Search for: All records

A<sc>bstract</sc> Euclidean path integrals for UV-completions ofd-dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors$$ {\mathcal{H}}_{\mathcal{B}} $$ $H_B$of the resulting Hilbert space were then defined for any (d− 2)-dimensional surface$$ \mathcal{B} $$ $B$, where$$ \mathcal{B} $$ $B$may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where$$ \mathcal{B} $$ $B$includes the specification of appropriate boundary conditions for bulk fields. Cases where$$ \mathcal{B} $$ $B$was the disjoint unionB⊔Bof two identical (d− 2)-dimensional surfacesBwere 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$$ {\mathcal{A}}_L^B $$ $A_L^B$,$$ {\mathcal{A}}_R^B $$ $A_R^B$that act respectively at the left and right copy ofBinB⊔B. Below, we consider the case of general$$ \mathcal{B} $$ $B$, and in particular for$$ \mathcal{B} $$ $B$=B_L⊔B_RwithB_L,B_Rdistinct. For anyB_R, we find that the von Neumann algebra atB_Lacting on the off-diagonal Hilbert space sector$$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ $H_{B_L\bigsqcup B_R}$is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space$$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ $H_{B_L\bigsqcup B_L}$. As a result, the von Neumann algebras$$ {\mathcal{A}}_L^{B_L} $$ $A_L^{B_L}$,$$ {\mathcal{A}}_R^{B_L} $$ $A_R^{B_L}$defined in [1] using the diagonal Hilbert space$$ {\mathcal{H}}_{B_L\bigsqcup {B}_L} $$ $H_{B_L\bigsqcup B_L}$turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors$$ {\mathcal{H}}_{\mathcal{B}} $$ $H_B$). A second implication is that, for any$$ {\mathcal{H}}_{B_L\bigsqcup {B}_R} $$ $H_{B_L\bigsqcup B_R}$, 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 ofB_LandB_R.