A New Spectral Cancellation in Quantum Gravity

TL;DR

This paper investigates a spectral anomaly in Euclidean quantum gravity on the four-ball, showing that certain scalar perturbation sectors are affected by the lack of strong ellipticity, but the zeta(0) value remains well-defined.

Contribution

It introduces a spectral identity that ensures the regularity of the zeta(0) value despite the lack of strong ellipticity in some scalar sectors.

Findings

Scalar perturbation sectors are affected by non-ellipticity

Remaining sectors of scalar perturbations are elliptic

The zeta(0) value remains well-defined on the four-ball

Abstract

A general method exists for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry. In the de Donder gauge, however, the resulting boundary-value problem for the Laplace type operator acting on h is known to be self-adjoint but not strongly elliptic. The present paper shows that, on the Euclidean four-ball, only the scalar part of perturbative modes for quantum gravity is affected by the lack of strong ellipticity. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is confined to the remaining fourth sector. The integral representation of the resulting zeta-function asymptotics on the Euclidean four-ball is also obtained; this remains regular at the origin by virtue of a peculiar spectral identity obtained by the authors. There is therefore encouraging evidence in favour of the zeta(0) value with fully diff-invariant boundary conditions remaining well defined, at least on the four-ball, although severe technical obstructions remain in general.