Three examples of quantum dynamics on the half-line with smooth bouncing

TL;DR

This paper introduces a quantization method for the half-plane using affine coherent states, demonstrating its ability to remove singularities and handle boundary conditions through three illustrative quantum dynamics examples.

Contribution

It presents a novel affine coherent state quantization approach for the half-plane, with applications to quantum bouncing and cosmology models.

Findings

Singularity at the origin is systematically removed.

Affine symmetry is preserved in the quantization process.

Three elementary quantum dynamics examples demonstrate the method's effectiveness.

Abstract

This article is an introductory presentation of the quantization of the half-plane based on affine coherent states (ACS). The half-plane is viewed as the phase space for the dynamics of a positive physical quantity evolving with time, and its affine symmetry is preserved due to the covariance of this type of quantization. We promote the interest of such a procedure for transforming a classical model into a quantum one, since the singularity at the origin is systematically removed, and the arbitrariness of boundary conditions can be easily overcome. We explain some important mathematical aspects of the method. Three elementary examples of applications are presented, the quantum breathing of a massive sphere, the quantum smooth bouncing of a charged sphere, and a smooth bouncing "dust" sphere as a simple model of quantum Newtonian cosmology.