Generalization of Wigner overlap formula I want to generalize the Wigner overlap formula,
$Tr( F G ) = 2 \pi \int_{-\infty}^{\infty} dq \int_{-\infty}^{\infty} dq W_F(q,p) W_G(q,p)$,
where $W_F(q,p)$ and $W_G(q,p)$ are the Wigner functions of the operators $F$ and $G$, respectively.
This formula is stated in literature for two operators $F,G$ (see e.g. Measuring the quantum states of light by Ulf Leonhardt) and some sources state that it is easy to generalize it to more than two operators.
My approach was to proof the statement for two operators and try to adapt the proof for three operators, hoping to find some pattern.
Thus, first my approach for two operators $F$ and $G$:
\begin{align}
Tr[FG] &= \int_{-\infty}^{\infty} dq_1 \langle q_1 | F G |q_1 \rangle = \int_{-\infty}^{\infty}dq_1 \int_{-\infty}^{\infty}dq_2 \langle q_1 | F | q_2 \rangle \langle q_2 |G |q_1 \rangle \\ 
&= \int_{-\infty}^{\infty}dq \int_{-\infty}^{\infty}dx_1 \langle q- \frac{x_1}{2} | F | q + \frac{x_1}{2} \rangle \langle q + \frac{x_1}{2} |G |q -\frac{x_1}{2} \rangle \\
&= \int_{-\infty}^{\infty}dq \int_{-\infty}^{\infty}dx_1 \int_{-\infty}^{\infty}dx_2 \langle q- \frac{x_1}{2} | F | q + \frac{x_1}{2} \rangle \langle q - \frac{x_2}{2} |G |q +\frac{x_2}{2} \rangle \delta(x_1+x_2) \\
&= \int_{-\infty}^{\infty}dq \int_{-\infty}^{\infty}dx_1 \int_{-\infty}^{\infty}dx_2 \int_{-\infty}^{\infty}dp \frac{1}{2\pi} e^{ip(x_1+x_2)} \langle q- \frac{x_1}{2} | F | q + \frac{x_1}{2} \rangle \langle q - \frac{x_2}{2} |G |q +\frac{x_2}{2} \rangle \\
&=2 \pi\int_{-\infty}^{\infty}dq \int_{-\infty}^{\infty}dp \int_{-\infty}^{\infty}dx_1 \frac{1}{2\pi} e^{ipx_1} \langle q- \frac{x_1}{2} | F | q +  \frac{x_1}{2} \rangle \frac{1}{2\pi} \int_{-\infty}^{\infty}dx_2 e^{ip x_2}\langle q - \frac{x_2}{2} |G |q +\frac{x_2}{2} \rangle \\
&= 2 \pi \int_{-\infty}^{\infty} dq \int_{-\infty}^{\infty} W_F(q,p) W_G(q,p)
\end{align}
Now I try to generalize this as similar as possible for three operators $F, G, H$:
\begin{align}
Tr[FGH] &= \int_{-\infty}^{\infty} dq_1 \langle q_1|FGH |q_1\rangle = \int_{-\infty}^{\infty} dq_1 \int_{-\infty}^{\infty} dq_2 \int_{-\infty}^{\infty} dq_3 \langle q_1|F|q_2 \rangle \langle q_2|G|q_3 \rangle \langle q_3| H |q_1\rangle\\
&= \int_{-\infty}^{\infty} dq \int_{-\infty}^{\infty} dx_1 \int_{-\infty}^{\infty} dx_3 \langle q - \frac{x_1}{2}|F|q + \frac{x_1}{2} \rangle \langle q + \frac{x_1}{2} |G|q - \frac{x_3}{2} \rangle \langle q - \frac{x_3}{2} | H |q - \frac{x_1}{2}\rangle\\
&= ...
\end{align}
Well, one observes that the trick from the proof for two operators doesn't work here, because if I choose $x_2$ to be $-x_1$ for the middle part, we require $x_3 = - x_2$, hence $x_3 = x_1$. So, the last part would have the form $|q-\frac{x_1}{2} | H |q - \frac{x_1}{2}\rangle$ and not those we require to proceed.
Is there anything (stupid?) that I oversee?
Does anyone have an idea how one can prove the formula for three operators?
Thank you in advance for your help!
 A: You seem to be profoundly misunderstanding the fundamental isomorphism of phase-space quantum mechanics. What you call "Wigner functions" are but Weyl symbols,
$$f(x,p) = \hbar\int\!\!dy ~ e^{-iyp}\langle x+\hbar y/2| F |  x-\hbar y/2 \rangle ,$$
c-number functions of phase space, so that
$$
h\operatorname{Tr} F = \int\!\! dx dp ~ f(x,p),\\
h\operatorname{Tr} (F G) = \int\!\! dx dp ~ f(x,p)\star g(x,p), \\
h\operatorname{Tr} (FGH) = \int\!\! dx dp ~ f(x,p)\star g(x,p)\star h(x,p),\\
h\operatorname{Tr} (FGHK) = \int\!\! dx dp ~ f(x,p)\star g(x,p)\star h(x,p)\star k(x,p), ...
$$
etc, utilizing the fundamental isomorphism of the Wigner map,
$$
FG\mapsto f\star g   =  f \, \exp{\left( \frac{i \hbar}{2} \left(\overleftarrow{\partial }_x \overrightarrow{\partial }_p -\overleftarrow{\partial}_p \overrightarrow{\partial}_x \right) \right)}  \, g  \\
= \hbar^2\int\!\! dy dy'~~e^{-ip(y+y')}
\langle x+\hbar(y+y')/2  |F|x-\hbar(y-y')/2  \rangle   \\  \times \langle x+\hbar(y'-y)/2  | G |x-\hbar (y+y')/2   \rangle .
$$
The star product is associative, like the QM operators on the left, so no grouping parentheses are warranted.
However, you may convince yourself of a basic fact of phase space QM, that only one star  inside a phase space integral may be dismissed (integrated out by parts), never more. Check this.
So you have, indeed,
$$
 \int\!\! dx dp ~ f(x,p)\star g(x,p) 
= \int\!\! dx dp ~ f(x,p) g(x,p), 
$$
but that's as far as the starless train goes. From this point on,
$$
h\operatorname{Tr} (FGH) = \bbox[yellow,5px]{ \int\!\! dx dp ~ f(x,p)\star g(x,p)\star h(x,p)  \\ =  \int\!\! dx dp ~ f(x,p)~~ \Big ( g(x,p)\star h(x,p)\Big  )\\ = \int\!\! dx dp ~ \Big ( f(x,p)\star g(x,p) \Big ) ~~ h(x,p)  },
$$
and so on. Your text should have taught you this.
