Local Quantum Fields for Anyons on the Circle leading to Non-Relativistic Anyons in Two Dimensions

TL;DR

This paper constructs a local quantum field theory for Anyons on the circle using Bogoliubov transformations, and extends it to non-relativistic Anyons in two dimensions, revealing their braid group statistics and localization properties.

Contribution

It introduces a method to explicitly define local covariant fields for Anyons on the circle and extends this framework to non-relativistic two-dimensional Anyons.

Findings

Explicit construction of local covariant Anyon fields on the circle

Demonstration of braid group statistics depending on winding number

Extension to non-relativistic two-dimensional Anyons

Abstract

Using the method of implementable one-particle Bogoliubov transformations it is possible to explicitly define a local covariant net of quantum fields on the (universal covering of the) circle $S_1$ with braid group statistics. These Anyon fields transform under a representation of $\widetilde{U(1)}$ for arbitrary real-valued spin and their commutation relations depend on the relative winding number of localization regions. By taking the tensor product with a local covariant field theory on $\mathbb{R}^2$ one can obtain a non-relativistic cone-localized field net for Anyons in two dimensions.