Foundational Problems in Quantum Gravity

TL;DR

This paper investigates boundary conditions in quantum gravity and gauge theories, establishing conditions for strong ellipticity and discussing implications for Euclidean quantum gravity.

Contribution

It develops a general scheme for boundary conditions invariant under diffeomorphisms and proves a criterion for strong ellipticity in gauge theories on manifolds with boundary.

Findings

Established a general scheme for boundary conditions in gauge theories.

Proved a criterion for strong ellipticity of boundary value problems.

Discussed implications for Euclidean quantum gravity.

Abstract

Boundary conditions play a crucial role in the path-integral approach to quantum gravity and quantum cosmology, as well as in the current attempts to understand the one-loop semiclassical properties of quantum field theories. Within this framework, one is led to consider boundary conditions completely invariant under infinitesimal diffeomorphisms on metric perturbations. These are part of a general scheme, which can be developed for Maxwell theory, Yang-Mills Theory, Rarita-Schwinger fields and any other gauge theory. A general condition for strong ellipticity of the resulting field theory on manifolds with boundary is here proved, following recent work by the authors. The relevance for Euclidean quantum gravity is eventually discussed.