Generalized entropy for general subregions in quantum gravity

TL;DR

The paper develops a stringent framework for quasilocal observables in semiclassical quantum gravity by associating subregion algebras with type II von Neumann factors in the $G_N\to 0$ limit. Through a crossed-product construction that enforces gravitational constraints via an observer-dressed setup, it shows that the algebraic entropy of these subregions is UV finite and coincides with the generalized entropy $S_{gen}$ up to a universal additive constant. A key ingredient is a conjectured modular flow that is geometric near the entangling surface, enabling density matrices and a renormalized trace to be defined. The bounded-subregion case yields a type II$_1$ algebra with a maximal entropy state (relating to Jacobson’s entanglement equilibrium), while subregions extending to infinity give a type II$_\infty$ algebra. The results generalize previous de Sitter and AdS constructions and open avenues for holographic interpretations, edge modes, and a Lorentzian formulation of quantum extremal surfaces.

Abstract

We consider quantum algebras of observables associated with subregions in theories of Einstein gravity coupled to matter in the $G_N\rightarrow 0$ limit. When the subregion is spatially compact or encompasses an asymptotic boundary, we argue that the algebra is a type II von Neumann factor. To do so in the former case we introduce a model of an observer living in the region; in the latter, the ADM Hamiltonian effectively serves as an observer. In both cases the entropy of states on which this algebra acts is UV finite, and we find that it agrees, up to a state-independent constant, with the generalized entropy. For spatially compact regions the algebra is type II$_1$, implying the existence of an entropy maximizing state, which realizes a version of Jacobson's entanglement equilibrium hypothesis. The construction relies on the existence of well-motivated but conjectural states whose modular flow is geometric at an instant in time. Our results generalize the recent work of Chandrasekaran, Longo, Penington, and Witten on an algebra of operators for the static patch of de Sitter space.

Figures (3)

• Figure 1: Two examples of a partial Cauchy slice $\Sigma,$ its causal development $\mathcal{S}$, the complementary region $\mathcal{S}',$ and the entangling surface $\partial \Sigma.$ In example (\ref{['fig:Sbd']}), $\mathcal{S}$ is bounded while $\mathcal{S}'$ is unbounded, while in example (\ref{['fig:Sunbd']}), both $\mathcal{S}$ and $\mathcal{S}'$ are unbounded.
• Figure 2: A vector field $\xi^a$ that is future-directed in $\mathcal{S},$ past-directed in $\mathcal{S}',$ tangent to the null boundaries, and approximates a boost near the entangling surface $\partial \Sigma.$
• Figure 3: Several types of "kinematical" operators in scalar-Maxwell theory that are invariant under compact gauge transformations supported within a region. There are charged scalars dressed to each other, charged scalars dressed to the boundary of the region, local field strength operators, and Wilson lines with no endpoints in the interior of the region. Any kinematical operator including a Wilson line with an endpoint on the boundary is not invariant under the global constraints of the theory.