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

Abstract

A bstract 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 anti-de Sitter 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 ⊔ B R where both B L and B R are compact manifolds without boundary. Our main result is then that (the UV-completion of) the quantum gravity path integral defines type I von Neumann algebras $$ {\mathcal{A}}_L^{B_L} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>L</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:msubsup> </mml:math> , $$ {\mathcal{A}}_R^{B_R} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:msubsup> </mml:math> of observables acting respectively at B L , B R such that $$ {\mathcal{A}}_L^{B_L} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>L</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:msubsup> </mml:math> , $$ {\mathcal{A}}_R^{B_R} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:msubsup> </mml:math> are commutants. The path integral also defines entropies on $$ {\mathcal{A}}_L^{B_L} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>L</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:msubsup> </mml:math> , $$ {\mathcal{A}}_R^{B_R} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:msubsup> </mml:math> . 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 ∈ 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 Ryu-Takayanagi 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.

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