Algebras, Entanglement Islands, and Observers

TL;DR

This paper demonstrates that entanglement islands in holography correspond to emergent Type II$_{ ext{infty}}$ von Neumann algebras, linking observer construction, symmetry breaking, and algebraic structures in gravitational systems.

Contribution

It shows that entanglement islands are associated with emergent Type II$_{ ext{infty}}$ von Neumann algebras, using the Goldstone mode from broken diffeomorphisms and the bath as an external system.

Findings

Entanglement islands correspond to Type II$_{ ext{infty}}$ von Neumann algebras.

Observers constructed from Goldstone modes relate to the external bath.

The geometric modular flow conjecture underpins the algebraic structure.

Abstract

Some recent work has postulated the existence of an "observer" for a consistent definition of subregion algebras in gravitational universes. The subregion algebras consist of operators dressed to this "observer" and are typically Type II von Neumann algebras. Nevertheless, as opposed to standard physical systems, such an "observer" was postulated to have a Hamiltonian $\hat{H}_{\text{obs}}$ linear in phase space variable. This linear form suggests that the complete dynamics of such an "observer" should also be controlled by an external system or some underlying degrees of freedom within the system. In this paper, we show that this is exactly the case in the island model. In the island model, we have a gravitational asymptotically anti-de Sitter (AdS) spacetime coupled with a non-gravitational bath, and the diffeomorphism symmetries in the gravitational AdS are spontaneously broken due to the bath coupling. In this setup, the "observer" is constructed using the Goldstone vector field associated with the spontaneously broken diffeomorphism symmetry, and the external system that also controls the dynamics of the "observer" is the non-gravitational bath. The basic consistency of the entanglement wedge reconstruction requires operators in the entanglement island to be dressed to this "observer". Thus, we establish the result that entanglement islands correspond to emergent Type II$_{\infty}$ von Neumann algebras from the holographic dual perspective. This result relies on assuming the geometric modular flow conjecture. Our study also raises a question for earlier constructions of Type II$_{1}$ von Neumann algebras.