Einstein–Hilbert action, with quantum corrections, from the Planck scale coarse-graining of the spacetime microstructure

Abstract

The gravitational dynamics of the coarse-grained spacetime geometry should emerge from extremizing the number of microscopic configurations, [Formula: see text], of the pre-geometric variables corresponding to a given geometry. This [Formula: see text] will be the product over all events [Formula: see text] of the density, [Formula: see text], of microscopic configurations associated with each event [Formula: see text]. I show how [Formula: see text] can be computed, in terms of the van Vleck determinant, and thus obtain directly the gravitational effective action [Formula: see text] at mesoscopic scales. The leading term of this, nonperturbative, effective action gives the Einstein–Hilbert action, thereby providing its microscopic derivation. The higher-order corrections are finite without any need for regularization and I demonstrate how they can be computed in a systematic manner.