Talk:Clebsch–Gordan coefficients for SU(3)

Hi, this is a page dealing specifically with the Clebsch-Gordan coefficient of SU(3) group. SU(3) is an important group as it represents flavour symmetry, and the Clebsch-Gordan coefficients of SU(3) are used to understand the decay of many particles which has flavour symmetry. The page, will deal with the following:-

Groups, Symmetry Groups and their properties.
Find out the symmetry group and their algebra of a 3-D isotropic Harmonic Oscillator.
Consider two coupled 3-D Harmonic oscillators, and try to decouple the representation of their combined Hilbert space into direct sum of some subspaces.
The expansion is the Clebsch-Gordan series for SU(3). And correspondingly the coefficients can be found out.
Use a different approach and solve the CG series from Young tableux.
Finally try to find the CGC from simple symmetries.
Answer the question, where are the CGC needed?

Thank you,
P.S.- If any one has anything to say about the plan or the page, please feel free to say it. — Preceding unsigned comment added by Arkadipta Sarkar (talk • contribs) 04:25, 1 November 2014‎ (UTC)[reply]

Review of this article?[edit]

I was asked to review this article by Arkadipta Sarkar, a person with whom I have had no contact. I have never reviewed an article for Wikipedia, so this "review" is a purely informal affair.

I like the over-all look of the article, and strongly endorse the use of in-line Latex for such articles. It look very professional to me.
I haven't thought seriously about Clebsch-Gordan coefficients since my prelims at U.C. Berkeley back in 1977. Somebody else needs to verify that the equations are correct. I have made a deliberate decision to focus my efforts on Wikiversity and work primarily at the introductory college level.
The only weakness I could see was that there is virtually no connection to how the CG coefficients are used in quantum mechanics. I vaguely remember them being associated with rotations of angular momentum eigenstates, and unfortunately, the article completely failed to refresh that memory. Could somebody please add that application and explain exactly how they are used? (If you want, we could make a link to Wikiversity that includes a few examples. I would be glad to facilitate such an effort.) Note added later: I just realized that WP has an article on CG coefficients--see next topic -guyvan52 (talk) 01:10, 15 November 2014 (UTC)[reply]

By the way, was this article properly accepted by WP, or did the author simply change the title and re-post? If so, please forgive the author (assuming that the article is what it seems to be). --guyvan52 (talk) 16:39, 14 November 2014 (UTC)[reply]

@Guy vandegrift: Regarding the angular momentum interpretation, perhaps you are thinking of the Clebsch-Gordan coefficients for $SU(2)\cong Spin(3)$ . Or is there a higher-dimensional generalization of this idea? --Sammy1339 (talk) 01:32, 15 November 2014 (UTC)[reply]

@Sammy1339: The apex of my understanding of C-G coefficients occurred circa 1977 when I studied them for my prelims, convinced myself that I understood them but also hoped that the subject would not appear on the test. My interest in this project is different, but not entirely unimportant. I write at a much lower level, on Wikiversity, because I am a tenured professor (i.e., not obligated to publish) who is alarmed at the high cost of education. We not only make students buy $100 books, but charge thousands of dollars per course to pay a PhD to write equations and lecture on the board. During this effort I have come to the following realization:

Wikitext exposition that is highly mathematical needs to be cast in somewhat unconventional format.

This includes the use of inline-Latex (in spite of its "ugliness"), and this includes avoiding the need to use equation numbers. It also involves hidden textboxes, and (IMHO) links to Wikiveristy for specialized discussion. I am not qualified to speak to where SU(3) C-G coefficients belong. But if they belong on Wikiversity, I'm here to help.

--guyvan52 (talk) 13:32, 15 November 2014 (UTC) I might add that my involvement in this article was an accident. I was invited to review it, and I did not realize that the article covered a specialized subset of the C-G coeffiecients. Seeing that appeared to be quality prose that was above my head, I got involved, not realizing the a lower level WP article on the subject already exists. Cheers! --guyvan52 (talk) 13:45, 15 November 2014 (UTC)[reply]

Actually

SU(2)

group CG-coefficients are related to angular momenta addition. in case of

SU(3)

we cannot say the same thing. These CG-coefficients are used in coupling of three state systems like colour.Arkadipta Sarkar (talk) 08:02, 23 November 2014 (UTC)[reply]

A couple of things[edit]

First off, I should say that I appreciate the presence of the article. I'll read it in detail at some point. It certainly fills in a need topic-wise.

