) 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

100

200

(a) Rule ρ1

300

400

0

0

100

200

300

400

(b) Rule ρ2

Fig. 4.3 Time evolutions of F1 (t) and G 1 (t) for the 3-Agent network of 3 agents, and rules ρ1 (a) and ρ2 (b)

here, exhibits similar oscillations). This is clearly seen by comparing Figs. 4.2 and 4.3. We can explain these behaviors with the way A1 interacts with A3 : initially A3 receives good news N , converts the nature of N that now becomes fake (because of the high value of λ3 ), f and induces A1 to change the nature of N (as a consequence of the high values of p1,3 and f

w1 (3)). On the other side, A2 tends to preserve the nature of N (low λ2 ), clarifying why the news N remains on average good (G(t) > F(t), ∀ t), regardless the uncertainty induced by A3 . When considering the active rule ρ2 , the agents are inclined to be less influenced by the other agents. We can observe from Figs. 4.2b and 4.3b that the amplitudes of the oscillations are weakened as compared to the case ρ1 , and the perception of the goodness of N , given by G(t), is higher. We conclude the analysis of this simple network by considering the effects of lowering the parameter λ3 : we should expect a decreasing variability in the perception of N by A3

4.3

Applications

61

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

100

200

300

400

(a) Rule ρ1

500

0

0

100

200

300

400

(b) Rule ρ2

Fig. 4.4 Time evolutions of G(t), G 1 (t) and F1 (t) for the case of three agents for different value of the interaction parameter λ3 , and for rules ρ1 (a) and ρ2 (b)

and, in turn, for A1 . In Fig. 4.4, we plot the time evolution of G(t), G 1 (t) and F1 (t) for different values of λ3 . As expected, lowering λ3 determines a more stable evolution, since A3 is less inclined to change the nature of N , and initial news is globally perceived good, as both G(t) and G 1 (t) get very close to the maximum value 1.

4.3.2

A Network with 7 Agents

We now consider a new application with a little bit larger network of agents. A schematic representation of this network is shown in Fig. 4.5, where we can observe the presence of different levels of agents. The first level consists in two non-interacting main agents transmitting the news. We suppose that agent A1 transmits a good news Ng , while the agent A2 transmits a fake news N f . The second level consists of three agents interacting with the main transmitting agents, and with two final receivers. Finally, the third level is made by two receivers communicating between themselves too, so that they can influence the perception of news of the other receiver. From the schematic representation of the interaction paths, we can observe that the first receiver A6 is more influenced by the agent A1 , whereas the second receiver A7 by A2 : therefore, the two final receivers are likely to perceive Ng and N f , respectively, as sent by the transmitters, at least if the strengths of the various interactions are similar. The initial conditions for this model are the following: G 01 = F20 = 1, while G 0j = 0 for j  = 1, and F j0 = 0, for j  = 2. The parameters ω f ,g are taken equal to 0.5, and the other parameters are chosen to strengthen or weaken the influence of the other mechanisms with respect to the free dynamics. The interaction parameters are different from zero only for the related agents having connections, as shown in Fig. 4.5, and depend also on the kind f f f g g g of initial news; in particular, we set p1,3 = p1,5 = p2,4 = p2,5 = 1, p3,6 = p4,7 = 0.5,

62

4 Spreading of Information in a Network

Fig. 4.5 Schematic representation of the network with 7 agents

Ng ↓

Nf ↓

A1

A2

A3

A4

A5

A6

f ,g

f ,g

A7

p5,6 = p5,7 = 0.25, and p f ,g (6, 7) = 1, and pi, j = pi, j for all i  = j. The above choices reflect a certain strength among the agents of the first two levels, and between the final receivers which could represent a real situation in which people transmit or modify news using social media. We also assume that only the final receivers can modify the way news is diffused (this could mimic the way the people distort news in real life), so that we put λ7 = 0.05, whereas λ6 is varied: when λ6 > λ7 , it results that the agent A6 is more inclined to change the nature of the news Ng in fake. As far as the application of the rules ρ1 and ρ2 is concerned, we take as before T = 60, and the weights used in (4.13) are set equal to the related interaction f f f ,g g g parameters: wi ( j) = pi, j , wi ( j) = pi, j ; only for the rule ρ2 we also take wi (i) = 1, ∀ i. The results for the rule ρ1 are shown in Fig. 4.6, where we plot the time evolution of the functions G 6 , F6 , G 7 and F7 for different values of λ6 . We can observe that for the low value λ6 = 0.025, after the application of the rule at T = 60, we have high amplitude oscillations in G 6 , whereas F6 shows lower amplitudes, meaning that the agent A6 considers news transmitted by A1 as good. At the same time, due to the interactions with the agent A6 , A7 changes its perception of fake news N f , and both G 7 and F7 show oscillations which express the uncertainty of the agent A7 . Increasing further λ6 has the effect of increasing the uncertainty of A6 , and decreasing that of A7 . The oscillations of F7 and G 7 are instead damped, and G 7 ≈ 0, F7 ≈ 1, so that A7 perceives news as fake. This could be explained by the fact that for large λ6 , the agent A6 is more inclined to distort Ng and change it in fake, reinforcing the perception of news by A7 as fake. The results concerning the application of the rule ρ2 are shown in Fig. 4.7 and compared with the outcomes of the rule ρ1 . The overall outcome is that all the uncertainties are damped for A6 , while are reinforced for the agent A7 . f

f

4.4

Conclusions

63

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

50

100

150

200

250

0

0

50

(a) G6 (t), rule ρ1 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

50

100

150

(c) G7 (t), rule ρ1

150

200

250

200

250

(b) F6 (t), rule ρ1

1

0

100

200

250

0

0

50

100

150

(d) F7 (t), rule ρ1

Fig. 4.6 The network with 7 agents agents and three levels for the rule ρ1 . The evolutions of the functions G 6 (t), F6 (t), G 7 (t) and F7 (t) are shown for different values of the parameters λ6

4.4

Conclusions

Our quantum-like approach based on ladder operators has been used to describe the spreading of news in a network. The inclusions of the rules and the (H , ρ)-induced dynamics is not only a mathematical expedient, but it allows to add effects which are plausible within the system considered, and allow to retrieve interesting dynamics of the spreading. This analysis opens the way to other applications in the field of information spreading dynamics, like the possibility of modeling special classes of agents (e.g., the so-called influencers) in the network.

64

4 Spreading of Information in a Network 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0

50

100

150

200

0

50

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

50

100

(c) λ6 = 0.8

150

200

150

200

(b) λ6 = 0.4

(a) λ6 = 0.4

0

100

150

200

0

0

50

100

(d) λ6 = 0.8

Fig. 4.7 The network with 7 agents agents and three levels. The evolutions of the functions G 6 (t) and F7 (t) are shown for different values of the parameters λ6 and for the two rules ρ1 , ρ2

References 1. S. Abdullah and X. Wu. An epidemic model for news spreading on Twitter. In IEEE 23rd International Conference on Tools with Artificial Intelligence, pages 163–169, 2001. 2. F. Bagarello. Quantum dynamics for classical systems: with applications of the Number operator. John Wiley & Sons, New York, 2012. 3. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 4. F. Bagarello, F. Gargano, and F. Oliveri. Spreading of competing information in a network. Entropy, 22:1169, 2020. 5. B. Doerr, M. Fouz, and T. Friedrich. Why rumors spread so quickly in social networks. Communications of the ACM, 55:70–75, 2012. 6. F. Jin, E. Dougherty, P. Saraf, Y. Cao, and N. Ramakrishnan. Epidemiological modeling of news and rumors on Twitter. In SNA-KDD ’13: Proceedings of the 7th Workshop on Social Network Mining and Analysis, pages 1–9, New York, NY, 2013. ACM. 7. K. Lerman. Social information processing in news aggregation. IEEE Internet Computing, 11:16– 28, 2007.

5

Population Dynamics in Large Domains

5.1

Introduction

In the previous chapters, we have seen that the relevant aspect of an operatorial model is to derive the time evolution of some observables of the macroscopic system using the Schrödinger or the Heisenberg equations. In particular, one usually tries to compute, in an exact or in an approximate way, the expected values which are phenomenologically associated to some macroscopic quantities like densities, decision functions, and so on. Here, we describe how to apply our operatorial approach to derive a model for the dynamics of some populations moving on a bounded, topologically non-trivial, 2D region. In some seminal works [4, 5] (see also the recent paper [12] where (H , ρ)-induced dynamics have been used), the framework based on fermionic operators was adopted although this introduced serious technical difficulties. In fact, using fermionic operators, we need to work in a Hilbert space possessing a very large dimension; this makes the problem hard to face from a computational point of view. For instance, considering the migration dynamics for an operatorial fermionic model describing a system of M populations on a region divided in N cells, the dimension of the Hilbert space is 2 M N ; moreover, the unknowns are matrices with 4 M N complex entries. This is because each site of a specific population is seen as a fermionic mode interacting with all other modes. Although this approach is reasonable and it has been successfully adopted, it is evident that the dimension of the Hilbert space grows quite rapidly with N and M. Using sparse matrices to write the operators does not simplify much the analytical or the numerical treatment of the problem. Hence, only when self-adjoint quadratic Hamiltonian are involved, an explicit formulation for the densities can be derived. However, as already stressed several times in these notes and in previous works, self-adjointness implies reversible mechanisms that could be unphysical: for instance, a self-adjoint Hamiltonian with a death process requires the reverse birth process, which is in general not reliable. Hence, one should consider non Hermitian Hamiltonians, as done in

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_5

65

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5 Population Dynamics in Large Domains

[3], although formal solution is generally not easy to determine: in this case, only numerical computations are possible since one needs to limit the number of sites or populations to control the Hilbert space dimension. Clearly this forces a lack of resolution in the derived dynamics, and here we want to propose an approach that tries to overcome this issue by using what we an call truncated bosonic operators instead of the fermionic ones, and a suitable construction of the Hilbert space and of the operators. The main application presented in this paper is the dispersal process of a marine population in a very large basin, that, as we shall see, requires a large number of degrees of freedom. Two different scenarios are chosen, and others can be found in the original paper [9] on which this chapter is based. In Sect. 5.2, we introduce the mathematical settings in which we define the main operators used to construct the Hamiltonian of the system, and we derive the time evolution of the system. In Sect. 5.3, we propose a pedagogical example on a 1D domain for the dynamics of two populations. In Sect. 5.4, we explain how to extend the approach to a 2D domain, and propose two applications: a toy-model based on prey-predator dynamics, and a more realistic model meant to describe a marine transport dynamics in a very large domain.

5.2

The Mathematical Framework

We present the mathematical framework based on the truncated bosonic operators satisfying the (truncated) CCR, and we refer the reader to [7] for further details. Consider a finite-dimensional Hilbert space H N , with 2 ≤ dim(H N ) = N < ∞, endowed √ with the scalar product ·, · and norm v = v, v, ∀v ∈ H N . Let E N = {e j , j = 0, 1, . . . , N − 1} be the canonical o.n. basis of H N , and, for simplicity, suppose that 1 , . . . , 0), ∀ j. We define the operators a and its adjoint a † via the fole j = (0, . . . ,  ( j+1)th

lowing ladder equations on the e j ’s (each of them should be intended as column vector):   ae0 = 0, ae j = je j−1 , a † e j = j + 1 e j+1 , a † e N −1 = 0. (5.1) These operators work essentially as the truncated versions of the lowering and raising bosonic operators in quantum mechanics (see Sect. 1.1): a decreases the jth level to the ( j − 1)th level (destroying the ground e0 ), whereas a † works in the opposite way by increasing the jth level to the ( j + 1)th level (destroying the most excited state e N −1 ). Notice that, if N=2, this is exactly what CAR produce. The truncated version of the CCR now reads as [a, a † ] = aa † − a † a = 11 N − N PN −1 ,

(5.2)

where 11 N is the identity operator in H N , and PN −1 = |e N −1 e N −1 | is the projection operator on e N −1 : PN −1 v = e N −1 , v e N −1 ,

5.2 The Mathematical Framework

67

for all v ∈ H N . We now extend the above definition to a macroscopic system S described by several truncated bosonic modes. Let P1 , P2 , . . . , P K be the K populations of our system S , and let a j and a †j the truncated bosonic operators attached to the generic population P j . ( j)

The Hilbert space related to the population P j is denoted with H N j , having dimension N j ( j)

( j)

and basis E N j = {el , l = 0, 1, . . . , N j − 1}. The whole Hilbert space of the system S is now given by the tensor product (K ) HS = H(1) N1 ⊗ · · · ⊗ H N K ,

and has dimension dim(HS ) = N =

 j=1,...,L

N j . Its o.n. basis is clearly defined as

 (1) (K ) E = ϕn := en 1 ⊗ · · · ⊗ en K , n = (n 1 , n 2 , . . . , n K ), n α = 0, 1, . . . , Nα − 1, α = 1, . . . , K ,

and hence any vector  ∈ HS can be expressed as

= c n ϕn ,

(5.3)

n

where, for the application we shall consider, the complex scalars cn do not necessarily satisfy  ( j) 2 ˆ n |cn | = 1. The action of a generic operator X j on H N j is extended on HS to the operator Xˆ S defined by the following tensor product: j

Xˆ Sj = 11 N1 ⊗ · · · ⊗ Xˆ j ⊗ · · · ⊗ 11 N K , ( j)

while the actions of multiple operators on the various H N j is given by

Xˆ 1 ⊗ Xˆ 2 ⊗ · · · ⊗ Xˆ K (v1 ⊗ v2 ⊗ · · · ⊗ v K ) = ( Xˆ 1 v1 ) ⊗ ( Xˆ 2 v2 ) ⊗ · · · ⊗ ( Xˆ K v K ), ( j)

for all v j ∈ H N j , j = 1, . . . , K . From now on, when no confusion arises, we will just write Xˆ j instead of Xˆ Sj , and the action is obviously intended on the whole HS , if not differently specified.

