Canonical Quantisation vs the Dirac Field, why does it even work? 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)
 A: TL;DR: The same logic still applies to Grassmann-odd variables with pertinent sign factors.


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*The Dirac prescription $\{\cdot,\cdot\}_{PB}\to [\cdot,\cdot]/i\hbar$ in principle also work for supervariables (up to higher-order quantum corrections, cf. the deformation quantization paradigm). The relevant (graded) bracket structures $\{\cdot,\cdot\}_{PB}$ and $[\cdot,\cdot]$ are a super-Poisson bracket and a supercommutator, respectively. 

*The Schrödinger representation$^1$
$$ \hat{q}~=~q  , \qquad \hat{p}~=~\frac{\hbar}{i}\frac{\partial_R}{\partial q}~=~(-1)^{|q|}\frac{\hbar}{i}\frac{\partial_L}{\partial q}, \qquad[\hat{q},\hat{p}]~=~i\hbar{\bf 1},\tag{1}$$
also works for supervariables &  supercommutators.

*For further details, see e.g. this, this & this related Phys.SE posts.
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$^1$ Here the subscript $R$ & $L$ denote left & right derivatives, and $|q|$ denotes the Grassmann-parity of $q$. NB: Apart from the explicitly written sign factor in eq. (1), there may be additional sign factors in accordance with the Koszul sign rule/convention, stemming from permuting Grassmann-odd objects.
