From classical to quantum models: the regularising role of integrals, symmetry and probabilities

TL;DR

This paper explores how integral calculus and symmetry principles facilitate the transition from classical to quantum models, demonstrating their regularizing effects through phase space quantization and probability distributions.

Contribution

It introduces well-defined quantization methods based on integrals and symmetry, illustrating their role in deriving quantum models from classical phase space distributions.

Findings

Quantization via integrals regularizes classical models.

Construction of quantum probability and quasi-probability distributions.

Application to particle motion with variable mass and step potential.

Abstract

In physics, one is often misled in thinking that the mathematical model of a system is part of or is that system itself. Think of expressions commonly used in physics like "point" particle, motion "on the line", "smooth" observables, wave function, and even "going to infinity", without forgetting perplexing phrases like "classical world" versus "quantum world".... On the other hand, when a mathematical model becomes really inoperative with regard to correct predictions, one is forced to replace it with a new one. It is precisely what happened with the emergence of quantum physics. Classical models were (progressively) superseded by quantum ones through quantization prescriptions. These procedures appear often as ad hoc recipes. In the present paper, well defined quantizations, based on integral calculus and Weyl-Heisenberg symmetry, are described in simple terms through one of the most basic examples of mechanics. Starting from (quasi-) probability distribution(s) on the Euclidean plane viewed as the phase space for the motion of a point particle on the line, i.e., its classical model, we will show how to build corresponding quantum model(s) and associated probabilities (e.g. Husimi) or quasi-probabilities (e.g. Wigner) distributions. We highlight the regularizing role of such procedures with the familiar example of the motion of a particle with a variable mass and submitted to a step potential.