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)

[edit] Wigner Function and Probability Distributions

Some additional comments should be added along the following lines.

First, there is a simpler characterization of the Wigner function in relation to Weyl quantization that gets across what it is really about: the Wigner function is just the expectation of the delta function, understood in the sense of a weak operator limit. Weyl quantization may therefore be thought of as an operator whose kernel is just the Weyl quantization of the delta function. In comparison, if you take the expectation of the delta function on a classical state, you get its probability distribution.

Second, there is a close link between Wigner functions and probability distributions, which in turn provides a link to coherent state quantization. Namely, the Gaussian convolution of a Wigner function with a spread in each (p,q) sector equal to Planck's constant or larger is a probability distribution. This convolution is related to the transition probability taken with a coherent state. Conversely, given the transition probability taken with a family of coherent state |p,q> as (p,q) range over phase space, the Wigner function can be reproduced. Therefore, though Wigner functions may not be probability distributions in themselves, they are characterized as the "inverse Gaussian convolutions" of probability distributions.

Comment in response by Cuzkatzimhut 00:53, 12 June 2007 (UTC): I would reluctantly concur with the first point, provided no confusing statements were made, and the comments were properly relegated to the Weyl Quantization pages, instead, where they would appear more apposite. What is refered to as a "weak operator limit delta function", centering an operator around its classical limit in Weyl ordering, is really the quantizer: the Fourier transform of a generic element of the Heisenberg group. Its expectation, in Moyal's original 1946 language, is the Fourier transform of the characteristic (moment-generating) function. I suspect that few readers would learn much from operator-valued distribution functions, as suggested, but I may be convinced otherwise. I suspect property 7 covers the basics, and A Royer's classic 1977 paper (Phys Rev A15 , pp 449-450), interpreting the Wigner function as the expectation value of the (parity) reflection operator in phase space should suffice, instead of formal excursions on operator-valued delta functions.

I fear the second point needs to be finessed with far too much work or detail to be made sound. Indeed, the convolution mentioned, as per de Bruijn's (1967) and Cartwright's (1976) theorems, is positive semidefinite; but it cannot serve as a plain probability measure in phase space, as physicists have long known, but electrical engineers often miss, possibly due to lack of attention in the marginals. (This problem may be remedied by proper account of the *-product within integrals, not ignorable as in the case of the Wigner function, but at the cost of further complication.) Conversely, the inverse Husimi transform of an arbitrary positive semidefinite function viewed as a "probability" in phase space rarely satisfies the highly constrained ancillary conditions for a Wigner function: the outcome rarely fits into the form of the first formula of the article for some psi, as readily demonstrated in standard texts on the subject. Thus, connecting to the Husimi function as blithely as suggested merely adds delicate elements of potential confusion to the non-expert, and may well be deprecated.