This is a popular question on this site but I haven't found the answer I'm looking for in other questions. It is often stated that charge conservation in electromagnetism is a consequence of local gauge invariance, or perhaps it is due to some global phase symmetry. Without talking about scalar or spinor fields, the EM Lagrangian that we're familiar with is: L=−14F2−A⋅J+Lmatter(J) The equation of motion for A is simply ∂μFμν=Jν From which it follows that J is a conserved current (by the antisymmetry of the field strength). But what symmetry gave rise to this? I'm not supposing that my matter has any global symmetry here, that I might be able to gauge. Then so far as I can tell, the Lagrangian given isn't gauge invariant. The first term is, indeed, but the second term only becomes gauge invariant on-shell (since I can do some integrating by parts to move a derivative onto J). If we demand that our Lagrangian is gauge-invariant even off-shell, then we can deduce that ∂⋅J=0 off-shell and hence generally. But we can't demand that this hold off-shell, since J is not in general divergenceless!
For concreteness, suppose that Lmatter(J)=12(∂μϕ)(∂μϕ)Jμ≡∂μϕ Then we find that ϕ (a real scalar) satisfies some wave equation, sourced by A. The equations of motion here constrain the form of J, but off-shell J is just some arbitrary function, since ϕ is just some arbitrary function. Then it is clear that the Lagrangian is not gauge-invariant off-shell.
And this is a problem, because when we derive conserved quantities through Noether's theorem, it's important that our symmetry is a symmetry of the Lagrangian for any field configuration. If it's only a symmetry for on-shell configurations, then the variation of the action vanishes trivially and we can't make any claims about conserved quantities.
So here's my question: what symmetry does the above Lagrangian have that implies the conservation of the quantity J, provided A satisfies its equation of motion? Thank you.
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