发表机构
School of Physics and Astronomy, Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University; School of Science, Beijing Jiaotong University（北京师范大学物理与天文学院，多尺度自旋物理教育部重点实验室; 北京交通大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单纯流形上的离散几何，通过边向量与和乐构造带挠率的 Regge 计算作用量，变分得到运动方程，并证明其与连续爱因斯坦-嘉当理论一致，从而建立其离散类比。
AI 中文摘要
为了将挠率纳入 Regge 计算，研究了 n 维单纯流形上的离散几何。在每个单纯形中，边向量被分配给各边，以编码其长度和方向的信息。此外，穿过两个相邻单纯形界面的曲线上的和乐（holonomy）由内部规范群元素表示。挠率表现为：属于一个单纯形的界面上的边向量与通过和乐平行移动的、属于相邻单纯形的同一边的边向量之间的差异。随后，在三维和四维中，分别针对欧几里得和洛伦兹号差，构造了作为边向量和和乐函数的单纯爱因斯坦-嘉当（Einstein--Cartan）作用量。结果表明，在无挠率情形下，它们回归到相应的 Regge 作用量。分别关于边向量和和乐对四维离散作用量进行变分，得到两个运动方程。结果表明，前者与连续理论中的相应方程一致，且无挠率和乐满足后者方程，如同连续情形。因此，在单纯流形上得到的理论可视为爱因斯坦-嘉当理论的离散类比。
英文摘要
The discrete geometry on an $n$-dimensional simplicial manifold are studied, in order to incorporate torsion into Regge calculus. In each simplex, the edge vectors are assigned to the edges to encode the information of their lengths and directions. In addition, the holonomies along the curves across the interfaces of two adjacent simplices are represented by the internal gauge group elements. The torsion manifests itself as the difference between an edge vector on an interface belonging to one simplex and the parallel transported edge vector, via the holonomy, of the same edge but belonging to the adjacent simplex. The simplicial Einstein--Cartan actions are then constructed as functions of the edge vectors and holonomies in three and four dimensions with Euclidean and Lorentzian signatures, respectively. It is shown that they return to the corresponding Regge actions for the torsion-free cases. The variations of the 4-dimensional discrete action with respect to the edge vectors and holonomies, respectively, give two equations of motion. It is shown that the former is consistent with the corresponding equation in the continuum theory, and the torsion-free holonomies satisfy the latter equation as in the continuous case. Thus, the resulting theory on the simplicial manifold can be regarded as a discrete analogue of Einstein--Cartan theory.
