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

At mesoscopic scales, the quantum corrected field equations of gravity should arise from extremizing, $Ω$, the number of microscopic configurations of pre-geometric variables consistent with a given geometry. This $Ω$, in turn, is the product over all events P of the density, $ρ(P)$, of microscopic configurations associated with each event P. One would have expected $ρ\propto\sqrt{g}$ so that $ρd^4x$ 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 $\lnρ\propto R$. 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 $ρ= Δ^{-1} = \sqrt{g}_{RNC}$ where $Δ$ 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.

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