Anyons in 1+1 Dimensions

TL;DR

This paper investigates the theoretical possibility of anyonic excitations with fractional spin and statistics in 1+1 dimensions, proposing a framework that interpolates between bosons and fermions and could aid in deriving propagators for 1D anyons.

Contribution

It introduces a novel approach to model anyons in 1+1 dimensions using boundary conditions and a relativistic ansatz, connecting spin parameters with particle statistics.

Findings

The Hamiltonian's self-adjointness depends on boundary conditions parametrized by γ.

The parameter γ interpolates between bosonic and fermionic behavior.

The approach reproduces the Polyakov spin factor for spinning particles.

Abstract

The possibility of excitations with fractional spin and statististics in $1+1$ dimensions is explored. The configuration space of a two-particle system is the half-line. This makes the Hamiltonian self-adjoint for a family of boundary conditions parametrized by one real number $\gamma$. The limit $\gamma \rightarrow 0, (\infty$) reproduces the propagator of non-relativistic particles whose wavefunctions are even (odd) under particle exchange. A relativistic ansatz is also proposed which reproduces the correct Polyakov spin factor for the spinning particle in $1+1$ dimensions. These checks support validity of the interpretation of $\gamma$ as a parameter related to the ``spin'' that interpolates continuously between bosons ($\gamma =0$) and fermions ($\gamma =\infty$). Our approach can thus be useful for obtaining the propagator for one-dimensional anyons.