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So, I understand the construction "on-shell", once we solve Maxwell's equations to find [itex]F^{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}[/itex], and how we can then write an action which is both gauge and Lorentz invariant.

But before we know anything about gauge symmetry, which comes about once we solve for F and express it in terms of A, what tells us that our action in not good enough (i.e. there is no local action to be constructed)?

Thanks