Why a Quantum Tool in Classical Contexts? (Part II)

The reader may wonder where Part I is. In fact, there is no Part I here. Part I is in [1]. When I wrote that book I felt strongly the responsibility to justify my approach, since it was, in fact, rather unusual, and the reaction of most referees, when submitting a research paper of mine, was quite often the same: “Why are you adopting this technique? What is wrong with a classical approach?” However, since 2012, I realized that this approach was not so crazy, and I discovered that many people, in many different fields of research, were adopting similar strategies, using quantum ideas and, in particular, mathematical tools deeply connected with quantum mechanics, to deal with problems that are not necessarily related to the microscopic world. For this reason I do not really feel anymore the necessity of justifying myself. However, I think that giving some words of explanation can still be useful for readers, and this is what the next few sentences are about. The driving idea behind my approach is that the lowering and raising operators related to the canonical (anti-)commutation relations (CCR or CAR) can be used in the description of processes where some relevant quantities change discontinuously. To cite a few examples, stock markets, migration processes or some biological systems: multiples of share are exchanged, in a market; one, two or more people (and not just half of a person) move from one place to another; and one cell duplicates producing two cells. This suggests that objects labeled by natural numbers are important, in some situations. People with a background in quantum mechanics know that nonnegative integers can be seen as eigenvalues of some suitable number operator constructed, in a natural way, using ladder operators. Also, ladder operators can be efficiently used to describe systems where discrete quantities of some kind are exchanged between different agents. Hence ladder operators, and some combinations of them, can be used in the description of particular systems. We refer to [1] for some results and models in this direction. But, due to the fact that the observables of a system S are now operators, one of the main questions to be answered is the following: How should we assign a time evolution