Initial conditions and unitarity in unimodular quantum cosmology

We consider quantization of the positive curvature Friedmann cosmology in the unimodular modification of Einstein's theory, in which the spacetime four-volume appears as an explicit time variable. The Hamiltonian admits self-adjoint extensions that give unitary evolution in the Hilbert space associated with the Schr\"odinger equation. The semiclassical estimate to the no-boundary wave function of Hartle and Hawking is found. If this estimate is accurate, there is a continuous flux of probability into the configuration space from vanishing three-volume, and the no-boundary wave function evolves nonunitarily. Generalizations of these results hold in a class of anisotropic cosmologies. (Talk given at the 5th Canadian Conference on General Relativity and Relativistic Astrophysics, Waterloo, Ontario, Canada, May 1993.)