A Quantum Formalism for Abstract Dynamical Systems
Joan C. Micó

Institut Universitari de Matemàtica Multidisciplinar, Universitat Politècnica de València, Camí de Vera s/n,
46022 Valencia, Spain; jmico@mat.upv.es

Institut Universitari de Matemàtica Multidisciplinar, Universitat Politècnica de València, Camí de Vera s/n,
46022 Valencia, Spain; jmico@mat.upv.es
Abstract: This paper presents a quantum formulation for classical abstract dynamical systems
(ADS), defined by coupled sets of first-order differential equations. They are referred to as “abstract”
because their dynamical variables can be of different interrelated natures, not necessarily
corresponding to physics, such as populations, socioeconomic variables, behavioral variables, etc.
A classical linear Hamiltonian can be derived for ADS by using Dirac’s dynamics for singular
Hamiltonian systems. Also, a corresponding first-order Schrödinger equation (which involves the
existence of a system Planck constant particular of each system) can be derived from this
Hamiltonian. However, Madelung’s reinterpretation of quantum mechanics, followed by the Bohm
and Hiley work, produces no further information about the mathematical formulation of ADS.
However, a classical quadratic Hamiltonian can also be derived for ADS, as well as a corresponding
second-order Schrödinger equation. In this case, the Madelung reinterpretation of quantum
mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS,
which provides new quantum variables interrelated dynamically with the classical variables. An
application case is presented: the one-dimensional autonomous system given by the logistic
dynamics. The differences between the classical and the quantum trajectories are highlighted in the
context of this application case.

Keywords: abstract dynamical systems; classical dynamics; Dirac Hamiltonian; Schrödinger
equation; Madelung quantum interpretation; Bohm and Hiley interpretation; quantum dynamics;
logistic function

MSC: 81Q65; 93A10

Citation: Micó, J.C. A Quantum
Formalism for Abstract Dynamical
Systems. Mathematics 2024, 12, 1076.
h􀁇ps://doi.org/10.3390/
math12071076
Academic Editor: Huaizhong Zhao
Received: 1 March 2024
Revised: 21 March 2024
Accepted: 28 March 2024
Published: 2 April 2024