### Abstract

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

### Fingerprint

### Keywords

- Non-commutative
- Borel resummation
- Quantum field theory

### Cite this

*Non-commutativity in polar coordinates*.

**Non-commutativity in polar coordinates.** / Edwards, James.

Research output: Working paper

}

TY - UNPB

T1 - Non-commutativity in polar coordinates

AU - Edwards, James

PY - 2016/7/18

Y1 - 2016/7/18

N2 - We reconsider the fundamental commutation relations for non-commutative$\mathbb{R}^{2}$ described in polar coordinates with non-commutativityparameter $\theta$. Previous analysis found that the natural transition fromCartesian coordinates to polars led to a representation of $\left[\hat{r},\hat{\varphi}\right]$ as an everywhere diverging series. We compute the Borelresummation of this series, showing that it can subsequently be extendedthroughout parameter space and hence provide an interpretation of thiscommutator. Our analysis provides a complete solution for arbitrary $r$ and$\theta$ that reproduces the earlier calculations at lowest order. We compareour results to previous literature in the (pseudo-)commuting limit, finding asurprising spatial dependence for the coordinate commutator when $\theta \ggr^{2}$. We raise some questions for future study in light of this progress.

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

KW - Non-commutative

KW - Borel resummation

KW - Quantum field theory

UR - http://arxiv.org/abs/arXiv:1607.04491

M3 - Working paper

BT - Non-commutativity in polar coordinates

ER -