The causal set and Wolfram model approaches to discrete quantum gravity both permit the formulation of a manifestly covariant notion of entanglement entropy for quantum fields. In the causal set case, this is given by a construction (due to Sorkin and Johnston) of a 2-point correlation function for a Gaussian scalar field from causal set Feynman propagators and Pauli-Jordan functions, from which an eigendecomposition, and hence an entanglement entropy, can be computed. In the Wolfram model case, it is given instead in terms of the Fubini-Study metric on branchial graphs, whose tensor product structure is inherited functorially from that of finite-dimensional Hilbert spaces. In both cases, the entanglement entropies in question are most naturally defined over an extended spacetime region (hence the manifest covariance), in contrast to the generically non-covariant definitions over single spacelike hypersurfaces common to most continuum quantum field theories. In this article, we show how an axiomatic field theory for a free, massless scalar field (obeying the appropriate bosonic commutation relations) may be rigorously constructed over multiway causal graphs: a combinatorial structure sufficiently general as to encompass both causal sets and Wolfram model evolutions as special cases. We proceed to show numerically that the entanglement entropies computed using both the Sorkin-Johnston approach and the branchial graph approach are monotonically related for a large class of Wolfram model evolution rules. We also prove a special case of this monotonic relationship using a recent geometrical entanglement monotone proposed by Cocchiarella et al.
翻译:分解量重力的因果设定和沃尔夫拉姆模型方法,都允许形成一个明显共变的概念,即量字段的缠绕性激素。在因果组合中,这是由(由于索金和约翰斯顿)因因因果组合Feynman 传播器和Pauli-Job 函数组成的高斯曲线场的2点相关函数构成的。从这些模型中,可以计算出一种异异因变形,并由此形成一种缠绕性激流。在沃尔夫拉姆模型中,它被用分形图中的富比尼-Study 度指标来表示。在分形图中,其强产产品结构是从有限维度的希尔伯特空间中继承的。在这两种情况中,有关缠绕的纠结性运动场最自然地在延长的时段区域(即表变异性)中定义,与类似单一空间模型的超额表面法则常见于最连续的量场理论。在本文章中,我们展示了一种特殊的亚化场理论,它是如何将亚化的机变法系理论用来用来显示一个自由的直径的直径阵列。