Talk:Wigner quasiprobability distribution

message: 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
A(q,p)=\int_{-\infty}^{\infty}ds\langle q-s/2|\hat A|q+s/2\rangle e^{ips/\hbar}
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.
By Special:Contributions/130.102.2.60 .
: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)
thread_title: First post , 4 July 2006

message: 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)
thread_title:  Wigner Function and Probability Distributions 

message: Wigner-Ville? Who is Ville? I'm presuming someone just hasn't connected the spelling and pronunciation of Weyl? Cesiumfrog (talk) 06:50, 1 April 2010 (UTC)
replies: text: See the 6th reference. A French information scientist and most definitely not Weyl. He is not known in physics, but the signal processing community pays homage to his independent introduction of that function on the time-frequency plane.Cuzkatzimhut (talk) 11:19, 1 April 2010 (UTC)
thread_title:  Ville 

message: I think if that minimum info goes away, the essence of P(x,p) will be impenetrable. Much better to 
merge it with the next section, which it explains, and refer to the Main article of the W-W transform. 
In that case, the title of the subsection can cover both. Cuzkatzimhut (talk) 13:15, 9 June 2012 (UTC)
thread_title: Split proposal for '''The Wigner–Weyl transformation section'''

message: In the 4th Formula (preceded by "In the general case, which includes mixed states", 
the Sign in the Exponential is Not Consistent With the Sign Used in the Definition.
Indeed 
ψ^*(x+y) ψ(x-y) = <x-y|ρ|x+y> if ρ=|ψ><ψ| (pure Case)
89.92.57.32 (talk) 01:07, 14 November 2012 (UTC)
replies: text: Good point! Now fixed, but you must flip the sing of the dummy variable y in the definition to see it. Interestingly, the error is replicated in the Wigner-Weyl transform article, as well! Cuzkatzimhut (talk) 02:20, 14 November 2012 (UTC)
thread_title:  Wrong sign in 4th formula 

message: There have been suggestions made to make sure that mathematical details regarding the time evolution of the Wigner function should not clutter this article, so that this article remains readable for non-specialized readers.
Currently, there exist sections Wigner quasiprobability distribution#Evolution equation for Wigner function and Density matrix#"Quantum Liouville", Moyal's equation and Phase space formulation#Time evolution, plus the very mathematically detailed Method of quantum characteristics. In order to create a place for more detailed information on the time evolution aspect, I suggest to create a dedicated new article "Moyal equation of motion" (as it is called for ex. here), or alternatively shorter "Moyal equation", for an article on the Wigner function's time evolution equation
Sources for such a new article could include on the one hand the original articles for the historical overview, plus on the other hand also more recent but free-access articles such as hep-th/9409120 or others.
Then this article's section "Evolution equation for Wigner function" could be brief and concise, referring to that new article as main article. And the new article could contain greater detail, without being quite as technical and detailed as the article on the method of quantum characteristics. Any comments, objections or support at this point? --Chris Howard (talk) 22:52, 15 November 2012 (UTC)
replies: text: I sense some resistance by just about everybody in discussing Moyal's equation of motion anywhere but in the Method of quantum characteristics article, which is both extensive and poorly written and in need of help. Odd priority disputes, philosophical divergences, etc.. emerge all over the place, except there, the designated arena for this evolution. I think if you had the gumption to rename the "Method of quantum characteristics" page into something like "Time-evolution of the Wigner function" (no superfluous "equation" please! of course evolution is controlled by Moyal's equation), reorganize it, rewrite it, compare and contrast methods, path integral techniques, etc, without multiplexing into mini-articles and stubs summarizing particular papers, that would be greatly appreciated by the interested reader.  But, the issue is technical, not encyclopedic, and after 60+ years of work has produced little explicit progress, beyond endless hyper-formal reformulations. The interested reader has aleady hit the books and abundant papers and reviews. Both ref [6] here (scribd), W Schleich's excellent book, and Wong's numerous papers so studiously effaced from this article could have a dimension to contribute in it. It is the study of phase space propagators, initiated by Moyal in 1949. Incidentally, virtually all the refs in all the articles you mentioned are available for free in google scholar and google books, so having people rediscover the wheel because they have no access to the original papers that did it all is a bit rich.Cuzkatzimhut (talk) 01:26, 16 November 2012 (UTC)
replies: text: "Wong's numerous papers" offer classical trajectories to calculate the quantum evolution of Wigner function. This is a mistake. Mi Tatara Buela (talk) 10:12, 29 March 2013 (UTC)
thread_title:  Proposal to create a new article "Moyal equation of motion" 

