One-loop quantum cosmology: ζ-Function technique for the Hartle-Hawking wave function of the universe

Abstract

We develop the one-loop calculational technique for the cosmological wave function in the no-boundary proposal of Hartle and Hawking. This proposal is reformulated as a path integral over true physical variables in the unitary quantum gravity theory analytically continued from the Lorentzian spacetime to the compact Euclidean one. Then the one-loop calculation of this integral and its covariant renormalization reduce to the construction of the generalized ζ-functions for differential operators on the ball-like Euclidean manifold. For ζ-functions of such operators with the explicitly unknown spectra we work out a special calculational algorithm based only on the knowledge of their basis functions and discuss in much detail their general properties in field theories as well as in quantum mechanical problems. This general technique is applied to the calculation of the one-loop graviton wave function of the DeSitter universe. Together with its anomalous scaling behaviour, which generalizes the old flat-space results to the case of the DeSitter spacetime, we also find the finite renormalized part of the one-loop graviton contribution and analyze its asymptotic behaviour at small Euclidean and large Lorentzian times.