  A quantum mechanical description of particle propagation on the discrete
spacetime of a causal set is presented. The model involves a discrete path
integral in which trajectories within the causal set are summed over to obtain
a particle propagator. The sum-over-trajectories is achieved by a matrix
geometric series. For causal sets generated by sprinkling points into 1+1 and
3+1 dimensional Minkowski spacetime the propagator calculated on the causal set
is shown to agree, in a suitable sense, with the causal retarded propagator for
the Klein-Gordon equation. The particle propagator described here is a step
towards quantum field theory on causal set spacetime.
