METAPLECTIC FORMULATION OF THE WIGNER TRANSFORM AND APPLICATIONS
We show that the cross Wigner function can be written in the form $W(\psi, \phi) = \hat S (\psi \otimes \overline{\hat\phi})$ where ${\hat\phi}$ is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrodinger and the Bopp representations of quantum mechanics.