Talk:Representation theory of the Lorentz group

message: * I added a subsection, The covering group, to the article. This particular description of the covering group is not the most common one, but I think it fits very well into the construction of the projective representations, and the 2 to 1 covering map  becomes obvious. The historical reference used, Wigner, 1937, is very detailed and thorough (and long). It contains fairly elementary and detailed proofs of other factoids as well, like that is simple, and that there are no finite dimensional unitary reps, that are otherwise hard to come by. YohanN7 (talk) 17:09, 22 February 2013 (UTC)
thread_title:  The covering group + Other edits 

message: I wrote a paragraph or two on how operators in QM transform under LT. This seemed to be easy enough to describe. It really is easy, but, as it turned out, not easy at all to describe. I did my best for now, using only word, no formulas.
In the future, I'll rewrite it provided there is some supporting material elsewhere, like how one formally handles tensor products of representations. Such things should not be developed in this article.
I'm reasonably happy with the added paragraphs, but not exactly full of joy. YohanN7 (talk) 18:32, 10 November 2013 (UTC)
replies: text: Can somebody help me fix the link to Greiners book in this section? (After "An algebraic proof of this fact is fairly lengthy ...) I don't understand what's wrong with it. YohanN7 (talk) 14:25, 11 November 2013 (UTC)
replies: text: Some anonymous hero did fix it. Thanks. YohanN7 (talk) 19:11, 12 November 2013 (UTC)
thread_title:  [[Representation theory of the Lorentz group#Induced representations|Induced representations]] 

message: Made a major addition. Small problems:
*Somebody please convert * to a dagger (for hermitean conjugate) in the obvious places. I don't know how to.
*Only Weinberg is a reference so far. He uses this example to show the non-simple connectedness of SO(3,1)+, but is not explicit with the formulas. Some formulas (and the notation) can be found in "Lie Groups, an introduction through linear groups" by Wulf Rossmann, but the article has enough refs as it is.
*I introduced a red link, namely the main theorem of compactness, saying that the continuous image of a connected set is connected. YohanN7 (talk) 17:36, 20 November 2013 (UTC)
replies: text: I'll try and fix the dagger problem. In future, at the top of the edit window, click "special characters", "symbols", and you should find the dagger in the top line of the character palette. M∧Ŝc2ħεИτlk 17:43, 20 November 2013 (UTC)
replies: text: Thanks. YohanN7 (talk) 17:58, 20 November 2013 (UTC)

text: By "main theorem of connectedness", are you referring to Zariski's main theorem or Zariski's connectedness theorem or something else? M∧Ŝc2ħεИτlk 17:51, 20 November 2013 (UTC)
replies: text: Something else, and simpler. If f:X->Y is continuous and X is connected, then f(X) is connected. YohanN7 (talk) 17:58, 20 November 2013 (UTC)

text: Maybe you could start a stub. Let's ask at WikiProject Mathematics. M∧Ŝc2ħεИτlk 13:02, 11 December 2013 (UTC)
thread_title:  [[Representation theory of the Lorentz group#The covering group|The covering group]] 

message: I wrote three new sections,Action of function spaces, The Möbius group and The Riemann P-functions. The first is supposed to make the transition to infinite-dimensional reps a little easier. The second and third gives what always has been promised in the lead, an action on the Riemann P-functions. 
As usual, there is the problem with references... YohanN7 (talk) 15:54, 22 December 2013 (UTC)
thread_title:  Three new sections 

message: IMHO one style should be chosen within the article. Most instances follow the latter syntax, but there are several in the former. Incnis Mrsi (talk) 08:34, 16 February 2014 (UTC)
replies: text: walked several times through the  formula in Representation theory of the Lorentz group #The Möbius group, and only now noticed that the orthochronous sign is missing. There may be more such eggs in the whole article. By the way, is there some reason to use  as the isomorphism sign instead of ? Incnis Mrsi (talk) 08:53, 16 February 2014 (UTC)
replies: text: Yes, it looks better. On the [S]O+(3;1) or [S]O(3;1)+ issue, of course the article should be internally consistent, but I don't know what is "right". YohanN7 (talk) 17:29, 27 February 2014 (UTC)
thread_title:  [S]O<sup>+</sup>(3;1) or [S]O(3;1)<sup>+</sup>? 

