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

Abstract

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.

Geometric representation of interacting nonrelativistic open strings using extended objects

“…where T i ( x, γ) and and T ij ( x, Σ) are the form factors that describes the open paths γ and open surfaces Σ, respectively [2,6,7]. We can see, using ∂ j T ji ( x, Σ) = −T i ( x, ∂Σ) and…”

“…We say in this case that the surface "emanates" or "starts" from the string, in analogy with the theory of self-interacting non-relativistic particles coupled through a Maxwell field [6,8]. It could happen instead, that the boundary of the surface and the string have opposite orientations; in that case the constraint would be satisfied if φ = −1, and we say that the surface "enters" or "arrives" at the string position.…”

“…Our discussion starts reviewing the case of a closed non-relativistic string in selfinteraction [7,9], which is described by an action that generalizes the theory of the self-interacting point particle [6] …”

“…Stückelberg gauge invariant version of the Proca model [6,8]. The vector field A µ has dimensions of mass, as in Maxwell theory, and mediates the interaction between the ends of the string.…”

“…A "surface representation" was considered years ago to study the free-field case [3,4], but it has to be adapted to include the particularities that the coupling with the string requires. This study can be seen as a generalization of the theory of charged non-relativistic point particles in electromagnetic interaction, quantized within the LR , for which it was found that electric charge must be quantized in order to the LR formulation be consistent [5][6][7][8][9][10]. As we shall see, both for the open and closed strings models coupled with the Kalb-Ramond field the "charge" of the string must be also quantized, if the geometric representation adapted to the model is going to be consistent.…”