A Diffeomorphism - Invariant Cut - off Regularization of the Determinant of a Scalar Particle in a Euclidean Gravitational Field

Continuing the thrust of our recent work, but with an important new idea, we find a cutoff 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. 2 1 Introduction.