Non-commutativity in polar coordinates

Abstract

Abstract We reconsider the fundamental commutation relations for non-commutative $$\mathbb {R}^{2}$$ R 2 described in polar coordinates with non-commutativity parameter $$\theta $$ θ . Previous analysis found that the natural transition from Cartesian coordinates to the traditional polar system led to a representation of $$[\hat{r}, \hat{\varphi }]$$ [ r ^ , φ ^ ] as an everywhere diverging series. In this article 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 $$\theta $$ θ that reproduces the earlier calculations at lowest order and benefits from being generally applicable to problems in a two-dimensional non-commutative space. We compare our results to previous literature in the (pseudo-)commuting limit, finding a surprising spatial dependence for the coordinate commutator when $$\theta \gg r^{2}$$ θ ≫ r 2 . Finally, we raise some questions for future study in light of this progress.

Keywords

• Black Hole
• Fuzzy Sphere
• Undetermined Constant
• Minimal Length Scale
• Weyl Symbol