Type I von Neumann Algebras from Gravitational Path Integrals: Ryu–Takayanagi as Entropy Without Holography

Recent works by Chandrasekaran, Penington and Witten have shown in various special contexts that the quantum-corrected Ryu-Takayanagi (RT) formula can be understood as computing an entropy on an algebra of bulk observables. These arguments do not rely on the existence of a holographic dual field theory. We show that analogous-but-stronger results hold in any UV-completion of asymptotically anti-de Sitter quantum gravity with a Euclidean path integral satisfying a simple and familiar set of axioms. In particular, the path integral defines type I von Neumann algebras of bulk observables acting on compact closed codimension-2 asymptotic boundaries, as well as entropies on these algebras. Such entropies can be written in terms of standard density matrices and standard Hilbert space traces, and in appropriate semiclassical limits are computed by the RT-formula with quantum corrections. Our work thus provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. Since our axioms do not severely constrain UV bulk structures, they may be expected to hold equally well for successful formulations of string field theory, spin-foam models, or any other approach to constructing a UV-complete theory of gravity.