Partition Function of a Volume of Space in a Higher Curvature Theory

TL;DR

This paper extends the calculation of the quantum partition function for a finite spatial volume to higher curvature theories in three dimensions, confirming the entropy interpretation and supporting holography.

Contribution

It generalizes the previous Einstein gravity results to theories with infinite curvature terms, showing the partition function relates to boundary entropy in a broader context.

Findings

Partition function equals exponential of boundary entropy.

Results hold for zero and finite cosmological constant.

Supports holographic interpretation of gravity.

Abstract

Recently, [Phys. Rev. Lett. 130, 221501 (2023)] Jacobson and Visser calculated the quantum partition function of a fixed, finite volume of a region with the topology of a ball in the saddle point approximation within the context of Einstein's gravity with or without a cosmological constant. The result can be interpreted as the dimension of Hilbert space of the theory. Here we extend their computation to a theory defined in principle with infinitely many powers of curvature in three dimensions. We confirm their result: The partition function of a spatial region in the leading saddle point approximation is given as the exponential of the Bekenstein-Hawking or the Wald entropy of the boundary of the finite spatial region both in the case of zero and finite cosmological constant. In the latter case, the effective Newton's constant appears in the entropy formula. The calculations lend support to the holographic nature of gravity for finite regions of space with a boundary.