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[tex]F_{\mu\nu}F^{\mu \nu}[/tex]

where

[tex]F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu[/tex]

since the equations of motion for

[tex]A_{\mu}[/tex]

are the inhomogenous Maxwell equations.

However, here comes my problem:

If one expresses this action in terms of the electric and magnetic

field E and B

[tex]F_{\mu\nu}F^{\mu \nu}=B^2-E^2[/tex]

the equations of motion for those fields

would be

E=0

and

B=0.

So, where is the trick and what is the correct action

for the fields E and B?

Thanks in advance for your ideas and comments!