From my understanding, the normal ordering of Klein-Gordon fields in QFT is valid because of the ambiguity that comes with quantizing a classical theory, in the sense that the conmutator of fields is negligible as a classical quantity. So, in principle, one could promote e.g. the hamiltonian as an operator with any given order of the operators $a_p$ and $a^\dagger_p$, and one chooses that which gives energies bounded from below.
In the case of the Dirac lagrangian, though, in order to get rid of negative energies, one ought to implement a different kind of normal ordering, with an additional change of sign, i.e. one that comes from an anticommutator:
$$\{ \psi_i(\mathbf{x},t), \psi_j^\dagger(\mathbf{y},t) \}=\delta^3(\mathbf{x-y})\delta_{ij}$$
The hamiltonian of the Dirac field is
$$\mathcal{H}=\int{\frac{d^3k}{(2\pi)^3}m\sum_\alpha{(b^\dagger_{k,\alpha}b_{k,\alpha}}-d_{k,\alpha}d^\dagger_{k,\alpha})}$$
My understanding is that this is valid for a non-quantized Dirac field as well. But isn't it the case that, since $d$ and $d^\dagger$ aren't operators in the classical picture, one could just commute them? I don't see how anything changes with respect to the case of scalar fields.
Is there, then, a similar argument to introduce normal ordering of Fermi fields? Or maybe, in other words, are anticommutators negligible in the classical regime for spinors, just as commutators are for scalar fields, or is the whole thing more subtle?
