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 delocalization 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 representations of canonical quantum gravity.