On Coordinate Transformations in Planar Noncommutative Theories

TL;DR

This paper explores coordinate transformations in planar noncommutative theories with space-dependent commutation relations, enabling the use of standard Moyal methods in certain cases, including curved interfaces.

Contribution

It introduces a method to find coordinate transformations that convert space-dependent noncommutativity into constant commutators, facilitating field theory construction.

Findings

Coordinate transformations can simplify noncommutative relations.

Standard Moyal approach applies in transformed coordinates.

Method extends to more general noncommutative geometries.

Abstract

We consider planar noncommutative theories such that the coordinates verify a space-dependent commutation relation. We show that, in some special cases, new coordinates may be introduced that have a constant commutator, and as a consequence the construction of Field Theory models may be carried out by an application of the standard Moyal approach in terms of the new coordinates. We apply these ideas to the concrete example of a noncommutative plane with a curved interface. We also show how to extend this method to more general situations.