Lorenz Gauge in Quantum Cosmology

TL;DR

This paper develops a Green-function method to analyze the Lorenz gauge in quantum cosmology, providing a calculational scheme for one-loop semiclassical wave function evaluation with gauge fields on bounded manifolds.

Contribution

It introduces a novel approach to handle coupled eigenvalue equations in Euclidean Maxwell theory within quantum cosmology, addressing gauge independence on manifolds with boundary.

Findings

Derived an integro-differential equation for longitudinal modes.

Established a complete calculational scheme for one-loop wave function evaluation.

Enhanced understanding of gauge independence in quantum cosmology.

Abstract

In a path-integral approach to quantum cosmology, the Lorenz gauge-averaging term is studied for Euclidean Maxwell theory on a portion of flat four-space bounded by two concentric three-spheres, but with arbitrary values of the gauge parameter. The resulting set of eigenvalue equations for normal and longitudinal modes of the electromagnetic potential cannot be decoupled, and is here studied with a Green-function method. This means that an equivalent equation for longitudinal modes is obtained which has integro-differential nature, after inverting a differential operator in the original coupled system. A complete calculational scheme is therefore obtained for the one-loop semiclassical evaluation of the wave function of the universe in the presence of gauge fields. This might also lead to a better understanding of how gauge independence is actually achieved on manifolds with boundary, whose consideration cannot be avoided in a quantum theory of the universe.