Using the "Dirac Prescription", we can preserve the format of a differential equation in its QM form. If we define the canonical variables s.t. they have the same commutation relations times $i$ as the Poisson brackets, Heisenberg and Hamiltons equations will have the same form.
This gives us a very simple way to quantise a boson field: Just set the commutation relations in the right way, and Heisberg's equation will just be the wave equation. From this point everything will follow: Heisenberg equation will be Lorentz invariant if the Lagrangian is, the Noether current is conserved in the QM version of the theory. Birds will sing.
my question is: What about the Dirac field?
Since we are defining the anti-commutator instead of the commutator, the same logic do not apply. Anti-commutator cannot (as far as I can tell) act as a derivative on a formal series like the commutator.
So why are we justified in quantising the Dirac Field like that?
(If you can, I would appreciate justifications that don't have anything to do with Feynman integrals)