Non-commutativity in polar coordinates

TL;DR

This paper addresses the challenge of defining the non-commutative coordinate commutator in polar coordinates by applying Borel resummation to a divergent series, providing a comprehensive interpretation across parameter space.

Contribution

It introduces a Borel resummation approach to interpret the polar coordinate commutator in non-commutative space, extending previous lowest-order analyses to arbitrary parameters.

Findings

Successfully resummed the divergent series for the commutator

Extended the interpretation of the commutator across all parameter space

Identified a surprising spatial dependence when the non-commutativity parameter dominates

Abstract

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.