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This is probably a trivial question, but I just wanted to check my understanding.

Is the following expression for the Dirac Lagrangian correct?

[itex]\mathcal{L}=\frac{i}{2}\overline{\Psi}\gamma^{\mu}\overleftrightarrow{∂_{μ}}\Psi-\overline{\Psi}m\Psi=\frac{i}{2}\overline{\Psi}\gamma^{\mu}\partial_{μ}\Psi-\frac{i}{2}\overline{\left(\partial_{\mu}\Psi\right)}\gamma^{\mu}\Psi-\overline{\Psi}m\Psi-\overline{\Psi}m\Psi[/itex]

which can alternatively be expressed as,

[itex]\cal{L}=\frac{i}{2}\overline{\Psi}\gamma^{\mu}\partial_{\mu}\Psi+\left(\frac{i}{2}\overline{\Psi}\gamma^{\mu}\partial_{\mu}\Psi\right)^{\dagger}-\overline{\Psi}m\Psi[/itex]

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# Dirac Lagrangian density

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