Lack of strong ellipticity in Euclidean quantum gravity

TL;DR

This paper demonstrates that the boundary value problem in Euclidean quantum gravity with certain gauge-invariant boundary conditions is not strongly elliptic, raising important interpretative issues for the theory on manifolds with boundary.

Contribution

It proves that the boundary value problem in Euclidean quantum gravity with invariant boundary conditions and de Donder gauge is not strongly elliptic, highlighting foundational issues.

Findings

Boundary value problem is not strongly elliptic.

Invariance under diffeomorphisms leads to mixed derivatives in boundary conditions.

Raises interpretative challenges for Euclidean quantum gravity with boundaries.

Abstract

Recent work in Euclidean quantum gravity has studied boundary conditions which are completely invariant under infinitesimal diffeomorphisms on metric perturbations. On using the de Donder gauge-averaging functional, this scheme leads to both normal and tangential derivatives in the boundary conditions. In the present paper, it is proved that the corresponding boundary value problem fails to be strongly elliptic. The result raises deep interpretative issues for Euclidean quantum gravity on manifolds with boundary.