New kernels in quantum gravity

Recent work in the literature has proposed the use of non-local boundary conditions in Euclidean quantum gravity. The present paper studies first a more general form of such a scheme for bosonic gauge theories, by adding to the boundary operator for mixed boundary conditions of local nature a 22 matrix of pseudo-differential operators with pseudo-homogeneous kernels. The requirement of invariance of such boundary conditions under infinitesimal gauge transformations leads to non-local boundary conditions on ghost fields. In Euclidean quantum gravity, an alternative scheme is proposed, where non-local boundary conditions and the requirement of their complete gauge invariance are sufficient to lead to gauge-field and ghost operators of pseudo-differential nature. The resulting boundary conditions have a Dirichlet and a pseudo-differential sector, and are pure Dirichlet for the ghost. This approach is eventually extended to Euclidean Maxwell theory