5.2.1

Transport Operators

The derivation of the relevant macroscopic quantities is due to the density operators which  allows to retrieve the density of a state  in a particular level. Namely, let  = m cm ϕm , and for each possible choice n let us define the following projector operator via the usual action: Pn  = (|ϕn ϕn |)  = ϕn ,  ϕn = cn ϕn .

(5.4)

68

5 Population Dynamics in Large Domains

We can now introduce the following expected values of the system:   |cn |2   = . (5.5) , Pn Pn  = 2   m |cm |  Of course, 0 ≤ Pm  ≤ 1 and n Pm  = 1. The expected values Pn , in the spirit of the number operators approach, can be phenomenologically interpreted as a measure of the density of  in the level n: this is the reason why we call Pn  density function. Adopting the probabilistic point of view of quantum mechanics, Pn  is nothing more than the probability of the state  to be ϕn , so that if n = (n 1 , n 2 , . . . , n K ), Pn  is the probability that the generic population P j has quantum number n j . ˜ of  on a generic subspace In our application, we shall consider suitable projections  ˜ ˜ H  H having basis E , and take this projection to compute the expected values as in (5.5). In particular, if χE˜ is the projection operator such that χE˜ ϕn = ϕn , ϕn ∈ E˜ , χE˜ ϕn = 0, ϕn ∈ / E˜ , then

  P˜n  =

χE˜ 

χE˜ 

, Pn

χE˜ 

χE˜ 

 .

(5.6)

The justification for this projection is that, as explained in the applications, we will be considering both levels that pertain to actual physical cells, and levels that pertain to other mechanisms, such as resource production or utilization. These latter levels, while potentially important in their own right, are not directly relevant for the calculation of the population density, and as such we will not include them in this computation. This explain the appearance of the projector in (5.6). We finally define the transport operators, which are used to mimic a transport-like effect by projecting a specific level into another. In particular these operators act on a vector in the following way: Pm,n  = (|ϕm ϕn |)  = ϕn ,  ϕm = cn ϕm .

(5.7)

The meaning of this action is straightforward: it projects on the level m only if the density in the level n is non zero (cn  = 0), and destroys the other levels. It is useful to describe the transport from the level n to the level m, for instance in an advection process, but also for the birth and death processes.  The time evolution (t) of an initial vector (0) = n cn (0)ϕn , and the related densities in each level at time t are obtained through the Schrödinger equation i ∂(t) ∂t = H (t) (t) (setting  = 1), where H (t) is the Hamiltonian of the system (see next section). We assume here that H (t) explicitly depends on time and on the local densities of the various populations, and in general it is not self-adjoint. Using the orthogonality conditions of the basis vectors

5.3 The Hamiltonian Operator

69

ϕn , we obtain the following system of ordinary differential equations for the coefficients cn (t) in (5.3): ∂cn (t) i (5.8) = ϕn , H (t), ∂t whose solution allows us to retrieve the various densities through (5.5). As the Hamiltonian is in general non self-adjoint, it follows that the evolution is not unitary, (t)  = (0), and this explains why an overall normalization is needed in the computation of the densities in (5.5) (see [3, 19] for other applications and further details). In the following section, we shall discuss how the Hamiltonian operator can be constructed by considering different mechanisms driving the evolution of our system.

5.3

The Hamiltonian Operator

We start this section by considering an oversimplified system in order to understand how to construct the Hamiltonian. The system is made by only one population moving on a domain R made by just two separate physical cells, and we assume that the population can move from the first to the second cell. We also assume that the population dies in the second cell and that the production of the population is possible only in the first cell: this situation can be interpreted with the presence of resources feeding the population in this cell, but not in the other. As we shall see, we can implement either the condition where resources are limited or can be continuously generated due to an infinite reservoir. The whole system is now constructed with a single bosonic mode having different levels. Two levels are attached to the cells 1 and 2, and two other levels are attached to the death process in cell 2 (the dummy level 3 which contains the dead population) and to the reservoir for the production in the cell 1 (the dummy level 4). We also consider the ground level that is connected with the empty system, that is there is no population or resource at all. Actually, this ground level is connected the ground state ϕ0 , that is used here to build the other elements of the basis through the ladder equations (5.1). Hence, as shown in Sect. 5.2.1, the Hibert space H of the system has dimension 5, with basis {ϕ j , j = 0, . . . , 4}, where the meaning of the various ϕ j is the one just described: for instance, ϕ1 is phenomenologically interpreted as the state in which the first physical cell 1 is fully populated, while the second cell and the dummy levels are all empty (no resources, no dead population). Similar interpretation for the other states. With this construction we avoid to introduce a second bosonic or fermionic mode attached to the resources: although this second mode could be reasonable (and natural) to introduce, it will increase the dimension of the Hilbert space, and hence the complexity of the system. With few cells this is not a big deal, but in applications to large domains this could represent a severe issue from a computational viewpoint. The construct the Hamiltonian, we make use of the transport and density operators described in Sect. 5.2.1. The advection from cell 1 to 2 is described by the transport operator |ϕ2 ϕ1 | which destroys the level 1 and creates the level 2; the exploitation of the resources

70

5 Population Dynamics in Large Domains

Transport from cell1 to cell 2

|ϕ2 ϕ1 |

1

2

|ϕ1 ϕ4 |

|ϕ3 ϕ2 |

Production in cell 1

Death in cell 2

4

|ϕ4 ϕ4 |

3

Subsistence of the resources

Fig. 5.1 Schematic representation of the 1D advection from cell 1 to cell 2, of the death process in cell 2, of the resources exploitation in the cell 1, and the subsistence of the resources

in level 1 is described by the operator |ϕ1 ϕ4 |, which destroys part of the resources in the level 4 and creates population in the level 1; the death process is described by the operator |ϕ3 ϕ2 | which destroys the level 2 and creates (dead population) in the level 3. We also construct an operator H p to be included if one requires a continuous sustenance over time for the resources in the reservoir. In fact, the operator |ϕ1 ϕ4 | continuously destroys the resources, without generating them. Therefore, we consider an operator of the form H p = ω|ϕ4 ϕ4 |, which behaves as the standard free dynamics operator in quantum mechanics, and it is responsible for inertial or gain processes if ω is real or purely imaginary, respectively. As we will see, to require a continuous sustenance we must take ω purely imaginary, a common way of describing a gain process in quantum system [6, 8, 16]. A schematic representation of the dynamics and the related operators is shown in Fig. 5.1. It follows that the Hamiltonian describing the dynamics can be written as H = λ|ϕ2 ϕ1 | + α|ϕ3 ϕ2 | + β|ϕ1 ϕ4 | + H p ,

(5.9)

the parameters λ, α and β being positive reals, and ω ∈ C. Looking at the dynamics of the system through (5.8), we consider an initial state representing a distribution initially concentrated in the cell 1, and the presence of resources: (0) = c1 ϕ1 + c4 ϕ4 with c1 , c4 ∈ C. As the Hamiltonian is time independent, the solution is easily written, (t) = exp (−i H t) (0), as well as cn (t) = ϕn , (t), n = 0, 1, 2, 3, 4 (c0 (t) = 0 ∀t because no terms in H works by changing it). The net densities of the population is computed through (5.6), adopting the Hilbert space H˜ generated by the elements of the basis E˜ = {ϕ˜ 1 = ϕ1 , ϕ˜ 2 = ϕ2 , ϕ˜ 3 = ϕ3 }: the reason is that we are only interested in

5.3 The Hamiltonian Operator

71

the densities of the population, including both those alive and dead, so that we project into the subspace that does not include the reservoir. We now write the solution considering two different scenarios depending on ω. If we suppose ω > 0, meaning that the term H p behaves like an inertial term, we describe the situation in which there is only a finite amount of resources and we should expect, as time goes on, the gradual decrease of the resources and, as a consequence, of the alive population in the two cells. The formal solution can be written but its expression is quite cumbersome: in general, the various densities  P˜k (t), k = 1, 2, 3 have the form k=4

f k (λ, α, β, ω, c1 , c4 , exp(iωt), exp(−iωt))t k ,

(5.10)

k=0

the f k being functions depending on the parameters and on the exponentials exp(±iωt). For simplicity, we report only the higher order asymptotic terms in time of the densities in the various levels: sin2 (tω)(βc4 − c1 ω)2 + (cos(tω)(c1 ω − βc4 ) + βc4 )2 ,   t 4 α2 λ2 41 c12 ω 2 − 21 βc1 c3 ω + 41 β 2 c4   2 2 c1 ω − 2βc1 c4 ω + β 2 c4  P˜2 (t) ≈  , t 2 α2 14 c13 ω 2 − 21 βc1 c4 ω + 41 β 2 c4  P˜3 (t) ≈ 1.  P˜1 (t) ≈

(5.11) (5.12) (5.13)

We deduce that, for t → ∞, the densities in the cells 1 and 2 decay to zero but with different speeds, while the density of dead population reaches asymptotically 1. This is somewhat expected because no generation of resources was considered in the Hamiltonian (5.9), and the term H p determines only how fast the densities in the cells decay to zero. In the second scenario, we consider ω = iω0 , ω0 > 0, to have a continuous sustenance through a gain effect. With this choice the densities (again we show the higher order asymptotic terms) are the following:  P˜1 (t) ≈

ω04 α 2 λ2

+ λ2 ω02

+ ω04

,  P˜2 (t) ≈

λ2 ω02 α 2 λ2

+ λ2 ω02

+ ω04

,  P˜3 (t) ≈

α 2 λ2

α 2 λ2 . + λ2 ω02 + ω04

(5.14) The densities reach an asymptotic equilibrium which is dependent only on the parameters of the Hamiltonian. In particular, a strong effect of resource production, corresponding to a value of ω0 larger than α and λ (ω0  α, λ), ensures a high density in the cell 1; a strong advection, λ  α, ω0 , induces greater density in the cell 2, the densities of the dead population being dependent on the balance of the parameters α and ω; finally, a strong death process, given by the condition α  λ, ω0 , leads to a large amount of dead population.

72

5.4

5 Population Dynamics in Large Domains

Application to Large Domain: The 2D Model

We now explain how to apply the operatorial approach to a population living in a 2D region R. Let us consider initially a very simple configuration where R is a 3 × 3 lattice (see Fig. 5.2). We suppose that the population living in R can move only leftward, for instance from cell 5 to cell 4 but not in the opposite way, and death and birth processes are possible. As done in previous section, we associate to the population P a single ladder operator a, and each cell occupied can be seen as a specific excited bosonic level of P . To add some birth/death process, we can also add various dummy levels as in the 1D case. Then, as far as the Hilbert space of the system is concerned, taking into account the 9 physical cells of R, the related dummy levels for the dead population and the resources in each cell, and the ground level 0, we conclude that dim(H) = 28, see below. Hence, a generic state   = k=27 k=0 ck ϕk of the system represents a specific configuration made of contributions given by the distribution of the population in R, as well as the presence of dead population and resources for the production. The meaning of the various element ϕk of the basis is again easily interpreted: {ϕ j , j = 1, . . . , 9} are related to specific configurations in which the population is concentrated only in the j−th cell, whereas a combination of them gives a distribution on R with densities depending on the coefficients c j according to (5.5); the elements {ϕ j , j = 10, . . . , 18} are related to the presence of dead population in the physical cells of R due to the death process; finally, the elements {ϕ j , j = 19, . . . , 27} are related to the presence of resources in the cells. To construct the Hamiltonian we focus on a specific cell, deriving the main operators for the dynamics in that cell: the operators for the other cells are deduced similarly. In the central cell (level number 5), the population can move from 5 to the cells 1, 4, 7, and can arrive from cells 9, 6, 3. In order to simulate the birth/death process we add two dummy levels for the death (level 14) and birth (level 23): the death can be seen as a transport from 5 to 14, while birth as a transport from 23 to 5. Because of these processes the Hamiltonian for the dynamics related to the cell 5 is written as

Fig. 5.2 Schematic representation of the dynamics in the central cell for a 3 × 3 region

7

8

9

4

6

5 1

14

2

3

23

5.4

Application to Large Domain: The 2D Model (5)

H (5) = Ho(5) + Hi

73 (5)

(5)

+ Hb + Hd ,

Ho(5) = λ1 |ϕ1 ϕ5 | + λ4 |ϕ4 ϕ5 | + λ7 |ϕ7 ϕ5 |, (5)

Hi

= λ9 |ϕ5 ϕ9 | + λ6 |ϕ5 ϕ6 | + λ3 |ϕ5 ϕ3 |,

(5) Hd Hb(5)

= λ14 |ϕ14 ϕ5 |, = λ5 |ϕ5 ϕ23 |,

where the various coefficients λi tune the strength of the related operatorial effect. The meaning of the various operators |ϕi ϕ j | is straightforward and follows the construction adopted in the 1D example in Sect. 5.3: it destroys the jth level, and creates the ith level, modifying the densities in the two levels. The full Hamiltonian is clearly the sum of the contributions of the various H ( j) , j = 1, . . . , 9. The extension to several populations Pk , k > 1 is straightforward and requires more bosonic modes, each attached to each population, and the use of the tensorial product to construct the Hilbert space and the operators as already shown in Sect. 5.2. Next section will be concerned with a system made by two populations (interacting with a prey-predator mechanism) in a non-trivial region R.

