  Continuing the thrust of our recent work, but with an important new idea, we
find a cut-off regularization of the determinant of a scalar particle in a
classical Euclidean gravitational field. The field is assumed asymptotically
flat, and the regularization is diffeomorphism-invariant under coordinate
changes that are the identity at infinity. The scalar field is expanded in term
of variables that depend on the gravitational field, with wavelet localization
of each variable. A renormalization group structure is thus automatically
present. A similar construction is carried out for the determinant of a scalar
field in a background Yang-Mills field.
