Wheeler-DeWitt equation in 3+1 dimensions

Physical properties of the quantum gravitational vacuum state are explored by solving a lattice version of the Wheeler-DeWitt equation. The constraint of diffeomorphism invariance is strong enough to uniquely determine part of the structure of the vacuum wave functional in the limit of infinitely fine triangulations of the three-sphere. In the large fluctuation regime, the nature of the wave function solution is such that a physically acceptable ground state emerges, with a finite nonperturbative correlation length naturally cutting off any infrared divergences. The location of the critical point in Newton's constant G(c), separating the weak from the strong coupling phase, is obtained, and it is inferred from the general structure of the wave functional that fluctuations in the curvatures become unbounded at this point. Investigations of the vacuum wave functional further suggest that for weak enough coupling, G<G(c), a pathological ground state with no continuum limit appears, where configurations with small curvature have vanishingly small probability. One would then be lead to the conclusion that the weak coupling, perturbative ground state of quantum gravity is nonperturbatively unstable and that gravitational screening cannot be physically realized in the lattice theory. The results we find tend to be in general agreement with the Euclidean lattice gravity results and would suggest that the Lorentzian and Euclidean lattice formulations of gravity ultimately describe the same underlying nonperturbative physics.