{\bf b}^{*}(\vec{p}\ ',r') \}_{P} = -{i
\over \hbar} \delta (\vec{p} - \vec{p}\ ') \delta_{rr'},$$ $$\{ {\bf b}(\vec{p},r), {\bf b}(\vec{p}\ ',r')
\}_{P} = 0, \ \ \ \ \ \{ {\bf b}^{*}(\vec{p},r), {\bf
b}^{*}(\vec{p}\ ',r') \}_{P}= 0. \label{ocommrela}$$ Multiplying Eq. (\[fpone\]) by $w_{\beta}^{\dag}(\vec{p}\ ',r')
\exp \big(-i\vec{p}\ '\cdot\vec{x}/\hbar\big)$ and integrating over $\mathbb{R}^3$ we get $$\int d^3x \ w_{\beta}^{\dag}(\vec{p\
}',r')\psi_{\alpha}(\vec{x},t) \exp \big( -i\vec{p}\
'\cdot\vec{x}/\hbar \big)$$ $$= \sum_{r=1}^4 \int {d^3x \ d^3p \over (2 \pi\hbar)^{3/2}}
\sqrt{mc^2 \over E_{\vec{p}}} w_{\beta}^{\dag} (\vec{p}\ ',r')
{\bf b}(\vec{p},t,r)w_{\alpha}(\vec{p},r) \exp \big(
i(\vec{p}-\vec{p}\ ')\cdot\vec{x}/\hbar \big). \label{comput}$$ Using now the relations (\[ident\]), after some computations we obtain $${\bf b}(\vec{p},t,r)=\sqrt{m c^2 \over E_{\vec{p}}}
\sum_{\alpha}\int {d^3x \over (2\pi\hbar)^{3/2}}\
w_{\alpha}^{\dag}(\vec{p},r) \psi_{\alpha}(\vec{x},t) \exp
\big(-i\vec{p}\cdot\vec{x}/\hbar \big), \label{annihi}$$ $${\bf b}^{*}(\vec{p},t,r)=\sqrt{mc^2 \over E_{\vec{p}}}
\sum_{\alpha} \int {d^3x \over (2\pi\hbar)^{3/2}}\
\psi_{\alpha}^{\dag}(\vec{x},t) w_{\alpha}(\vec{p},r)
 \exp \big(i\vec{p}\cdot\vec{x}/\hbar \big).
\label{creat}$$ Thus the Grassmann variables ${\bf b}$ and ${\bf b}^{*}$, determine precisely the canonical conjugate variables which we will use to describe the Dirac field in the WWM formalism.

Substituting (\[fpone\]) and (\[fptwo\]) into the Hamiltonian (\[dhamilton\]) yields $$H_D[{\bf b}^*,{\bf
b}]=  \sum_{r=1}^4 \int {d^3p} \varepsilon_r E_{\vec{p}} {\bf
b}^*(\vec{p},r) {\bf b}(\vec{p},r), \label{hamione}$$ or in terms of the standard variables, $$H_D[b,b^*,d,d^*]= \sum_{i=1}^2 \int {d^3p \over (2\pi\hbar)^{3}}
mc^2 \bigg[ {b}_i^*(\vec{p}) {b}_i(\vec{p}) - {d}_i(\vec{p})
{d}_i^*(\vec{p}) \bigg]. \label{hamitwo}$$

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The Stratonovich-Weyl 