I am interested in computing the Wigner transform in $R^{2n}$ of $$e^{-\beta H}$$ where $$H = \sum_{k=1}^n P_k^2 + V(x_1, \ldots,x_n)$$ I am assuming that $V$ is a polynomial bounded below and obviously $$P_k= -i\hbar \frac{\partial}{\partial x^k}$$ I am not necessarily interested in the exact result. In particular, I would like to know if the Wigner function tends to the corresponding classical exponential for $\hbar \to 0$. Some references on these issues would be welcome.

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    $\begingroup$ My gut says that if Balescu's book doesn't have anything, nobody does. Andre Voros' papers must address such things, but I don't know which one... Even for one dumb oscillator, the most "classical potential", the Wigner transform is a star exponential, a virtual monster when it comes to multiplex and study the classical limit thereof....It is the example that made a fool of von Neumann's book by his acolyte and admirer Groenewold. $\endgroup$ Jan 22 at 20:35
  • $\begingroup$ Background? $\endgroup$ Jan 22 at 20:41
  • $\begingroup$ I am dealing with some topics of strict quantization maps and ssb and I am trying to construct an example. Thanks for the reference! $\endgroup$ Jan 22 at 21:36

Your question completely outranges my competence, but Balescu's famous book might be addressing it, at some level. In your shoes, I might choose an oscillator potential and resolve your hamiltonian to decoupled normal modes, which should be tractable with the celebrated (by dint of its non-triviality!) result for just one oscillator, I thought I might review below. It would only be useless if it were universally in mind.

Work in natural units of the problem, $m=1, ~~\omega=1$, i.e. absorb $\sqrt{m\omega}$ into x and 1/p, and $1/\omega$ into H, rendering it as $H=(p^2+x^2)/2$. The quantum hamiltonian is then $\hat H=(\hat p^2+\hat x^2)/2$, whose Wigner transform is $H=(p^2+x^2)/2$.

The Groenewold-Moyal correspondence of QM operators to phase space variables x and p dictates that the Weyl transform of $\exp(-\beta \hat H)$ is $\exp_\star (-\beta H)$, the star exponential representing formally the power expansion of the exponential with powers supplanted by $\star$-powers.

$$ \exp_\star \left ( -\beta H \right ) = \exp_\star \left ( -\beta H \right ) \star 1 \nonumber \\ =\exp_\star \left ( -\beta H \right )\star 2\pi\hbar \sum_n f_{n} =2\pi\hbar \sum_n e^{-\beta E_n }f_{n} ~, \label{spectral} $$ which is thus a generating function for the stargenfunctions $f_n$. Of course, for $\beta =0$, the obvious identity resolution is recovered.

This Wigner transform, then, is directly seen (Bartlett-Moyal) to sum to \begin{equation} \exp_\star \left (-\beta H \right )= \left ( \cosh(\frac{ \beta \hbar}{2})\right ) ^{-1} \exp\left ( \frac{-2}{\hbar} \tanh(\frac{\beta\hbar}{2}) H\right ) , \end{equation} which is to say just a Gaussian in phase phase, with the proper classical $\hbar\to 0$ limit, the starting point.

Details and clarifications could be found in our short booklet, section 0.14 (time evolution).


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