\widehat{\Omega}[{\bf b}^*,{\bf b}] \widehat{\rho}^{phys}\bigg\}$$ $$= 2^{-\infty} \int {\cal D} \xi \exp
\bigg\{ -{2}\sum_{r=1}^4 \int d^3p \ {\bf b}^*(\vec{p},r)
\xi(\vec{p},r) \bigg\} \langle {\bf b} - \xi |
\widehat{\rho}^{phys}| {\bf b} + \xi \rangle. \label{wignerf}$$ In the case when $\widehat{\rho}^{phys} = |\Phi \rangle \langle
\Phi^\dag|$ the above equation gives $$\rho_{_W}[{\bf b}^*,{\bf b}] = 2^{-\infty} \int {\cal D} \xi \exp
\bigg\{-{2} \sum_{r=1}^4 \int d^3p \ {\bf b}^*(\vec{p},r)
\xi(\vec{p},r)  \bigg\} \Phi[{\bf b} - \xi] \Phi^{\dag}[{\bf b} +
\xi], \label{wignerftwo}$$ where $\Phi[{\bf b}]=\langle {\bf b} | \Phi \rangle$ and $\Phi^\dag[{\bf b}]=\langle  \Phi^\dag |{\bf b}  \rangle$.

\[Note that since $r$ runs through $1,2,3$ and $4$ i.e., through an even number of indices the Jacobian of transformation leading from (\[ttres\]) or (\[tcinco\]) to (\[wignerf\]) or (\[wignerftwo\]), respectively, is equal to $1$.\]

1truecm [*Example: The Ground State*]{}

In the Schrödinger representation the ground state corresponding to the system described by the Dirac equation can be easily found. In order to do that we use the wave functional defined by $\Phi[\psi] = \langle \psi | \Phi \rangle$.Then the canonical conjugate momentum to $\psi$ is represented by $i \hbar
\widehat{\psi}^{\dag}= i \hbar{\delta \over \delta \psi}$. Substituting this momentum into the Hamiltonian (\[dhamilton\]) we get the time independent Schrödinger equation $$\int 