{i\over \hbar}
(2\pi\hbar)^3 \sum_{i=1}^{2}\int {d^3p } \bigg( {E_{\vec{p}} \over
mc^2} \bigg) \bigg \{ \bigg( {{\buildrel{\leftarrow}\over
{\delta}}\over \delta {b}_i(\vec{p})}
{{\buildrel{\rightarrow}\over {\delta}}\over \delta
{b}_i^*(\vec{p})} + {{\buildrel{\leftarrow}\over {\delta}}\over
\delta {b}_i^*(\vec{p})} {{\buildrel{\rightarrow}\over
{\delta}}\over \delta {b}_i(\vec{p})} \bigg)
 +  \bigg( {{\buildrel{\leftarrow}\over {\delta}}\over
\delta {d}_i^*(\vec{p})} {{\buildrel{\rightarrow}\over
{\delta}}\over \delta {d}_i(\vec{p})} +
{{\buildrel{\leftarrow}\over {\delta}}\over \delta {d}_i(\vec{p})}
{{\buildrel{\rightarrow}\over {\delta}}\over \delta
{d}_i^*(\vec{p})} \bigg)\bigg\}. \label{dpoissonbs}$$ The operator $\buildrel{\leftrightarrow}\over {\cal P}_D$ determines the Poisson bracket $\{F,G\}_P=F\buildrel{\leftrightarrow} \over {\cal P}_D G$ defined by the symplectic structure $$\omega_D=\int d^3x \sum_{\alpha} \delta \psi_{\alpha}(\vec{x})
\wedge \delta \pi_{\psi}(\vec{x})$$ $$= i \hbar \int d^3x \sum_{\alpha} \delta \psi_{\alpha}(\vec{x})
\wedge \delta \psi_{\alpha}^{\dag}(\vec{x}). \label{dsymplstru}$$ This symplectic structure defines the symplectic supermanifold $({\cal Z}_D,\omega_D)$ which represents the phase space. In terms of the original field variables $\psi_{\alpha}^{\dag},\psi_{\alpha}$ one has $$\buildrel{\leftrightarrow}\over {\cal P}_D := - {i \over \hbar}
\sum_{\alpha}\int d^3x \bigg({{\buildrel{\leftarrow}\over
{\delta}}\over \delta \psi_{\alpha}(\vec{x})}
{{\buildrel{\rightarrow}\over {\delta}}\over \delta
\psi_{\alpha}^{\dag}(\vec{x})} + {{{\buildrel{\leftarrow}\over
{\delta}}\over \delta \psi_{\alpha}^{\dag}(\vec{x})}
{{\buildrel{\rightarrow}\over {\delta}}\over \delta
\psi_{\alpha}(\vec{x})}}\bigg). \label{anotherpoiss}$$ .5truecm

The Wigner Functional
---------------------

The Wigner functional for the Dirac free field is defined in analogy to the scalar field case. Let $\widehat{\rho}^{phys}$ be the density operator corresponding to the quantum physical state of the Dirac field. Then the Wigner functional corresponding to this state is given by $$\rho_{_W}[{\bf b}^*,{\bf b}] = {\rm tr} \bigg\{
