This is probably a stupid question, but when I apply the Euler-Lagrange equation to the Lagrangian density of the Dirac field I get for the conjugate field(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\bar{\psi} (-i \partial_\mu \gamma^{\mu} -m) = 0[/tex] (derivative acts to the left).

But when I take a hermitian conjugate of the Dirac equation for the field I get an extra [tex]\gamma^0[/tex]:

[tex]0 = \left[ (i \partial_\mu \gamma^{\mu} -m)\psi \right]^\dagger = \psi^\dagger (-i \partial_\mu (\gamma^{\mu})^\dagger -m) = \psi^\dagger (-i \partial_\mu \gamma^0 \gamma^{\mu} \gamma^0 -m) = \psi^\dagger \gamma^0(-i \partial_\mu \gamma^{\mu} \gamma^0 -m) = \bar{\psi} (-i \partial_\mu \gamma^{\mu} \gamma^0 -m)[/tex].

What am I missing?

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# Dirac equation for the conjugated field

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