  Non-local boundary conditions for Euclidean quantum gravity are proposed,
consisting of an integro-differential boundary operator acting on metric
perturbations. In this case, the operator P on metric perturbations is of
Laplace type, subject to non-local boundary conditions; by contrast, its
adjoint is the sum of a Laplacian and of a singular Green operator, subject to
local boundary conditions. Self-adjointness of the boundary-value problem is
correctly formulated by looking at Dirichlet-type and Neumann-type realizations
of the operator P, following recent results in the literature. The set of
non-local boundary conditions for perturbative modes of the gravitational field
is written in general form on the Euclidean four-ball. For a particular choice
of the non-local boundary operator, explicit formulae for the boundary-value
problem are obtained in terms of a finite number of unknown functions, but
subject to some consistency conditions. Among the related issues, the problem
arises of whether non-local symmetries exist in Euclidean quantum gravity.
