Wheeler-DeWitt Equation in 3 + 1 Dimensions

Abstract: 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 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 non-perturbative 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 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 is then lead to the conclusion that the weak coupling, perturbative ground state of quantum gravity is non-perturbatively unstable, and that gravitational screening cannot be physically realized in the lattice theory. The results we find are in general agreement with the Euclidean lattice gravity results, and lend further support to the claim that the Lorentzian and Euclidean lattice formulations for gravity describe the same underlying non-perturbative physics.