Anyons in 1+1 Dimensions

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 $γ$. The limit $γ\rightarrow 0, (\infty$) 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 $γ$ as a parameter related to the ``spin'' that interpolates continuously between bosons ($γ=0$) and fermions ($γ=\infty$). Our approach can thus be useful for obtaining the propagator for one-dimensional anyons.