y')
\psi (x - x', y - y') dx' dy' .\hspace*{2cm}\end{aligned}$$ If the wave function $\psi(x,y)$ is the Gaussian form of Eq.(\[gau00\]), the Wigner function becomes $$\label{wigf00}
W(x,y:p,q) = \left(\frac{1}{\pi}\right)^{2}
          \exp{\left\{-\left(x^2 + p^2 + y^2 + q^2\right)\right\} } .$$

The Wigner function is defined over the four-dimensional phase space of $(x, p, y, q)$ just as in the case of classical mechanics. The unitary transformations generated by the operators of Eq.(\[dir10\]) are translated into Wigner transformations [@knp91; @hkn95jmp; @kn13]. As in the case of Dirac’s oscillators, there are ten corresponding generators applicable to the Wigner function. They are $$\begin{aligned}
\label{rotphase}
L_{1} &=& +{i\over 2}\left\{\left(x{\partial \over \partial q} -
q{\partial \over \partial x} \right) +
\left(y{\partial \over \partial p} -
p{\partial \over \partial y} \right)\right\}, \nonumber \\[3mm]
L_{2} &=& -{i\over 2}\left\{\left(x{\partial \over \partial y} -
y{\partial \over \partial x}\right) +
\left(p {\partial \over \partial q} -
q{\partial \over \partial p}\right)\right\} ,\nonumber \\[3mm]
L_{3} &=& +{i\over 2}\left\{\left(x{\partial \over \partial p} -
p{\partial \over \partial x}\right) -
\left(y{\partial \over \partial q} -
q{\partial \over \partial y}\right)\right\} , \nonumber \\[3mm]
S_{3} &=& -{i\over 2}\left\{\left(x{\partial \over \partial p} -
p{\partial \over \partial x}\right) +
\left(y{\partial \over \partial q} -
q{\partial \over \partial y}\right)\right\} ,\end{aligned}$$ and 