On the relation between causality and topology in the semiclassical universe

Publisher Website
Google Scholar
Abstract
We show that the in-in path-integrel formalism is very effective for systematic, straightforward investigations of semiclassical gravity, in contrast with the traditional formalism, which takes the Wheeler-DeWitt equation as a starting point. In particular, we show that we can derive directly from the in-in formalism a series of validity conditions for the semiclassical treatment of quantum gravity, by investigating the second variation of the path integral. We begin with the investigation of the fundamental setting of quantun gravity. We show that, in order to obtain the causal semiclassical Einstein equation at the late stage of the Universe, we have to regard the in-in path-integral formalism as fundamental in quantum gravity. We then perform the stationary phase approximation for the gravity mode. From the first variation of the phase, we obtain the semiclassical Einstein equation. From the second variation, we obtain a series of validity conditions of semiclassical gravity in a completely general manner. We show that one of the conditions is that the dispersion in the energy-momentum tensor should be negligible. This condition has been inferred by several authors so far but only on the basis of special models. We show that it is a completely general condition required for the consistency of the semiclassical approximation of quantum gravity. Other conditions have been overlooked so far, since they are not easy to infer in the traditional formalism. As an application of the formalism, we propose a natural formulation of quantum cosmology in terms of the in-in path integral, and examine its consequences. By investigating the stationary phase configurations, we find out that the semiclassical universe should be the one which admits at least one totally geodesic spatial surfce. Assuming a natural energy condition, this means that the Universe should be a Wheeler universe, i.e., the spacetime which begins from and ends in a singularity. Furthermore, this also means that the possible topology of the Universe is very strongly restricted. In this way, we realize the connection between causality and topology in the semiclassical universe.