When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions
Abstract Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors H B $$ {\mathcal{H}}_{\mathcal{B}} $$ of the resulting Hi...
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2024-08-01
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| author | Donald Marolf Daiming Zhang |
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| description | Abstract Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors H B $$ {\mathcal{H}}_{\mathcal{B}} $$ of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface B $$ \mathcal{B} $$ , where B $$ \mathcal{B} $$ may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where B $$ \mathcal{B} $$ includes the specification of appropriate boundary conditions for bulk fields. Cases where B $$ \mathcal{B} $$ was the disjoint union B ⊔ B of two identical (d − 2)-dimensional surfaces B were studied in detail and, after the inclusion of finite-dimensional ‘hidden sectors,’ were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras A L B $$ {\mathcal{A}}_L^B $$ , A R B $$ {\mathcal{A}}_R^B $$ that act respectively at the left and right copy of B in B ⊔ B. Below, we consider the case of general B $$ \mathcal{B} $$ , and in particular for B $$ \mathcal{B} $$ = B L ⊔ B R with B L , B R distinct. For any B R , we find that the von Neumann algebra at B L acting on the off-diagonal Hilbert space sector H B L ⊔ B R $$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space H B L ⊔ B L $$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ . As a result, the von Neumann algebras A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B L $$ {\mathcal{A}}_R^{B_L} $$ defined in [1] using the diagonal Hilbert space H B L ⊔ B L $$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors H B $$ {\mathcal{H}}_{\mathcal{B}} $$ ). A second implication is that, for any H B L ⊔ B R $$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ , including the same hidden sectors as in the diagonal case again provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. We also show the above central projections to satisfy consistency conditions that lead to a universal central algebra relevant to all choices of B L and B R . |
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| institution | Kabale University |
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| spelling | doaj-art-09f379f118b1480d9f98f7d0e2e5f8952024-11-24T12:08:15ZengSpringerOpenJournal of High Energy Physics1029-84792024-08-012024813510.1007/JHEP08(2024)010When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditionsDonald Marolf0Daiming Zhang1Department of Physics, University of CaliforniaTsinghua UniversityAbstract Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors H B $$ {\mathcal{H}}_{\mathcal{B}} $$ of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface B $$ \mathcal{B} $$ , where B $$ \mathcal{B} $$ may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where B $$ \mathcal{B} $$ includes the specification of appropriate boundary conditions for bulk fields. Cases where B $$ \mathcal{B} $$ was the disjoint union B ⊔ B of two identical (d − 2)-dimensional surfaces B were studied in detail and, after the inclusion of finite-dimensional ‘hidden sectors,’ were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras A L B $$ {\mathcal{A}}_L^B $$ , A R B $$ {\mathcal{A}}_R^B $$ that act respectively at the left and right copy of B in B ⊔ B. Below, we consider the case of general B $$ \mathcal{B} $$ , and in particular for B $$ \mathcal{B} $$ = B L ⊔ B R with B L , B R distinct. For any B R , we find that the von Neumann algebra at B L acting on the off-diagonal Hilbert space sector H B L ⊔ B R $$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space H B L ⊔ B L $$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ . As a result, the von Neumann algebras A L B L $$ {\mathcal{A}}_L^{B_L} $$ , A R B L $$ {\mathcal{A}}_R^{B_L} $$ defined in [1] using the diagonal Hilbert space H B L ⊔ B L $$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors H B $$ {\mathcal{H}}_{\mathcal{B}} $$ ). A second implication is that, for any H B L ⊔ B R $$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ , including the same hidden sectors as in the diagonal case again provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. We also show the above central projections to satisfy consistency conditions that lead to a universal central algebra relevant to all choices of B L and B R .https://doi.org/10.1007/JHEP08(2024)010AdS-CFT CorrespondenceModels of Quantum Gravity |
| spellingShingle | Donald Marolf Daiming Zhang When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions Journal of High Energy Physics AdS-CFT Correspondence Models of Quantum Gravity |
| title | When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions |
| title_full | When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions |
| title_fullStr | When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions |
| title_full_unstemmed | When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions |
| title_short | When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions |
| title_sort | when left and right disagree entropy and von neumann algebras in quantum gravity with general alads boundary conditions |
| topic | AdS-CFT Correspondence Models of Quantum Gravity |
| url | https://doi.org/10.1007/JHEP08(2024)010 |
| work_keys_str_mv | AT donaldmarolf whenleftandrightdisagreeentropyandvonneumannalgebrasinquantumgravitywithgeneralaladsboundaryconditions AT daimingzhang whenleftandrightdisagreeentropyandvonneumannalgebrasinquantumgravitywithgeneralaladsboundaryconditions |