霍奇星算子与外微分算符的组合规律

Combination rules of Hodge star and the exterior derivative

系统探讨了霍奇星算子与外微分算符作用于任意微分形式场时二者的一般组合规律.首先,找到了保持微分形式场的次不变的2个组合算符,并通过二者的线性组合得到了一个新算符.其次,当由任意数目的霍奇星算子与外微分算符进行组合时,导出了所有形式上彼此互异的组合算符的统一表达式.这些表达式由单个霍奇星算子与外微分算符以及二者的任选2个的非零组合构成.在此基础上,分析了所有算符之间的相互作用关系,并根据这些算符对微分形式的次的改变情况,对它们进行了具体分类.最后,作为一个应用,详细讨论了如何由次相同的微分形式的线性组合来构造电磁场的麦克斯韦方程.

In this paper,we have systematically explored the general rules for all kinds of combination of Hodge star and exterior differentiation operators.We have derived the unified forms of the non-vanishing and independent operators made up of arbitrary numbers of Hodge star and exterior differentiation operators.What’s more,we have explicitly investigated the interactions of all the combined operators.All the operators have been classified according to the ranks of the newly generated differential forms.It has been demonstrated that the Maxwell’s equations for U(1)gauge field can be constructed from the linear combinations of two(n-1)-forms.