  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 two-by-two
matrix of pseudo-differential operators with pseudo-homogeneous kernels. The
request 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 request 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.
