Noncommutative Planar Particle Dynamics with Gauge Interactions

TL;DR

This paper explores two methods of introducing gauge interactions into noncommutative planar particle models, revealing their relation via a classical Seiberg-Witten map and analyzing implications for two-body problems and quantum Hall systems.

Contribution

It compares standard and generalized gauge theories in noncommutative planar particle models, showing their connection through a classical Seiberg-Witten map and phase space transformations.

Findings

Both gauge approaches are related by a classical Seiberg-Witten map.

The generalized gauge theory involves time-dependent area-preserving transformations.

Application to two-body problems and fractional quantum Hall effect contexts.

Abstract

We consider two ways of introducing minimal Abelian gauge interactions into the model presented in [1]. They are different only if the second central charge of the planar Galilei group is nonzero. One way leads to standard gauge transformations and second to a generalized gauge theory with gauge transformations accompanied by time-dependent area-preserving coordinate transformations. Both approaches, however, are related to each other by a classical Seiberg-Witten map supplemented by the noncanonical transformation of the phase space variables for planar particles. We also formulate the two-body problem in the model with a generalized gauge symmetry and consider the case with both CS and background electromagnetic fields, as it is used in the description of fractional quantum Hall effect.