Journal Article

Non-commutativity in polar coordinates

Edwards, James P.

2017

Overview

We reconsider the fundamental commutation relations for non-commutative R 2 described in polar coordinates with non-commutativity parameter θ . Previous analysis found that the natural transition from Cartesian coordinates to the traditional polar system led to a representation of [ 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 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 θ ≫ r 2 . Finally, we raise some questions for future study in light of this progress.

• Publisher: Springer Berlin Heidelberg,Springer,Springer Nature B.V,SpringerOpen
• Subject: Astronomy / Astrophysics and Cosmology / Black Hole / Cartesian coordinates / Commutation / Elementary Particles / Fuzzy Sphere / Hadrons / Heavy Ions / Letter / Mathematical analysis / Measurement Science and Instrumentation / Minimal Length Scale / Nuclear Energy / Nuclear Physics / Physics / Physics and Astronomy / Polar coordinates / Quantum Field Theories / Quantum Field Theory / String Theory / Undetermined Constant / Weyl Symbol