Non-commutativity in polar coordinates

We reconsider the fundamental commutation relations for non-commutative $\mathbb{R}^{2}$ described in polar coordinates with non-commutativity parameter $\theta$. Previous analysis found that the natural transition from Cartesian coordinates to polars led to a representation of $\left[\hat{r}, \hat{\varphi}\right]$ as an everywhere diverging series. We compute the Borel resummation of this series, showing that it can subsequently be extended throughout parameter space and hence provide an interpretation of this commutator. Our analysis provides a complete solution for arbitrary $r$ and $\theta$ that reproduces the earlier calculations at lowest order. We compare our results to previous literature in the (pseudo-)commuting limit, finding a surprising spatial dependence for the coordinate commutator when $\theta \gg r^{2}$. We raise some questions for future study in light of this progress.