Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT

TL;DR

This paper demonstrates that in UV-complete quantum gravity with a Euclidean path integral, the Ryu-Takayanagi entropy can be understood as a standard quantum entropy on von Neumann algebras, providing a Hilbert space interpretation without relying on holographic duality.

Contribution

It proves that the RT entropy arises from von Neumann algebras in UV-complete quantum gravity, extending previous special-case results to a general axiomatic framework.

Findings

Von Neumann algebras of bulk observables are commutants.

RT entropy can be expressed as standard quantum entropy.

Entropy quantization condition: $ abla N$ for some positive integer $N$.

Abstract

Recent works by Chandrasekaran, Penington, and Witten have shown in various special contexts that the quantum-corrected Ryu-Takayanagi (RT) entropy (or its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization) can be understood as computing an entropy on an algebra of bulk observables. These arguments do not rely on the existence of a local holographic dual field theory. We show that analogous-but-stronger results hold in any UV-completion of asymptotically AdS quantum gravity with a Euclidean path integral satisfying a simple and (largely) familiar set of axioms. We consider a quantum context in which a standard Lorentz-signature classical bulk limit would have Cauchy slices with asymptotic boundaries $B_L \sqcup B_R$ where both $B_L$ and $B_R$ are compact manifolds without boundary. Our main result is that (the UV-completion of) the quantum gravity path integral defines type I von Neumann algebras ${\cal A}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ of observables acting respectively at $B_L$, $B_R$ such that ${\cal A}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ are commutants. The path integral also defines entropies on ${\cal A}^{B_L}_L, {\cal A}^{B_R}_R$. Positivity of the Hilbert space inner product then turns out to require the entropy of any projection operator to be quantized in the form $\ln N$ for some $N \in {\mathbb Z}^+$ (unless it is infinite). As a result, our entropies can be written in terms of standard density matrices and standard Hilbert space traces. Furthermore, in appropriate semiclassical limits our entropies are computed by the RT-formula with quantum corrections. Our work thus provides a Hilbert space interpretation of the RT entropy. Since our axioms do not severely constrain UV bulk structures, it is plausible that they 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.