Noncommutative spacetime symmetries: Twist versus covariance

TL;DR

This paper demonstrates the covariance of the Moyal product under affine transformations, introduces an extended space for noncommutative representations, and compares this with twist-deformed algebra approaches.

Contribution

It provides a new noncommutative representation of affine symmetries and clarifies the limitations of covariance extension beyond linear transformations.

Findings

Moyal product is covariant under linear affine transformations

Introduces an (x,Θ)-space for noncommutative symmetry representation

Covariance cannot be extended to higher-order polynomial transformations

Abstract

We prove that the Moyal product is covariant under linear affine spacetime transformations. From the covariance law, by introducing an $(x,\Theta)$-space where the spacetime coordinates and the noncommutativity matrix components are on the same footing, we obtain a noncommutative representation of the affine algebra, its generators being differential operators in $(x,\Theta)$-space. As a particular case, the Weyl Lie algebra is studied and known results for Weyl invariant noncommutative field theories are rederived in a nutshell. We also show that this covariance cannot be extended to spacetime transformations generated by differential operators whose coefficients are polynomials of order larger than one. We compare our approach with the twist-deformed enveloping algebra description of spacetime transformations.