Page:Grundgleichungen (Minkowski).djvu/34

(54)	$\Psi _{4}=i({\mathfrak {w}}_{x}\Psi _{1}+{\mathfrak {w}}_{y}\Psi _{2}+{\mathfrak {w}}_{z}\Psi _{3})$

The vector $\Psi$ is perpendicular to w; we can call it the Magnetic rest-force.

Relations analogous to these hold among the quantities $iwF^{*},{\mathfrak {M,E,w}}$ and Relation (D) can be replaced by the formula

{D}	$wF^{*}=\mu wf^{*}$

We can use the relations (C) and (D) to calculate F and f from $\Phi$ and $\Psi$, we have

$wF=-\Phi ,\ wF^{*}=-i\mu \Psi ,\ wf=-\epsilon \Phi ,\ wf^{*}=-i\Psi$

and applying the relation (45) and (46), we have

(55)	$F=[w,\Phi ]+i\mu [w,\Psi ]^{*}$,

(56)	$f=\epsilon [w,\Phi ]+i[w,\Psi ]^{*}$,

i.e.

$F_{12}=(w_{1}\Phi _{2}-w_{2}\Phi _{1})+i\mu (w_{3}\Psi _{4}-w_{4}\Psi _{3})$, etc. $f_{12}=\epsilon (w_{1}\Phi _{2}-w_{2}\Phi _{1})+i(w_{3}\Psi _{4}-w_{4}\Psi _{3})$, etc.

Let us now consider the space-time vector of the second kind $[\Phi \Psi ]$, with the components

$\Phi _{2}\Psi _{3}-\Phi _{3}\Psi _{2},\ \Phi _{3}\Psi _{1}-\Phi _{1}\Psi _{3},\ \Phi _{1}\Psi _{2}-\Phi _{2}\Psi _{1}$, $\Phi _{1}\Psi _{4}-\Phi _{4}\Psi _{1},\ \Phi _{2}\Psi _{4}-\Phi _{4}\Psi _{2},\ \Phi _{3}\Psi _{4}-\Phi _{4}\Psi _{3}$,

Then the corresponding space-time vector of the first kind

$w[\Phi ,\Psi ]=-(w{\overline {\Psi }})\Phi +w({\overline {\Phi }})\Psi$

vanishes identically owing to equations 49) and 53).

Let us now take the vector of the 1st kind

(57)	$|\Omega =iw[\Phi ,\ \Psi ]^{*}$

with the components

$\Omega _{1}=-i\left|{\begin{array}{ccc}w_{2},&w_{3},&w_{4}\\\Phi _{2},&\Phi _{3},&\Phi _{4}\\\Psi _{2},&\Psi _{3},&\Psi _{4}\end{array}}\right|$, etc.

Then by applying rule (45), we have