标题： 当左右意见不一致时：量子引力中熵与冯·诺依曼代数的一般AdS边界条件 显示英文标题

标题： When left and right disagree: Entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions

摘要： 欧几里得路径积分对于$d$维体积量子引力的紫外完备性在[1]中被研究，假设它们满足有限性、现实性、连续性、反射正定性和分解的公理。 结果希尔伯特空间的${\cal H}_{\cal B}$部分对于任何$(d-2)$维曲面${\cal B}$定义，其中${\cal B}$可以看作是相应洛伦兹描述中体积柯西面的边界$\partial\Sigma$，而${\cal B}$包括体积场的边界条件的指定。 当${\cal B}$是两个相同$(d-2)$维曲面的不相交并集$B\sqcup B$时，进行了详细研究，并在包含有限维“隐藏扇区”之后，被证明提供了相关 Ryu-Takayanagi 熵的希尔伯特空间解释。 分析是通过构造分别作用于$B$在$B\sqcup B$中的左和右副本上的类型-I冯·诺依曼代数$\mathcal A_L^B,\mathcal A_R^B$来完成的。 下面，我们考虑一般情况下的${\cal B} = B_L\sqcup B_R$，其中$B_L,B_R$是不同的。 对于任何$B_R$，我们发现由$B_L$在${\cal H}_{B_L\sqcup B_R}$上作用的冯诺依曼代数是对应类型-I 冯诺依曼代数在“对角”希尔伯特空间${\cal H}_{B_L\sqcup B_L}$上的中心投影。 因此，使用“对角”希尔伯特空间定义的冯诺依曼代数$\mathcal A_L^{B_L},\mathcal A_R^{B_L}$与使用理论的全希尔伯特空间定义的那些代数完全一致。 第二个含义是，对于任何${\cal H}_{B_L\sqcup B_R}$，包括与对角情况相同的隐藏部分再次提供了Ryu-Takayanagi熵的希尔伯特空间解释。 我们还证明上述中心投影满足导致与所有$B_L,B_R$选择相关的普遍中心代数的一致性条件。 显示英文摘要

摘要： Euclidean path integrals for UV-completions of $d$-dimensional bulk quantum gravity were studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors ${\cal H}_{\cal B}$ of the resulting Hilbert space were defined for any $(d-2)$-dimensional surface ${\cal B}$, where ${\cal B}$ may be thought of as the boundary $\partial\Sigma$ of a bulk Cauchy surface in a corresponding Lorentzian description, and where ${\cal B}$ includes the specification of boundary conditions for bulk fields. Cases where ${\cal B}$ was the disjoint union $B\sqcup B$ of two identical $(d-2)$-dimensional surfaces 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 $\mathcal A_L^B,\mathcal A_R^B$ that act respectively at the left and right copy of $B$ in $B\sqcup B$. Below, we consider the case of general ${\cal B} = B_L\sqcup B_R$ with $B_L,B_R$ distinct. For any $B_R$, we find that the von Neumann algebra at $B_L$ acting on ${\cal H}_{B_L\sqcup B_R}$ is a central projection of the corresponding type-I von Neumann algebra on the `diagonal' Hilbert space ${\cal H}_{B_L\sqcup B_L}$. As a result, the von Neumann algebras $\mathcal A_L^{B_L},\mathcal A_R^{B_L}$ defined in [1] using the diagonal Hilbert space coincide precisely with those defined using the full Hilbert space of the theory. A second implication is that, for any ${\cal H}_{B_L\sqcup B_R}$, 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 $B_L,B_R$.