Non-Local Boundary Conditions in Euclidean Quantum Gravity

TL;DR

This paper introduces non-local boundary conditions for Euclidean quantum gravity, exploring their mathematical formulation, self-adjointness, and implications for gravitational perturbations on the Euclidean four-ball.

Contribution

It formulates a general class of non-local boundary conditions for gravitational perturbations and analyzes their properties, including explicit solutions for specific cases.

Findings

Explicit boundary-value problem solutions for certain non-local operators

Conditions for self-adjointness of the boundary-value problem

Discussion on the existence of non-local symmetries in Euclidean quantum gravity

Abstract

Non-local boundary conditions for Euclidean quantum gravity are proposed, consisting of an integro-differential boundary operator acting on metric perturbations. In this case, the operator P on metric perturbations is of Laplace type, subject to non-local boundary conditions; by contrast, its adjoint is the sum of a Laplacian and of a singular Green operator, subject to local boundary conditions. Self-adjointness of the boundary-value problem is correctly formulated by looking at Dirichlet-type and Neumann-type realizations of the operator P, following recent results in the literature. The set of non-local boundary conditions for perturbative modes of the gravitational field is written in general form on the Euclidean four-ball. For a particular choice of the non-local boundary operator, explicit formulae for the boundary-value problem are obtained in terms of a finite number of unknown functions, but subject to some consistency conditions. Among the related issues, the problem arises of whether non-local symmetries exist in Euclidean quantum gravity.