# Wigner transform of a thermal density matrix (Schroedinger Hamiltonian)

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.

• 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. – Cosmas Zachos Jan 22 at 20:35
• – Cosmas Zachos Jan 22 at 20:41
• I am dealing with some topics of strict quantization maps and ssb and I am trying to construct an example. Thanks for the reference! – Valter Moretti Jan 22 at 21:36

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.