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ME in Majorana algebras

The Majorana algebra denoted by $C\ell_{3,1}$ is defined as follows::

For the basis $\{e_0,e_1,e_2,e_3\}$ in $Cl_{3,1}$ we can take

\begin{eqnarray*}e_0=\left(i\quad 0\atop 0\quad i\right)\!,\quade_k=\left(0\quad\sigma _k\atop\sigma _k\quad0\right)\quad(k=1,2,3),\end{eqnarray*}
where $\sigma _k$ are the PAULI matrices. Let
\begin{eqnarray*}D=\sum_{k=1}^3e_k\partial _k\quad\mbox{ and}\quad \partial =D-e_0\partial _0,\end{eqnarray*}
where $e_0^2=-1$ and $e_k^2=1\;\; (k=1, 2, 3)$. It can easily be seen that the basis elements $e_k\;\; (k=1,2,3)$ fulfil anticommutator relations. We now introduce the electro-magnetic field${\bf F}$, which is given by the bivector
\begin{eqnarray*}{\bf F}:={\bf E}e_0-{\bf B}e_{123}\end{eqnarray*}
with $e_{123}^2=(e_1e_2e_3)^2=-1$. Hereby ${\bf E}$ is defined as the bivector
\begin{eqnarray*}{\bf E}=E_1e_1e_0+E_2e_2e_0+E_3e_3e_0\end{eqnarray*}
and ${\bf B}$ is just the vector ${\bf B}=B_1e_1+B_2e_2+B_3e_3$. E and B are functions of the position ${ X}=x_1e_1+x_2e_2+x_3e_3$ (GIBBS vector) and the time $t$. Thus we have the coordinates
\begin{eqnarray*}(X,t)=\sum_{k=1}^3 e_kx_k+ te_0.\end{eqnarray*}
Now MAXWELL equations can be written in one single equation namely
\begin{eqnarray*}\partial {\bf F}={\bf J},\end{eqnarray*}
where the right-hand side is the sum of the trivector $J_0=\rho$ (charge density) and the vector ${\bf J}$ (current vector). We obtain
\begin{eqnarray*}\partial \cdot {\bf F}={\bf J}\quad\mbox{and}\quad\partial \wedge {\bf F}=\rho.\end{eqnarray*}
For the electric field yields now
\begin{eqnarray*}{\bf E}=({\bf F}\cdot e_0)e_0^{-1}.\end{eqnarray*}
Often one will have the field in the form of a vector potential ${\bf A}$. Then the equation
\begin{eqnarray*}\partial {\bf A}={\bf F}\end{eqnarray*}
yields the relations
\begin{eqnarray*}\partial \wedge{\bf A}=-{\bf F}\quad\mbox{and}\quad\partial \cdot{\bf A}=0.\end{eqnarray*}
The latter relation is called LORENZ condition.

From the electro-magnetic field ${\bf F}$ one can also deduce other important quantities such as

\begin{eqnarray*}{\bf T}_k=-\frac{1}{2} {\bf F}e_k{\bf F}\quad (k=0,1,2,3).\end{eqnarray*}
For $k=0$ follows
\begin{eqnarray*}{\bf T}_0=\frac{1}{2}({\bf E}^2+{\bf B}^2)e_0-e_1e_2e_3{\bf E}\cdot{\bf B}.\end{eqnarray*}
The coefficient of the first term reads energy density and the second one POYNTING vector. By scalar multiplication of ${\bf T}_k$$(k=1,2,3)$ with $e_\ell\: (\ell=1,2,3)$ from the right we get the so-called MAXWELL stress tensor:
\begin{eqnarray*}{\cal T}:=\left(T_{k\ell}\right)\quad\mbox{with}\quad {T}_{k\ell}={\bf T}_k\cdot e_\ell,\end{eqnarray*}
which had been introduced by M. RIESZ in 1947.

When we look for the flow of the energy momentum through a surface with$\alpha ={\alpha }$ as the unit vector of the normal at the point $x$, then it is given by the energy momentum tensor

\begin{eqnarray*}{\bf T}(\alpha ):=-\frac{1}{2}{\bf F}\alpha {\bf F}.\end{eqnarray*}
It is nothing other than as a vector valued linear function on the tangent space at each space point .

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