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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 involve both normal and tangential derivatives of the metric perturbations $h_{00}$ and $h_{0i}$, while the $h_{ij}$ perturbations and the whole ghost one-form are set to zero at the boundary. The corresponding one-loop divergency 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 fields and gravitation to the one-loop effective action for problems with boundaries. The functional determinant has a non-local nature, independently of boundary conditions. Moreover, the analysis of one-loop divergences for supergravity with non-local boundary conditions has not yet been completed and is still under active investigation.