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Homework Statement
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Show that in order for the free Lagrangian to be invariant when ## A^\mu ## is transformed by a transformation U, it has to transform as below:
## A'^{\mu}=\frac i g (\partial^\mu U) U^{-1}+U A^\mu U^{-1} ##
Homework Equations
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The wording of the problem is a bit awkward I think, but it seems it means the free EM Lagrangian!
## L_{free \ EM}=-\frac 1 4 F_{\mu \nu} F^{\mu \nu} ##
## F_{\mu \nu}=\partial_\mu A_\nu - \partial_\nu A_\mu ##
The Attempt at a Solution
I've never seen it like this before. It was always figuring out the transformation rule for ## A^\mu ## somehow that it leaves another field's Lagrangian invariant!
The only method that comes into my mind is substituting ## A'^\mu ## in ## F'_{\mu \nu} F'^{\mu \nu} ## and expanding it. But it doesn't show any sign of convinient cancellations early enough and it seems I have to expand it all the way. But it would be a giant with 100 terms in it! So I think there has to be a better way for doing this. But I just can't figure it out!
Any small hint is appreciated.
Thanks