Derivation of Dirac equation using the Lagrangian density for Dirac field How can I derive the Dirac equation from the Lagrangian density for the Dirac field?
 A: The Lagrangian density for a Dirac field is
$$
\mathcal{L} = i\bar\psi\gamma^\mu\partial_\mu\psi -m \bar\psi\psi
$$
The Euler-Lagrange equation reads
$$
\frac{\partial\mathcal{L}}{\partial\psi} - \frac{\partial}{\partial x^\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\right] = 0
$$
We treat $\psi$ and $\bar\psi$ as independent dynamical variables. In fact, it is easier to consider the Euler-Lagrange for $\bar\psi$
$$
\frac{\partial\mathcal{L}}{\partial\bar\psi} - \frac{\partial}{\partial x^\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\bar\psi)}\right] = 0\\
\Rightarrow i\gamma^\mu\partial_\mu\psi -m\psi - \frac{\partial}{\partial x^\mu}[ 0] = 0\\
\Rightarrow i\gamma^\mu\partial_\mu\psi -m\psi=0
$$
The partial differentiation is trivial - remember that $\bar\psi$ and $\partial_\mu\bar\psi$ are treated as though independent. We recover the Dirac equation as expected. If we had instead chosen the Euler-Lagrange for $\psi$, we would have found the conjugate Dirac equation.
