I find the Lagrangian associated with the Dirac equation given in texts as(adsbygoogle = window.adsbygoogle || []).push({});

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

or

[tex]\mathcal{L}=i\bar{\psi}\gamma^\mu \partial_\mu \psi- m\bar{\psi}\psi[/tex]

[tex]\mathcal{L}=i \psi^{\dagger}\gamma^0\gamma^\mu \partial_\mu \psi- m\psi^{\dagger}\gamma^0\psi[/tex]

Taking the hermitian conjugate of the expression on the right, we get

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

[tex]=-i \partial_\mu\psi^{\dagger}\gamma^0\gamma^\mu \psi- m\psi^{\dagger}\gamma^0\psi[/tex]

[tex]=-i \partial_\mu\bar{\psi}\gamma^\mu \psi- m\bar{\psi}\psi[/tex]

which, as far as I can tell, is not equal to [tex]\mathcal{L}[/tex]

So if it is not hermitian, is that ok?

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# Shouldn't Lagrangians be real (hermitian)?

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