Maxwell equations

The Maxwell equations are the special case of the Yang-Mills equations when the gauge group is just the abelian circle group U(1). These equations take the form

$\partial ^{\alpha }F_{\alpha \beta }=0$

where F is the curvature of a U(1) connection $A_{\alpha }:R^{1+d}\to R$:

$F_{\alpha \beta }:=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }.$

Note that the Bianchi identity also gives the equation

$\partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\gamma }F_{\beta \alpha }=0$

which is also traditionally given as one of the Maxwell equations.

In the presence of a current $j_{\beta }$, the Maxwell equations now take the form

$\partial ^{\alpha }F_{\alpha \beta }=j_{\beta }$

(after normalizing various physical constants).

The Maxwell equations are linear. However, by coupling the Maxwell equations to a scalar field one obtains the nonlinear Maxwell-Klein-Gordon equations.