Euclidean Quantum Gravity

Abstract

This chapter studies the linearized gravitational field in the presence of boundaries. For this purpose, zeta-function regularization is used to perform the mode-by-mode evaluation of Faddeev-Popov amplitudes in the case of flat Euclidean four-space bounded by two concentric three-spheres, or just one three-sphere. On choosing the de Donder gauge-averaging term, the resulting ζ(0) value is found to agree with the space-time covariant calculation of the same amplitudes, which relies on the recently corrected geometric formulae for the asymptotic heat kernel in the case of mixed boundary conditions. Two sets of mixed boundary conditions for Euclidean quantum gravity are then compared in detail. The analysis proves that one cannot restrict the path-integral measure to transverse-traceless perturbations. By contrast, gauge-invariant amplitudes are only obtained on considering from the beginning all perturbative modes of the gravitational field, jointly with ghost modes. Unlike the mixed boundary conditions involving (complementary) projectors, one knows from chapter six that boundary conditions completely invariant under infinitesimal diffeomorphisms involve both normal and tangential derivatives of metric perturbations. The corresponding ζ(0) value is obtained, and the proof of symmetry of the Laplace operator in such a case is obtained. Mixed boundary conditions are also considered which lead to Robin conditions on spatial metric perturbations, and Dirichlet conditions on normal metric perturbations. Last, a review of Hawking’s proposal to consider smooth simply connected four-manifolds as the building blocks of Euclidean quantum gravity is presented. This makes it necessary to study physical processes in S ² x S ², K3 and CP ² geometries. Yet another open problem is a consistent formulation of quantum supergravity on manifolds with boundary.