Remarks on the geometric quantization of a class of harmonic oscillator type potentials

TL;DR

This paper investigates the mathematical conditions under which geometric quantization can be applied to certain harmonic oscillator-like potentials, analyzing integrable systems and providing new examples including Lennard-Jones potentials.

Contribution

It identifies specific mathematical criteria involving momentum map critical points for applying geometric quantization to these systems and introduces new examples illustrating these conditions.

Findings

Conditions for geometric quantization are linked to momentum map critical points.

Two new systems are successfully quantized using the proposed criteria.

Local potential isomorphism does not necessarily imply observable algebra isomorphism.

Abstract

The conditions that must be fulfilled by a certain physical system to apply geometric quantization prescription on it are investigated. These terms are sought as mathematical requirements, which can be traced in an analysis of integrable systems, from the perspective of both potential function and Hamiltonian vector field. The answer is found in momentum map critical points. Basically, a certain disposal of points that allow a momentum map $C^*$ isomorphism, of observables, with harmonic oscillator momentum map enforce geometric quantization rules. Following the general theory, two newly presented examples, which exhibits these properties, are quantified through geometric quantization prescription. The Lennard-Jones' type potential is one of the examples, it is known as describing molecules in interaction. We end with a third example that shows the local isomorphism of potentials do not induce $C^*$ isomorphism of observables.