Maxwell equations

From DispersiveWiki

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 $j β$ , 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.

Maxwell equations

From DispersiveWiki

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