5.4.1

First Application: Prey-Predator Dynamics

We now consider a toy model for the prey-predators dynamics (two populations) inside a square lattice R made by L × L cells, where not all the cells are accessible. We suppose that both populations can move only rightward and upward, the prey-predator dynamics occurs in every cell, preys can proliferate, and predators die in absence of preys. Following the operator construction adopted in Sects. 5.3 and 5.4, the basis of the Hilbert space is E = {ϕn 1 ,n 2 , n 1 = 0, . . . , 2L 2 , n 2 = 0, . . . , 3L 2 }. In particular, the first index n 1 is related to the L 2 levels attached to each physical cell of R for the predators, and to the L 2 dummy levels for the dead predators; the second index n 2 is related to the second population, the preys, with the same interpretation for the first 2L 2 levels, but with the additional L 2 levels for the prey production due to the resources. Of course, we have also the ground ϕ0,0 which represents the empty system. The Hamiltonian containing the operators describing all the mechanisms is H = H1 + H2 + H3 + H4 ,

(1) (2) λ j,k |ϕ j,m ϕk,m | + λ j,k |ϕm, j ϕm,k |, H1 =

(5.15) (5.16)

j,k=1,...,L 2 , m=0,...,2L 2

H2 =



j=1,...,L 2 , m=0,...,2L 2

α j |ϕ j+L 2 ,m ϕ j,m |,

(5.17)

74

5 Population Dynamics in Large Domains



H3 =

ω j |ϕm, j ϕm, j+2L 2 |

(5.18)

j=1,...,L 2 , m=0,...,2L 2 2

H4 =

L

δ j |ϕ j, j+L 2 , ϕ j, j |.

(5.19)

j=1

The meaning of the various terms are again easily understood and written in terms of the transport and density operators. H1 is responsible for the advection of the two populations, (1) (2) and the non negative parameters λ j,k , λ j,k are different form zero only if the related popula(1)

tion can move from cell k to cell j: in particular λ j,k |ϕ j,m ϕk,m | moves the population P1 (the predators) from the cell k to j leaving the prey in the generic cell m (similar interpreta(2) tion for λ j,k |ϕm, j ϕm,k |). The operator H2 , with α j non negative, is related to the death of the predators, destroying the population in the physical cell j and creating dead population in the level j + L 2 . The operator H3 , with ω j > 0 or purely imaginary, is responsible for the prey production, destroying the level j + 2L 2 which is the reservoir for the cell j, and creating population in the cell j. Finally H4 , with δ j non negative, describes the prey-predator dynamics in each physical cell of R, destroying prey in the cell j and creating dead prey in j + L 2 : of course this effect is null if no predators are in j (this is the reason why we use the generic operator |ϕ j, j+L 2 , ϕ j, j |). (1)

(2)

The densities Pn 1 (t), Pn 2 (t) in all the physical cells n 1 , n 2 = 1, . . . , L 2 , are computed by means of (5.6):

Pn(1) (t) = Pn 1 ,n 2 (t), n 1 = 1, . . . , L 2 , (5.20) 1 n 2 =0,...2L 2



(t) = Pn(2) 2 n1

Pn 1 ,n 2 (t),

n 2 = 1, . . . , L 2 ,

(5.21)

=0,...2L 2

where the various contributions Pn 1 ,n 2  are obtained considering the subspace E = {ϕn 1 ,n 2 , n 1 = 0, . . . , 2L 2 , n 2 = 0, . . . , 2L 2 } related to the populations levels and hence excluding the resources for the prey. We show a numerical experiment for the dynamics of this model considering purely imaginary values of the parameters ω j that, as explained in Sect. 5.3, induce a continuous sustainment for the prey in each cell. Figures 5.3 and 5.4 display the density distributions in R for different times, assuming that the cells corresponding to the lattice [10 : 14] × [10 : 14] are (1) (2) not accessible. The parameters of the model are as follows: L = 20, λ j,k = λ j,k = 0.2 if the cell k and j are connected, and 0 otherwise (we recall that the movements are rightward and upward); α j = 0.05, ω j = 0.05i, δ j = 0.3 for all j. The initial conditions are spots of preys and predators distributed around the left bottom corner, and such that the global densities are initially both equal to 1/2. The outcome of the simulations shows that prey and predators move towards the top right corner, and a crowding effect is observed for the predator; due to the prey-predator dynamics, the preys survive only where the predators have low density.

5.4

Application to Large Domain: The 2D Model

75 10 -3

20

0.08

20 9

0.07 0.06

15

8

15

7 6

0.05 0.04

10

5

10

4

0.03

3 0.02

5

5

2

0.01

5

10

15

20

1

0

5

10

15

20

0

(1)

(1)

(b) Pn1 (t), t = 10

(a) Pn1 (t), t = 0

20

0.035

20

0.15

0.03

15

0.025

15 0.1

0.02

10

10 0.015 0.05 0.01

5

5 0.005

5

10

(c)

15

(1) Pn1 (t), t

20

0

= 20

5

10

(d)

(1) Pn1 (t), t

15

20

0

= 40

(1)

Fig. 5.3 The densities Pn 1 (t), n 1 = 1, . . . , L 2 at various times for the first population (predator)

 (1) The global density of the predators P (1) (t) = n 1 Pn 1 (t) is shown in Fig. 5.5a (the global density of the prey is simply P (2) (t) = 1 − P (1) (t)), while in Fig. 5.5b the global density is shown for different values of the parameter δ: as expected, if no prey-predator dynamics exists (i.e., δ = 0), all the predators die, P (1) (t) asymptotically goes to zero, whereas for increasing values of δ their global density is in average higher.

5.4.2

Second Application: Marine Dynamics

The aim of this second application is to apply our framework to the dynamics of a population in a very large domain, usually too hard to be numerically solvable with fermionic modes. We suppose that the population P is made by drifters passively transported on the surface of Mediterranean basin1 through the marine current. We divide the whole domain in squared cells of amplitude 1/8°×1/8°(≈ 14 km × 14 km), and to each cell j we asso1 The longitudinal-latitudinal domain is [−6, 36.25] × [30.185, 45.9375].

76

5 Population Dynamics in Large Domains

20

0.08

20 0.01

0.07 0.06

15

15

0.008

0.05 0.006 0.04

10

10

0.03 0.02

5

0.004

5

0.002

0.01

5

10

15

20

0

5

(2)

10

15

20

0

(2)

(a) Pn2 (t), t = 0

(b) Pn2 (t), t = 10 10 -3

20

20 0.012 0.01

15

6 5

15

0.008

10

0.006

4

10

3

0.004

2

5

5 0.002

5

10

15

(2)

(c) Pn2 (t), t = 20

20

0

1

5

10

15

20

0

(2)

(d) Pn2 (t), t = 40

(2)

Fig. 5.4 The densities Pn 2 (t), n 2 = 1, . . . , L 2 at various times for the second population (prey)

ciate a steady marine current field u( j) = (u( j), v( j)) corresponding to the real monthly mean field (measured in meter per second) on the surface in August 2015. This velocity field is obtained from the Mediterranean Sea Forecasting System dataset (freely available at https://www.copernicus.eu/en/myocean), and it is shown in Fig. 5.6. The total number of cells is N = 27247, counted following the latitude order starting from the south–west corner, and we have excluded the inland cells to avoid unwanted increment of the Hilbert space dimension. To write the Hamiltonian we consider two transport effects acting on two different spacetime scales. The first one is responsible for the large scale circulation induced by the largescale velocity field u that transports the populations according to the real marine currents. The second one is introduced to partially restore the small scale variability of the marine current circulation which is unavoidably removed from the averaging nature of velocity field u. In particular, following [10, 17, 18], we consider a further velocity field usc = (u sc , vsc ) , that permits the transport from one cell to all the contiguous cells. This velocity field is obtained from a stream function ψsc , i.e., ∂x ψsc = vsc , ∂ y ψsc = −u sc ,

5.4

Application to Large Domain: The 2D Model

77

0.9

0.9

0.8

0.8

0.7

0.7 0.6

0.6

0.5

0.5

0.4

0.4 0.3

0.3

0.2

0.2 0.1

0.1

0

20

40

60

(a)

80

100

0

0

20

40

60

80

100

(b)

Fig. 5.5 (a) The time evolutions of the global densities P (1) (t) (predator) and P (2) (t) (prey): parameters as in Figs. 5.3 and 5.4. (b) The time evolutions of the global density of the predators for different values of the parameter δ

Fig. 5.6 The Mediterranean basin divided in 27247 (non-inland) cells. The direction of monthlyaverage marine current velocity field u at the surface in the month August 2015 is superimposed

ψsc =

A sin[k(x − ε sin(ωt)] sin[k(y − ε sin(ωt)] k

where A = 0.1 ms−1 , ε = 0.1I0 , ω = 2πA0 /I0 , k = 2π/I0 , I0 = 20 km, and x, y are the longitude and latitude coordinates of the center of the generic cell. This stream function defines an incompressible flow of non-steady 2D lattice of vortices subject to time-periodic oscillations around their mean positions. The choice of the parameters is based on the typical hydrological features of the basin, in particular I0 has been chosen twice the Rossby radius in the Mediterranean basin. The main difference as compared to the applications in [10, 18] is that our model can be considered an Eulerian-like model, and no quasi-Lagrangian approach is adopted (see the aforementioned papers for details). The Hamiltonian we consider is therefore made by two main terms, and takes into account the two passive transport dynamics:

78

5 Population Dynamics in Large Domains

H = Hlc + Hsc , Hlc =

N

j,k=1

λlc k, j |ϕk ϕ j |,

(5.22) Hsc =

N

λsc k, j |ϕk ϕ j |.

(5.23)

j,k=1

The term Hlc contains the parameters λlc k, j that are zero if the cell j and k are not contiguous; if they are contiguous we construct λlc k, j by checking for the angle formed by the cell k and the direction of the velocity field u( j). In particular (see Fig. 5.7), let k1 , k2 be the two contiguous cells pointing to the direction of u( j). Given α(k1 ), α(k2 ) (the two angles formed between the direction of the cells k1 , k2 and the direction of u( j)), we define lc λlc k1 , j = wk1 |u( j)|, λk2 , j = wk2 |u( j)|,

(5.24)

where wk1 = α(k2 )/αmax , wk2 = α(k1 )/αmax are the weights of the cells, where the maximum value admissible of the two angles is αmax = π/4. This construction allows for the transport in the two cells, instead of only one. Of course, there are some special cases to consider, for instance when one of the two angles is 0: in that case, the velocity field points exactly towards one cell (say k1 ), and it results α(k1 ) = 0, α(k2 ) = π/4, wk1 = 1, wk2 = 0, lc and consequently we put λlc k1 , j = |u( j)|, λk2 , j = 0. The other special case is when u( j) points toward one or two inland cells (it actually happens in very few cases): in this case we search for the two (or one) non inland cells, and we define the parameters as before but with the weights defined as the angle formed between these two cells and the direction of u( j). The parameters λsc k, j in Hsc are retrieved in the same way already described for the lc parameters λk, j in (5.24) by replacing u with usc . We have performed two numerical simulations starting from two different initial conditions, E1 and E2: E1 covers the large subregion close to the Strait of Gibraltar, E2 the coastal subregion of Algeria and Tunisia. The densities for the density distributions are shown in Figs. 5.8 and 5.9 for different times. The results show that, in all cases, the main transport is

Fig. 5.7 Schematic representation of the two main cells for the transport from the cell j used for the determination coefficients λlc,sc k, j of Hlc in (5.23)

α(k1 )

u(j) α(k2 )

k1

j

k2

(d) Experiment E1, t = 370

(c) Experiment E1, t = 150

Fig. 5.8 The density distribution for the initial condition E1. Colormaps are nonlinear to highlight the low density cells

(b) Experiment E1, t = 50

(a) Experiment E1, t = 2

5.4 Application to Large Domain: The 2D Model 79

(d) Experiment E2, t = 370

(c) Experiment E2, t = 150

Fig. 5.9 The density distribution for the initial condition E2. Colormaps are nonlinear to highlight the low density cells

(b) Experiment E2, t = 50

(a) Experiment E2, t = 2

80 5 Population Dynamics in Large Domains

5.5

Conclusions

81

dictated by the large scale field u with dispersive effects due to the small scale field usc . In the experiment with E1, densities change over time following the African coastal areas where the velocity field is intense and rightward directed. In the experiment with E2, densities are mainly concentrated in the north side of the strait of Sicily.

