When left and right disagree: entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions
Feb 14, 2024Citations per year
Abstract: (Springer)
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 of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface , where may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where includes the specification of appropriate boundary conditions for bulk fields. Cases where 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 , that act respectively at the left and right copy of B in B ⊔ B.Below, we consider the case of general , and in particular for = B ⊔ B with B, B distinct. For any B, we find that the von Neumann algebra at B acting on the off-diagonal Hilbert space sector is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space . As a result, the von Neumann algebras , defined in [1] using the diagonal Hilbert space turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors ). A second implication is that, for any , 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 and B.Note:
- 35 pages, 4 figures; typos corrected, reference added, comments added in section 1, section 5 and Discussion
- AdS-CFT Correspondence
- Models of Quantum Gravity
- algebra: von Neumann
- path integral: Euclidean
- Hilbert space
- surface
- entropy
- quantum gravity
- boundary condition
- hidden sector
References(30)
Figures(6)
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