Extracting the Maxwell charge from the Wheeler-DeWitt equation

We consider the Wheeler-DeWitt 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.