# Page:Grundgleichungen (Minkowski).djvu/35

 (58) $|\Phi \Psi] = i[w, \Omega]^{*}$,

i,e.

$\Phi_{1}\Psi_{2} - \Phi_{2}\Psi_{1} = i(w_{3}\Omega_{4} - w_{4}\Omega_{3})$, etc.

.

The vector $\Omega$ fulfills the relation

 (59) $(w\bar{\Omega})=w_{1}\Omega_{1}+w_{2}\Omega_{2}+w_{3}\Omega_{3}+w_{4}\Omega_{4}=0$,

which we can write as

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

and $\Omega$ is also normal to w. In case $\mathfrak{w} =0$, we have $\Phi_{4} = 0,\ \Psi_{4} = 0,\ \Omega_{4} = 0$, and

 (60) $\Omega_{1} = \Phi_{2} \Psi_{3} - \Phi_{3} \Psi_{2},\ \Omega_{2} = \Phi_{3} \Psi_{1} - \Phi_{1} \Psi_{3},\ \Omega_{3} = \Phi_{1} \Psi_{2} - \Phi_{2} \Psi_{1}$,

I shall call $\Omega$, which is a space-time vector 1st kind the Rest-Ray.

As for the relation E), which introduces the conductivity $\sigma$, we have

$-w\bar{s}=-(w_{1}s_{1}+w_{2}s_{2}+w_{3}s_{3}+w_{4}s_{4})=\frac{-\left|\mathfrak{w}\right|s_{\mathfrak{w}}+\varrho}{\sqrt{1-\mathfrak{w}^{2}}}=\varrho'$

This expression gives us the rest-density of electricity (see §8 and §4). Then

 (61) $s+(w\bar{s})w$

represents a space-time vector of the 1st kind, which since $w\bar{w}=1$, is normal to w, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the x-, y-, z co-ordinates of the space-vector, then the component in the direction of $\mathfrak{w}$ is

$\mathfrak{s_{w}}-\frac{\left|\mathfrak{w}\right|\varrho'}{\sqrt{1-\mathfrak{w}^{2}}}=\frac{\mathfrak{s_{w}}-\left|\mathfrak{w}\right|\varrho}{\sqrt{1-\mathfrak{w}^{2}}}=\frac{\mathfrak{F_{w}}}{1-\mathfrak{w}^{2}}$

and the component in a perpendicular direction is $\mathfrak{s_{\bar{w}}}=\mathfrak{F_{\bar{w}}}$.

This space-vector is connected with the space-vector $\mathfrak{F}=\mathfrak{s}-\varrho\mathfrak{w}$, which we denoted in § 8 as the conduction-current.

Now by comparing with $\Phi = -wF$, the relation (E) can be brought into the form

 (E) $s+(w\bar{s})w=-\sigma wF$.