We describe a systematic mathematical approach to the geometric discretization of gauge field theories that is based on Dirac and multi-Dirac mechanics and geometry, which provide a unified mathematical framework for describing Lagrangian and Hamiltonian mechanics and field theories, as well as degenerate, interconnected, and nonholonomic systems. Variational integrators yield geometric structure-preserving numerical methods that automatically preserve the symplectic form and momentum maps, and exhibit excellent long-time energy stability. The construction of momentum-preserving variational integrators relies on the use of group-equivariant function spaces, and we describe a general construction for functions taking values in symmetric spaces. This is motivated by the geometric discretization of general relativity, which is a second-order covariant gauge field theory on the symmetric space of Lorentzian metrics.

中文翻译：

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使用群等变插值的规范场理论的变分离散化

我们描述了一种基于Dirac，多狄拉克力学和几何学的规范场论几何离散的系统数学方法，该方法为描述拉格朗日和汉密尔顿力学和场论以及退化，互连，和非完整系统。变分积分器提供了保留几何结构的数值方法，该方法可自动保留辛形式和动量图，并具有出色的长期能量稳定性。保持动量的变分积分器的构造依赖于组等变函数空间的使用，我们描述了在对称空间中取值的函数的一般构造。这是由广义相对论的几何离散化引起的，