The gravitational dynamics of the coarse-grained spacetime geometry should emerge from extremizing the number of microscopic configurations, Ω, of the pre-geometric variables corresponding to a given geometry. This Ω will be the product over all events 𝒫 of the density, ρ(𝒫), of microscopic configurations associated with each event 𝒫. I show how ρ can be computed, in terms of the van Vleck determinant, and thus obtain directly the gravitational effective action ℒE 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.

中文翻译：

---

爱因斯坦-希尔伯特作用，带有量子校正，来自时空微观结构的普朗克尺度粗粒度

粗粒度时空几何的引力动力学应该来自于微观配置的数量的极端化，Ω, 对应于给定几何的前几何变量。这Ω将成为所有活动的产物𝒫的密度，ρ(𝒫), 与每个事件相关的微观配置𝒫. 我展示了如何ρ可以根据 van Vleck 行列式计算，从而直接获得引力有效作用ℒ乙在介观尺度上。这种非微扰有效作用的主要术语给出了爱因斯坦-希尔伯特作用，从而提供了它的微观推导。高阶校正是有限的，不需要正则化，我演示了如何以系统的方式计算它们。