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ME and Dirac equation

In the time-harmonic case in [#!KS!#] is to find a further approach by the so-called quaternionic MAXWELL operator. For MAXWELL equations in the vacuum
\begin{eqnarray*}&&{\rm rot}\:{\bf H}=\sigma {\bf E},\quad{\rm rot}\:{\bf E}=i\......H}\\ [2ex] &&{\rm div}\:{\bf E}=0,\quad\quad{\rm div}\:{\bf E}=0\end{eqnarray*}
with the complex electrical conductivity$\sigma :=\rho^{-1}-i\omega \varepsilon$ one obtains the
HELMHOLTZ equations
\begin{eqnarray*}\Delta {\bf E}+\lambda {\bf E}=0\quad\mbox{and}\quad\Delta{\bf H}+\lambda {\bf H}=0\end{eqnarray*}
with $\lambda :=i\omega \mu\rho^{-1} +\omega ^2\mu\varepsilon$. Here $\sqrt{\lambda }=\alpha $ is called medium wave number. The MAXWELL operator is defined by
\begin{eqnarray*}{M}:=\left(\sigma \quad \quad {\rm div-rot}\atop-{\rm div+rot}\:-i\omega \mu\right).\end{eqnarray*}
V.V KRAVSHENKO and M.V. SHAPIRO worked out the following connection between MAXWELL equations and the DIRAC equation. It reads as follows:
\begin{eqnarray*}AMB=\left(\begin{array}{cc}\overline {D-\alpha }& 0\\0 & D-\alpha \end{array}\right)\end{eqnarray*}
with
\begin{eqnarray*}A:=\left(\:\alpha \:-\sigma \atop-\alpha \:-\sigma \right),\qu......a ^{-1}\:-\sigma ^{-1}\atop\alpha ^{-1}\:\:\alpha ^{-1}\right).\end{eqnarray*}


collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg