Boundary Operators in Euclidean Quantum Gravity

TL;DR

This paper investigates gauge-invariant boundary conditions in Euclidean quantum gravity, analyzing boundary operators, elliptic properties, and 1-loop divergences, with implications for boundary value problems and quantum corrections.

Contribution

It introduces a detailed analysis of boundary operators with projection and nilpotent components, and evaluates 1-loop divergences in specific gauges, advancing understanding of boundary conditions in quantum gravity.

Findings

Boundary operators involve projection and nilpotent operators.

The elliptic operator on metric perturbations is symmetric.

1-loop divergence in axial gauge matches that of transverse-traceless perturbations.

Abstract

Gauge-invariant boundary conditions in Euclidean quantum gravity can be obtained by setting to zero at the boundary the spatial components of metric perturbations, and a suitable class of gauge-averaging functionals. This paper shows that, on choosing the de Donder functional, the resulting boundary operator involves projection operators jointly with a nilpotent operator. Moreover, the elliptic operator acting on metric perturbations is symmetric. Other choices of mixed boundary conditions, for which the normal components of metric perturbations can be set to zero at the boundary, are then analyzed in detail. Last, the evaluation of the 1-loop divergence in the axial gauge for gravity is obtained. Interestingly, such a divergence turns out to coincide with the one resulting from transverse-traceless perturbations.