$$H_D[\pi_{\psi}, \psi] = \int d^3x \ \overline{\psi}(\vec{x},t)
\cdot \hbar c \big(-i\gamma^j\partial_j + {mc\over\hbar}
\big)\psi(\vec{x},t)$$ $$= \int d^3x \ \psi^{\dag}(\vec{x},t) \cdot i \hbar
{\partial \over
\partial t} \psi(\vec{x},t)$$ $$= \int d^3x \ \pi_{\psi}(\vec{x},t) \cdot {\partial \over
\partial t} \psi(\vec{x},t),
\label{dhamilton}$$ where as usual $ \pi_\psi \cdot \partial_t \psi \equiv
\sum_{\alpha=1}^4 \pi_{\psi_\alpha} \partial_t \psi_\alpha, $ and $\gamma^i = \beta \alpha^i$, $\gamma^0 = \beta,$ with $i=1,2,3$. Note that we use the Weyl (or chiral) representation of the Dirac matrices $\gamma^\mu,$ $\mu=0,1,2,3$ $$\begin{array}{ccc}
  \gamma^0 = \left(\begin{array}{cc}
  1 & 0 \\
  0 & -1 \\
\end{array}\right), & \hspace{0.5cm}  \gamma^j = \left(\begin{array}{cc}
  0 & \sigma_j \\
  -\sigma_j & 0 \\
\end{array}\right), &\hspace{0.5cm} j=1,2,3 \\
\end{array}$$ with $\sigma_j$ being Pauli’s matrices. The Dirac matrices have the following properties: $ {\gamma^j}^\dag = -\gamma^j,$ ${\gamma^0}^\dag=\gamma^0,$ $ \gamma^\mu \gamma^\nu + \gamma^\nu
\gamma^\mu = 2\eta^{\mu\nu}.$ According to the definition of the Poisson bracket for Grassmann fields given by (\[poissonsusy\]), the Poisson bracket corresponding to the Dirac free field takes the form $$\{F,G\}_P = F \ {\buildrel \leftrightarrow \over {\cal P}}\ G$$ where $$\begin{aligned}
{\buildrel \leftrightarrow \over {\cal P}} &=& \int d^3x \bigg(
{{\buildrel \leftarrow \over \delta}\over \delta \psi(\vec{x},t)}
{{\buildrel \rightarrow \over \delta}\over \delta
\pi_{\psi}(\vec{x},t)} + {{\buildrel \leftarrow \over \delta}\over
\delta \pi_{\psi}(\vec{x},t)} 