麦克斯韦方程组的新纽结解
New knotted solutions of Maxwell's equations
摘要：
本文进一步发展了真空电动力学拓扑非平凡解的研究。我们已经发现了一种新的方法产生这样的解决方案，通过应用保角变换与复杂的参数对已知的解决方案表示的贝特曼变量。这使我们能够从基本的配置，如恒定的电磁场和平面波得到广泛的解决方案。我们已经介绍了贝特曼的建设的协变配方，并讨论了保守的费用与保形群以及一组四种类型的保守螺旋。我们也给出了一个公式的四元数。这导致了一个简单的映射之间的电磁打结和链接到$SU(2)$规范理论的平面连接的解决方案。我们计算了一类解中相应的CS荷，它取整数值。
英文摘要：
In this note we have further developed the study of topologically non-trivial solutions of vacuum electrodynamics. We have discovered a novel method of generating such solutions by applying conformal transformations with complex parameters on known solutions expressed in terms of Bateman's variables. This has enabled us to get a wide class of solutions from the basic configuration like constant electromagnetic fields and plane-waves. We have introduced a covariant formulation of the Bateman's construction and discussed the conserved charges associated with the conformal group as well as a set of four types of conserved helicities. We have also given a formulation in terms of quaternions. This led to a simple map between the electromagnetic knotted and linked solutions into flat connections of $SU(2)$ gauge theory. We have computed the corresponding CS charge in a class of solutions and it takes integer values.
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