  As a prototypical massive field theory we study the scalar field on the
recently introduced Finsler spacetimes. We show that particle excitations exist
that propagate faster than the speed of light recognized as the boundary
velocity of observers. This effect appears already in Finsler spacetime
geometries with very small departures from Lorentzian metric geometry. It
switches on for a sufficiently large ratio of the particle four-momentum and
mass, and is the consequence of a modified version of the Coleman-Glashow
velocity dispersion relation. The momentum dispersion relation on Finsler
spacetimes is shown to be the same as on metric spacetimes, which differs from
many quantum gravity models. If similar relations resulted for fermions on
Finsler spacetimes, these generalized geometries could explain the potential
observation of superluminal neutrinos claimed by the Opera Collaboration.
