In S.Weinberg [QFT V1][1] section 7.1, in eq (7.1.15) and (7.1.16), he states that in order to be consistent with the previous-derived anti-commutator relation, we should take $\psi_{\text{n}}$ and $\psi^{\dagger}_{\text{n}}$ as canonical variables, $$q^{\text{n}}(x) \equiv\psi_{\text{n}}(x)\tag{7.1.15}$$ $$p_{\text{n}}(x)\equiv i\psi^{\dagger}_{\text{n}}(x)\tag{7.1.16}$$
However, if we focus on Lagrangian of spinor fields first and ignore Weinberg's formalism, $$\mathcal{L}\left[\psi(x), \bar\psi(x), ...\right] = \bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi(x)\tag{1}$$ and treat $\psi$ and $\bar\psi$ as independent canonical variables, I found that there are constraints, $$\frac{\partial \mathcal{L}}{\partial(\partial_{0}\bar\psi)} = 0$$ which is primary constraint. and thus, $$\frac{\partial\mathcal{L}}{\partial\bar\psi} = (i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$$ which is secondary constraint.
However, I haven't seen any refs. discussing the constraints in Dirac Spinor Field theory, does anyone know if it is proper to conduct the following Dirac algorithm and promote Dirac Bracket for spinors? [1]: https://www.amazon.com/-/zh_TW/Steven-Weinberg/dp/0521670535
