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

In physics, when a mathematical model becomes really inoperative  in regard with 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 Classical models were (progressively) superseded by quantum ones through quantization prescriptions.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. I will describe  well defined quantizations,  based on integral calculus and  Weyl-Heisenberg symmetry. They are described in simple terms through one of the most basic examples of mechanics. Starting from  (probability or not) 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, I will show how to build corresponding quantum model(s) and associated probabilities (e.g. Husimi) or quasi-probabilities (e.g. Wigner) distributions. I will 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. In quantum cosmology, I will show how the procedure allows to consider the quantum dynamics of the anisotropic part of the Mixmaster universe as a perturbation of the explicitly integrable periodic 3-particle Toda system.