I have removed two categories (rotation + 3d rotation). I know that you can speak of rotation in flavor space and the like, but the categories in question are meant for $SO(n)$ and things related to ordinary rotations.

Then there is the LaTeX issue. It's totally possible to write math-heavy articles without inline TeX. If you can't, then it is badly designed to begin with. This article is nicely designed, but has a lot of inline TeX code making it look like shit on some set-ups. WP articles are supposed to be read on any machine. Some editors just don't give a damned because things look fine for them. Well, either that or they are simply to lazy to make it look good. Below is some utterly unnecessary TeX

Closure: For every pair of elements $x$ and $y$ in $G$ , the product $x*y$ is also in $G$ ( in symbols, for every two elements $x,y\in G,x*y$ is also in $G.$

Try instead

Closure: For every pair of elements $x$ and $y$ in $G$ , the product $x * y$ is also in $G$ ( in symbols, for every two elements $x, y \in G, x * y$ is also in $G$ .

If you don't see much difference, try Chrome with PNG rendering on a large screen. It will make you scream of horror. The fact that some editors are of the opinion that TeX should be used inline (for some reason) does not justify its use. Minor typographical differences, like between the two versions of an asterisk above are acceptable (if the alternative is inline TeX).

Incidentally, I'm of the opinion that all text should be in TeX, as a long term goal, because it produces the best overall quality, including of ordinary text. This is not realistic today and probably never will be. YohanN7 (talk) 19:11, 1 January 2015 (UTC)[reply]

Well, I prefer the inline TeX. It makes the math stands out and not be confused with the text. I don't see why the main point should be for the article to LOOK good? — Preceding unsigned comment added by 198.58.159.147 (talk) 10:43, 6 July 2020 (UTC)[reply]

Bad grammar[edit]

The second sentence seems to be ungrammatical. The decomposition is not the subject or object of a verb. There seems to be no main verb in the sentence.

Then so fix it. And please sign your posts. YohanN7 (talk) 10:57, 13 November 2015 (UTC)[reply]

External links modified[edit]

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Section 1 - groups[edit]

The section on groups is unnecessary (because anyone reading this knows what a group is, and a link on the first occurrence of the word "group" would be enough), doesn't follow Wikipedia standard notation, is not concise, uses bad grammar, and is inaccurate. (The sentence "For each element (x ) in G, there must be an element y in G such that product of x and y is the identity element e" is inaccurate. The equation afterwards clarifies, but it is absolutely necessary to have both xy=e and yx=e, and not just one or the other.) I'm just going to delete it.

199.249.110.156 (talk) 15:31, 26 June 2018 (UTC)[reply]

Do not, please. Even in the interest of fixing notation, it is useful. Yes, it is bad, so improve it, and possibly condense it (There is more on groups than on Lie groups). But reduplication in WP is not a minus, it is a plus. Links do not always keep the reader focussed. It need not be concise, as long as it teaches somebody something. If you can see its flaws, it is for somebody other than you. I have fielded dozens of complaints, in another venue, about the elliptical terse style of WP technical pages. Deletion is the last option. Bona-fide efforts to improve need to be exhausted first. Remember, you are not a critic or a taskmaster, you are an editor. Why don't you get a WP account for accountability? Cuzkatzimhut (talk) 18:51, 26 June 2018 (UTC)[reply]

I issued a good faith revert due to content blanking/removal. It seems this user isn't interested in the conversation. JeremiahY (talk) 01:54, 27 June 2018 (UTC)[reply]

