因果图上的离散微分几何
Discrete Differential Geometry on causal graphs
摘要：
本文用非交换几何的方法研究离散空间上的微分学，将微分学表示为一个合适的Krein空间上的算子代数。将（伪）黎曼度量的离散模拟编码为该空间上的内积的变形，这是本文的关键技术。我们研究的一般情况下，但发现，激烈的和可能至关重要的简化发生时，基本的晶格拓扑超立方，在这种情况下，我们明确构建模仿的体积形式，霍奇星算子，霍奇内积任意离散几何的类似物。事实证明，形式主义挑出一个伪黎曼度量拓扑超三次图的所有边都是轻的。我们详细研究了这种因果图复合体，并考虑了它们的一些可能的物理应用，如晶格杨-米尔斯理论和晶格费米子。
英文摘要：
Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric is encoded in a deformation of the inner product on that space, which is the crucial technique of this paper. We study the general case but find that drastic and possibly vital simplifications occur when the underlying lattice is topologically hypercubic, in which case we explicitly construct mimetic analogues of the volume form, the Hodge star operator, and the Hodge inner product for arbitrary discrete geometries. It turns out that the formalism singles out a pseudo-Riemannian metric on topologically hypercubic graphs with respect to which all edges are lightlike. We study such causal graph complexes in detail and consider some of their possible physical applications, such as lattice Yang-Mills theory and lattice fermions.
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