{{\buildrel
\rightarrow \over \delta}\over \delta \psi(\vec{x},t)} \bigg)\nonumber\\
&=&-{i \over \hbar} \int d^3x \bigg( {{\buildrel \leftarrow \over
\delta}\over \delta \psi(\vec{x},t)} {{\buildrel \rightarrow \over
\delta}\over \delta \psi^\dag(\vec{x},t)} + {{\buildrel \leftarrow
\over \delta}\over \delta \psi^\dag(\vec{x},t)} {{\buildrel
\rightarrow \over \delta}\over \delta \psi(\vec{x},t)} \bigg),
\label{PB}\end{aligned}$$ and ${{\buildrel \leftarrow \over \delta}\over \delta
\psi(\vec{x},t)} {{\buildrel \rightarrow \over \delta}\over \delta
\pi_{\psi}(\vec{x},t)} \equiv \sum_{\alpha =1}^4 {{\buildrel
\leftarrow \over \delta}\over \delta \psi_{\alpha}(\vec{x},t)}
{{\buildrel \rightarrow \over \delta}\over \delta
\pi_{\psi_{\alpha}}(\vec{x},t)},$ etc.

Consequently, for $\psi_{\alpha}(\vec{x},t)$ and ${\pi_{\psi}}_{\alpha} (\vec{x},t)$ one gets $$\{ \psi_{\alpha} (\vec{x},t), {\pi_{\psi}}_{\beta} (\vec{y},t)
\}_{P} =  \delta (\vec{x}-\vec{y}) \delta_{\alpha\beta},$$ $$\{ \psi_{\alpha} (\vec{x},t), \psi_{\beta} (\vec{y},t) \}_{P} = 0,
\hspace{1.0cm} \{
{\pi_{\psi}}_{\beta}(\vec{x},t),{\pi_{\psi}}_{\beta}(\vec{y},t)
\}_{P}=0. \label{comm}$$ (Remember that $\alpha=1,2,3,4$ runs over the components of the Dirac spinor).

Usually quantization is done by the substitution $
{\{\cdot,\cdot\}}_P \mapsto {1\over i\hbar} {[\cdot,\cdot]}_+$. Thus we obtain $$\begin{array}{l}
{[\widehat{\psi}_\alpha(\vec{x},t),
\widehat{\psi}_\beta(\vec{y},t)]}_+ =
{[\widehat{\pi}_{\psi_\alpha}(\vec{x},t),
\widehat{\pi}_{\psi_\beta}
(\vec{y},t)]}_+ = 0, \\
{[\widehat{\psi}_\alpha(\vec{x},t), \widehat{\pi}_{\psi_\beta}
(\vec{y},t)]}_+
= i\hbar \delta_{\alpha\beta} \delta(\vec{x}-\vec{y}),\\
{[\widehat{\psi}_\alpha(\vec{x},t),
\widehat{\psi}_\beta^\dag(\vec{y},t)]}_+ =
 \delta_{\alpha\beta} \delta(\vec{x}-\vec{y}).\\
\end{array}$$

The field variables $\psi_{\alpha}(\vec{x},t)$ and ${\pi_{\psi}}_{\alpha} (\vec{x},t)$ can be expanded into the plane waves $$\psi_{\alpha}(\vec{x},t) = \sum_{i=1}^2 \int {d^3p
\over (2 \pi\hbar)^{3}} {m c^2\over E_{\vec{p}}} \bigg[ {b}_i
(\vec{p},t) u_{\alpha}^i(\vec{p}) e ^{i\vec{p}\cdot\vec{x}/ \hbar
} + {d}_i^* (\vec{p},t) {v_{\alpha}^i}(\vec{p})
e^{-i\vec{p}\cdot\vec{x}/\hbar} \bigg], \label{fieldone}$$ $${\pi_{\psi}}_{\alpha} (\vec{x},t) = i \hbar
{\psi}_{\alpha}^{\dag}(\vec{x},t) = \sum_{i=1}^2 \int {d^3p 