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

• Published in: Physics letters. B 2021-03, Vol.814, p.136109, Article 136109
• Main Author: Padmanabhan, T.
• Format: Article
• Language: English
• ISSN: 0370-2693; 1873-2445
• DOI: 10.1016/j.physletb.2021.136109

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.