Classical limits of quantum mechanics on a non-commutative configuration space

TL;DR

This paper investigates how non-commutative quantum mechanics, modeled by two harmonic oscillators on a non-commutative space, approaches classical mechanics, revealing that removing non-commutativity in different ways yields non-commuting results.

Contribution

It analyzes the classical limit of non-commutative quantum mechanics and demonstrates that removing non-commutativity in configuration space and operators are non-commuting procedures.

Findings

Removal of non-commutativity from configuration space and operators are not equivalent.

The classical limit process depends on the order of removing non-commutativity.

Non-commutative effects persist differently depending on the approach to the classical limit.

Abstract

We consider a model of non-commutative Quantum Mechanics given by two harmonic oscillators over a non-commutative two dimensional configuration space. We study possible ways of removing the non-commutativity based on the classical limit context known as anti-Wick quantization. We show that removal of non-commutativity from the configuration space and from the canonical operators are not commuting operations.