We present a formulation of Regge calculus where arbitrary coordinates are associated to each vertex of the simplicial complex and the fundamental degrees of freedom are given by the metric g _μν (α) on each simplex α. The lengths of the edges, which are the usual degrees of freedom of Regge calculus, are thus determined and are left invariant under arbitrary transformations of the discrete set of coordinates, provided the metric transforms accordingly. Invariance under coordinate transformations entails tensor calculus and our formulation then follows closely the usual formalism of the continuum theory. This includes a definition of partial derivative which stems from a generalization to simplicial lattices of the symmetric finite difference operator on a cubic lattice. The definitions of parallel transport, Christoffel symbol, covariant derivatives and Riemann curvature tensor follow in a rather natural way establishing a kind of dictionary between continuum and simplicial lattice quantities. In this correspondence Einstein action becomes Regge action with the deficit angle θ replaced by sin θ. The correspondence with the continuum theory can be extended to actions with higher powers of the curvature tensor, to the vielbein formalism and to the coupling of gravity with matter fields (scalars, fermionic fields including spin 3/2 fields and gauge fields) which are then determined unambiguously and discussed in the paper. An action on the simplicial lattice for N = 1 supergravity in four dimensions is derived in this context. Another relavant result is that Yang–Mills actions on a simplicial lattice consist, even in absence of gravity, of two plaquettes terms, unlike the one plaquette Wilson action on the hypercubic lattice. An attempt is also made to formulate a discrete differential calculus to include differential forms of higher order and the gauging of free differential algebras in this scheme. However this leads to form products that do not satisfy associativity and distributive law with respect to the d operator. A proper formulation of theories that contain higher order differential forms in the context of Regge calculus is then still lacking.

我们提出了 Regge 演算的公式，其中任意坐标与单纯复形的每个顶点相关联，基本自由度由每个单纯形α上的度量g _μν ( α )给出 . 边的长度是 Regge 演算的通常自由度，因此被确定并且在离散坐标集的任意变换下保持不变，提供相应的度量变换。坐标变换下的不变性需要张量演算，我们的公式紧跟连续统理论的通常形式主义。这包括偏导数的定义，该定义源于对三次晶格上的对称有限差分算子的单纯晶格的推广。平行输运、克里斯托夫符号、协变导数和黎曼曲率张量的定义以一种相当自然的方式在连续体和单纯晶格量之间建立了一种字典。θ由 sin θ代替 。与连续统理论的对应关系可以扩展到具有更高曲率张量幂的作用、vielbein 形式主义以及引力与物质场（标量、费米子场，包括自旋 3/2 场和规范场）的耦合，然后明确确定并在论文中讨论。对N的单纯格的作用= 1 四个维度的超重力是在这种情况下推导出来的。另一个相关结果是，即使在没有重力的情况下，简单格子上的杨-米尔斯作用也由两个小块项组成，这与超立方晶格上的一个小块威尔逊作用不同。还尝试制定离散微分，以在该方案中包括高阶微分形式和自由微分代数的测量。然而，这导致形成不满足关于d算子的结合律和分配律的乘积。然后仍然缺乏在 Regge 微积分的上下文中包含高阶微分形式的理论的正确表述。