I think the original poster makes a good point. Including the definition of a group in this article does not improve readability, and I don't see any notation in it that is used in the rest of the article. Saying that other articles on wikipedia are too terse is not a compelling argument to make this one too verbose. On the other hand, it also doesn't hurt to have this section in there. I'm still working through the rest of the article, but I can try to edit it and address the faults that the original poster pointed out (Also: listing "Commutative" as a "property that a group has to satisfy" is misleading, even though the item correctly states that it may or may not be commutative). 50f61674674d7a04461a081b25bee6b2584a5e1d (talk) 12:30, 30 September 2019 (UTC)[reply]

Unnecessary Acronyms[edit]

The Acronym CSCO is introduced but then never used. I am guilty of doing that as well with names that I'd rather only type out once, but if it really only appears once it makes the text less readable. I'll delete it. 50f61674674d7a04461a081b25bee6b2584a5e1d (talk) 12:47, 30 September 2019 (UTC)[reply]

Hello unknown wikipedian,

it is a pity that this CSCO acronym does not appear any more. I have been in contact with the French school around Claude Cohen-Tannoudji who introduced the concept quite frequently in their classic quantum mechanics textbook. They would really love to read that the mathematical object called Cartan subalgebra is nearly precisely what an CSCO is: a “complete set of commuting operators”. The only problem is that it is actually not complete if one has to distinguish between points in the isospin-hypercharge (I_3-Y) plane that have more than one multiplicity, or when in a product representation, an irrep appears twice.

I do not like the current shape of the article (too many details, some of them irrelevant), and I will comment on that if I find time. Eventually, I would propose a streamlined version where at least this CSCO-Cartan connection is spelled out. DieHenkels (talk) 18:27, 26 October 2023 (UTC)[reply]

A few technical errors[edit]

There seems to be a few technical errors and unclear sentences: For instance "Thus U(3) can be decomposed into a direct product of U(1)⊗SU(3)". This is false, the intersection of both subgroups is not trivial. It is a semi-direct product. At the level of the Lie algebra, they will be, which might have confused people regarding the representation theory. Similarly, the notation "U(1)⊗SU(3)" is quite weird, as that would normally imply bilinearity which is meaningless on the groups themselves. Is what is meant here the Lie algebras? Or the group algebras?

"A slightly differently normalized standard basis consists of the F-spin operators, which are defined as F i ^ = 1 2 λ i {\displaystyle {\hat {F_{i}}}={\frac {1}{2}}\lambda _{i}} \hat{F_i}=\frac{1}{2}\lambda_i for the 3, and are utilized to apply to any representation of this algebra." What is that supposed to mean? F-spin operators are not well known thing. I supposed the "3" refers to the fundamental representation? I honestly don't understand what "and are utilized to apply to any representation of this algebra." is supposed to mean and I work with closely related tools. — Preceding unsigned comment added by 198.58.159.147 (talk) 11:12, 6 July 2020 (UTC)[reply]

Not well known to you, perhaps, for the Fs. They are a standard change of basis to fit in with the standard conventions and normalizations for su(N) generators employed in physics. If you kept reading, you'd see in the Casimir section that they apply to all reps, not just the fundamental. This is a resolutely physics article. Indeed, most discussions on the direct product are meant to apply to the algebra, and for better or for worse, this is standard notation in the physics literature. You did not, exactly propose an improvement here, on this talk page. Topology issues are deprecated in Clebsching algorithms, of course. This is not a forum. Cuzkatzimhut (talk) 20:21, 6 July 2020 (UTC)[reply]

I can certainly edit the page, but I thought it would be more appropriate to discuss it first. Also, technical commentaries can help by themselves. Isn't it something worthy to be pointed out to other editors that, as a mathematical physicist with published work on the Clebsch-Gordan problem for other Lie groups, I find many sections of this article confusing an vague? I disagree that this is a resolutely physics article. In fact, I object with the first sentence of the article. The second sentence gives the meaning of "Clebsch-Gordan coefficient" generally accepted in mathematical physics. I don't think that following the common habit in physics of abusing mathematical notations is best for an encyclopedic treatment of the topic. This yields incoherent articles, as a whole, on Wikipedia. However, I agree it can be relevant to mention the typical notations used in physics, but I would not write technically false mathematical statements. I see that you have actually updated that section of the article and took the quotient by the center.198.58.159.147 (talk) 00:14, 8 July 2020 (UTC)[reply]