5.5

Conclusions

In this chapter, we have derived a model for the dynamics of populations in closed regions. In the very first attempt to solve population dynamics with an operatorial approach [5], the fermionic operators were used to describe a migratory dynamics. The main drawback of using the fermionic approach is the impossibility to solve the problem in a very large domain, due to the computational request needed to deal with the large dimension of the Hilbert space. On the contrary, the present approach, based on truncated bosons, can be effective to handle this problem. In particular, we can successfully use boson levels attached to the various compartments (dead or alive population, and resources) in a suitable way. We have seen that it is possible to easily add different kinds of mechanisms in the Hamiltonian of the system with the aid of the transport and density operators: transport, birth/death process, prey-predator dynamics, resources maintenance. The applications we have proposed are intended to show how the operatorial approach can be successfully adopted to mimic different kinds of population dynamics, especially in very large domains. With this in mind the next step is to apply the methodology proposed in this chapter to real problems as done in [11] for a small domain. One key point is of course the correct definition of the values of the various parameters and the time scale of the system. We conclude by discussing another possible extension of our approach. For instance, the approach based on the quantum master equation, the Gorini–Kossakowski–Sudarshan– Lindblad (GKSL), and truncated bosonic modes and non self-adjoint Hamiltonian. This approach was already used to derive macrosystem dynamics with fermions [1, 2, 13–15]: obviously this would require the definition of the correct master equation for the density operators in the hypothesis of a non self-adjoint Hamiltonian, and this is part of the future applications of the proposed operatorial approach.

82

5 Population Dynamics in Large Domains

References 1. M. Asano, I. Basieva, A. Khrennikov, M. Ohya, Y. Tanaka, and I. Yamat. A model of epigenetic evolution based on theory of open quantum systems. Systems and Synthetic Biology, 7:161–173, 2013. 2. M. Asano, M. Ohya, Y. Tanaka, I. Basieva, and A. Khrennikov. Quantum-like dynamics of decision-making. Physica A: Statistical Mechanics and its Applications, 391:2083–2099, 2012. 3. F. Bagarello and F. Gargano. Non-Hermitian operator modelling of basic cancer cell dynamics. Entropy, 20:270, 2018. 4. F. Bagarello, F. Gargano, and F. Oliveri. A phenomenological operator description of dynamics of crowds: escape strategies. Applied Mathematical Modelling, 39:2276–2294, 2015. 5. F. Bagarello and F. Oliveri. An operator description of interactions between populations with applications to migration. Mathematical Models and Methods in Applied Sciences, 23:471–492, 2013. 6. C. M. Bender and S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PTSymmetry. Physical Review Letters, 80:5243, 1998. 7. H. A. Buchdahl. Concerning a kind of truncated quantized linear harmonic oscillator. American Journal of Physics, 35:210, 1967. 8. J. Doppler, A. Mailybaev, J. Bohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter. Dynamically encircling an exceptional point for asymmetric mode switching. Nature Letters, 537:76–79, 2016. 9. F. Gargano. Population dynamics based on ladder bosonic operators. Applied Mathematical Modelling, 96:39–52, 2021. 10. F. Gargano, G. Garofalo, and F. Fiorentino. Exploring connectivity between spawning and nursery areas of Mullus barbatus (L., 1758) in the Mediterranean through a dispersal model. Fishery Oceanography, 26:476–497, 2017. 11. F. Gargano, L. Tamburino, F. Bagarello, and G. Bravo. Large-scale effects of migration and conflict in pre-agricultural groups: Insights from a dynamic model. PLoS ONE, 12:3, 2017. 12. G. Inferrera and F. Oliveri. Operatorial formulation of a model of spatially distributed competing populations. Dynamics, 2:414–433, 2022. 13. P. Khrennikova. Application of quantum master equation for long-term prognosis of asset-prices. Physica A: Statistical Mechanics and its Applications, 450:253–263, 2016. 14. P. Khrennikova. Modeling behavior of decision makers with the aid of algebra of qubit creationannihilation operators. Journal of Mathematical Psychology, 78:76–85, 2017. 15. P. Khrennikova, E. Haven, and A. Khrennikov. An application of the theory of open quantum systems to model the dynamics of party governance in the us political system. International Journal of Theoretical Physics, 53:1346–1360, 2014. 16. S. Klaiman, U. Gunther, and N. Moiseyev. Visualization of branch points in PT-symmetric waveguides. Physical Review Letters, 101:080402, 2008. 17. G. Lacorata, L. Palatella, and R. Santoleri. Lagrangian predictability characteristics of an ocean model. Journal of Geophysical Research: Oceans, 119:8029–8038, 2014. 18. L. Palatella, F. Bignami, F. Falcini, G. Lacorata, A. S. Lanotte, and R. Santoleri. Lagrangian simulations and interannual variability of anchovy egg and larva dispersal in the sicily channel. Journal of Geophysical Research: Oceans, 119:1306–1323, 2014. 19. F. G. Scholtz, H. B. Geyer, and F. J. Hahne. Quasi-hermitian operators in quantum mechanics and the variational principle. Annals of Physics, 213:74–101, 1992.

6

Political Dynamics

6.1

Introduction

Political events constitute an extremely complex topic including several aspects and agents. Mathematical modeling of a variety of political issues, among which creation of coalitions or cooperations between different political parties, political decision making, voting rules and vote maximizers, is a currently active research area. A broader insight into some questions and methods of the mathematical approaches which are used in political science, together with comparative static and dynamic path predictions, can be found in [9]. Also, either a standard epidemiological approach or the classical game theory tools may be exploited to build mathematical models for the analysis of spread of political parties, as well as of coalition formation and change of voters’ opinion over time due to self-reflection, communication, random external events, and noise, respectively (see [12, 14, 16], and references therein). Moreover, the methods of quantum information theory (to reduce essentially the complexity of the classical stochastic models), and the formalism of the theory of open quantum systems have been used to describe the dynamics of the voters’ mental states and the approaching of a stable state of a decision equilibrium [10, 11]. In [8], an approach, based on operator algebras, typical of quantum theory [13, 15], is considered to model the dynamics of a political system affected by a “turncoat-like” behavior of part of their members [6–8]. As widely discussed in [1, 3], and reviewed in the other chapters of this monograph, the description of the dynamics of macroscopic complex systems may be profitably carried out by using raising and lowering operators and the number representation. In particular, operatorial techniques have been used to set up dynamical models describing alliances in politics [2, 4, 5] with specific reference to Italian political parties. In this chapter, in particular, fermionic annihilation and creation operators, together with their associated number operators, are used to define local densities of observables of a

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_6

83

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6 Political Dynamics

macroscopic political system affected by “turncoat-like” behaviors. We are interested to model the fluxes of disloyal politicians among different compartments (representing political groups), and the use of fermionic number operators works very well for this purpose. Essentially, a turncoat is a person who changes allegiance from one loyalty or ideal to another in an unscrupulous way to get maximum benefits and personal gain. The reason of the shift is frequently self-interest, and the faithless attitude in politics may come upon in a manifest way through an open and declared change of side, or in a veiled and subtle way in terms of incoherence in the voting secrecy (just think of those politicians from fragmented sub-factions who declare a certain ideology, but secretly give electoral support to a different cause). With respect to the political landscape in place in Italy in the recent years, a look at the behavior of politicians from various parties reveals a high level of disloyal attitude and openness towards accepting chameleons from other political groups in both the two houses of the Italian Parliament. Such kind of phenomena clearly emerges from the official data available in the institutional web pages of the Chamber of Deputies (http://www.camera.it) and the Senate of the Republic (http://www.senato.it). Focusing in particular on the political elections of 2013 in Italy, we start observing that there was a result where three parties (the Partito Democratico, PD, the Popolo della Libertà, PdL, and the Movimento 5 Stelle, M5S) took more or less the same number of votes, while other political groups gained few seats in the Parliament. PD was really the first party in that election, but due to the different electoral rules for the two houses of Parliament, it got the majority of parliamentary seats only in the Chamber of Deputies. Then, the formation of a stable government required forming some alliances between different parties. This determined a fragmentation of some political parties, the birth of new parliamentary groups, and a paroxysmal occurrence of (also multiple) changes of side of groups of members of the parliament. The model proposed here involves a finite number of fermionic modes, which means that the Hilbert space in which the system S lives will result finite dimensional, and the dynamics is assumed to be ruled by a time-independent self-adjoint quadratic Hamiltonian operator, that, despite its simplicity, works very well in describing the dynamics of political party groups affected by turncoat-like behaviors of part of their members. The choice of a quadratic Hamiltonian H is also motivated by the necessity of describing a system where the total amount of the actors stays constant in time, and this is guaranteed by our choice of H . Nevertheless, in order to obtain an interesting time evolution,1 we assume the dynamics to be driven, besides the Hamiltonian, by the periodic action of certain rules on the system able to modify the values of some of the parameters on the basis of the state variations of the system. Stated differently, we will use here a quadratic H , but in connection with a rule ρ, as discussed in Sect. 1.6.

1 It may be useful to recall that, as we have seen in, for instance, Sect. 1.8, a finite-dimensional system

with a self-adjoint Hamiltonian can only have a periodic or quasi-periodic dynamics.

6.2 The Basic Model

85

In Sect. 6.2, we describe the preliminary case in which the dynamics is ruled only by a strictly quadratic Hamiltonian; in this simplifying situation we propose a detailed study of the influence of the parameters entering the model on the dynamics of the system. In Sect. 6.3 we present an alternative approach consisting in the introduction of specific rules repeatedly acting on the parameters involved in the model. In this way we enrich the standard Heisenberg description of the dynamics by taking into account the subsequent states reached by the system (the so called (H , ρ)-induced dynamics, as in Sect. 1.6). In such a way, we express a sort of dependence of the parameters (which describe in some sense the political style of the various parties) upon the variations of the mean values of the observables. In Sect. 6.4, by considering a more realistic model and the data available on the official web pages of the Italian Parliament, a case study concerning the dynamics of turncoats in the Italian XVII Legislature is presented; as a result, a good agreement between real data and the model is obtained. More applications of the proposed model can be found in the original paper [8] which this chapter is based on.

6.2

The Basic Model

The model initially presented (schematized in Fig. 6.1) consists of nine groups of politicians Pi j (i, j = 1, 2, 3), represented by fermionic operators, identified on the basis of three possible attitudes in relation to three main factions f i (i = 1, 2, 3) expressing three possible simplified political strategies. More precisely, at the initial time, f 1 represents a moderate faction, characterized by scarce openness to exchanges with other parties (external flows) and small propensity to transformations in ideological direction (internal flows), f 2 represents a fickle faction, highly permissive towards transitions between political groups and other ideologies sympathizers in the party itself, while f 3 stands for an extremist faction, intransigent against contaminations and influences from others. The members of each faction f j are divided in three other compartments, P j1 , P j2 and P j3 ; the compartment P j j refer to the politicians of the faction f j who are loyal to the their faction, P jk to the politicians who belong to the faction f j but are drawn towards faction f k . The formulation adopted in the following is based on the assumption that to the compartment P jk ( j, k = 1, 2, 3) † , and a number we associate a fermionic annihilation operators P jk , a creation operators P jk † operator  p jk = P jk P jk ; of course, these annihilation and creation operators satisfy the CAR (see Sect. 1.2), that in our case read as † {P jk , Pm } = δ j, δk,m 11,

† † {P jk , Pm } = {P jk , Pm } = 0,

j, k, , m = 1, . . . , 3. The mean values of the number operators, evaluated on a given initial condition, are interpreted as a measure of the number of members in each compartment. The states of the system (involving a finite number of fermionic states) are vectors in the 29 -dimensional Hilbert space H constructed as the linear span of the vectors

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Fig. 6.1 A schematic view to the simplified model of three parties

† p11 † p12 † p13 † p21 † p22 † p23 † p31 † p32 † p33 ϕ p11 , p12 ,..., p33 := (P11 ) (P12 ) (P13 ) (P21 ) (P22 ) (P23 ) (P31 ) (P32 ) (P33 ) ϕ0 ,

see (1.5), which therefore turn out to form an o.n. basis for H, where p jk ∈ {0, 1} for j, k = 1, 2, 3, and the vector ϕ0 is the vacuum of the theory, i.e., a vector annihilated by all the operators P jk . The dynamics is assumed to be governed by a self-adjoint time independent Hamiltonian operator H embedding the main effects deriving from the interactions among the compartments of the system, according to the equation Heisenberg equation X˙ (t) = i[H , X (t)], see Sect. 1.3.2. Once defined a vector state ϕ p11 , p12 ,..., p33 representing the initial configuration of the system, we are interested in computing the mean values p jk (t) = ϕ p11 , p12 ,..., p33 ,  p jk (t)ϕ p11 , p12 ,..., p33 ,

j, k = 1, 2, 3,

(6.1)

and such averages are interpreted as the densities of the various compartments (i.e., a measure of the number of politicians belonging to the various political groups) of the model. Of course,  p jk (t) is the time evolution of the number operator  p jk , still to be deduced. The evolution of the density of the faction f j ( j = 1, 2, 3) is easily obtained, say f j (t) =

3  k=1

p jk (t),

j = 1, 2, 3.

