Lagrange Formalism for Electromagnetic Field in Terms of Complex Isotropic Vectors

Abstract

In previous works, Weyl's equation for neutrino has been written in tensor form, in the form of non-linear Maxwell's like equations, through complex isotropic vector $\overrightarrow{F}=\overrightarrow{E}+i\overrightarrow{H}$. It has been proved, that the complex vector $\overrightarrow{F}=\overrightarrow{E}+i\overrightarrow{H}$ satisfies non-linear condition $\overrightarrow{F}^{2}=0$, equivalent to two conditions for real quantities $\overrightarrow{E}^{2}-\overrightarrow{H}^{2}=0$ and $\overrightarrow{E}. \overrightarrow{H}=0$, obtained by separating real and imaginary parts in the equality $\overrightarrow{F}^{2}=0$. Further, it has been proved, that Maxwell's equations can also be written through complex vector $\overrightarrow{F}=\overrightarrow{E}+i\overrightarrow{H}$. However, in the general case, the solution of Maxwell's equations does not satisfy non-linear condition$\overrightarrow{F}^{2}=0$. In this work, in the development of this new tensor formalism, we elaborated the Lagrange formalism for electromagnetic field in terms of complex isotropic vectors.