I would like to perform a Wigner transform of an object that depends on 4-different coordinates, and in addition, might satisfy a periodicity condition like $A(x_{1}+X,x_{2}+X,x_{3}+X,x_{4}+X)=A(x_{1},x_{2},x_{3},x_{4})$. I am a bit confused on how to perform the transformation for dimensions $d>2$.
Let's start with the 2D case. We take the function $A(x_{1},x_{2})=A(x_{1}+X,x_{2}+X)$ and define the average $\tau=(x_{1}+x_{2})/2$ and relative $z=x_{1}-x_{2}$ coordinates. In this coordinate basis $(\tau,z)$, the function is only periodic in the $\tau$ variable. Then, Fourier transform against $z$ and get: \begin{eqnarray} \int_{-\infty}^{+\infty}dze^{ipz}A\left(\tau+\frac{z}{2},\tau-\frac{z}{2}\right)= A(\tau,p) \end{eqnarray} Now, since $A(\tau,p)$ is periodic in $\tau$, we can further transform to get Fourier components as: \begin{eqnarray} \frac{1}{X}\int_{0}^{X}d\tau e^{im\frac{2\pi}{X}\tau}A(\tau,p)=A_{m}(p) && m\in\mathbb{Z},p\in(-\infty,+\infty) \end{eqnarray} In the literature, these are the Fourier components of the Wigner distribution of $A(x_{1},x_{2})$. Thus, we have mapped the two coordinate function to a function depending on a integer index $m$ and a continuous variable $p$. This is fine and I understand.
The real question comes on how to generalise such transformation in the nD case. What I have tried so far, but I am still not very convinced is the following: Consider a set of coordinates $(x_{1},x_{2},x_{3},x_{4})$ and define the following coordinates from them: \begin{eqnarray} \tau = \frac{x_{1}+x_{2}+x_{3}+x_{4}}{4}; && z_{1}=\frac{(x_{1}-x_{2})-(x_{3}-x_{4})}{2}\\ z_{2}=\frac{(x_{1}-x_{3})-(x_{2}-x_{4})}{2}; && z_{3}=\frac{(x_{1}-x_{4})-(x_{2}-x_{3})}{2} \end{eqnarray} Then, write $A(x_{1},x_{2},x_{3},x_{4})$ in terms of these coordinates. Then FT on the coordinates $z_{1},z_{2},z_{3}$ and we are left with a function depending on $\tau$, which is again periodic with period $X$, i.e. \begin{eqnarray} \frac{1}{X}\int_{0}^{X}d\tau e^{im\frac{2\pi}{X}\tau}\int_{-\infty}^{+\infty}\prod_{j=1,2,3}dz_{j}e^{ip_{j}z_{j}}A(x_{1},x_{2},x_{3},x_{4})\equiv A_{m}(p_{1},p_{2},p_{3}) \end{eqnarray} My question is; is this the correct way to generalise the Wigner transform to $d>2$? It seems very natural in the 2D case but I have my doubts about my approach for the 4D case. Does this mean that for $\mathbb{R}^{n}$ we have: \begin{eqnarray} \mathcal{W}^{(n)}\left[A(x_{1},...,x_{n})\right]=A_{m}(p_{1},...,p_{n-1}) && \text{if} A(\vec{x}+\vec{X})=A(\vec{x});x\in \mathbb{R}{n} \end{eqnarray} i.e., one can always define the average and lag coordinates in such a way to perform the transformation as indicated above? Any help/references is greatly appreciated.