The geometrical origin of dark energy

Abstract

The geometrical formulation of the quantum Hamilton–Jacobi theory shows that the quantum potential is never trivial, so that it plays the rôle of intrinsic energy. Such a key property selects the Wheeler–DeWitt (WDW) quantum potential $Q[g_{\mathit{jk}}]$ as the natural candidate for the dark energy. This leads to the WDW Hamilton–Jacobi equation with a vanishing kinetic term, and with the identification $$\begin{array}{c}\hfill \mathrm{\Lambda }=-\frac{{\kappa }^2}{\sqrt{\overline{g}}}Q[g_{\mathit{jk}}].\end{array}$$This shows that the cosmological constant is a quantum correction of the Einstein tensor, reminiscent of the von Weizsäcker correction to the kinetic term of the Thomas–Fermi theory. The quantum potential also defines the Madelung pressure tensor. The geometrical origin of the vacuum energy density, a strictly non-perturbative phenomenon, provides strong evidence that it is due to a graviton condensate. Time independence of the regularized WDW equation suggests that the ratio between the Planck length and the Hubble radius may be a time constant, providing an infrared/ultraviolet duality. We speculate that such a duality is related to the local to global geometry theorems for constant curvatures, showing that understanding the universe geometry is crucial for a formulation of Quantum Gravity.