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The conserved current in KG is

$$j^{\mu}=i(\phi^*\partial^{\mu}\phi-\phi\partial^{\mu}\phi^*) =2p^{\mu}|N|^2$$

where N is a normalization factor. This current can't be understood as a probability current because $j^0$ can be negative. This problem is solved thinking of $j^{\mu}$ as a charge current and not probability. Then negative and positive energy solutions have opposite charges.

On the other hand, the conserved current in Dirac equation is

$$j^{\mu}=\bar{\psi}\gamma^{\mu}\psi$$

which is definite positive. This is okay because we think of this current, again, as a probability current.

Which is the correct interpretation? are conserved currents of classical equations charge currents or probability currents? Is this solved in any way in QFT?

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