## 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.

$\mathbb{R}^{2}$ 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.

Original language | English |
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Publication status | E-pub ahead of print - 18 Jul 2016 |

## Keywords

- Non-commutative
- Borel resummation
- Quantum field theory