Partition function for a volume of space

TL;DR

This paper computes the quantum gravity partition function for a finite spatial volume, revealing its connection to entropy and holography, and generalizing previous de Sitter entropy calculations.

Contribution

It introduces a saddle point approximation for the quantum gravity partition function at fixed volume, linking it to Bekenstein-Hawking entropy and extending holographic insights.

Findings

Partition function exponential of boundary entropy

Generalizes Gibbons-Hawking de Sitter entropy

Supports holographic nature of quantum gravity

Abstract

We consider the quantum gravity partition function that counts the dimension of the Hilbert space of a spatial region with topology of a ball and fixed proper volume, and evaluate it in the leading order saddle point approximation. The result is the exponential of the Bekenstein-Hawking entropy associated with the area of the saddle ball boundary, and is reliable within effective field theory provided the mild curvature singularity at the ball boundary is regulated by higher curvature terms. This generalizes the classic Gibbons-Hawking computation of the de Sitter entropy for the case of positive cosmological constant and unconstrained volume, and hence exhibits the holographic nature of nonperturbative quantum gravity in generic finite volumes of space.