We review some recent results on Sorkin’s spacetime formulation of the entanglement entropy (SSEE) for a free quantum scalar field both in the continuum and in manifold-like causal sets. The SSEE for a causal diamond in a 2d cylinder spacetime has been shown to have a Calabrese–Cardy form, while for de Sitter and Schwarzschild de Sitter horizons in dimensions \(d>2\), it matches the mode-wise von-Neumann entropy. In these continuum examples the SSEE is regulated by imposing a UV cut-off. Manifold-like causal sets come with a natural covariant spacetime cut-off and thus provide an arena to study regulated QFT. However, the SSEE for different manifold-like causal sets in \(d=2\) and \(d=4\) has been shown to exhibit a volume rather than an area law. The area law is recovered only when an additional UV cut-off is implemented in the scaling regime of the spectrum which mimics the continuum behaviour. We discuss the implications of these results and suggest that a volume-law may be a manifestation of the fundamental non-locality of causal sets and a sign of new UV physics.

我们回顾了 Sorkin 对自由量子标量场的纠缠熵 (SSEE) 时空公式的一些最新结果，包括在连续体和流形类因果集中。二维圆柱时空中因果钻石的 SSEE 已被证明具有 Calabrese-Cardy 形式，而对于维度\(d>2\)中的 de Sitter 和 Schwarzschild de Sitter 视界，它与模态的 von-Neumann 相匹配熵。在这些连续的例子中，SSEE 是通过施加 UV 截止来调节的。流形类因果集具有自然协变时空截止，因此为研究调节的 QFT 提供了一个舞台。然而，在\(d=2\)和\(d=4\)中不同类流形因果集的 SSEE已被证明表现出体积而不是面积定律。只有在模拟连续谱行为的光谱缩放方案中实施额外的 UV 截止时，才能恢复面积定律。我们讨论了这些结果的含义，并提出体积定律可能是因果集基本非局域性的表现，也是新紫外物理学的标志。