6.2 The Basic Model

87

Within the framework of operatorial models for macroscopic systems [1], let us introduce the Hamiltonian ruling the dynamics of the political system, ⎧ H = H0 + H I , with ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪  ⎪ ⎪ ⎪ H0 = ωi j Pi†j Pi j , ⎪ ⎨ i, j=1 (6.2) ⎪ ⎪ 3   ⎪   ⎪ ⎪ ⎪ λi j Pii Pi†j + Pi j Pii† + μi j Pi j P ji† + P ji Pi†j , HI = ⎪ ⎪ ⎪ ⎪ i, j=1 1≤i< j≤3 ⎩ j=i

describing the main effects deriving from the interactions among the nine actors of the system. The real constants ωi j , appearing in the first standard part H0 , are related to the tendency of each degree of freedom to stay constant in time [1], and so, in some sense, they are a measure of the inertia of the various compartments. Moreover, the real parameters λi j in H I are used to describe the internal flows that occur within each faction, namely the minor ideological positions representing in some sense early warning of disloyalty, whereas the parameters μi j are related to the external flows stemming from the real changes of side. Notice that, requiring λi j = μi j = 0 (i, j = 1, 2, 3), meaning that the components of the p jk ] = 0: even if the operators P jk and system do not interact, implies that [H ,  p jk ] = [H0 ,  † P jk have a non-trivial time dependence, the densities of all the compartments of the system stay constant in time. Moreover, it is ⎡ ⎤ 3  ⎣H ,  p jk ⎦ = 0, (6.3) j,k=1

 whereupon the quantity 3j,k=1 p jk (t) is a constant of motion, whatever the values of the parameters are: thus, we have the conservation of all the parliamentary seats. Adopting the Heisenberg representation, we are led to consider the following (operator) differential equations ⎧ ˙ P ⎪ ⎪ 11 ⎪ ⎪ ⎪ P˙12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P˙13 ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ⎪ ⎨ P21 P˙22 ⎪ ⎪ ⎪ ⎪ P˙23 ⎪ ⎪ ⎪ ⎪ ⎪ P˙31 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P˙32 ⎪ ⎪ ⎩ P˙33

= i (−ω11 P11 + λ12 P12 + λ13 P13 ) , = i (−ω12 P12 + λ12 P11 + μ12 P21 ) , = i (−ω13 P13 + λ13 P11 + μ13 P31 ) , = i (−ω21 P21 + λ21 P22 + μ12 P12 ) , = i (−ω22 P22 + λ21 P21 + λ23 P23 ) , = i (−ω23 P23 + λ23 P22 + μ23 P32 ) , = i (−ω31 P31 + λ31 P33 + μ13 P13 ) , = i (−ω32 P32 + λ32 P33 + μ23 P23 ) , = i (−ω33 P33 + λ31 P31 + λ32 P32 ) .

(6.4)

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6 Political Dynamics

0 , j, k = 1, 2, 3. Because that have to be solved with suitable initial conditions P jk (0) = P jk 9 each operator P jk is a square matrix of order 2 , in principle we have to solve a system of 9 · 49 linear ordinary differential equations. Nevertheless, because of the linearity, we can adopt a reduced approach. Let us introduce  † † T , . . . , P33 (the superscript T stands for transpoa formal vector A ≡ P11 , . . . , P33 , P11 sition) and the square matrix of order 18    0 09 , = 09 −0

where 09 is a zero matrix of order 9, and ⎛

⎞ −ω11 λ12 λ13 0 0 0 0 0 0 ⎜ λ12 −ω12 0 ⎟ μ12 0 0 0 0 0 ⎜ ⎟ ⎜ λ13 ⎟ 0 −ω13 0 0 0 μ13 0 0 ⎜ ⎟ ⎜ 0 ⎟ μ12 0 −ω21 λ21 0 0 0 0 ⎜ ⎟ ⎜ ⎟ 0 = ⎜ 0 0 0 0 ,⎟. 0 0 λ21 −ω22 λ23 ⎜ ⎟ ⎜ 0 ⎟ μ23 0 0 0 0 λ23 −ω23 0 ⎜ ⎟ ⎜ 0 0 0 0 −ω 0 λ 0 μ 13 31 31 ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 μ23 0 −ω32 λ32 ⎠ 0 0 0 0 0 0 λ31 λ32 −ω33 The system of linear differential Eq. (6.4), coupled with their adjoint version, can be expressed in matrix form as ˙ = iA, A A(0) = A0 , (6.5) so that the solution may be clearly expressed as A(t) = exp(it)A(0) = B (t)A0 .,

(6.6)

where B (t) = exp(it). Now, let us define the vector ϕ=

9   k=1

n 0k ϕ0,..., 1 ,...,0 ,  kth

and (n 01 , n 02 , . . . , n 09 ) represent the initial values of the mean values of the number operators associated to the political groups of the system. Let B j,k be the generic entry of matrix B (t); relation (6.6), using the CAR, provides the mean values of the number operators at time t:

6.2 The Basic Model

pmn (t) =

9  i=1

+

89

ϕi2

9 

B2m+n, f (,2m+n) B2m+n+9,g(,2m+n)

=1

8  9 

 ϕi ϕ j B2m+n,i B2m+n+9, j+9 + B2m+n, j B2m+n+9,i+9

(6.7)

i=1 j=i+1

 −B2m+n,i+9 B2m+n+9, j − B2m+n, j+N B2m+n+9,i ,

where

 f (, i) =

i if i = , i + 9 if i  = ,

 g(, i) =

i + 9 if i = , i if i  = .

Due to the quadratic form of the Hamiltonian H , the solution will exhibit a never ending oscillatory behavior. This aspect has already been stressed several times all along these notes, and also in [1, 3]. In particular, if we choose, as a start, the following values of the parameters (consistent with the ideological and attitudinal interpretation of the parties above specified), ω11 = 0.7, ω23 = 0.1, λ12 = 0.5, λ32 = 0.3,

ω12 = 0.5, ω31 = 0.8, λ13 = 0.4, μ12 = 0.1,

ω13 = 0.4, ω32 = 0.7, λ21 = 0.6, μ13 = 0.2,

ω21 = 0.3, ω22 = 0.2, ω33 = 1, λ23 = 0.7, λ31 = 0.4, μ23 = 0.3,

(6.8)

and we suppose that the initial densities of the three factions under consideration are essentially the same (n 01 = n 05 = n 09 = 0.33, n 02 = n 03 = n 04 = n 06 = n 07 = n 08 = 0), we get the evolution of the densities f 1 , f 2 and f 3 shown in Fig. 6.2. In this case, we can observe (at least in the time interval considered) how the model reproduces a dynamics characterized in that the moderate party’s approach proves successful, the fickle party results scarcely promoted, whereas the strategy of the extremist party reveals to be losing.

6.2.1

Different Behaviors Depending on the Cross Interactions

It is useful, for a better understanding of the model, to perform a study of the parameters entering the model. This is what we will do now. Different assignments of values to the set of parameters involved in the model, apart from the initial data, of course play a significant role in the evolution of the system giving rise to slightly different dynamical behaviors. The fact that each compartment of the model described in this chapter is associated to a specific political fringe (identified on the basis of the declared ideology and openness towards the other political groups) implicitly forces the assignment of the values to the inertia parameters ωi j related to the tendency of each group to keep the same amount of members during the evolution as well as to the internal interaction parameters λi j , which need to be inversely proportional to the loyalty to the corresponding faction. Moreover, it

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6 Political Dynamics

Fig. 6.2 Linear model: evolution of the densities of politicians in each faction up to t = 100, with the parameters fixed as in (6.8)

is clear from (6.2) that the parameters μi j definitely influences different factions. We may therefore think to minimize the exchanges between only two of the main factions. For these reasons, in the numerical simulations shown in Fig. 6.3, we fix the parameters ω jk and λ jk entering the model as follows, ω11 = 0.7, ω12 = 0.6, ω13 = 0.5, ω22 = 0.4, ω21 = 0.3, ω23 = 0.2, ω33 = 1, ω31 = 0.8, ω32 = 0.9, λ12 = 0.25, λ13 = 0.3, λ21 = 0.7, λ23 = 0.75, λ31 = 0.15, λ32 = 0.1,

(6.9)

whereas three sets of the parameters μi j (as shown in Fig. 6.3) are chosen. This means that the third party is intended as the most severe and unwilling to betray or become contaminated with the other parties, the first one is slightly more open towards the other ideologies, while the second one is characterized by a markedly open and fickle attitude. Of course, the parameters ω j j ( j = 1, 2, 3) are slightly larger than the other parameters ω jk ( j  = k) referring to the disloyal fringes of the three parties. Figure 6.3 shows different behaviors related to the choice of the values for the cross interaction parameters μ jk (determining the transitions among factions) as indicated in the figure: the first two subframes illustrate the alternative time evolutions of the system corresponding to the cases in which either f 1 or f 3 is characterized by a marked attitude towards switching side compared to the other groups (having very few interactions instead), while the last subframe refers to a situation of uniform disloyal tendency. Nevertheless, we observe a wide variation for f 3 compared with those of f 1 and f 2 .

6.3 The Advanced Model: Rule Induced Dynamics

(a) µ12 = 0.1, µ13 = 0.09, µ23 = 0.01

91

(b) µ12 = 0.01, µ13 = 0.1, µ23 = 0.09

(c) µ12 = 0.04, µ13 = 0.04, µ23 = 0.04

Fig. 6.3 Linear model: time evolution of the densities of the three party groups up to t = 100 corresponding to the parameters in (6.9) and different choices of the cross interaction parameters

6.3

The Advanced Model: Rule Induced Dynamics

The simulations shown in Figs. 6.2 and 6.3, as expected, provide an evolution of the densities not admitting any asymptotic limit, that is the densities of the parties always oscillate in time, even if the sum of the densities of all parties remains constant. Since for finite-dimensional Hilbert space H no time independent, self-adjoint quadratic Hamiltonian H makes the dynamics different from being periodic or quasi-periodic, in order to describe some different kind of dynamics, let us adopt the (H , ρ)-induced dynamics approach, see Sect. 1.6. The rules we will consider are able to modify the inertia parameters, simulating in this way the effects of a change of the attitudes of the members of the various

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6 Political Dynamics

political parties,2 and/or changing also some of the interaction parameters ruling the fluxes among the compartments of the model. Thus, let us consider again a general political system in which three major factions are outlined, each one of them internally subdivided according to the political attitudes of its members into three fringes as schematically shown in Fig. 6.1. We assume that the nature of the factions is characterized by the choice of values for the parameters of the model as in (6.9) along with μ12 = 0.1, μ13 = 0.09, μ23 = 0.01; moreover, let us take the initial densities such that the first faction is predominant, but not significantly, compared to the sum of the remaining two. In other words, let us imagine that at the beginning of a parliamentary term there is a moderate party ( f 1 ) which struggles to keep the majority of seats and has not the right numbers to govern alongside a fickle faction ( f 2 ) and an extremist one ( f 3 ). Taking into account the turncoat attitudes of the various actors of our model, reasonable estimates of the densities of the parliamentary majority and opposition to the faction f j are obtained by considering the densities Mj = fj +

3 

pk j ,

Oj =

k=1 k= j

3 

pk ,

k,=1 k,= j

say, the possible parliamentary majority (based on the jth faction) and opposition. The description of the evolution of such a political system by means of the linear model (6.5) in the case where no rule is applied produces the results shown in Fig. 6.4. As a result, none of the three majorities M j (based on factions f 1 , or f 2 or f 3 ) is higher than the corresponding oppositions for all times. In order to account for the changes in political views and party affiliations due to the evolution and the mutual interactions between the compartments of the political system, we now enrich the description of the dynamics by considering two different rules (ρ1 and ρ2 ) periodically acting on some of the parameters of the model in such a way to have remarkable effects on the evolution of the system. More explicitly, once the variations of the densities of the main factions D j = f j (kτ ) − f j ((k − 1)τ ),

k ≥ 1,

j = 1, 2, 3,

(6.10)

have been computed after a period of length τ of the Heisenberg-like evolution of the political system, the first rule ρ1 consists in the conditions  ρ1 (ω j j ) = ω j j (1 + D j ), j,  = 1, 2, 3, (6.11) ρ1 (ω j ) = ω j (1 − D j ),   = j, which are intended to modify, according to the evolution of each faction, the tendency of the loyal fringes to maintain their attitude in time on the contrary of those of the disloyal 2 This is in the same spirit of what we have seen in Sect. 2.2.

