Loop representation of charged particles interacting with Maxwell and Chern-Simons fields

TL;DR

This paper explores the loop-space formulation of non-relativistic charged particles interacting with Maxwell and Chern-Simons fields, highlighting similarities, differences, and the role of charge quantization in each case.

Contribution

It demonstrates that Maxwell and Chern-Simons models share structural similarities in loop representation, with charge quantization playing a key role in their geometric formulations.

Findings

Loop representations of Maxwell and Chern-Simons models are structurally similar.

Charge quantization is necessary for the consistency of the geometric representation.

A unitary transformation in the Chern-Simons case links to anyonic interpretations.

Abstract

The loop representation formulation of non-relativistic particles coupled with abelian gauge fields is studied. Both Maxwell and Chern-Simons interactions are separately considered. It is found that the loop-space formulations of these models share significant similarities, although in the Chern-Simons case there exists an unitary transformation that allows to remove the degrees of freedom associated with the paths. The existence of this transformation, which allows to make contact with the anyonic interpretation of the model, is subjected to the fact that the charge of the particles be quantized. On the other hand, in the Maxwell case, we find that charge quantization is necessary in order to the geometric representation be consistent.