Dynamics of Classical Systems in the Operator Algebra of Quantum Mechanics

Abstract

The use of the formalism and the operatorial techniques typical of quantum mechanics proved in recent years to be an effective approach for the description of even macroscopic systems characterized by complex interactions in several contexts with interesting applications both in the bioecological and in the socioeconomic area. The operatorial approach, based on raising and lowering operators, especially using the number representation, provides useful tools for modeling collective dynamics of spatially distributed physical systems in a completely general and precise way. Quantum-like models (not necessarily related to the microscopic approach) offer an interesting mathematical insight into phenomena and processes even at macroscopic scales in several situations in which some quantities changing discontinuously are well described in terms of the integer eigenvalues of certain self-adjoint operators useful for a complete description of the system under consideration (the so-called observables of the system itself). Moreover, according to the innovative approach called rule-induced dynamics, the derivation of the dynamics in the operator algebra of quantum mechanics from a time-independent Hamiltonian operator may be enriched by means of the repeated application of specific ''rules'' including in the dynamics meaningful effects occurring during the time evolution of the system and, therefore, producing an adjustment of the model itself as a consequence of its evolution. This method is of great interest to describe systems for which a nontrivial and sufficiently regular asymptotic behavior is expected. The original contributions of this thesis, besides the construction and the numerical investigation of operatorial models to describe complex systems of interest in many areas (mathematics, physics, ecology, social sciences), are concerned with the introduction and exploitation of the so called (H, rho)-induced dynamics. The combined action of the Hamiltonian and of some rules allowed to take into account in the model relevant effects that can not be described by a time independent self-adjoint Hamiltonian. This strategy, which provides a powerful strategy to simulate the effect of using a time-dependent Hamiltonian, revealed capable of greatly enriching the dynamics of the considered models still with simple quadratic Hamiltonians without additional computational costs.