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I'm trying to get the current vector $J^\mu$ of a Klein-Gordon equation:

$$\Psi^* \Box \Psi =\Psi^* \partial^{\mu} \partial_\mu \Psi= \partial^{\mu}(\Psi^*\partial_\mu \Psi)-\partial^\mu \Psi^*\partial_\mu \Psi $$

$$\Psi \Box \Psi^* =\Psi\partial^{\mu} \partial_\mu \Psi^*= \partial^{\mu}(\Psi\partial_\mu \Psi^*)-\partial^\mu \Psi\partial_\mu \Psi^*. $$

Now I have to substract (1)-(2) so the term in the right is the same and it dissapears. My question is why are theose terms the same, I cant' see it.

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I will assume, for simplicity, that your fields are defined in a Euclidean or Minkowski space, with metric $\eta_{\mu \nu}$.

Then, remember that $\partial^{\mu} = \eta^{\mu \nu} \partial_{\nu}$. Thus, both terms in of the RHS of your equations are actually equal.

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