Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT
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...
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2024-10-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP10(2024)063 |
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| author | Eugenia Colafranceschi Xi Dong Donald Marolf Zhencheng Wang |
| author_facet | Eugenia Colafranceschi Xi Dong Donald Marolf Zhencheng Wang |
| author_sort | Eugenia Colafranceschi |
| collection | DOAJ |
| description | 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 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 A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B R $$ {\mathcal{A}}_R^{B_R} $$ of observables acting respectively at B L , B R such that A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B R $$ {\mathcal{A}}_R^{B_R} $$ are commutants. The path integral also defines entropies on A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B R $$ {\mathcal{A}}_R^{B_R} $$ . 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 ∈ 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. |
| format | Article |
| id | doaj-art-c9ffde99a1f644fb9543c51b975dda3d |
| institution | OA Journals |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | SpringerOpen |
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| series | Journal of High Energy Physics |
| spelling | doaj-art-c9ffde99a1f644fb9543c51b975dda3d2025-08-20T02:20:38ZengSpringerOpenJournal of High Energy Physics1029-84792024-10-0120241017110.1007/JHEP10(2024)063Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFTEugenia Colafranceschi0Xi Dong1Donald Marolf2Zhencheng Wang3Department of Physics, University of California Santa BarbaraDepartment of Physics, University of California Santa BarbaraDepartment of Physics, University of California Santa BarbaraDepartment of Physics, University of California Santa BarbaraAbstract 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 A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B R $$ {\mathcal{A}}_R^{B_R} $$ of observables acting respectively at B L , B R such that A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B R $$ {\mathcal{A}}_R^{B_R} $$ are commutants. The path integral also defines entropies on A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B R $$ {\mathcal{A}}_R^{B_R} $$ . 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 ∈ 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.https://doi.org/10.1007/JHEP10(2024)063AdS-CFT CorrespondenceModels of Quantum Gravity |
| spellingShingle | Eugenia Colafranceschi Xi Dong Donald Marolf Zhencheng Wang Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT Journal of High Energy Physics AdS-CFT Correspondence Models of Quantum Gravity |
| title | Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT |
| title_full | Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT |
| title_fullStr | Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT |
| title_full_unstemmed | Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT |
| title_short | Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT |
| title_sort | algebras and hilbert spaces from gravitational path integrals understanding ryu takayanagi hrt as entropy without ads cft |
| topic | AdS-CFT Correspondence Models of Quantum Gravity |
| url | https://doi.org/10.1007/JHEP10(2024)063 |
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