Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation
Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation
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DOI：
10.1007/s10208-019-09420-4
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发表时间：
2019-10
影响因子：
3
通讯作者：
M. Leok
中科院分区：
文献类型：
--
作者：
M. Leok
我们描述了一个系统的数学方法的几何离散化的规范场理论，是基于狄拉克和多狄拉克力学和几何，它提供了一个统一的数学框架，描述拉格朗日和哈密顿力学和场论，以及退化，互连，和非完整系统。变分积分器产生几何结构保持数值方法，自动保持辛形式和动量映射，并表现出良好的长期能量稳定性。动量保持变分积分器的建设依赖于使用组等变函数空间，我们描述了一个一般的构造函数在对称空间中取值。这是由广义相对论的几何离散化激发的，广义相对论是洛伦兹度量的对称空间上的二阶协变规范场理论。
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.