Interacting particles and strings in path and surface representations

Abstract

Non-relativistic charged particles and strings coupled with abelian gauge fields are quantized in a geometric representation that generalizes the Loop Representation. We consider three models:the string in self-interaction through a Kalb-Ramond field in four dimensions, the topological interaction of two particles due to a BF term in 2+1 dimensions, and the string-particle interaction mediated by a BF term in 3 + 1 dimensions. In the first case one finds that a consistent "surfacerepresentation" can be built provided that the coupling constant is quantized. The geometrical setting that arises corresponds to a generalized version of the Faraday's lines picture: quantum states are labeled by the shape of the string, from which emanate "Faraday's surfaces". In the other models, the topological interaction can also be described by geometrical means. It is shown that the open-path (or open-surface) dependence carried by the wave functional in these models can be eliminated through an unitary transformation, except by a remaining dependence on the boundary of the path (or surface). These feature is closely related to the presence of anomalous statistics in the 2 + 1 model, and to a generalized "anyonic behavior" of the string in the other case.

Geometric representation of interacting nonrelativistic open strings using extended objects

“…Given an open path γ starting at z i and ending at (37) is solved if γ is attached to the endpoints of the string. If e is positive the path should start at z i and end at z f , which is consistent with the geometrical representation of non-relativistic particles in electromagnetic interaction [7], and the constraint gives e = 1. In a more general sense γ may consist of a bundle of open paths starting at z i and ending at z f .…”

“…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…”

“…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] …”

“…In the next section we consider the model of a closed string in self-interaction by means of an Abelian Kalb-Ramond field [1,[7][8][9]; the more interesting open string model is treated in an independent section. Final remarks and some discussions are made at the end of the paper.…”

“…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.…”