A Theory of Quantum Gravity from First Principles

TL;DR

This paper develops a first-principles approach to quantum gravity by exploring boundary conditions for gauge fields, allowing pseudo-differential operators to ensure strong ellipticity and gauge invariance.

Contribution

It introduces a novel framework for one-loop quantum gravity based on boundary conditions involving pseudo-differential operators, ensuring mathematical consistency and physical invariance.

Findings

Established conditions for pseudo-differential boundary operators

Developed a strongly elliptic boundary-value problem for quantum gravity

Provided a mathematically rigorous foundation for gauge invariance in quantum gravity

Abstract

When quantum fields are studied on manifolds with boundary, the corresponding one-loop quantum theory for bosonic gauge fields with linear covariant gauges needs the assignment of suitable boundary conditions for elliptic differential operators of Laplace type. There are however deep reasons to modify such a scheme and allow for pseudo-differential boundary-value problems. When the boundary operator is allowed to be pseudo-differential while remaining a projector, the conditions on its kernel leading to strong ellipticity of the boundary-value problem are studied in detail. This makes it possible to develop a theory of one-loop quantum gravity from first principles only, i.e. the physical principle of invariance under infinitesimal diffeomorphisms and the mathematical requirement of a strongly elliptic theory.