Non-local Properties in Euclidean Quantum Gravity

In the one-loop approximation for Euclidean quantum gravity, the boundary conditions which are completely invariant under gauge transformations of metric perturbations cannot be written in terms of complementary projection operators. By contrast, they express the h00 and h0i perturbations at the boundary as integrals at the boundary of the action of a set of di erential operators on metric perturbations. Hence they are non-local. The corresponding trace anomaly for pure gravity has been recently evaluated by means of analytic techniques. It now remains to compute the contribution of all perturbative modes of gauge elds and gravitation to the one-loop e ective action for problems with boundaries. The functional determinant has a non-local nature, independently of boundary conditions. Moreover, the analysis of trace anomalies for pure gravity and supergravity with non-local boundary conditions has not yet been completed and is still under active investigation. To appear in Proceedings of the Third Workshop on Quantum Field Theory under the In uence of External Conditions, Leipzig, September 1995.