\over
(2 \pi\hbar)^{3}} \ i \hbar {mc^2 \over E_{\vec{p}}}
\bigg[{b}_i^{*}(\vec{p},t) {{u}_{\alpha}^i}^{\dag}(\vec{p})
e^{-i\vec{p}\cdot\vec{x}/\hbar } + {d}_i(\vec{p},t)
{{v}_{\alpha}^i}^{\dag}(\vec{p}) e^{i\vec{p}\cdot\vec{x}/ \hbar
}\bigg], \label{fieldtwo}$$ where $b_i(\vec{p},t)=b_i(\vec{p})\exp{(-iE_{\vec{p}}t/\hbar)},$ $d_i^*(\vec{p},t)=d_i^*(\vec{p})\exp{(iE_{\vec{p}}t/\hbar)}$ and $E_{\vec{p}} =\sqrt{c^2\vec{p}^2 + m^2c^4}$. Here the index $i=1,2$ stands for the spin degrees of freedom. Therefore, $u^i$ and $v^i$ are solutions of the Dirac equation of momentum $p$ with positive or negative energies, respectively $$(\not \! \! p-mc)u^i=0,\ \ \ \ \ \ \ \ \ \ (\not \! \!
p+mc)v^i=0, \label{pnspinor}$$ with $\not \! \! p= \gamma^{\mu}p_{\mu}$; and they are restricted to satisfy the following relations $${u^i}^{\dag}(\vec{p})u^j(\vec{p})=
{v^i}^{\dag}(\vec{p})v^j(\vec{p})={E_{\vec{p}}
\over mc^2}\delta_{ij},$$ $$\overline{v}^i(\vec{p})u^j(\vec{p})=0.
\label{relations}$$ Classically Dirac fields are described by Grassmann variables since they are functions of the anti-commuting Grassmann variables $b_i(\vec{p}), b_i^*(\vec{p}) ,d_i(\vec{p})$ and $d_i^*(\vec{p})$. Poisson brackets for these variables are

$$\{b_i(\vec{p}),b_j^*(\vec{p}\ ') \}_P =
\{d_i(\vec{p}),d_j^*(\vec{p}\ ')\}_P= -{i \over \hbar} (2
\pi\hbar)^3\bigg({E_{\vec{p}}\over mc^2}
\bigg)\delta_{ij}\delta(\vec{p}-\vec{p}\ '). \label{grass}$$

.5truecm [*Oscillator Variables*]{}

The variables $b$ and $d$ are quite asymmetric and it is not easy to express these variables in terms of the field variables $\psi$ and $\pi_{\psi}$. To carry out the deformation quantization in the simplest possible way we introduce the variables ${\bf
b}(\vec{p},t,r)$ and ${\bf b}^{*}(\vec{p},t,r)$ [@hatfield] which are related to the variables $b$ and $d$ as 