|WikiProject Physics||(Rated C-class, Mid-importance)|
I believe the original - sign in the definition of the * product was superfluous, and did not agree with that of the single PB; and so needed to be deleted, together with the sign of the n-th power PB kernel, so as to agree with standard conventions, and with the article on the Moyal product. Cuzkatzimhut 15:57, 17 January 2007 (UTC)
I did not mean to be unconstructive in taking out the last two comments of the Deformation Quantization section of Henry Delform, but the point of the article is to utilize the Weyl map to eliminate any and all reference to Hilbert space, so as to describe Classical Mechanics and Quantum mechanics on the same footing and with the same variables: it is then that the deformation nature of the latter appears most natural, and it is in this context and picture (Phase-space quantization) that the uncertainty principle appears most appealing and intuitive. Cuzkatzimhut (talk) 20:16, 2 April 2009 (UTC)
Well, the point is that nobody in his right mind thinks of "Weyl quantization" to actually be Weyl's original 1927 proposal, anymore, after Groenewold's and Kubo's insights (the ħ-dependence of the phase-space kernel functions). And most understand it as, really, well, the Wigner-Weyl transform. It would be a source of confusion to have a meaningful split: my best recommendation is to completely lift this and merge it with the Wigner-Weyl transform article, reserving a short dismissive/reassuring section on Weyl's dream like the couple of sentences in the introduction of this. Then, have the remaining included arguably reduplicative info, like the star-product and deformation quantization (which provides the connection to the systematic correction of Weyl's vision) , stay in there, as the heart of the W-W transform is its application to the phase space formulation—beyond the important formal connection to the reps of the Heisenberg group, of course.
But unless the merge were done thoughtfully, the end result might well look incoherent. This article itself is a flotsam of similar tweaks and merges. I hope the proposer of the split should highjack the whole article and merge it with the Wigner-Weyl transform one, and the tastefully pare out the reduplicative parts and uniformize the conventions... Another "cheaper" alternative is to rename this one into Wigner-Weyl transform, something that should have been done a while ago, so the aliases and links are inherited, and rearrange the intro, etc... merging the few items of the developing Wigner-Weyl transform article into this! Lots of work, though, and it is not for me to proffer "unfunded mandates"! Cuzkatzimhut (talk) 14:55, 12 June 2012 (UTC)
nth poisson bracket
Shouldn't the operation be "-" again instead of "x"?
- No. "Again" is unsound. Check for n=1, and then n=3. Cuzkatzimhut (talk) 20:22, 8 October 2013 (UTC)
Add information about point operators?
Would it be worth making explicit the way the Wigner map corresponds to projecting operators onto a particular basis of operators (phase-space point operators)? When I was first trying to learn this subject, the boxed definitions did not seem very motivated, and I think mentioning that these transforms may be derived by finding a basis of operators that makes the wigner map have desired properties may make this article more accessible. Rjones122 (talk) 22:01, 8 May 2015 (UTC)
"Weyl quantization is not always well defined and sometimes gives unphysical answers"
This seams to be a very important statement questioning a complete fork of quantization processes. A citation and and example of such an ill behaved outcome should be given.
This seems to contradict the statement that "There results a complete phase space formulation of quantum mechanics, completely equivalent to the Hilbert-space operator representation, with star-multiplications paralleling operator multiplications isomorphically."
I think this problem arises from a mismatch of Weyl quantization and phase space quantum pysics, but an expert should clarify this seeming contradiction.
- ?? I am not sure what you find unclear here. It is stated quite clearly in the introduction, and an example is provided in the last sentence of the next section, "Example". As a quantization procedure, the Weyl map is a complete failure, as exemplified. The correct Phase space formulation is not "Weyl quantization", that is, the Wigner transform of a correct QM operator is not always the classical quantity--only its ħ→0 limit is. What you call a "mismatch" is but the star product of section 3, explained in the main article on the phase-space formulation. Properly speaking there is no "fork" on your notional road: the *-product-less road is a dead end. What, exactly, would you propose? Cuzkatzimhut (talk) 16:27, 17 May 2015 (UTC)
I thought that Weyl quantization IS quantum mechanic on the phase space. So if I understand you correctly, Weyl quantization is a correspondence of classical mechanics on phase space (function algebra with dot product) and Operator algebra. And this correspondence does not work.
What works is a correspondence between the function algebra on phase space with a star product and the QM operator algebra.
If that's correct it should eventually be made more clear in the article.
- I'm not sure how to improve what is being said. To me it appears clear, and all the crucial contradistinctions are made early on––and no bad question is asked, which would lead to even worse answers. (The one task that it skips is indicating how the baby statement one learns in school, "classical mechanics = c-number functions; QM = operators" is hopelessly off... But there is no way to beat a dead horse without confusing the reader...)
- In the lead of the article, it is emphasized that Weyl quantization most definitely is not quantum mechanics in phase space. "Quantization" is a bunch of techniques extending classical mechanics to QM, but this is an obscure art, and no naive recipe of the Weyl-map sort can possibly work: ħ information cannot be automatically generated out of its ħ→0 limit! As a quantization, recipe, the Weyl map is no good, and is not related to this article, so, then, dismissed right away, in the introduction.
- As a map to operators however, the subject of this article, it is an unimpeachable tool: Despite its hapless history, the Weyl-Wigner map is one of the cornerstones of the phase space formulation of QM, which simply relies on this map and the *-product to re-express operator Hilbert space QM to c-number functions in phase space, invertibly, (once the correct quantum theory has been found, i.e. once the quantization problem has already been solved!).
- The article emphasizes this mere change of representation aspect ––still not be be scoffed at.
- (Obiter dictum: actually this formulation and the Weyl map can assist the quantization problem, by more intuitive deformation of c-number functions in phase space, from classical to quantum, but that is such a subtle problem that, I insist, wisely, it is not even being touched upon here.) Cuzkatzimhut (talk) 18:28, 18 May 2015 (UTC)
- In fact, your comment was salutary. "Weyl quantization" was the original name of the article, as you see in item 2, above. Since the article dismisses this in favor of the transform, which undergirds the phase space formulation, there is no confusion here. However, the old name is still linked in other articles sending the reader here, presumably to learn about the transform, and thus the phase-space formulation. The reader might end up getting the wrong impression, until this article is read more carefully. I did a secondary sweep of the wikilinks to Weyl quantization and replaced the most dangerously misleading ones. Cuzkatzimhut (talk) 14:14, 20 May 2015 (UTC)