An Optimal Control Perspective on Classical and Quantum Physical Systems

TL;DR

This paper investigates classical and quantum systems through the lens of optimal control theory, revealing that both can be described by closed-loop feedback mechanisms, highlighting the natural occurrence of feedback in physical phenomena.

Contribution

It demonstrates that classical and quantum dynamics can be modeled as closed-loop feedback control problems, providing a unified control-theoretic perspective on physical systems.

Findings

Classical systems can be described by closed-loop feedback via canonical transformations.

Quantum systems exhibit closed-loop strategies due to Heisenberg relations.

Feedback mechanisms are inherent in the fundamental dynamics of physical systems.

Abstract

In this paper, we analyze classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that it can be described by a closed-loop feedback problem. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be thought of as constraints in a non-commutative space, then the momentum must be dependent on the position for any generic wave function. This implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. For the sake of exposition, we give a short review of control theory, and some familiar examples at the classical and quantum levels are analyzed.