# Naive first-quantization of the Dirac field?

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 ?

• Are you asking whether you can regard ψ as a single-particle wavefunction instead of a field, as Dirac did? Commented May 16, 2023 at 18:03

It would be correct and is often done. Just like in "normal" QM, you can define operators which correspond to physical quantities, such as $$\vec p = - i \vec \nabla$$. You can even discuss symmetries and conservation laws and stuff like that. If you're interested in that, there is a chapter in Messiah QM2 about that.