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## Main Question or Discussion Point

It is clear that a conserved current [tex]\partial_{\mu} J^\mu = 0[/tex] implies the existence of a conserved charge [tex]Q= \int d^3x J^0 [/tex]. Now I want to go the other way round: Suppose we have a basis of momentum eigenstates, such that these states are also eigenstates of the charge. Then clearly the charge commutes with the energy operator and is thus conserved but can we say anything else about the 4-vector current by for example invoking Lorentz invariance? It would be nice if there was a way to deduce [tex]\partial_{\mu} J^\mu = 0[/tex] but I do not see how that is possible