Non-commutativity as a measure of inequivalent quantization

TL;DR

This paper investigates how non-commutativity influences boundary conditions in quantum systems, using a non-commutative inverse square potential as a model, revealing a family of boundary conditions linked to non-commutativity strength.

Contribution

It introduces a novel analysis of boundary conditions in non-commutative quantum mechanics, showing their dependence on the non-commutativity parameter and identifying a family of self-adjoint extensions.

Findings

Non-commutativity affects boundary conditions in quantum problems.

A family of self-adjoint extensions is found for the non-commutative inverse square problem.

Boundary conditions are parametrized by the non-commutativity scale .

Abstract

We show that the strength of non-commutativity could play a role in determining the boundary condition of a physical problem. As a toy model we consider the inverse square problem in non-commutative space. The scale invariance of the system is known to be explicitly broken by the scale of non-commutativity \Theta. The resulting problem in non-commutative space is analyzed. It is shown that despite the presence of higher singular potential coming from the leading term of the expansion of the potential to first order in \Theta, it can have a self-adjoint extensions. The boundary conditions are obtained, belong to a 1-parameter family and related to the strength of non-commutativity.