The intention, in this chapter, is to illustrate something of the versatility of natural number dynamics as a means of modelling time dependent populations of items that occur in systems of interacting categories and to demonstrate the universality of the quantum nature of things that flavours the whole of this book. The applicability of the natural number dynamics to systems outside those that are normally treated in physics, is made possible, to a certain degree, by the way natural number dynamics was introduced in Chapter 3 , without any reference to physical systems, nor with any reliance on preconceived physical notions. The examples we will look at come from a variety of scientific fields including ecology, and molecular biology as well as physics. The examples from ecology and molecular biology involve bosonic natural number operators. The example from ecology is a type of predator-prey model and the molecular biology example involves an original model of cell division [ 106 ]. The application of the number operator in an extra-physical context is a very recent development of the use of quantum techniques. Quantum population dynamics in a social science context that has been pioneered by Bagarello and his co-workers [ 5 , 6 , 8 , 7 ] has received a lot of attention recently.
The intention, in this chapter, is to illustrate something of the versatility of natural number dynamics as a means of modelling time dependent populations of items that occur in systems of interacting categories and to demonstrate the universality of the quantum nature of things that flavours the whole of this book. The applicability of the natural number dynamics to systems outside those that are normally treated in physics, is made possible, to a certain degree, by the way natural number dynamics was introduced in Chapter 3 , without any reference to physical systems, nor with any reliance on preconceived physical notions. The examples we will look at come from a variety of scientific fields including ecology, and molecular biology as well as physics. The examples from ecology and molecular biology involve bosonic natural number operators. The example from ecology is a type of predator-prey model and the molecular biology example involves an original model of cell division [ 106 ]. The application of the number operator in an extra-physical context is a very recent development of the use of quantum techniques. Quantum population dynamics in a social science context that has been pioneered by Bagarello and his co-workers [ 5 , 6 , 8 , 7 ] has received a lot of attention recently.
Mathematics is increasingly applied to classical problems in finance, biology, economics, and elsewhere. Quantum Dynamics for Classical Systems describes how quantum tools—the number operator in particular—can be used to create dynamical systems in which the variables are operator-valued functions and whose results explain the presented model. The book presents mathematical results and their applications to concrete systems and discusses the methods used, results obtained, and techniques developed for the proofs of the results. The central ideas of number operators are illuminated while avoiding excessive technicalities that are unnecessary for understanding and learning the various mathematical applications. The presented dynamical systems address a variety of contexts and offer clear analyses and explanations of concluded results. Additional features in Quantum Dynamics for Classical Systems include: Applications across diverse fields including stock markets and population migration as well as a unique quantum perspective on these classes of models Illustrations of the use of creation and annihilation operators for classical problems Examples of the recent increase in research and literature on the many applications of quantum tools in applied mathematics Clarification on numerous misunderstandings and misnomers while shedding light on new approaches in the field Quantum Dynamics for Classical Systems is an ideal reference for researchers, professionals, and academics in applied mathematics, economics, physics, biology, and sociology. The book is also excellent for courses in dynamical systems, quantum mechanics, and mathematical models.
Quantum mechanics is traditionally associated with microscopic systems; however, quantum concepts have also been successfully applied to a diverse range of macroscopic systems both within and outside of physics. This book describes how complex systems from a variety of fields can be modelled using quantum mechanical principles; from biology and ecology, to sociology and decision-making. The mathematical basis of these models is covered in detail, furnishing a self-contained and consistent approach. This book provides unique insight into the dynamics of these macroscopic systems and opens new interdisciplinary research frontiers. It will be an essential resource for students and researchers in applied mathematics or theoretical physics who are interested in applying quantum mechanics to dynamical systems in the social, biological or ecological sciences.
Introducing a unified framework for describing and understanding complex interacting systems common in physics, chemistry, biology, ecology, and the social sciences, this comprehensive overview of dynamic critical phenomena covers the description of systems at thermal equilibrium, quantum systems, and non-equilibrium systems. Powerful mathematical techniques for dealing with complex dynamic systems are carefully introduced, including field-theoretic tools and the perturbative dynamical renormalization group approach, rapidly building up a mathematical toolbox of relevant skills. Heuristic and qualitative arguments outlining the essential theory behind each type of system are introduced at the start of each chapter, alongside real-world numerical and experimental data, firmly linking new mathematical techniques to their practical applications. Each chapter is supported by carefully tailored problems for solution, and comprehensive suggestions for further reading, making this an excellent introduction to critical dynamics for graduate students and researchers across many disciplines within physical and life sciences.
