A Remark on Polar Noncommutativity

TL;DR

This paper investigates the proper formulation of polar noncommutativity in two-dimensional space, clarifying conditions for its validity and potential extensions to more complex spacetime models.

Contribution

It provides a detailed analysis of the conditions under which polar noncommutativity can be correctly formulated in 2^{2}, enhancing the theoretical foundation for noncommutative geometry in gravity theories.

Findings

Identifies the correct form of polar noncommutativity in 2^{2}

Clarifies conditions for the validity of noncommutative spacetime approaches

Suggests potential for extending to more complex spacetime parametrizations

Abstract

Noncommutative space has been found to be of use in a number of different contexts. In particular, one may use noncommutative spacetime to generate quantised gravity theories. Via an identification between the Moyal $\star$-product on function space and commutators on a Hilbert space, one may use the Seiberg-Witten map to generate corrections to such gravity theories. However, care must be taken with the derivation of commutation relations. We examine conditions for the validity of such an approach, and determine the correct form for polar noncommutativity in $\mathbb{R}^{2}$. Such an approach lends itself readily to extension to more complicated spacetime parametrisations.