message: User:Jheald proposed merger of the Signal processing/ transform  article Wigner distribution function into this one.  The relative ratios of the Traffic stats in the last 30 days of the respective articles are WDF/WQD ~ 0.42, at this point in time.  The two articles, are, of course, related in mathematical detail, but there are conceptual and cultural chasms all but impossible to bridge. 
I can imagine an extra section with WDF here, but it looks like a major task, if it were to be done right. 
In particular, unless the issue of negative values, ritually frowned upon by the signal-processing community,  with little reason (leading to creative "remedies"), were dealt with responsibly, the merger could easily lead to a bonfire of confusions, eye-rolling,  and pseudo-science. Enterprising wikipedians may give it a shot, but fretting about cross-terms without considering expectation values and the uncertainty principle as it subtly manifests itself in signal processing can only lead to grief and avoidable pointless mass debate, I suspect. Cuzkatzimhut (talk) 14:30, 2 July 2013 (UTC)
thread_title: Proposed merger of [[Wigner distribution function]]

message: I am pretty certain that the 3D and 1D definitions of the Wigner function are not consistent with each other. See the factors of tow in the exponent and in the arguments of the bras/kets.  — Preceding unsigned comment added by 66.207.203.214 (talk) 15:27, 18 December 2017 (UTC)
replies: text: It is correct, if you properly change variables. What, specifically, in formulas, is your problem? Reassure yourself of the sign change from bra-ket to the 1d wavefunction expression, and then from that to the 3d one. New comments go to the bottom of the talk page, not the top. Cuzkatzimhut (talk) 16:11, 18 December 2017 (UTC)
thread_title: new comment

message: If I'm not mistaken, the equation:
: W(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^\infty \varphi^*(p+q)\varphi(p-q)e^{-2ixq/\hbar}\,dq
should be:
: W(x,p)=2\int_{-\infty}^\infty \varphi^*(p+q)\varphi(p-q)e^{-2ixq/\hbar}\,dq
Else, the units turn wrong. Can someone check me and fix it? Shohamjac (talk) 08:53, 10 August 2019 (UTC)
thread_title:  Constant in the momentum symmetrical Wigner function 

message: I am not tempted by serial reversions of the sentence discussed  here ( last one of section 5 : "The truncated Wigner approximation is a semiclassical approximation to the dynamics obtained by replacing Moyal's equation with the classical Liouville's equation.")  which has reemerged. I strongly believe, nevertheless, that the sentence adds nothing to the small section inserted into. 
Truncations are, of course, hinted at in the generic first sentence and its references, in that section, and the relation to Liouville dynamics in the very first section of the article.  
I can only see trouble ahead, if anyone took the bait and provided the redlinked wished-for article, which would then have to be linked. But as it stands, the sentence provides no meaningful information, beyond a misguided clarion call. Cuzkatzimhut (talk) 17:56, 10 November 2019 (UTC)
thread_title: Removal of insistent insert on "truncations"

message: The present article does not quite correctly state that "the Wigner distribution holds a privileged position among all these distributions, since it is the only one whose required star-product drops out". The Glauber-Sudarshan P representation mentioned in the article leads to an average value of the operator G, which coincides formally with that in classical mechanics. In this representation, the density matrices are diagonal in the basis of coherent states, so one can write
:  ρ = ∫dαdα*/(2πi) exp(-αα*) |α〉P(α,α*)〈α|.
The average value of the operator G is an integral over the phase space:
:  〈G〉= Tr[ρG] = ∫dαdα*/(2πi) exp(-αα*) P(α,α*)〈α|G|α〉.
The usual dot-product works inside the integral. --Edehdu (talk) 07:14, 20 May 2022 (UTC)
replies: text: I fear you misread the statement. The star product of the Glauber ordering does not integrate itself out by parts! The expression you provide has a Gaussian optical phase-space measure  . Converting to the phase space discussed, the Glauber-Sudarshan Cohen measure is distinctly nontrivial, \exp ( -\hbar(\tau^2+\sigma^2)/4), as you may check in standard texts, specifies a nontrivial star product, and prevents its elimination inside the integral. Please review optical phase space. Cuzkatzimhut (talk) 13:33, 21 May 2022 (UTC)
thread_title: Average value of operator in the Glauber-Sudarshan P representation