Wheeler-DeWitt equation in $3\mathbf{+}1$ 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.

Figure 1

A tetrahedron with labels.

Figure 2

A small section of a suitable spatial lattice for quantum gravity in $3+1$ dimensions.

Figure 3

Wave function of Eq. (135) squared, $|\psi (V,R){|}^2$, plotted as a function of the total volume $V$ and the total curvature $R$, for coupling $g=\sqrt{G}=1$ and $N_3=10$. One notes that for strong enough coupling $g$ the distribution in curvatures is fairly flat around $R=0$, giving rise to large fluctuations in the curvature. These become more pronounced as one approaches the critical point at $g_c$.

Figure 4

Same wave function of Eq. (135) squared, $|\psi (V,R){|}^2$, plotted as a function of the total volume $V$ and the total curvature $R$, but now for weaker coupling $g=\sqrt{G}=0.5$ and still $N_3=10$. For weak enough coupling $g$, the distribution in curvature is such that values around $R=0$ are almost completely excluded, as these are associated with a very small probability. Note that, unless the total volume $V$ is very small, the probability distribution is markedly larger toward positive curvatures.

Figure 5

Curvature distribution in $R$ as a function of the coupling $g=\sqrt{G}$. The strong coupling relationship between the average volume and the coupling $g$ [Eq. (64)] allows one to plot the wave function of Eq. (135) squared as a function of the coupling $g$ and the total curvature $R$ only (we use again here $N_3=10$ for illustrative purposes). Then, for strong enough coupling $g=\sqrt{G}$, the probability distribution $|\psi {|}^2$ is again fairly flat around $R=0$, giving rise to large fluctuations in the curvature. The latter are interpreted here as signaling the presence of a massless particle. On the other hand, for weak enough coupling $g$, one notices that curvatures close to $R=0$ have essentially vanishing probability. The distribution shown here points therefore toward a pathological ground state for weak enough coupling $g<g_c$ [given in Eq. (119)], with no sensible continuum limit.