Quantum gravity and turning points in the semiclassical approximation

The wave functional in quantum gravity gives an amplitude for three-geometries and matter fields. The four-space is usually recovered in a semiclassical approximation where the gravity variables are taken to oscillate rapidly compared to matter variables; this recovers the Schrödinger evolution for the matter. We examine turning points in the gravity variables where this approximation appears to be troublesome. We investigate the effect of such a turning point on the matter wave function, in simple quantum-mechanical models and in a closed minisuperspace cosmology. We find that after evolving sufficiently far from the turning point the matter wave function recovers to a form close to that predicted by the semiclassical approximation, and we compute the leading correction (from "back reaction") in a simple model. We also show how turning points can appear in the gravitational sector in dilation gravity. We give some remarks on the behavior of the wave functional in the vicinity of turning points in the context of dilaton gravity black holes.