An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III$_{1}$, in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II$_∞$ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag's assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.

中文翻译：

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子区域的交叉积代数和广义熵

代数量子场论的一个早期结果是，QFT 中任何子区域的代数都是 III$_{1}$ 型冯·诺依曼因子，其中熵无法明确定义，因为此类代数不允许存在迹或迹。密度状态。然而，与代数相关的是一个自同构的模群，它表征了该区域自由度的局部动态，并且代数与其模群的叉积产生了一个 II$_∞$ 类型因子，其中有迹，因此冯·诺依曼熵可以被明确定义。在这项工作中，我们原则上将 TFD 的交叉积代数的最新构造推广到任意 QFT 中的任意时空区域，正式为研究没有 UV 发散的纠缠熵铺平了道路。与之前的作品相比，我们强调这种结构独立于重力。从这个意义上说，交叉积构造代表了 Haag 通过提供 II 型因子的自然构造将可观测代数网分配给时空区域的改进。我们提出了几个具体的例子：Rindler 空间中的 QFT、Minkowski 空间开球中的 CFT 以及 AdS/CFT 中的任意边界子区域。在全息环境中，我们提供了一个新颖的论据来解释为什么体对偶必须是纠缠楔，并讨论激发态和不相交区域的因果楔和纠缠楔之间的边界模流所产生的区别。