Boundary Operators in Quantum Field Theory

TL;DR

This paper investigates the conditions under which pseudo-differential boundary operators can be used in one-loop Euclidean quantum gravity, ensuring invariance and mathematical well-posedness, thus broadening boundary condition options.

Contribution

It establishes compatibility conditions for pseudo-differential boundary operators with invariance and ellipticity in quantum gravity, expanding the theoretical framework.

Findings

Compatible boundary operators can be constructed under specific kernel assumptions.

Overcoming local boundary condition limitations enhances mathematical consistency.

Framework supports broader boundary condition choices in quantum gravity.

Abstract

The fundamental laws of physics can be derived from the requirement of invariance under suitable classes of transformations on the one hand, and from the need for a well-posed mathematical theory on the other hand. As a part of this programme, the present paper shows under which conditions the introduction of pseudo-differential boundary operators in one-loop Euclidean quantum gravity is compatible both with their invariance under infinitesimal diffeomorphisms and with the requirement of a strongly elliptic theory. Suitable assumptions on the kernel of the boundary operator make it therefore possible to overcome problems resulting from the choice of purely local boundary conditions.