Talk:Generalized Clifford algebra

Overlap

Why is this stub here? These are the standard Sylvester clock and shift matrices, of the main article, except in idiosyncratic conventions and referencing. They have been used in mathematical physics since the 1870s and are definitely not traceable to Weyl, who merely applied them. The stub is an embarrassment. Cuzkatzimhut (talk) 18:17, 7 May 2013 (UTC)

I think the material looks interesting and is worthy of an article, if there's room for improvement then it should be done. I notified the author of your message. M∧Ŝc²ħεИ_τlk 05:05, 8 May 2013 (UTC)

As I indicated, and now appended as the main article wikilink conduit, the Pauli matrix generalization articlet briefly summarizes what any quantum optics text or discrete Hilbert space quantum information text covers quite adequately. I cannot sink time to redeem the article, nor do I have the patience to usefully merge it to the main article, obviously the ideal solution. I just jotted down the bizarrely missing refs of Weyl and Schwinger, which have introduced generations of physicists to this GL(N) basis, and algebra. (Discrete) Quantum Mechanics around the clock is an over-ripe subject, with a groaning literature, and quite beautiful, but conconcting a bogus abstract math field and an even weirder "history" for it is pretty shabby. The WP readers just want the goods in a summary language that makes some sense to them, and not a pseudo-scholarly marginal disquisition. I can attest to several physicists' usual sadly dismissive comments on WP standards when they stumbled on it. Cuzkatzimhut (talk) 14:59, 8 May 2013 (UTC)

Your efforts are really appreciated. Thank you! It doesn't have to all be you though, (hopefully) someone else will eventually come along.... I'd edit myself if I had enough knowledge on Clifford algebras. Regards, M∧Ŝc²ħεИ_τlk 21:02, 8 May 2013 (UTC)

Your words "an embarrassement", "bogus", "shabby" are pretty strong language, Cuzkatzimhut. I am replying solely because I was notified on my talk page that you have raised doubt as to the article's purpose.

Actually in case you aim at proposing it for deletion or merge into the "Generalization of Pauli matrices" article, I really do not see the point, given that Generalized Clifford Algebra (GCA) is a well-established terminology, and possibly even more so than the expression "generalization of Pauli matrices".

Also, it is by no means unusual to have two corresponding articles in Wikipedia: one for the algebra, and one for its matrix representation. See for example Dirac algebra and Gamma matrices. And none is a priori the "main article" compared to the other. Are you intending to establish priority status for the terminology that you refer to as the "Sylvester clock and shift matrices"? Google currently just shows one article with that exact phrase. And the terminology "shift matrices" does not appear to be entirely standardized either: at least it would be very necessary that the "sift matrices" introduced here are not to be confused with the other well-known shift matrices which are nilpotent.

Concerning the GCA history, your edits have been quite contradictory. On the one hand, you add references behind the statement that GCA goes back to Weyl, on the other hand you delete the entire history section without improving it (nor tagging it for improvement first) and declare in the talk page that Weyl did nothing but apply it.

Much as I appreciate your mathematical expertise in general, I do not find this particular way of proceeding very constructive. But that's aside, let's look at the contents about this point too.

Concerning the role of Weyl, in the book edited by Abramovicz and Lounesto (certainly two experts on Clifford algebras), there is the clear statement that "Further development [...] by Weyl resulted in the introduction of a more general commutation relation incorporating both commuting and anticommuting operators as particular cases. This new relation formed the foundation of an algebra which is called a generalized Clifford algebra (GCA). The name is justifiable if we recall that the anticommutation rule is one of the basic relations of a conventional Clifford algebra", and "Weyl has developed his original idea of GCA for purely applied purposes". I have found one further reference which supports your statement as to Sylvester, but not as to Weyl having only done an application. As that article appears to be quite well-balanced about the actual facts, I summarize the statement here [1]: The algebras for p=2 were first investigated by Sylvester and Cartan, and rediscovered by Weyl. I presume you could subscribe to this statement, as a constructive edit to the article. There is also a rather tentative statement in an arxiv publication "apparently, Sylvester [..] was the first to study these [..] generators". Of high interest is a statement by Weyl himself about the work of Sylvester (see Weyl cited in: Andreas Karachalios (2010). Erich Huckel (1896-1980). Springer. pp. 120 ff. ISBN 978-90-481-3560-8. Retrieved 9 May 2013.).

