How the Hilbert space of two-sided black holes factorises

TL;DR

This paper demonstrates that non-perturbative spacetime wormholes in AdS/CFT cause the bulk Hilbert space of two-sided black holes to factorise, offering a potential resolution to the factorisation puzzle in holography.

Contribution

It shows how non-perturbative wormhole effects lead to the factorisation of the bulk Hilbert space, resolving the factorisation puzzle from the gravitational path integral perspective.

Findings

Hilbert space factorises due to non-perturbative wormholes

Trace over states factorises into left and right components

Persistence of factorisation at all orders in 1/G_N

Abstract

In AdS/CFT, two-sided black holes are described by states in the tensor product of two Hilbert spaces associated with the two asymptotic boundaries of the spacetime. Understanding how such a tensor product arises from the bulk perspective is an important open problem in holography, known as the factorisation puzzle. In this paper, we show how the Hilbert space of bulk states factorises due to non-perturbative contributions of spacetime wormholes: the trace over two-sided states with different particle excitations behind the horizon factorises into a product of traces of the left and right sides. This precisely occurs when such states form a complete basis for the bulk Hilbert space. We prove that the factorisation of the trace persists to all non-perturbative orders in $1/G_N$, consequently providing a possible resolution to the factorisation puzzle from the gravitational path integral. In the language of von Neumann algebras, our results provide strong evidence that the algebra of one-sided observables transitions from a Type II or Type III algebra, depending on whether or not perturbative gravity effects are included, to a Type I factor when including non-perturbative corrections in the bulk.