  We consider in detail the analytic behaviour of the non-interacting massless
scalar field two-point function in H.S. Snyder's discretized non-commuting
spacetime. The propagator we find is purely real on the Euclidean side of the
complex $p^2$ plane and goes like $1/p^2$ as $p^2\to 0$ from either the
Euclidean or Minkowski side. The real part of the propagator goes smoothly to
zero as $p^2$ increases to the discretization scale $1/a^2$ and remains zero
for $p^2>1/a^2$. This behaviour is consistent with the termination of
single-particle propagation on the ultraviolet side of the discretization
scale. The imaginary part of the propagator, consistent with a
multiparticle-production branch discontinuity, is finite and continuous on the
Minkowski side, slowly falling to zero when $1/a^2<p^2<\infty$. Finally, we
argue that the spectral function for the multiparticle states appears to
saturate as $p^2$ probes just beyond the $1/a^2$ discretization scale. We
speculate on the cosmological consequences of such a spectral function.
