Topology, decoherence, and semiclassical gravity.

We address the issue of recovering the time-dependent Schrödinger equation from quantum gravity in a natural way. To reach this aim it is necessary to understand the nonoccurrence of certain superpositions in quantum gravity. We explore various possible explanations and their relation. These are the delocalisation of interference terms through interaction with irrelevant degrees of freedom (decoherence), gravitational anomalies, and the possibility of θ states. The discussion is carried out in both the geometrodynamical and connection representation of canonical quantum gravity.