Tensor Formulation of Maxwell's Equations

Question

I've been reading up about the tensor formulation of Maxwell's Equations of Electromagnetism, and the derivations I have seen (found here: http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-2-3/maxwells-equations/) gives them in covariant form (in vacuo) as: $$\begin{array}{rcl}\partial_{\mu}F^{\mu \nu} &= &0 \\ \partial_{\mu}\tilde{F}^{\mu \nu} & = & 0 \end{array}$$ Where $F^{\mu \nu}$ is the electromagnetic tensor and $\tilde{F}^{\mu \nu}$ is a modified electromagnetic tensor in which: $$\begin{array}{rcl} E_m & \to & -B_m \\ B_m & \to & E_m \end{array}$$

I (just about) understand how these arose, however I'm sure I had previously seen them formulated as a single tensor equation (of a similar form): does anyone know how this looks/arises?

If possible could answers take a more component-based approach, as this is what I am able to work with easiest (don't worry if not though).

Answer 1

It is possible to combine the two equations into one by defining the Riemann-Silberstein field $\Psi= {\bf E}+i{\bf B}$. Then the time dependent Maxwell equations become $$ i\partial_t \Psi = -i (\Sigma_x \partial_y+ \Sigma_y \partial_y+\Sigma_z \partial_z)\Psi $$ where the $\Sigma_{x,y,z}$ are the spin-one version of the Pauli sigma matrices. The resulting equation is a spin-one version of the Weyl equation. There are a number of subtleties here though, and these equations do not describe static solutions. There is Wikipedia article on Riemann-Silberstein that contains the details.