Gravitational effective action at mesoscopic scales from the quantum microstructure of spacetime

Abstract

At mesoscopic scales, the quantum corrected field equations of gravity should arise from extremizing, $\Omega$, the number of microscopic configurations of pre-geometric variables consistent with a given geometry. This $\Omega$, in turn, is the product over all events P of the density, $\rho (P)$, of microscopic configurations associated with each event P. One would have expected $\rho \propto \sqrt{g}$ so that $\rho d^{4}x$ scales as the proper volume of a region. On the other hand, at leading order, we would expect the extremum principle to be based on the Hilbert action, suggesting $\ln \rho \propto R$. I show how these two apparently contradictory requirements can be reconciled by using the functional dependence of $\sqrt{g}$ on curvature, in the Riemann normal coordinates (RNC), and coarse-graining over Planck scales. This leads to the density of microscopic configurations to be $\rho ={\Delta }^{-1}={\sqrt{g}}_{RNC}$ where $\Delta$ is the coarse grained Van-Vleck determinant. The approach also provides: (a) systematic way of computing QG corrections to field equations and (b) a direct link between the effective action for gravity and the kinetic theory of the spacetime fluid.

arXiv commentver 2: minor edits; no figures; 9 pages

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