arXiv math-ph/05010522005-01math-phmath.MP

利用势的麦克斯韦方程组的对称性、守恒律和上同调

Symmetries, conservation laws, and cohomology of Maxwell's equations using potentials

Stephen C. Anco, Dennis The

备注：: Minor revisions, mainly in notation and discussion of the complex of differential forms on jet space. Accepted in Acta Applicandae Mathematicae
发表信息：: Acta Appl. Math. 89 (2005), 1-52

摘要：

利用电磁张量场及其对偶场的联合矢势协变系统，导出了麦克斯韦方程组的新的非定域对称性和守恒律。这个系统的一个关键属性，以及这类新的对称性和守恒定律，是他们的不变性下的对偶变换，交换电磁场与其对偶。麦克斯韦方程组的非定域对称性来源于洛伦兹规范中联合势系统所承认的某种几何形式的所有对称性的明确分类。除了缩放和对偶旋转对称性，以及众所周知的庞加莱和膨胀对称性，其中涉及位似Killing向量，分类产生新的几何对称性，涉及Killing-Yano张量相关的旋转/提升和反转。麦克斯韦方程组的非局部守恒定律是从这些几何对称性中构造出来的，通过应用一个公式，该公式使用联合势并直接从麦克斯韦方程组的任何（局部或非局部）对称性中产生守恒电流。这个公式是通过一系列映射产生的，这些映射分别与联合势系统的对称性/伴随对称性和麦克斯韦方程组的伴随对称性/对称性有关。映射是作为研究封闭的1-形式和2-形式的上同调局部构造的电磁场及其衍生物的一些有限阶麦克斯韦方程组的所有解决方案的副产品。唯一的非平凡的上同调是由电磁场（2-形式）本身以及它的对偶（2-形式），这2-形式的上同调被杀死的引进相应的潜力。

英文摘要：

New nonlocal symmetries and conservation laws are derived for Maxwell's equations using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincare and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing-Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a formula that uses the joint potentials and directly generates conserved currents from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed 1-forms and 2-forms locally constructed from the electromagnetic field and its derivatives to some finite order for all solutions of Maxwell's equations. The only nontrivial cohomology is shown to consist of the electromagnetic field (2-form) itself as well as its dual (2-form), and this 2-form cohomology is killed by the introduction of corresponding potentials.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文