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

The loop representation formulation of nonrelativistic 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 a unitary transformation that allows us to remove the degrees of freedom associated with the paths. The existence of this transformation, which allows us to make contact with the anyonic interpretation of the model, is subjected to the fact that the charge of the particles is quantized. On the other hand, in the Maxwell case, we find that charge quantization is necessary in order for the geometric representation to be consistent.