Non-commutativity in polar coordinates

James Edwards

Research output: Working paper

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.
Original languageEnglish
Publication statusE-pub ahead of print - 18 Jul 2016

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polar coordinates
commutators
Cartesian coordinates
commutation

Keywords

  • Non-commutative
  • Borel resummation
  • Quantum field theory

Cite this

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

2016.

Research output: Working paper

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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.

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