Regarding the standard basis section, my main issue is not in the introduction of the F-spin operators as a new basis, but is rather that I sincerely find it difficult to understand what "and are utilized to apply to any representation of this algebra." means. If you pick a basis for the algebra it has nothing to do with its representations. I think what is meant is that the association with the canonical generators is made through an identification via the fundamental representation. This can somewhat work as this representation is faithful but ONLY at the level of the Lie algebra relations, not its universal enveloping algebra, which is itself implicitly used in treating the CG problem (as the Casimirs are defined in it). This can easily confuse readers with a more mathematical background. Why not give first the standard presentation in terms of the Cartan-Weyl generators (H_i and E_alpha) and then define those F-spin operators in terms of those. Also, I went and checked my paper version of citation [2] and saw no reference to either the Cartan-Weyl or the F-spin operators, only a discussion of the Eightfold way. I compared the ISBN to be sure. (I could not find any mention of the F-spin operators through a quick check in the rest of the book, only the Gell-Mann matrices.

Some other comments: The sentence: "These Casimir operators serve to label the irreducible representations of the Lie group algebra SU(3), because all states in a given representation assume the same value for each Casimir operator, which serves as the identity in a space with the dimension of that representation." is rather vague. All states in an irrep are degenerate eigenvectors of the Casimirs with the same eigenvalue. A Casimir does not "serve as the identity" but rather, is a multiple of the identity on all irreps. This is Shur's lemma.

Apologies if I am not using the talk page for it's intended use. Aren't the talk pages precisely to discuss editorial issues? I am not well-versed on Wikipedia policies and modus operandi, but wish to improve it on my topics of expertise and help it grow beyond simply donating.198.58.159.147 (talk) 00:14, 8 July 2020 (UTC)[reply]

Here is the thing. There are plenty of mathematically elaborate articles in WP on several subjects, SU(3), Clebsches, and wonderful group theory. Unfortunately, they fell into the hands of mathematicians more interested in their compact abstract formulation of the subject, and thereby made them inaccessible and useless to physics students interested in performing a standard amplitude calculation in standard conventions. Arguably "redundantly", a student started this page as a compendium of "straight facts" necessary for these calculations, a place for these refugees to go to, instead of books. Please take a look at Greiner (a terrible book, I agree) which works at the level of these calculations. The F basis is handled well in pp 78-79 of Don Lichtenberg's standard text, pp 78-79, ISBN-13: 978-0123941992 : It would be pointless to seek them in a baby caricature undergraduate text such as Griffiths'. The basis fixes the structure constants, and hence the normalizations of all representations as utilized in routine applications in physics, such as the Casimirs following. Remember, students know group theory, and need to rush to fix some conventions; this is not a Lie Algebra tutorial. Other articles in WP beat the Cartan-Weyl basis to death. While it might be OK to insert small tweaks and explanations to alienate math readers less, here, it is extremely important to keep the basic cheat-sheet facts for these applications. My sense is you are really in the wrong place; you want to be in the SU(3) article, the Clebsch-Gordan article, or the Gell-Mann matrix article, or Eightforl way, before you descend here. Consider writing a self-standing article/stub to be linked here. Cuzkatzimhut (talk) 13:01, 8 July 2020 (UTC)[reply]

Renaming[edit]

Would it be worth renaming this article to 'representation theory of SU(3) so that it gets found together with representation theory of other important lie groups? and making the requisite changes to do this. Zephyr the west wind (talk) 21:29, 6 June 2022 (UTC)[reply]

You appear to be focussing on the standard/trivial introductory sections, and not the CG sections that the former lead to by establishing language. What is wrong with placing links on the mystery articles you are hinting at? Special unitary group#The group SU(3) certainly links here, no? Cuzkatzimhut (talk) 22:06, 6 June 2022 (UTC)[reply]

Hi, sorry I should have been more clear. I meant articles such as representation theory of SU(2), representation theory of the Lorentz group. I know general representation theory is not really the point of this article, and such an article ought really to be a new article rather than a renaming of this one. Zephyr the west wind (talk) 22:38, 7 June 2022 (UTC)[reply]