Introduction The central claim of this book is that all intentional phenomena are quantum mechanical. That goes both for the private thoughts inside our heads and for public or collective intentions like norms, culture, and language, which we might generically call institutions. I suggested in Chapter 1 that by virtue of their dependence on consciousness, a classical, materialist ontology cannot explain these phenomena, and as such from that perspective they must be epiphenomenal or illusions. In the rest of this book I show that a quantum, panpsychist ontology can provide a physical basis for what we all know to be true, which is that both private and collective intentions are part of the natural order. The elementary parts of social ontology are human beings in their biological individuality, which in this Part I consider in abstraction from their lived social contexts. By taking humans out of their natural habitat I aim to focus attention on what we bring to the social table simply by virtue of being organisms of a certain kind. The abstract individual is a common enough starting point in classical social theory (think Hobbes and the state of nature). However, it might seem an odd place to begin a quantum social theory, given the holism of quantum phenomena, both “all the way down” in having no elementary parts, and “all the way across” in universal non-locality. What look like separate organisms are just local decoherence effects of quantum fields; everything really is related to everything else. In that light the abstract individual appears not just as an abstraction, but as positively occluding what “the individual” really is. However, there is a difference between what is true of a universal ontology and what is practically relevant to specifically socialontologies, which are constrained by the properties of their constituent organisms. Unlike some organisms, humans live in highly interdependent societies, and most of what is interesting about us is socially constituted as well.
Since its very beginning, quantum mechanics has been developed to deal with systems on the atomic or sub-atomic scale. For many decades, there has been no reason to think about its application to macroscopic systems. Actually, macroscopic objects have even been used to show how bizarre quantum effects would appear if quantum mechanics were applied beyond its original realm. This is, for example, the essence of the so-called Schrödinger cat paradox (Schrödinger, 1935). How-ever, due to recent advances in the design of systems on the meso- and nanoscopic scales, as well as in cryogenic techniques, this situation has changed drastically. It is now quite natural to ask whether a specific quantum effect, collectively involving a macroscopic number of particles, could occur in these systems. In this book it is our intention to address the quantum mechanical effects that take place in properly chosen or built “macroscopic” systems. Starting from a very naïve point of view, we could always ask what happens to systems whose classical dynamics can be described by equations of motion equivalent to those of particles (or fields) in a given potential (or potential energy density). These can be represented by a generalized “coordinate” φ( r , t ) which could either describe a field variable or a “point particle” if it is not position dependent, φ( r , t ) = φ( t ).
This thesis discusses different aspects of time-dependent two-level quantum systems and their coherent dynamics. In the introductory chapters, we present some basic models used to study them and some basic results related to the level-crossing problems generally. As these systems are only rarely analytically solvable, some emphasis is put on the existing approximative theories. In particular, the generalization of the method of Dykhne, Davis and Pechukas (DDP), originally developed for problems in near-adiabatic region, is studied further. In the DDP theory, the final populations of different states are determined by the complex zero points of the eigenenergies. Superparabolic level-crossing models are introduced in this thesis and they prove to be simple but versatile test models for these studies. It is shown that, by considering all of the zero points, one can obtain accurate approximations also for highly non-adiabatic regions. In Chapter 4, a general differential geometric framework for time-dependent level-crossing models is developed and its basic character is discussed. A natural way of associating a plane curve to a time-dependent two-level model is introduced. This association allows one to use all of the mathematical results concerning plane curves to study these models. As an example, we use the Four-vertex theorem to discuss the adiabatic limit of certain type of systems. We also consider the prospect of enhancing the population transfer by transforming the time-dependent coupling pulse into a zero-area coupling. This is done either smoothly or by an abrupt jump in the phase of the coupling. It has been shown earlier, that one can obtain a complete population inversion (CPI) in a robust way with strong, non-resonant zero-area pulses. We study this in the more general time-dependent setting in Chapter 5, where also the energy levels are driven in a time-dependent fashion. This allows to transport the CPI phenomenon from the strong coupling region towards more moderate couplings. As an example of the smooth case, we consider the SechTanh-model, and show that even the nearest-zero DDP approximation can be used, despite the highly non-adiabatic origin of the phenomenon. We develop an approximative scheme also for the phase-jump case, using the parabolic model as an example. As a limiting case, we derive an approximative formula for the transition probability, which has some universal character and which proves to be useful in the most interesting case where the robust CPI is obtained. It is shown that even the highly suppressed transitions in tunnelling cases can be strongly enhanced in the phase-jump scenario.