  A recently-developed theory of quantum general relativity provides a
propagator for free-falling particles in curved spacetimes. These propagators
are constructed by parallel-transporting quantum states within a quantum bundle
associated to the Poincare frame bundle. We consider such parallel transport in
the case that the spacetime is a classical Robertson-Walker universe. An
explicit integral formula is developed which expresses the propagators for
parallel transport between any two points of such a spacetime. The integrals in
this formula are evaluated in closed form for a particular spatially-flat
model.
