  The fundamental laws of physics can be derived from the requirement of
invariance under suitable classes of transformations on the one hand, and from
the need for a well-posed mathematical theory on the other hand. As a part of
this programme, the present paper shows under which conditions the introduction
of pseudo-differential boundary operators in one-loop Euclidean quantum gravity
is compatible both with their invariance under infinitesimal diffeomorphisms
and with the requirement of a strongly elliptic theory. Suitable assumptions on
the kernel of the boundary operator make it therefore possible to overcome
problems resulting from the choice of purely local boundary conditions.
