Anyons in 1 + 1 dimensions

The possibility of excitations with fractional spin and statistics 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 → 0 (∞) 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 the validity of the interpretation of γ as a parameter related to the “spin” that interpolates continuously between bosons (γ = 0) and fermions (γ = ∞). Our approach can thus be useful for obtaining the propagator for one-dimensional anyons.