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I have a question about conserved currents. Suppose that we have a Lagrangian which is perfectly invariant under a certain infinitesimal transformation of the field ##\phi\to\phi+\alpha\Delta\phi##, i.e. ##\mathcal{L}\to\mathcal{L}##. Now, more generally, the Lagrangian should be invariant up to a total divergence, ##\mathcal{L}\to\mathcal{L}+\alpha\partial_\mu\mathcal{J}^\mu##. This means that in our case ##\partial_\mu\mathcal{J}^\mu=0## and so ##\mathcal{J}^\mu## could be any aribtrary function with vanishing divergence.

Now, the general form for the conserved current is:

$$

j_\mu(x)=\frac{\partial \mathcal{L}}{\partial(\partial^\mu\phi)}\Delta\phi-\mathcal{J}_\mu.

$$

Suppose now that the first term has zero divergence as well because of the equation of motion. Now, it seems to me that in this case, I could simply set the current to be zero by choosing ##\mathcal{J}^\mu=\frac{\partial \mathcal{L}}{\partial(\partial^\mu\phi)}\Delta\phi## since this indeed satisfies ##\partial^\mu\mathcal{J}_\mu=0##.

Am I doing something wrong or is this indeed the case?

Thanks!

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# Noether's theoem and conserved current arbitrariness

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