6.3 The Advanced Model: Rule Induced Dynamics

93

Fig. 6.4 Time evolution of the densities of the three party groups, (a), the three possible groupings of two opposite blocks, (b), (c) and (d), up to t = 500, corresponding to the parameters in (6.9) and μ12 = 0.1, μ13 = 0.09, μ23 = 0.01. No rule is applied

ones. Notice that this approach is quite close to the one described, in rather general terms, in Sect. 2.2. The time evolutions of the system and of the density of the parliamentary majority and opposition obtained by means of the stepwise linear model applying the rule ρ1 after every time step of length τ = 5 are shown in Fig. 6.5. The resulting dynamics can be further adapted in favor of the leading party f 1 through the set of conditions ρ2 , taking into account the variations d j = p j (kτ ) − p j ((k − 1)τ ),

k ≥ 1, j,  = 1, 2, 3,

(6.12)

of the densities of each compartment p j of the model, and introducing further conditions on the internal interaction parameters:  ρ2 (ω j ) = ω j (1 + d j ), j,  = 1, 2, 3. (6.13) ρ2 (λ j ) = λ j (1 − d j ),   = j

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6 Political Dynamics

Fig. 6.5 Time evolution of the densities of the three party groups, (a), the three possible groupings of two opposite blocks, (b), (c) and (d), up to t = 500, corresponding to the parameters in (6.9) and μ12 = 0.1, μ13 = 0.09, μ23 = 0.01. Rule 1 with τ = 5 is applied

Notice that, due to the fact that the system under consideration is conservative, at any given time the variations d j (as well as D j ), j,  = 1, 2, 3, will never have the same sign. This results in a consolidation of the factions whose density is growing associated with an incentive to variability for the other factions whose density is decreasing, i.e., ρ2 acts increasing (lowering, respectively) the value of the inertia parameters and lowering (increasing, respectively) the values of the internal interaction parameters in order to suppress or promote the internal disloyalty flows. Therefore, the periodic application of the rule ρ2 accounts for the tendency of the candidates to jump on the bandwagon and, as visible in Fig. 6.6, the stepwise model well describes the situation in which the leading faction reaches a value of the density greater than those of the other factions, and maintains a majority of overall seats in the parliament.

6.4

A Case Study: The Dynamics of Turncoats in the Italian XVII Legislature

95

Fig. 6.6 Time evolution of the densities of the three party groups, (a), the three possible groupings of two opposite blocks, (b), (c) and (d) up to t = 500, corresponding to the parameters in (6.9) and μ12 = 0.1, μ13 = 0.09, μ23 = 0.01. Rule 2 with τ = 5 is applied

6.4

A Case Study: The Dynamics of Turncoats in the Italian XVII Legislature

In this Section, along the lines of what done heretofore, we present a more realistic model in order to capture the dynamics of the Italian political system in the XVII Legislature. To this end, the political parties taking part to the Italian XVII Legislature have been grouped according to ideology and attitudes into five main factions: the center-left coalition ( f 1 ), the center-right coalition ( f 2 ), the center coalition ( f 3 ), the M5S ( f 4 ), and the set of all the minor parties, or Mixed Group ( f 5 ). This last compartment represents a mass of outcast politicians without any kind of internal cohesion or ideological identity. A graphical representation of the possible political interactions among these actors is shown in Fig. 6.7.

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6 Political Dynamics

Fig. 6.7 A schematic view to the model describing the main parties interacting in the Italian XVII Legislature

Fig.6.8 Desertions (normalized to the final densities) inside the Chamber of Deputies (a) and the Senate of the Republic (b) during the Italian XVII Legislature (data from http://www.camera.it and http://www.senato.it, accessed on April 14th, 2016)

The Italian political system has been sadly characterized, especially in recent years, by a tendency of the government to attempt to hold on to power by forming coalitions to prevent the formation of any credible opposition. Phenomena such as opportunism and easy acceptance of turncoats within the various parties have strongly influenced the XVII Italian Legislature, reaching even paroxysmal levels during the first thirty months of this term (see Figs. 6.8 and 6.9 for a display of the uniform disloyalty, exhibited by all political groups, in both the two houses of the Italian Parliament). The time series of the densities of the parties

6.4

A Case Study: The Dynamics of Turncoats in the Italian XVII Legislature

97

Fig. 6.9 Entries (normalized to the final densities) inside the Chamber of Deputies (a) and the Senate of the Republic (b) during the Italian XVII Legislature (data from http://www.camera.it and http://www.senato.it, accessed on April 14th, 2016)

within the Chamber of Deputies and the Senate of the Republic during the period under consideration are shown in Fig. 6.10. By observing the trends that emerge from the official data about the changes of side it springs to mind to interpret such evolutions as strongly driven by the lack of scruples. This key translates into the possibility of using the turncoat model presented in this paper (with a specific rule ρ P taking into account the dynamics within the Italian Parliament) in order to attempt to fit the actual data extracted from the Institutional web sites. The various subgroups of the political parties are described by fermionic operators and their evolution is ruled by a quadratic Hamiltonian operator in the Heisenberg representation. On the basis of the observed behavior of the various parties, we describe the political interactions by means of the following Hamiltonian: ⎧ H = H0 + H I , with ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ H =  ω P† P + ω P† P , ⎪ ⎪ 0 ij ij ij 5 5 5 ⎪ ⎪ ⎪ i, j=1 ⎪ ⎪ ⎪ ⎨ 4     ( j) (6.14) Pii Pi†j + Pi j Pii† + H = λi μi j Pi j P ji† + P ji Pi†j I ⎪ ⎪ ⎪ ⎪ i, j=1 1≤i< j≤4 ⎪ ⎪ j=i ⎪ ⎪ ⎪ ⎪ ⎪ 4  ⎪  ⎪ † † ⎪ ⎪ + P ν P + P P i j i j 5 ⎪ i j 5 ⎩ i, j=1

The equations of motion look similar to those obtained in (6.4), say

(b)

3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4

0

20

40

60

80

100

120

Center-left Center-right Center Five Star Mixed

Fig. 6.10 Time series of the seats of the political groups within the Chamber of Deputies (a) and within the Senate of the Republic (b) during the Italian XVII Legislature

(a)

3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 /2 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4

0

50

100

150

200

250

300

350

Center-left Center-right Center Five Star Mixed

98 6 Political Dynamics

6.4

A Case Study: The Dynamics of Turncoats in the Italian XVII Legislature

 (2) (3) (4) P˙11 = i −ω11 P11 + λ1 P12 + λ1 P13 + λ1 P14 + ν11 P5 ,  (2) P˙12 = i −ω12 P12 + λ1 P11 + μ12 P21 + ν12 P5 ,  (3) P˙13 = i −ω13 P13 + λ1 P11 + μ13 P31 + ν13 P5 ,  (4) P˙14 = i −ω14 P14 + λ1 P11 + μ14 P41 + ν14 P5 ,  (1) P˙21 = i −ω21 P21 + λ2 P22 + μ12 P12 + ν21 P5 ,  (1) (3) (4) P˙22 = i −ω22 P22 + λ2 P21 + λ2 P23 + λ2 P24 + ν22 P5 ,  (3) P˙23 = i −ω23 P23 + λ2 P22 + μ23 P32 + ν23 P5 ,  (4) P˙24 = i −ω24 P24 + λ2 P22 + μ24 P42 + ν24 P5 ,  (1) P˙31 = i −ω31 P31 + λ3 P33 + μ13 P13 + ν31 P5 ,  (2) P˙32 = i −ω32 P32 + λ3 P33 + μ23 P23 + ν32 P5 ,  (1) (2) (4) P˙33 = i −ω33 P33 + λ3 P31 + λ3 P32 + λ3 P34 + ν33 P5 ,  (4) P˙34 = i −ω34 P34 + λ3 P33 + μ34 P43 + ν34 P5 ,  (1) P˙41 = i −ω41 P41 + λ4 P44 + μ14 P14 + ν41 P5 ,  (2) P˙42 = i −ω42 P42 + λ4 P44 + μ24 P24 + ν42 P5 ,  (3) P˙43 = i −ω43 P43 + λ4 P44 + μ34 P34 + ν43 P5 ,  (1) (2) (3) P˙44 = i −ω44 P44 + λ4 P41 + λ4 P42 + λ4 P43 + ν44 P5 , P˙5 = i (−ω5 P5 + ν11 P11 + ν12 P12 + ν13 P13 + ν14 P14 + ν21 P21 +ν22 P22 + ν23 P23 + ν24 P24 + ν31 P31 + ν32 P32 + ν33 P33 +ν34 P34 + ν41 P41 + ν42 P42 + ν43 P43 + ν44 P44 ) . The values of the parameter involved in the model are initially selected as ω11 = 0.7, ω12 = 0.65, ω13 = 0.65, ω14 = 0.65, ω22 = 0.45, ω21 = 0.4, ω23 = 0.4, ω24 = 0.4, ω33 = 0.65, ω31 = 0.6, ω32 = 0.6, ω34 = 0.6, ω44 = 0.9, ω41 = 0.85, ω42 = 0.85, ω43 = 0.85, ω5 = 0.2, (2)

(3)

(4)

λ1 = 0.3, λ1 = 0.35, λ1 = 0.35, (1)

(3)

(4)

λ2 = 0.45, λ2 = 0.5, λ2 = 0.5,

99

100

6 Political Dynamics (1)

(2)

(4)

(1)

(2)

(3)

λ3 = 0.35, λ3 = 0.3, λ3 = 0.35, λ4 = 0.15, λ4 = 0.2, λ4 = 0.15, μ12 = 0.1, μ13 = 0.15, μ14 = 0.15, μ23 = 0.2, μ24 = 0.25, μ34 = 0.1, ν11 = ν22 = ν33 = ν44 = 0.1, ν12 = ν13 = ν14 = ν21 = ν23 = ν24 = ν31 = ν32 = ν34 = ν41 = ν42 = ν43 = 0. These values are then modified during the evolution of the system (after any time interval of length τ = 1) on the basis of the variations of the densities of the various compartments associated to the parties inside the Parliament: D j = f j (kτ ) − f j ((k − 1)τ ),

k ≥ 1, j = 1, . . . , 5

(6.15)

according to the rule ρ P , acting as ⎧ ⎪ ρ P (ω11 ) = ω11 (1 + D1 /100) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ P (ω22 ) = ω22 (1 − (D1 + D3 )/10) ρ P (ω33 ) = ω33 (1 + (D1 + D2 )/10) ⎪ ⎪ ⎪ ⎪ ρ P (ω44 ) = ω44 (1 − (D4 + D5 )/10) ⎪ ⎪ ⎪ ⎩ ρ (ω ) = ω (1 + (D + D )/10). P 5 4 5 5

(6.16)

The rule ρ P , which resembles the rule ρ1 defined in (6.11), has to be intended as a means to adjust the model to the observed phenomena in order to reasonably mimic the actual trends, as the evolutions in Fig. 6.11 show. We see that the model captures well the essential behaviours of the real data.

6.5

Conclusions

The model presented here, even if built with tools (operator algebras, number representation) typically used in quantum mechanics, is fully deterministic. Despite its simplicity, the model, whose evolution is governed by a time independent self-adjoint quadratic Hamiltonian operator enriched with the action of appropriate rules that modify some of the involved parameters, is able to provide satisfactory results. Remarkably, the numerical simulations we exhibit seem to capture some relevant features (reinforcement of moderate factions, jumps on the bandwagon, …) of the dynamics of political parties affected by turncoat-like behaviors. In particular, as shown in Sect. 6.4, a more realistic version of the model provides a good fitting with the official data extracted from the Institutional web pages of the Italian Parliament.

0

5

10

(a)

15

20

25

30

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

5

10

(b)

15

Center-left Center-right Center Five Star Mixed

20

25

30

Fig. 6.11 Time evolution of the political system according to the stepwise model with rule ρ P and τ = 1: the numerical simulations exhibit a good likeness to the actual data in both the case of the Chamber of Deputies (a) and of the Senate of the Republic (b). The x-axis is scaled to the number of months in the period under study

0

0.1

0.2

0.3

0.4

0.5

0.6

Center-left Center-right Center Five Star Mixed

6.5 Conclusions 101

102

6 Political Dynamics

Within the same framework it has been also studied a coupled model of political parties of both houses of the Italian Parliament, as well as a coupled model suitable to describe the influence of the opportunistic attitudes on voters’ opinion. Some results concerning these analyses are contained in [6, 7].

