To begin with, please note that I am fully aware of the differences between the confusingly named "first quantization" and "second quantization", and how they correspond to classical non-fields, and classical fields. (thanks for all the great contributions regarding these topics by the way !)
Take the Dirac partial differential equation
$$ (i\gamma^{\mu}\partial_{\mu}-m)\psi = 0. $$
Defining $\vec{\alpha}=\gamma^0\vec{\gamma}$ and $\beta=\gamma^0$, we get the Schrödinger-looking equation
$$ i\partial_t\psi = h_D\psi $$
where $$ h_D = -i\vec{\alpha}\cdot\vec{\nabla} + m\beta. $$
This is still a partial differential equation, $h_D$ being a matrix differential operator that can be thought of acting on the Hilbert space $\mathbb{C}^4 \otimes L^2(\mathbb{R}^3)$. Now, how right would it be to try and quantize this equation by using the $\hat{\vec{p}} \leftrightarrow -i\vec{\nabla}$ correspondence, and studying the quantum Hamiltonian operator
$$ \hat{h}_D = \sum_{k=1}^{3}\hat{\alpha}_k\otimes\hat{p}_k +m\hat{\beta}\otimes\hat{\mathbb{1}} \ \ \ ? $$
How mathematically relevant is it, besides the interpretation problems fixed by the proper 2nd quantization of the Dirac field carried out in textbooks ?
