Extracting the Maxwell charge from the Wheeler-DeWitt equation

TL;DR

This paper investigates the Maxwell charge as an eigenvalue of the Wheeler-DeWitt equation using a variational approach, focusing on gravitational fluctuations in a Schwarzschild background, and employs zeta function regularization and renormalization.

Contribution

It introduces a novel method to extract Maxwell charge eigenvalues from the Wheeler-DeWitt equation via a variational approach with Gaussian wave functionals.

Findings

Eigenvalues related to Maxwell charge can be derived from gravitational fluctuations.

One-loop approximation in Schwarzschild background is feasible with zeta function regularization.

A renormalization group approach effectively handles divergences.

Abstract

We consider the Wheeler-De Witt equation as a device for finding eigenvalues of a Sturm-Liouville problem. In particular, we will focus our attention on the electric (magnetic) Maxwell charge. In this context, we interpret the Maxwell charge as an eigenvalue of the Wheeler-De Witt equation generated by the gravitational field fluctuations. A variational approach with Gaussian trial wave functionals is used as a method to study the existence of such an eigenvalue. We restrict the analysis to the graviton sector of the perturbation. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation.