What is the Wigner representation of $\left(\hat{x}^2+\hat{p}^2\right)^n$? I would like to calculate the Wigner representation of the operators $\left(\hat{x}^2+\hat{p}^2\right)^3$ and $\left(\hat{x}^2+\hat{p}^2\right)^4$. I know at least two ways to do it, but both rely on brute force:

*

*The first would be to use the formula
$$ A(x,p) = \int \left< x+\frac{1}{2} \xi \biggr|\hat{A}\biggr| x-\frac{1}{2} \xi \right> e^{-i p \xi/\hbar} d \xi$$
where $A(x,p)$ is the Wigner representation of the operator $\hat{A}$ and $\left| x \right>$ is an eigenstate of $\hat{x}$.


*The second is to put the operators in the totally symmetric ordering by using the canonical commutation relations.
Both of these methods are feasible for $\left(\hat{x}^2+\hat{p}^2\right)^2$, and I was able to obtain the Wigner representation $(x^2+p^2)^2 - \hbar^2$, but they start to get too tiresome for higher powers. I was wondering if there was a smarter way to do it, that perhaps would generalize to $\left(\hat{x}^2+\hat{p}^2\right)^n$ with arbitrary $n$.
 A: Both of your methods are correct but impossibly inefficient. The fundamental theorem of phase-space quantization is that the Wigner image of operator products is the star-product of the Wigner image of the operators.
So, use the * product of Groenewold, eqns (13), (14) of this booklet.
Groenewold's $\star$-product is
$$
 \star \equiv e^{{i\hbar \over 2} ({\stackrel{\leftarrow}{\partial}}_x {\stackrel{\rightarrow}{\partial}}_p- {\stackrel{\leftarrow}{\partial}} _p {\stackrel{\rightarrow}{\partial}}_x )} ~,  $$
which boils down to the Bopp shift for your $A(x,p)\equiv x^2+p^2$,
$$A(x,p) \star g(x,p) = A\left( x+{i\hbar\over 2}{\stackrel{\rightarrow}{\partial}}_p ,~ 
p-{i\hbar\over 2}{\stackrel{\rightarrow}{\partial}}_x \right )~ g(x,p) ~.  
  $$
Note there is lots of symmetry (rotation al symmetry in phase-space), all outcomes are real, etc... You might try do do recursions: $A\star A=A^2-\hbar^2$, your result, then $A\star (A\star A)= A\star A \star A$, from the associativity of $\star$ and so on.
The easiest sequence follows from the recursive method below, since all arguments of g are functions of A, rotational scalars in phase-space, hence real. Thus, the Bopp-shifted star product reduces to a one variable recursion.
Define $w\equiv A/\hbar$, so, for $g_1=w$,
$$
g_{n+1} =\Bigl ( w(1-\partial_w^2) -\partial_w \Bigr ) g_n~~~\leadsto \\
g_2= w^2-1 ~~~~(\hbox {your answer}), \\
g_3= w^3-5w, ~~~~...$$
