Interacting Particles and Strings in Path and Surface Representations

TL;DR

This paper explores geometric representations of charged particles and strings coupled with abelian gauge fields, revealing quantization conditions, topological interactions, and generalized anyonic behaviors in various dimensions.

Contribution

It introduces a generalized surface-representation for string interactions, demonstrating quantization of coupling constants and geometric descriptions of topological effects.

Findings

Surface-representation requires quantized coupling constants

Wave functionals depend on boundary, not entire path or surface

Topological interactions lead to generalized anyonic statistics

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 "surface-representation" 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.