Change of coordinates in quantum phase space

[dft5]

Hello!

I was reading a paper on formulation of QM in phase space (https://arxiv.org/abs/physics/0405029) and I have some doubts related to chapter 5. It seems to me that there is a transformation to modified polar coordinates (instead of radius there is u which is square of radius multiplied by a constant). Although there is no φ coordinate explicitly, left side of eq. (44) looks exactly like derivative with respect to φ in old coordinates and that's why (as it's equal to zero) Wigner function of harmonic oscillator depends only on u.
I suppose that we can't treat it exactly like PDE in Euclidean space with Cartesian coordinates (x,p) as phase space has symplectic structure.
What are the rules for coordinates change in quantum phase space when new coordinates are mixture of position and momentum coordinates?
Also second term in eq. (45) looks like Laplacian and in Euclidean space there are formulas for differential operators like gradient, rotation, divergence and Laplacian in arbitrary coordinate system involving metric tensor and its determinant. Can something similar be applied here?
Could you resolve my doubts and maybe also point out some materials that I should read?

Thanks!

 

[jambaugh]

The phase space is still mathematically speaking the same classical phase space (It is the algebra defined over it which changes). Thus -I suppose- one should utilize canonical transformations to effect a "change of coordinates".