Quantum Field Theory from First Principles

TL;DR

This paper develops a first-principles approach to quantum field theory on manifolds with boundary, emphasizing the role of pseudo-differential boundary conditions to ensure strong ellipticity, which is crucial for consistent quantum gravity formulations.

Contribution

It introduces a novel framework allowing pseudo-differential boundary conditions in quantum field theory, enabling a consistent first-principles approach to quantum gravity.

Findings

Pseudo-differential boundary operators can be projectors with strong ellipticity.

The approach supports a non-local formulation of quantum field theory.

This framework facilitates a first-principles derivation of 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. It therefore seems that a non-local formulation of quantum field theory has some attractive features which deserve further investigation.