How to Count States in Gravity

TL;DR

This paper demonstrates that the Gibbons-Hawking Euclidean gravity path integral accurately computes the thermal trace over the Hilbert space, clarifying its role in calculating gravitational entropy and entanglement entropy.

Contribution

It provides a rigorous proof that the Euclidean gravity path integral equals a Hilbert space trace, clarifying its interpretation in gravitational thermodynamics.

Findings

The path integral equals an explicit Hilbert space trace.

Factorization of the Hilbert space supports the trace interpretation.

Replicated path integrals compute powers of the density matrix and entropy.

Abstract

Gibbons and Hawking proposed that the Euclidean gravity path integral with periodic boundary conditions in time computes the thermal partition sum of gravity. As a corollary, they argued that a derivative of the associated free energy with respect to the Euclidean time period computes gravitational entropy. Why is this interpretation is correct? That is, why does this path integral compute a trace over the Hilbert space? Here, we show that the quantity computed by the Gibbons-Hawking path integral is equal to an {\it a priori} different object -- an explicit thermal trace over the Hilbert space spanned by states produced by the Euclidean gravity path integral. This follows in two ways: (a) if the Hilbert space with two boundaries factorizes into a product of two single boundary Hilbert spaces, as we have previously shown; and (b) via explicit resolution of the trace by a spanning basis of states. We similarly show how a replicated Euclidean gravity path integral with a single periodic boundary computes a Hilbert space trace of powers of the density matrix, explaining why this approach computes the entropy of states entangled between two universes.