# Second quantization field derivative commutator

For the symmetrized quantum field theory Lagrangian of the free dirac field

$$\mathcal{L} = i[\overline{\psi}_a,({\partial_\mu}\gamma^\mu \psi)^a] -m[\overline{\psi}_a,\psi^a ]$$

the terms are symmetrized since $\overline{\psi}_a$ and $\psi_a$ do not anti-/commute. Is this the correct symmetrized Lagrangian? Now I tried to order the terms in the Lagrangian such that

$$\mathcal{L} = i\overline{\psi}{\partial_\mu}\gamma^\mu \psi +A -m \overline{\psi}\psi +B$$

Can I determine $A,B$ with the canonical commutation relations since I do not know something about the commutation relations about $\overline{\psi}$ and ${\partial_\mu}\gamma^\mu \psi$? I hope to finde that

$$\delta \mathcal{L} = \delta (i\overline{\psi}{\partial_\mu}\gamma^\mu \psi) -\delta(m \overline{\psi}\psi)$$

with $\delta A = 0 = \delta B$ where $\delta$ is the variation.