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This is at the current limit of my understanding of fermionic QFT. It's such a good question that it deserves an answer from an expert. a) Fourier analysing the second quantized Dirac operator field gives, $\hat{\psi}^{a}(x)$, $$\hat{\psi}^{a}(x)=\int \frac{d^{3}p}{(2\pi)^{3/2}}\exp(ix^{r}p^{r})\hat{\psi}^{a}(p)$$ where the label ...
One usually starts from the CCR for the creation/annihilation operators and derives from there the commutation rules for the fields. However, one can start from either (see for example here about this). Suppose we want then to start from the equal-time anticommutation rules for a Dirac field $\psi_\alpha(x)$:  \tag{1} \{ \psi_\alpha(\textbf{x}), ...