Friday, 31 July 2020

electromagnetism - Is $F_{munu}F^{munu}$ the only possible gauge-invariant Lagrangian for the electromagnetic field?


Maxwell's equations can be derived from a Lagrangian formulation using the Lagrangian term (modulo some constants) $$\mathcal L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A_\mu J^\mu.$$ Focusing on the free term for the moment, I've seen mentioned (though I can't find a source right now) that the $F^2$ term is the only possible gauge-invariant Lagrangian for electromagnetism.


Is this the case? How can we prove it?




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