Anyons in One Dimension

I give a non-technical account of fractional statistics in one dimension. In systems with periodic boundary conditions, the crossing of anyons is always uni-directional, and the fractional phase $\theta$ acquired by the anyons gives rise to fractional shifts in the spacings of the relative momenta, ${\Delta p =2\pi\hbar/L\, (|\theta|/\pi+n)}$. The fractional shift $\theta/\pi$ is a good quantum number of interacting anyons, even though the single particle momenta, and hence the non-negative integers $n$, are generally not.