As to mathematical content, in view of your own confidence it would be helpful if you would insert a statement as to the exact relation between the generalized Pauli matrices and representation(s) of GCA. Also, what "generalized Pauli algebra" means in this context compared to CGA.

So much for now, regards --Chris Howard (talk) 22:32, 8 May 2013 (UTC)

Thanks for your timely reply. I am afraid I can't back off from my characterizations---I did not wish to come across as more forgiving than I feel, even though I left out the dark motive aspersions that many of the people I deal with have heaped on this stuff, to spare needless acrimony.

A few points of fact, first, regardless of WP or editorial expediencies. One does not count references and run to secondary references to appraise Sylvester's original and influential ownership of the structures: one goes to the original online Sylvester papers and settles the issue. I expect all good faith workers on these matrices to have read and appreciated and refered to Sylvester's papers. I can send you to explicit private copies of them, if you wished, but I believe the links in the main article suffice. As you indicated, Weyl has been influential for at least three generations in popularizing these matrices, and I have been known to eponymize their braiding relations as "Weyl's relations" myself. Nevertheless, Schwinger's highly cited papers notwithstanding, clock-and-shift matrices are both a standard rubric in the field (of finite dimensional QM), and hardly confusable in this combination: are you asking for a truckload of references? I could provide them, but it would be as much a tendentious waste of time as it would be defocussing. Indeed, "Generalized Pauli matrices" is a suboptimal name, I never use it myself, and, were it not for the pedagogical linkage with the previous section, I would have rejected it. I am only talking about the celebrated "Sylvester's clock-and-shift matrices". I mean, the man found a representation of SU(3) almost a century before Gell-Mann, and a more structured one, at that! (p 649 in here.) Sylvester simply owns the gig. If you found a better home for them, cool....

I can tell you are keen to preserve a perception of GCA as a self-standing island and argue against deletion or merging. I am at peace with that; but, really, now, this article needs work. (Check out and contrast the traffic stats.) No, I need not spell out the exact relation to either the matrices or the braiding relation that characterizes them in the main article: behold! The reader may draw his or her own conclusions upon reading both, and if they still needed help, would such help them? As long as they are given a fair, helpful opportunity to. I made an extraordinary effort to limit the number of relevant references from eithr article, since a truckload of references would bury and obscure the main points. (Frankly, ref 3 in this article without Santhanam & Tekumalla's ref is precisely what I meant when judging it "an embarrassment". Please do try to convice me this was well meaning.) I would not even object to your softening the "main article" wikilink, or your transferring some of the material of that main article to this one, or otherwise linking the two. To my taste, both articles are awkwardly silent on the 500lb gorilla, namely QM around the clock. Marching in blithe isolation to the mainstream, however, and barricading behind algebra vs rep is a bit much to the astute, well-meaning reader. Good luck with the effort.Cuzkatzimhut (talk) 00:56, 9 May 2013 (UTC)

PS Thanks for leading me to Cartan's 1898 Les groupes.... with the above link. I was not aware of it. His ignorance of Sylvester's work, summarized by JJS in French (!!), in 1883, however, is not pretty to have to witness.Cuzkatzimhut (talk) 14:32, 9 May 2013 (UTC)

Nevertheless, a fact is a fact, and I linked Cartan's 1898 contribution, letting the reader reach her/his own indictments. Cuzkatzimhut (talk) 20:58, 10 November 2014 (UTC)