  Anyons and fractional statistics are by now well established in
two-dimensional systems. In one dimension, fractional statistics has been
established so far only through Haldane's fractional exclusion principle, but
not via a fractional phase the wave function acquires as particles are
interchanged. At first sight, the topology of the configuration space appears
to preclude such phases in one dimension. Here we argue that the crossings of
one-dimensional anyons are always unidirectional, which makes it possible to
assign phases consistently and hence to introduce a statistical parameter
theta. The fractional statistics then manifests itself in fractional spacings
of the single-particle momenta of the anyons when periodic boundary conditions
are imposed. These spacings are given by Delta p = 2 pi hbar/L
(|theta|/pi+non-negative integer) for a system of length L. This condition is
the analogue of the quantisation of relative angular momenta according to
l_z=hbar(-theta/pi+2integer) for two-dimensional anyons.
