标题： 子代数的指标作为体积 显示英文标题

标题： Volume as an index of a subalgebra

摘要： 我们提出了一种新的方法来理解AdS时空体内某些子区域的体积，这是通过将其与一个称为包含指标的代数量相关联。这个指标直观地衡量了嵌入在较大代数$\mathcal{M}$中的子代数$\mathcal{N}$的相对大小。根据子区域-代数对偶性，体内的子区域由边界上的冯·诺依曼代数描述。当一个因果完备的体内心区域对应于边界子代数的相对余核$\mathcal{N}' \cap \mathcal{M}$——即在$\mathcal{M}$中与$\mathcal{N}$对易的操作符集合——时，我们提出该子区域最大体积片的体积的指数等于包含指标。这种“体积-指标”关系为黑洞内部体积的增长提供了新的边界解释，将其重新表述为操作符代数相对大小的变化。它从海森堡绘景的角度提供了复杂度增长的互补视角，并有各种其他应用，包括量化与边界区域的纠缠楔和因果楔对偶的代数的相对大小，以及量化在大$N$极限下操作符代数的可加性违反。最后，它可能通过北极大和南极大观察者代数随时间的变化，为德西特空间的体积增长提供见解。 显示英文摘要

摘要： We propose a new way to understand the volume of certain subregions in the bulk of AdS spacetime by relating it to an algebraic quantity known as the index of inclusion. This index heuristically measures the relative size of a subalgebra $\mathcal{N}$ embedded within a larger algebra $\mathcal{M}$. According to subregion-subalgebra duality, bulk subregions are described by von Neumann algebras on the boundary. When a causally complete bulk subregion corresponds to the relative commutant $\mathcal{N}' \cap \mathcal{M}$ -- the set of operators in $\mathcal{M}$ that commute with $\mathcal{N}$ -- of boundary subalgebras, we propose that the exponential of the volume of the maximal volume slice of the subregion equals the index of inclusion. This ``volume-index'' relation provides a new boundary explanation for the growth of interior volume in black holes, reframing it as a change in the relative size of operator algebras. It offers a complementary perspective on complexity growth from the Heisenberg picture, and has a variety of other applications, including quantifying the relative size of algebras dual to the entanglement wedge and the causal wedge of a boundary region, as well as quantifying the violation of additivity of operator algebras in the large $N$ limit. Finally, it may offer insights into the volume growth of de Sitter space through the changes in North and South pole observer algebras in time.