General noncommuting curvilinear coordinates and fluid Mechanics

TL;DR

This paper explores how noncommutative geometry arises in charged particles constrained to the lowest Landau level and connects it to fluid mechanics through gauge fields and coordinate transformations.

Contribution

It introduces a general framework for noncommuting curvilinear coordinates and links gauge fields in this setting to fluid mechanics, including the Seiberg-Witten map as a quantum fluid correspondence.

Findings

Noncommutativity between coordinate operators in Landau levels.

Unified treatment of Cartesian, cylindrical, spherical coordinates.

Connection between gauge fields in noncommutative space and fluid dynamics.

Abstract

We show that restricting the states of a charged particle to the lowest Landau level introduces noncommutativity between general curvilinear coordinate operators. The cartesian, circular cylindrical and spherical polar coordinates are three special cases of our quite general method. The connection between U(1) gauge fields defined on a general noncommuting curvilinear coordinates and fluid mechanics is explained. We also recognize the Seiberg-Witten map from general noncommuting to commuting variables as the quantum correspondence of the Lagrange to Euler map in fluid mechanics.