The geometrical origin of dark energy

Alon E Faraggi,Marco Matone

doi:10.1140/epjc/s10052-020-08665-6

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_{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 Λ=-κ2g¯Q[gjk].\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Lambda =-\\frac{\\kappa ^2}{\\sqrt{{\\bar{g}}}}Q[g_{jk}]. \\end{aligned}$$\\end{document}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.

Highlights

This leads to the Wheeler– DeWitt (WDW) Hamilton–Jacobi equation with a vanishing kinetic term, and with the identification κ2
Such a key property selects the Wheeler–DeWitt (WDW) quantum potential Q[g 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 κ2
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

Summary

Introduction

Another consequence of the GQHJ theory is that if space is compact, there is no notion of particle trajectory [12]. It is immediate to show that the QHJE implies the Schrödinger equation In such a formulation, it has been shown that the quantum Hamilton characteristic function S is non-trivial even in the case of the free particle with vanishing energy. The basic duality, that is the Möbius symmetry, which extends to the QHJE in higher dimension [24], is the defining property of the Schwarzian derivative Such a duality, that relates small and large scales, and acts like the map between different fundamental domains, is at the heart of the proof of the energy quantization [5,6,7,8,9,10]. We argue that time independence of the regularized WDW equation would imply that K is a space-time constant

WDW Hamilton–Jacobi equation

QHJE and Einstein paradox

Cosmological constant from the quantum potential