Talk:Wigner quasi-probability distribution
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I'm pretty sure that the Wigner-Weyl Transform is a different thing from the Wigner Function. The Wigner function is the Weyl-Wigner Transform of the Density Matrix of Hilbert Space, but the Weyl-Wigner Transform in general is given by
Please correct me if I'm wrong, because it will mean that my thesis has to be re-written, and I've only got four months left.
As soon as I've finished, I'll write a Weyl-Wigner Transform article for Wikipedia.
A: Indeed, you are right, and can check the facts in the Book QMPS, which also includes the original seminal papers, adduced last in the references to this article. Conventionally, the Weyl transform maps phase-space (kernel) functions (sometimes called "symbols", a bit awkwardly) to hermitean operators; the reverse transform is the Wigner transform you write, which maps hermitean operators to phase-space kernel functions---which may or may not contain hbar, depending on whether these operators are Weyl-ordered or not (If not, the transform implicitly Weyl-orders them and generates hbar-dependence, in general, whence calling such kernels "classical" may be confusing).
I suspect people dubbed it "Wigner transform" since the Wigner function is the most celebrated example of it; and as you correctly point out, it is the Wigner transform of the Density Matrix (cf. QMPS).
This article has the Wigner transform in property 7, and the article on Weyl quantization details the Weyl transfrom, its inverse. Cuzkatzimhut 16:09, 18 January 2007 (UTC)
