The cosmological constant and the stability of Flat Space

We extend the Gross-Perry-Yaffe approach of hot flat space instability to Minkowski space. This is done by examining the massless gravitational perturbation in a saddle point approximation of the partition function. In particular, we show that a nonconformal instability appears in the transverse-traceless tensor sector (massless gravitons) if a Schwarzschild wormhole background is considered. The appearance of an instability in the whole manifold is here interpreted as a black hole pair creation. The instability of .at space via black hole pair creation is further explored evaluating the Casimir energy to one loop in a Hamiltonian approach. A variational approach associated to a Sturm-Liouville problem is used to evaluate the Casimir energy by means of trial wave functionals of Gaussian form. The Casimir energy is here interpreted as an induced cosmological constant associated to the Sturm-Liouville eigenvalue. To handle with divergences which appear to one loop approximation, we use a zeta function regularization. A renormalization and a renormalization group equation are involved to handle with ultra-violet divergences.