协变场论中的规范约化
Gauge reduction in covariant field theory
AI总结：
本文建立协变场论的 Lagrange 规范约化理论，利用逐纤维作用的广义主联络约化变分原理，给出重构条件并联系 Noether 定理，且以多个经典与规范场例子验证。
摘要：
在本工作中，我们发展了一种针对具有规范对称性的协变场论的 Lagrange 约化理论。这些对称性由一个 Lie 群纤维丛建模，该纤维丛在构形丛上逐纤维作用。为了约化变分原理，我们利用广义主联络，这是一类在逐纤维作用下等变的 Ehresmann 联络。在得到约化方程之后，我们给出重构条件，并将垂直约化方程与 Noether 定理联系起来。最后，我们通过若干例子阐明该理论，包括经典情形（Lagrange-Poincaré 约化）、电磁学、对称性破缺以及非 Abel 规范理论。
英文摘要：
In this work, we develop a Lagrangian reduction theory for covariant field theories with gauge symmetries. These symmetries are modeled by a Lie group fiber bundle acting fiberwisely on a configuration bundle. In order to reduce the variational principle, we utilize generalized principal connections, a type of Ehresmann connections that are equivariant by the fiberwise action. After obtaining the reduced equations, we give the reconstruction condition and we relate the vertical reduced equation with the Noether theorem. Lastly, we illustrate the theory with several examples, including the classical case (Lagrange-Poincaré reduction), Electromagnetism, symmetry-breaking and non-Abelian gauge theories.
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