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 extremising, 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∝g so that ρd4x 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⁡ρ∝R. I show how these two apparently contradictory requirements can be reconciled by using the functional dependence of 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=gRNC 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.