When people want to find a conserved current which is constructed from a Dirac spinor, they consider the Dirac equation and its "Hermitian conjugate". But the equations they consider are ## (i\gamma^\mu \partial_\mu -m)\psi=0 ## and ##\bar{\psi}(i\gamma^\mu \overleftarrow{\partial_\mu}+m)=0 ## where ## \bar\psi=\psi^\dagger \gamma^0 ##.(adsbygoogle = window.adsbygoogle || []).push({});

But its obvious that they're assuming that ## (i\partial_\mu)^\dagger=-i\partial_\mu ##. But this is actually the naive way of finding the Hermitian conjugate of an operator because we know that the definition of the Hermitian conjugate of an operator is ## \langle A x,y\rangle=\langle x,A^\dagger y \rangle ## and we also know that by this definition, the operator ## i\partial_k## is actually Hermitian. But I don't know about ## i\partial_0 ## because the inner product doesn't contain a time integral.

Can anybody clarify this?

Thanks

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# Do we really mean Hermitian conjugate here?

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