A Remark on Polar Noncommutativity

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.