Electromagnetic equations

Gauge equations

Electromagnetic field

$F^{\mu \nu }={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }$

$P^{\lambda \sigma }={\partial }^{\lambda }{\Lambda }^{\sigma }-{\partial }^{\sigma }{\Lambda }^{\lambda }$

from (4-3)

Field equation

${\partial }_{\mu }{F}^{\mu \nu }=\frac{4\pi }{c}{j}^{\nu }$

${\partial }_{\lambda }{P}^{\lambda \sigma }=J^{\sigma }$

from (4-5)

Field equation

${\partial }^{\alpha }{F}^{\mu \nu }+{\partial }^{\mu }{F}^{\nu \alpha }+{\partial }^{\nu }{F}^{\alpha \mu }=0$

${\partial }^{\alpha }{P}^{\lambda \sigma }+{\partial }^{\lambda }{P}^{\sigma \alpha }+{\partial }^{\sigma }{P}^{\alpha \lambda }=0$

from (4-8)

Continuous eq.

${\partial }_{\nu }{j}^{\nu }=0$

${\partial }_{\sigma }{J}^{\sigma }=0$

from (4-7)