Polar Coordinates and Noncommutative Phase Space

TL;DR

This paper explores the Weyl transform in noncommutative phase space, focusing on expressing polar coordinate transformations through explicit orthogonal function expansions to simplify understanding of these complex mathematical concepts.

Contribution

The paper provides a simplified, elementary approach to realizing polar transformations in noncommutative phase space using orthogonal function expansions, building on existing results.

Findings

Explicit orthogonal function expansions for polar transformations

Simplified derivation of noncommutative phase space transformations

Clarification of the Weyl transform's role in noncommutative geometry

Abstract

The so-called Weyl transform is a linear map from a commutative algebra of functions to a noncommutative algebra of linear operators, characterized by an action on Cartesian coordinate functions of the form $(x, y) \mapsto (X, Y)$ such that $XY -YX = i\epsilon I$, i.e. the defining relation for the Heisenberg Lie algebra. Study of this transform has been expansive. We summarize many important results from the literature. The primary goal of this work is to prove the final result: the realization of the polar transformation $(\rho, e^{i\theta}) \mapsto (R, e^{i\Theta})$ in terms of explicit orthogonal function expansions, while starting from elementary principles and utilizing minimal machinery. Our results are not strictly original but their presentation here is intended to simplify introduction to these subjects in a novel manner.