Partition function for a volume of space

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.

5 pages plus appendices, 2 figures; v4: reorganized and clarified some aspects, published version; v5: corrected equation references in supplemental material

References in corpus (3)

• Towards a derivation of holographic entanglement entropy
• Towards a quantum theory of de Sitter space
• Black hole entanglement entropy and the renormalization group

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