Instantons and unitarity in quantum cosmology with fixed four-volume

We find a number of complex solutions of the source-free Einstein equations in the so-called unimodular version of general relativity, and we interpret them as saddle points yielding estimates of a gravitational path integral taken over a space of almost everywhere Lorentzian metrics on a spacetime manifold with a topology of the “no-boundary” type. Within this interpretation, we address the compatibility of the no-boundary initial condition with the definability of the quantum measure, which reduces in this setting to the normalizability and unitary evolution of the no-boundary wave function ψ. We consider three spacetime topologies, $R^4,$ ${\mathrm{RP}}^4\#R^4,$ and $R^2\times T^2.$ (The corresponding truncated manifolds with boundary are respectively the closed 4-dimensional disk or ball, the closed 4-dimensional cross cap, and the product of the two-torus with the closed two-dimensional disk.) The first two topologies we investigate within a Taub minisuperspace model with a spatial topology $S^3,$ and the third within a Bianchi type I minisuperspace model with a spatial topology $T^3.$ In each of the three cases there exists exactly one complex solution of the classical Einstein equations (or combination of solutions) that, to the accuracy of our saddle point estimate, yields a wave function compatible with normalizability and unitary evolution. The existence of such solutions tends to bear out the suggestion that the unimodular theory is less divergent than traditional Einstein gravity. In the Bianchi type I case, moreover, the distinguished complex solution is approximately real and Lorentzian at late times, and appears to describe an explosive expansion from zero size at $T=0.\mathrm{}$ In this connection, we speculate that a fully normalizable ψ can result only from the imposition of an explicit short distance cutoff. (In the Taub cases, in contrast, the only complex solution with nearly Lorentzian late-time behavior yields a wave function that is normalizable but evolves nonunitarily, with the total probability increasing exponentially in the unimodular “time” in a manner that suggests a continuous creation of new universes at zero volume.) The issue of the stability of these results upon the inclusion of more degrees of freedom is raised.