Noncommutative theories and general coordinate transformations

TL;DR

This paper explores a class of noncommutative theories where spatial coordinates are related by smooth transformations to standard noncommutative coordinates with constant commutation relations, showing that in two dimensions, space-dependent relations can be mapped to constant ones.

Contribution

It demonstrates that noncommutative theories with space-dependent relations can be transformed into constant $ heta_{ij}$ theories in two dimensions, extending understanding of coordinate transformations in noncommutative geometry.

Findings

Any 2D space-dependent noncommutative relation can be mapped to a constant $ heta_{ij}$ theory.

The study characterizes properties of noncommutative theories under smooth coordinate changes.

Provides explicit examples of such transformations in two dimensions.

Abstract

We study the class of noncommutative theories in $d$ dimensions whose spatial coordinates $(x_i)_{i=1}^d$ can be obtained by performing a smooth change of variables on $(y_i)_{i=1}^d$, the coordinates of a standard noncommutative theory, which satisfy the relation $[y_i, y_j] = i \theta_{ij}$, with a constant $\theta_{ij}$ tensor. The $x_i$ variables verify a commutation relation which is, in general, space-dependent. We study the main properties of this special kind of noncommutative theory and show explicitly that, in two dimensions, any theory with a space-dependent commutation relation can be mapped to another where that $\theta_{ij}$ is constant.