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 $θ$ acquired by the anyons gives rise to fractional shifts in the spacings of the relative momenta, ${Δp =2π\hbar/L\, (|θ|/π+n)}$. The fractional shift $θ/π$ is a good quantum number of interacting anyons, even though the single particle momenta, and hence the non-negative integers $n$, are generally not.

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