Quantum gravity and turning points in the semiclassical approximation.

The wavefunctional in quantum gravity gives an amplitude for 3-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 wavefunction, in simple quantum mechanical models and in a closed minisuperspace cosmology. We find that after evolving sufficiently far from the turning point the matter wavefunction recovers to a form close to that predicted by the semiclassical approximation, and we compute the leading correction (from ‘backreaction’) in a simple model. We also show how turning points can appear in the gravitational sector in dilaton gravity. We give some remarks on the behavior of the wavefunctional in the vicinity of turning points in the context of dilaton gravity black holes. ∗ This work was supported in part by funds provided by the U.S. Department of Energy under cooperative agreement DE-FC02-94ER40818 and by a grant DE-FG03-92-ER40701.