Quantizer
-------------------------------

The Weyl-Wigner-Moyal correspondence for the Dirac field in the field variables reads $$\widehat{F} = W(F[\pi_{\psi},\psi])= W(F[{\psi}^{\dag},\psi])=
\big({i\hbar}\big)^{-\infty} \int F[{\psi}^{\dag},\psi]
\widehat{\Omega} [{\psi}^{\dag},\psi] \prod d{\psi}^{\dag} d\psi ,
\label{operatorone}$$ where $$\widehat{\Omega}[{\psi}^{\dag},\psi] = (-i)^\infty\int {\cal D}
\mu \exp \bigg \{ -{i } \int d^3x \psi(\vec{x}) \cdot \mu(\vec{x})
\bigg \} \bigg|i \hbar {\psi}^{\dag} - {\hbar\mu \over 2}
\bigg\rangle \bigg\langle i \hbar {\psi}^{\dag} + {\hbar\mu \over
2}\bigg|$$ $$= (i)^\infty \int {\cal D} \lambda \exp \bigg \{\hbar \int d^3x
 {\psi}^{\dag}(\vec{x}) \cdot \lambda(\vec{x})  \bigg \} \bigg|\psi
 -{\hbar \lambda \over 2} \bigg\rangle \bigg\langle \psi + {\hbar \lambda \over 2}\bigg|.
\label{stratono}$$ In terms of the oscillator variables we have $$\widehat{F} = W(F[{\bf b}^*,{\bf b}])=
\big({i\hbar}\big)^{-\infty} \int \prod {d\bf b}^* {d\bf b} F[{\bf
b}^*,{\bf b}] \widehat{\Omega} [{\bf b}^*,{\bf b}],
\label{operatorsb}$$ and $$\widehat{\Omega}[{\bf b}^*,{\bf b}] =(-i)^\infty \int {\cal D}
\chi  \exp \bigg \{ -i\sum_{r=1}^4 \int {d^3p} {\bf b}(\vec{p},r)
\chi(\vec{p},r) \bigg \} \bigg|i \hbar{\bf b}^* - {\hbar\chi \over
2} \bigg\rangle \bigg\langle  i \hbar {\bf b}^* + {\hbar\chi \over
2}\bigg|$$ $$= (i)^\infty \int {\cal D} \xi \exp \bigg \{ \hbar \sum_{r=1}^4\int
 {d^3p}{\bf b}^*(\vec{p},r) \xi(\vec{p},r) \bigg
\} \bigg|{\bf b} - {\hbar\xi \over 2} \bigg\rangle \bigg\langle
{\bf b} + {\hbar\xi \over 2}\bigg|, \label{stratobs}$$ where $\chi$ and $\xi$ are Dirac spinors.

\[Observe that here $\infty$ 