It's frequently said, that the Lagrangian of a Dirac field is
$$\mathcal{L}=i\bar{\psi}(\gamma^\mu\partial_\mu-m)\psi.$$
Applying the Euler-Lagrange equation we get the Dirac equation. Although, we can get a similar construction of Lagrangian, which leads to the same equation, e.g.
$$\mathcal{L}=\frac{i}{2}\left[\bar{\psi}\gamma^\mu\partial_\mu\psi-\left(\partial_\mu\bar{\psi}\right)\gamma^\mu\psi-m\bar{\psi}\psi\right].$$
Is there a way to derive the Lagrangian we normally use?
Best Answer
Both are perfectly valid. I actually prefer the second.