References 1. F. Bagarello. Quantum dynamics for classical systems: with applications of the Number operator. John Wiley & Sons, New York, 2012. 2. F. Bagarello. An operator view on alliances in politics. SIAM Journal on Applied Mathematics, 75:564–584, 2015. 3. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 4. F. Bagarello and F. Gargano. Modeling interactions between political parties and electors. Physica A: Statistical Mechanics and its Applications, 481:243–264s, 2017. 5. F. Bagarello and E. Haven. First results on applying a non-linear effect formalism to alliances between political parties and buy and sell dynamics. Physica A: Statistical Mechanics and its Applications, 444:403–414, 2016. 6. R. Di Salvo, M. Gorgone, and F. Oliveri. (H , ρ)-induced political dynamics: facets of the disloyal attitudes into the public opinion. International Journal of Theoretical Physics, 56:3912–3922, 2017. 7. R. Di Salvo, M. Gorgone, and F. Oliveri. Political dynamics affected by turncoats. International Journal of Theoretical Physics, 56:3604–3614, 2017. 8. R. Di Salvo and F. Oliveri. An operatorial model for complex political system dynamics. Mathematical Methods in the Applied Sciences, 40:5668–5682, 2017. 9. P. E. Johnson. Formal theories of politics. Mathematical modelling in political science. Elsevier Ltd, Pergamon, 1989. 10. P. Khrennikova. Quantum dynamical modeling of competition and cooperation between political parties: the coalition and non–coalition equilibrium model. Journal of Mathematical Psychology, 71:39–50, 2016. 11. P. Khrennikova, E. Haven, and A. Khrennikov. An application of the theory of open quantum systems to model the dynamics of party governance in the us political system. International Journal of Theoretical Physics, 53:1346–1360, 2014. 12. K. Lichtenegger and T. Hadzibeganovic. The interplay of self–reflection, social interaction and random events in the dynamics of opinion flow in two–party democracies. International Journak of Modern Physics C, 27:1650065, 2012. 13. E. Merzbacher. Quantum mechanics. John Wiley & Sons, New York, 1970. 14. A. K. Misra. A simple mathematical model for the spread of two political parties. Nonlinear Analysis: Modelling and Control, 17:343–354, 2012. 15. P. Roman. Advanced quantum mechanics. Addison-Wesley, New York, 1965. 16. I. Sened. A model of coalition formation: Theory and evidence. The Journal of Politics, 58:350– 372, 1996.

7

Phase Transitions and Decision Making

7.1

Introduction

In this chapter, we will describe an application of quantum tools to decision making, but in a way which is slightly different from the approaches described before, which were mostly focused on the analysis of the dynamics of some given system. Here, our interest will not be on the time evolution, but rather on the possibility of using tools from statistical mechanics to describe some sort of phase transition, where different phases describe different macroscopic situations for our system. This is in agreement with many recent applications, where many authors have considered the role of quantum in decision making and in social sciences [1, 4, 5, 13]. In recent years, Khrennikov introduced the notion of social laser [11–13] as a sort of general mechanism to describe the fact that, in social systems, some small event can be sufficient to produce a large effect, as we observe in real lasers. It is like a sort of phase transition occurring under certain conditions, bringing an originally disordered system into an ordered phase. This is the idea we want to consider in this chapter, by making use of some tools usually adopted in statistical mechanics and in operator algebras [7–9, 15–17]. More in detail, we will show here how to use a Heisenberg spin system in its mean field approximation to mimic the mechanism of a decision making when some special conditions on the parameters of the system are satisfied, like in ordinary ferromagnetic materials, for which the various spins align only if the temperature is below some critical value. A similar approach was considered, for instance, in [10] (see also references therein), where statistical (but not quantum or quantum-like) methods are used in the analysis of some decision making problems. This chapter is organized as follows. In Sect. 7.2, we introduce the essential ingredients of the mathematical settings needed in our analysis. Section 7.3 is devoted to the definition of the model, to its interpretation, and to the analysis of the results in connection with our original problem, which is essentially the following: we have a group of N agents, mutually © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_7

103

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7 Phase Transitions and Decision Making

interacting, which should come out with some binary decision. In general, for instance if the agents are different and if the interaction is weak, there is no particular reason to imagine that they reach a shared collective decision, at least if the agents form a closed system, i.e., they do not communicate with the external world. On the contrary, it is easy to imagine that such a decision could be reached if the interaction is strong enough or if we open the system to the environment. This is essentially what we will state in Sect. 7.3, using methods and ideas borrowed from algebraic statistical mechanics and quantum dynamics. Section 7.4 contains our conclusions. More details on this system can be found in [3].

7.2

Introducing the Framework

Let S be a group of N agents which have to take a decision, with two only possible choices: “yes” or “no”. Hence, it is natural to imagine that each agent τ j , j = 1, 2, . . . , N , is described by a linear combination of two orthogonal vectors, one representing the choice “yes”, e+j =     1 0 , and a second vector, e−j = , corresponding to the choice “no”. These two vectors 0 1 are an o.n. basis in the Hilbert space H j = C2 , endowed with its standard scalar product ., . j . A general vector describing τ j can therefore be written as ψ j = a j e+j + b j e−j ,

(7.1)

with |a j |2 + |b j |2 = 1, if we are interested in the normalization of the vector. Here, as before, j = 1, 2, . . . , N . We introduce the following Pauli matrices       1 0 01 00 + − 3 , σj = , σj = . σj = 0 −1 00 10 β

We observe that, ∀ j, the vectors eαj and the matrices σ j , α = ± and β = 3, ±, are all copies of the same vectors and matrices. That is why we do not use j also in the explicit expressions of these objects, trying to simplify the notation, when possible. It is clear that ∓ ± ± σ± σ± (7.2) σ 3j e±j = ±e±j , j ej = ej , j e j = 0. Because of these formulae, we can consider σ 3j as an operator measuring the decision of τ j , according to its eigenvalue: if τ j is choosing “yes”, then it is described by e+j , which is an eigenstate of σ 3j with eigenvalue +1. Similarly, if τ j is choosing “no”, then it is described by e−j , which is also an eigenstate of σ 3j , but with eigenvalue −1. Of course, σ ± j are operators which modify the original attitude of τ j , according to (7.2). For instance, after the action of + σ+ j on the agent τ j originally willing to choose “no”, τ j is moved to e j : he now is going to vote “yes”. However, if somebody insists too much, then it is like acting more than once − with σ + j (or with σ j ), and the result is that τ j is moved to the zero vector!

7.2

Introducing the Framework

105

Out of σ ± j we can also introduce the other two Hermitian Pauli matrices σ 1j

=

σ+ j

+ σ− j

 =

 01 , 10

σ 2j

=

i(σ − j

− σ+ j )

 =

0 −i i 0

 ,

and the following rules are easily checked: 2 (σ ± j ) = 0,

± 3 [σ ± j , σ j ] = ∓ 2σ j

− 3 [σ + j , σj ] = σj,

− {σ + j , σ j } = 11,

(7.3)

where, as always, [A, B] = AB − B A and {A, B} = AB + B A are the commutator and the anti-commutator of the operators A and B. In particular, we observe that σ ± j behave 3 as fermionic ladder operators, while σ j can be understood as a number-like operator (see Sect. 1.2). The Hilbert space for S is made of copies of C2 , one for each agent:

H = ⊗ Nj=1 C2j , and an o.n. basis of H consists of tensor products of states e+j and e−j for various j. For instance, the state describing a situation in which the first L 1 agents of S votes “yes” and the remaining L 2 agents vote “no”, L 1 + L 2 = N , is − −  = (e1+ ⊗ · · · ⊗ e+ L 1 ) ⊗ (e1 ⊗ · · · ⊗ e L 2 ).

Of course, the dimensionality of H increases with N . In fact, we have dim(H) = 2 N . An operator X 1 acting, for instance, on C21 , is identified with the tensor product X 1 ⊗ 112 ⊗ · · · 11 N , i.e., the tensor product of X 1 with N − 1 copies of the identity operator 11, acting on all the other single-agent Hilbert spaces. In this way we can relate X 1 with a (bounded) operator on H. The set of all the linear (bounded) operators on H defines an algebra A, the algebra of the observables. As discussed in Sect. 1.4, the vector  above, and all other normalized vectors on H, define a state on A, see [9, 16], as follows: ω (A) = , A ,

A ∈ A.

Here, ., . is the scalar product on H,  f 1 ⊗ · · · ⊗ f N , g1 ⊗ · · · ⊗ g N  =

N 

 f j, gjj,

j=1

f j , g j ∈ C j . The state ω is only one among all the possible states over A. Other examples of states, i.e., of positive normalized functionals on A, are Gibbs and KMS1 states, [8, 16]. These latter are particularly important because of their physical interpretation: they are both equilibrium states at a non zero temperature, while ω usually is used to describe 1 KMS stands for Kubo–Martin–Schwinger.

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7 Phase Transitions and Decision Making

a zero-temperature situation. While Gibbs states are used for finite dimensional systems, KMS states are their counterpart for systems with an infinite number of degrees of freedom. Both Gibbs and KMS states satisfy the so-called KMS-condition, which we give here in a simplified version. The KMS condition is ω(AB) = ω(B A(iβ)),

(7.4)

where A, B ∈ A, A(t) is the time evolution of A and A(iβ) is the analytic continuation of A(t) computed in t = iβ, where β = T1 is the inverse temperature (in units in which the Boltzmann constant is equal to one). In the Heisenberg picture, if H is the time-independent Hamiltonian of the system, A(t) is the usual operator A(t) = exp(i H t)A exp(−i H t). Also, ω can be written as exp(−β H ) . (7.5) ω(A) = tr(ρA), ρ= tr(exp(β H )) Here tr(X ) is the trace of the operator X [14]. We refer to [8, 16, 17] for more details and a deeper mathematical analysis of KMS states.

7.3

The Model

As we have briefly anticipated in the previous section, S is a group of N agents which have to decide between “yes” and “no”. The decision should be taken as a consequence of the interaction between the various agents. In this perspective, the system is closed: no input from outside will be considered. However, we will also comment briefly in the following about the possibility of opening the system. The single agent is described in terms of Pauli matrices: in view of what we have seen 3 before, we have changing opinion operators, σ ± j , and the counting opinion operator σ j . The full system S is made by N agents, and the description lives in H = ⊗ Nj=1 C2j , as discussed in Sect. 7.2. As everywhere in this book, see also [4, 5], the dynamics of S is given in terms of a Hamiltonian operator H N which is constructed in order to describe the (possibly, and hopefully) more relevant effects occurring in S . The first effect we want to consider is the following: N  (N ) Hcoop = Ji, j σi3 σ 3j , (7.6) i, j=1

where the label coop stands for cooperative. The reason for this name is the following: suppose that the two-body potential Ji, j is positive, for all i, j = 1, 2, . . . , N . Then the (N ) contribution to the eigenvalues of Hcoop is positive if τi and τ j are both up, ei+ and e+j , or both down, ei− and e−j . On the other hand, if Ji, j < 0, then the contribution of Ji, j σi3 σ 3j to (N )

the eigenvalues of Hcoop is positive only when τi and τ j are in opposite states, ei+ and e−j , or ei− and e+j . In what follows, the latter will be the relevant situation for us. This means that

7.3 The Model

107 (N )

(again, assuming that Ji, j < 0) Hcoop will be minimized if the various interacting agents (N ) are in the same state since, when this is the case, the corresponding eigenvalue of Hcoop is negative, and it is the largest possible one in absolute value. Then the spins are “energetically forced” to be all parallel (up or down, it does not matter). A second natural term to consider for the dynamics of S , and therefore in its Hamiltonian, is the following N    − + (N ) Hopp (7.7) = pi, j σi+ σ − + σ σ j i j . i, j=1

The meaning of this term is more psychological than energetic: it describes a situation in which, during an interaction between τi and τ j , the two agents tend to act in an opposite way: if τi moves from a “no” to a “yes” decision (because of σi+ ), τ j moves in the opposite (N ) direction, because of σ − j . Hence, Hopp describes an effect which is different with respect to that produced by (7.6). Of course, pi, j (when compared with Ji, j ) gives the strength of the mutual tendency of the agents to behave differently. Putting the two contributions together, we introduce the Hamiltonian hN =

(N ) Hcoop

+

(N ) Hopp

=

N 

Ji, j σi3 σ 3j

i, j=1

+

N 

  − + pi, j σi+ σ − j + σi σ j ,

(7.8)

i, j=1

which is the starting point of our analysis. Remark 7.1 It is not hard to imagine other terms to consider to define an improved version of N (N ) σi3 , where B ∈ R. If B > 0, then h N . For instance, we could add a term like Hext = B i=1 (N ) the lowest eigenvalue of Hext is obtained by a vector with all down vectors, e1− ⊗ · · · ⊗ e− N, and its corresponding eigenvalue is clearly −N B. Analogously, if we take B < 0, then the (N ) lowest eigenvalue of Hext is N B, with corresponding eigenvector e1+ ⊗ · · · ⊗ e+ N . A possible (N ) interpretation of Hext is that it can be used to model the presence of some information coming, for instance, from outside S , which is uniformly diffused in S , and contribute to the final decision of the agents. − + 1 1 1 2 2 Since σi+ σ − j + σi σ j = 2 (σi σ j + σi σ j ), the Hamiltonian h N can be rewritten only in α terms of σ j , with α = 1, 2, 3. Moreover, let us assume that both Ji, j and pi, j only depend on the number of the agents N , and not on the nature of the single τ j . Hence, as it is often discussed in statistical mechanics, [7, 15], we can consider the following mean field approximation, p J Ji, j → . pi, j → , N N

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7 Phase Transitions and Decision Making

If we further assume that J and p are related, and that in particular that2 p = 2J , we can replace h N with its mean field approximation H N , HN =

N 3 J   α α σi σ j , N

(7.9)

i, j=1 α=1

which is known as the mean-field Heisenberg model [15]. The rigorous existence of the dynamics for H N , and of the thermodynamical limit of this model (N → ∞), has been discussed in [6]. Here we report only some essential facts: 1. the operator σ αN = 2.

