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

TL;DR

This paper introduces a diffeomorphism-invariant cut-off regularization method for the determinant of a scalar particle in a Euclidean gravitational field, ensuring coordinate invariance and incorporating wavelet localization.

Contribution

It presents a novel regularization technique that maintains diffeomorphism invariance and employs wavelet localization, applicable to scalar fields in gravitational and Yang-Mills backgrounds.

Findings

Regularization preserves diffeomorphism invariance.

Wavelet localization enables automatic renormalization group structure.

Method extends to Yang-Mills fields.

Abstract

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.