Casimir energy, the cosmological constant and massive gravitons

The cosmological constant appearing in the Wheeler–De Witt equation is considered as an eigenvalue of the associated Sturm–Liouville problem. A variational approach with Gaussian trial wave functionals is used as a method to study such a problem. 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. The case of massive gravitons is discussed.