N

α i=1 σi converges (in the strong topology restricted to a suitable α which commutes with all the σ β , for all k and β; set of states) to an operator σ∞ k H N generates a time evolution of each σαi , αtN (σαi ) = exp(i H N t)σαi exp(−i H N t), which 1 N

converges (again, in the strong topology) to αt (σiα ) = cos2 (Ft)σiα +

i 1 sin(Ft) cos(Ft)[F · σ i , σiα ] + 2 sin2 (Ft)(F · σ i ) σiα (F · σ i ), F F

(7.10) where F = J σ ∞ and F = F = |J | σ ∞ (we adopt the following standard notation: v = (v 1 , v 2 , v 3 )). Incidentally we observe that (7.10) is well defined for all F  = 0, but it can also be extended continuously to the case F = 0; 3. if we are now interested to the expression of αt (σiα ) in a representation πω , arising as the GNS (Gelfand, Naimark and Segal)-representation3 of a state ω over A, formula (7.10) produces απt (σiα ) = cos2 ( f t)σiα +

i 1 sin( f t) cos( f t)[ f · σ i , σiα ] + 2 sin2 ( f t)( f · σ i ) σiα ( f · σ i ), f f

(7.11) where f = πω (F) = J m, m = πω (σ ∞ ). Here and in (7.10) α = 1, 2, 3, while i = 1, 2, 3, . . . , N ; 4. the effective dynamics απt described by (7.11) can be deduced by the following effective, representation-dependent, Hamiltonian Hπ = f ·

∞ 

σi .

i=1

Of course, using Hπ rather than H N simplifies a lot the analysis of the dynamics of S , [6], since it is linear (rather than quadratic) in the spin variables. For this reason, and 2 This imposes some relation between the strengths of the two contributions H (N ) and H (N ) in (7.8), coop opp

of course. 3 The mathematical details of this construction are not very relevant for us, and can be found in [6]. The relevant aspect here is that different representations correspond to different physics for the same system [8, 9].

7.3 The Model

109

since it does not change the form of the dynamics, Hπ will be used in the rest of our analysis. While in the standard Heisenberg model m is essentially the magnetization of the spin system, here m is a measure of how close our agents are to a common decision: if m 0, their decisions are so different that their mean value approaches zero. In fact, we can write N ω(σ i ), so that m = m is essentially a measure of the mean value m = lim N ,∞ N1 i=1 of the choices of each agent. On the other hand, the closer m is to one, the more all the agents (or most of them) share the same conclusion, even if we do not know which one is (N ) the preferred, between “yes” and “no”. In the first case, (m 0) the effect of Hopp wins (N ) over Hcoop , while, when m 1, the opposite happens. This result, as we will see, is not uniquely connected to the simplifying choice J = 2 p, but rather to other characteristics of the model, and in particular to the existence of a critical temperature. The next step of our analysis consists in using the KMS identity (7.4) with formula (7.11). Due to the properties of the GNS representation, it is known that m = πω (σ ∞ ) = ω(σ i ), ∀i. In particular, this shows that the dependence of the agent with index i disappears. This is because all our agents are equivalent in the treatment discussed here, as the analytic expression of H N in (7.9) clearly shows. Hence, after some algebra, using (7.3) and putting A = B = σ1i , from the equality in (7.4), ω(σ1i σ1i ) = ω(σ1i σ1i (iβ)), we get

1 1 2 2 (7.12) cosh(J mβ) + 2 sinh(J mβ) = 0. sinh(J mβ)(m 2 + m 3 ) m m Notice that, despite of the appearance of m = m in the denominators, formula (7.12) is well defined also in the limit m → 0. As discussed before, a solution m = 0 can be seen as the incapacity of the agents to find an agreement on the original “yes”-“no” question. We are more interested in finding if and under which condition a non zero solution of (7.12) does exist. Of course, at a first sight, one can imagine that any vector of the form m a = (m 1 , 0, 0) is such a solution, if m 1  = 0. And, in fact, it is clear that m a solves (7.12), and that it is non-zero. However, m a is not a solution of the KMS condition analogous to that in (7.12) when we replace σ1i with, say, σ2i . In this case, a non zero solution of the related KMS condition would be m b = (0, m 2 , 0), m 2  = 0. In the same way, m c = (0, 0, m 3 ), m 3  = 0, would solve the KMS condition in (7.4) with A = B = σ3i . Since the KMS condition should be satisfied for any possible choice of A and B in A, it is clear that the only possible common solution for these three different choices of A and B is m 0 = 0. This preliminary analysis implies that, if we are only interested to non zero solutions, the term sinh(J mβ)(m 22 + m 23 ) in (7.12) is not so important. What is important, on the contrary, is the other part, which can be rewritten as tanh(J mβ) = −m, (7.13) which appears often when dealing with many body systems, see e.g. [7, 15] and references therein. Here J is the parameter we introduced when going from the Hamiltonian h N to its

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mean field version H N in (7.9). We also recall that, in our derivation, J was also assumed to (N ) be proportional to p, and that J needs to be taken negative if we want the energy of Hcoop to be minimized by parallel spins (or, in our present situation, by agents sharing the same idea). This is in agreement with the analysis which follows, which are meant mainly for those readers which are not very familiar with statistical mechanics. Let us introduce the function (m) = m + tanh(J mβ), in which J and β are seen as parameters. It is clear that m ≥ 0 necessarily, since m is the norm of a vector. It is also clear that, since tanh(x) ∈] − 1, 1[ for all x ∈ R, (0) = 0 and limm,∞ (m) = ∞. Then, if  (0) < 0, we are sure that the function (m) acquires negative values in some right neighbourhood of m = 0. But (m) is a continuous function. Hence a value m c > 0 must exists such that (m c ) = 0. This m c is, of course, the non zero solution of (7.13) we were looking for. Of course, more than a single non trivial zero of (m) could exist. However, for what is relevant for us, one solution is enough. We have  (m) = 1 + cosh2J(Jβ βm) , and since cosh(0) = 1 it follows that  (0) = 1 + J β,

which is negative only if J β < −1. Since β = T1 is always positive, the inequality can only be satisfied if J is negative, which appears to be in agreement with our working condition. Now, calling Tc = −J , we can conclude that a non zero solution of (7.13) does exist if T < Tc , i.e., for temperatures less than a critical value which is proportional (with a minus sign) to the strength with which the agents are interacting. What we have deduced here suggests the following interpretation: the various agents can cooperatively come out with a common decision only in the presence of some lack of disorder in S . In other words, we can see at T (the temperature, in its original interpretation) as a measure of the order in the system, as in a standard thermodynamical interpretation: the higher the temperature, the higher the disorder. On the opposite side, when the temperature decreases the order increases, and then it is possible to get a common shared decision. Thus, we can look at T = β −1 in ω as a sort of order parameter, and at J = −Tc as the critical value of this parameter measuring a phase transition from order to disorder, or vice versa. A small change may produce large effects, like in the social laser approach [11–13]. Moreover, since we can rewrite −J β = TTc , it is not difficult to see that, if T  Tc , m approaches one. This is in agreement with our interpretation: the lower the temperature of S , the closer the result to ±1: the degree of disorder decreases and all the agents (or a large part of them) share the same decision.

7.4

Conclusions

What we have discussed in this chapter is, to our knowledge, one of the few existing applications of methods of algebraic quantum dynamics and algebraic statistical mechanics to social systems, and more specifically to decision making. A possible role of KMS states in similar contexts, and in particular to finance, was first proposed in [2], but never explored after that first attempt. Here, having in mind a rather abstract problem in decision making,

References

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we discuss how the KMS condition can be used to describe a sort of phase transition during the process of deciding on some binary question. In other words, we suggest a possible quantitative approach which produces a sort of collective decision, using a set of minimal (although mathematically not so simple) ingredients and techniques which, we believe, can be relevant for further studies and more applications in the area of decision making. Notice that there is (almost) no dynamics in our present analysis of S , even if its Hamiltonian still plays a crucial role. The next step is, of course, a deeper understanding of this kind of phase transitions in macroscopic systems, and in particular of the real role of the KMS condition. This understanding will be easier in presence of other models which, hopefully, should clarify the relevance of our approach in similar problems.

References 1. P. M. Agrawal and R. Sharda. OR Forum - Quantum mechanics and human decision making. Operations Research, 61:1–16, 2013. 2. F. Bagarello. An operatorial approach to stock markets. Journal of Physics A, 39:6823–6840, 2006. 3. F. Bagarello, Phase transitions, KMS-condition and Decision Making, Phil. Trans. A, in press. 4. F. Bagarello. Quantum dynamics for classical systems: with applications of the Number operator. John Wiley & Sons, New York, 2012. 5. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 6. F. Bagarello and G. Morchio. Dynamics of mean field spin models from basic results in abstract differential equations. Journal of Statistical Physics, 66:849–866, 1992. 7. R. J. Baxter. Exactly solved models in statistical mechanics. Academic Press, London, 1982. 8. O. Bratteli and D. W. Robinson. Operator algebras and quantum statistical mechanics 2. Springer-Verlag, New York, 2002. 9. O. Bratteli and D.W. Robinson. Operator algebras and quantum statistical mechanics 1. Springer-Verlag, New York, 2002. 10. S. Galam. Rational group decision making: A random field ising model at t = 0. Physica A: Statistical Mechanics and its Applications, 238:66–80, 1997. 11. A. Khrennikov. Towards information lasers. Entropy, 17:6969–6994, 2015. 12. A. Khrennikov. Social laser: action amplification by stimulated emission of social energy. Philosophical Transactions of the Royal Society, 374:20150094, 2016. 13. A. Khrennikov. Social laser. Jenny Stanford Publishing, Singapore, 2020. 14. M. Reed and B. Simon. Methods of modern mathematical physics I: Functional analysis. Academic Press, New York, 1980. 15. D. Ruelle. Statistical mechanics. Rigorous results. Imperial College Press, London, 1999. Reprint of the 1989 edition, World Scientific Publishing Co., Inc., River Edge, NJ. 16. G. L. Sewell. Quantum Theory of Collective Phenomena. Oxford University Press, Oxford, 1989. 17. G. L. Sewell. Quantum Mechanics and its Emergent Macrophysics. Princeton University Press, Princeton, 2002.

8

Conclusions

Our original aim was to discuss the possibility of using quantum ideas in contexts which are not necessarily those of the usual quantum world, and to convince the reader that the Schrödinger or the Heisenberg equations of motion, the ladder operators, or even the number or the Hamiltonian operators, can be efficiently used in the description of some macroscopic systems, with few or many agents, in various fields of research. With this in mind, we have considered several applications of these quantum-like ideas to many different systems. In particular, we have considered the spread of a pandemics, and of some (good or fake) news in a lattice. Also, applications have been described in politics and in population dynamics, as well as to a particular problem in decision making. All these applications, together with those already proposed in recent years by us and several other authors [1–9], enforce our belief that the approach we are considering and refining since some years, can be useful in several contexts, and can help to get some insight in situations and systems where the ordinary approaches, based on partial differential equations or stochastic models, do not work well. We should mention that the main obstacle in carrying on our analysis is due to the operatorial nature of the differential equations of motion which usually drive the dynamics of our systems. This imposes certain constraints on our ability to find exact numerical or analytical solutions. In Chap. 4, we have proposed a first way to overcome this difficulty, but this approach does not always work. We plan to consider again this problem since this is the main barrier to a rather rich list of applications that could be—we believe—efficiently discussed by means of ladder operators. Also, we feel the necessity to test the general strategy not only at a qualitative level, but also checking its quantitative aspects: it must be fully understood that our strategy is not

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Bagarello et al., Quantum Tools for Macroscopic Systems, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-30280-0_8

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only able to produce reasonable results. It should be checked if it is also able to reproduce data! This is what happens in some of the applications considered along the years, but it is still, to be honest, something to be deeply considered. In conclusion, what we did was already quite interesting for us, but we are even more interested and curious to understand what can be done next. So, this is not a conclusion!

References 1. B. E. Baaquie. Quantum Finance. Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press, New York, 2004. 2. B. E. Baaquie. Interest Rates and Coupon Bonds in Quantum Finance. Cambridge University Press, New York, 2009. 3. F. Bagarello. Quantum dynamics for classical systems: with applications of the Number operator. John Wiley & Sons, New York, 2012. 4. F. Bagarello. Quantum Concepts in the Social, Ecological and Biological Sciences. Cambridge University Press, Cambridge, 2019. 5. F. Bagarello. Pseudo-Bosons and their coherent states. Mathematical Physics Studies. Springer, New York, 2022. 6. J. R. Busemeyer and P. D. Bruza. Quantum Models of Cognition and Decision. Cambridge University Press, Cambridge, 2013. 7. E. Haven and A. Khrennikov. Quantum social science. Cambridge University Press, Cambridge, 2013. 8. A. Khrennikov. Ubiquitous Quantum Structure: from Psychology to Finance. Springer-Verlag, Berlin, 2012. 9. A. Khrennikov. Social laser. Jenny Stanford Publishing, Singapore, 2020.