INSTANTONS AND UNITARITY IN QUANTUM COSMOLOGY WITH FIXED 4-VOLUME - ART. NO. 084008

We find a number of complex solutions of the source-free Einstein equa tions in the so-called unimodular version of general relativity, and w e interpret them as saddle points yielding estimates of a gravitationa l path integral taken over a space of almost everywhere Lorentzian met rics on a spacetime manifold with a topology of the ''no-boundary'' ty pe. Within this interpretation, we address the compatibility of the no -boundary initial condition with the definability of the quantum measu re. which reduces in this setting to the normalizability and unitary e volution of the no-boundary wave function psi. We consider three space time topologies, R-4, RP4#R-4, and R-2 x T-2. (The corresponding trunc ated 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 top ologies we investigate within a Taub minisuperspace model with a spati al 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 ex ists exactly one complex solution of the classical Einstein equations (or combination of solutions) that, to the accuracy of our saddle poin t 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 tradi tional Einstein gravity. In the Bianchi type I case, moreover, the dis tinguished complex solution is approximately real and Lorentzian at la te times, and appears to describe an explosive expansion from zero siz e at T = 0. In this connection, we speculate that a fully normalizable psi can result only from the imposition of an explicit short distance cutoff. (In the Taub cases, in contrast, the only complex solution wi th nearly Lorentzian late-time behavior yields a wave function that is normalizable but evolves nonunitarily, with the total probability inc reasing exponentially in the unimodular ''time'' in a manner that sugg ests 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. [S0556-2821(98)08418-5].