message: So I threw in a bunch of pictures of the people involved. I think it looks okay, especially when the table of content is hidden.
In the process, I removed this monkey (to the right stupid :D):
It definitely does not blend well with old black and white photographs. If you think it looks awful, well, shuffle around, make the pics smaller, or delete them. It's Wikipedia.
I have also made major edits (mostly) on to how to rigorously obtain group representations from Lie algebra reps, putting Lies fundamental correspondence into the picture. Also there are some new clarifying remarks on the unitarian trick. The latter section is still admittedly hard to understand. YohanN7 (talk) 23:09, 9 April 2014 (UTC)
thread_title:  Pictures 

message: I have expanded the text on  and companions. There is also a commutative diagram showing most of the ingredients in the section. Structurally, the diagram is okay, but it seems virtually impossible to get the fonts right. It looks somewhat better on my machine (fraktur font for Lie algebras, non-fat greek letters, etc). I'd highly appreciate if someone could improve on the picture, or, at least, tell me which fonts to use. It is made in Incscape and uploaded to commons. Is it possible to download from there? Else I can email the source to anyone itching to fix this. YohanN7 (talk) 21:52, 15 April 2014 (UTC)
replies: text: Fixed. Unfortunately at the cost of making paths out of text. YohanN7 (talk) 21:35, 16 April 2014 (UTC)
thread_title:  The universal covering group and a commutative diagram 

message: Can someone verify/correct this:
I don't have a reference for it. It's not in the article, but it's supposed to be (if correct) in Representations of SL(2, C) and sl(2, C) after the group formula for the -representations. YohanN7 (talk) 00:32, 16 April 2014 (UTC)
thread_title:  Unverified formula 

message: I rewrote most (actually all) of it providing supporting arguments, proof outlines and references. There are now a few formulae ("formulae" looks so much more sophisticated than "formulas") without proper citations, including the above mentioned one. Apart from that, there is only one thing left that I can think of for the irreducible finite-dimensional representations. Which ones are faithful and which ones aren't?
When it comes to infinite-dimensional unitary representations, I think it is fairly complete. It needs detailed proof outlines with references to conform with the rest of the article. I'll get to that next.
Then there is a mountain to write about finite-dimensional representations that are not irreducible. How do you construct them? It isn't as simple as saying that all of them are direct sums of the irreps. That is a tautology that leads nowhere for the applications of the theory. See, for instance, here: The unitary representations of the Poincaré group in any spacetime dimension. This is the key to the derivation of relativistic wave equations which would form a neat Representation theory of the Lorentz group#Applications section. YohanN7 (talk) 05:02, 21 April 2014 (UTC)
thread_title:  [[Representation theory of the Lorentz group#Properties of the (m, n) representations|Properties of the (m, n) representations]] 

message: Endnote 101 has "both" misspelled. (It says "botyh".)
There was another spelling error I just corrected, but in this case I can't edit endnotes.
I rarely see spelling errors in Wikipedia articles. Yet here I saw two. Please proofread this article.
166.137.101.174 (talk) 21:49, 20 July 2014 (UTC)
replies: text: You'll find that you can edit the footnote: edit the section that the footnote applies to, not the footnotes section. I've corrected this particular error, but not proofread the article as whole. —Quondum 04:17, 21 July 2014 (UTC)
thread_title:  Spelling 

message: I originally wrote parts of this section. My intention now is to write a slightly more detailed account of the Plancherel theorem for L2(G / K) and L2(G). The former reduces to the theory of spherical functions, which in the case of G = SL(2,C) in turn reduces to the Fourier transform on R; a purely formal argument using elementary aspects of operator algebras (von Neumann, Gelfand, Naimark, Godement, Dixmier, et al) leads to the direct integral decomposition of L2(G / K) into irreducible representations (the spherical principal series). The first part of this material is described in more or less self-contained form in Plancherel theorem for spherical functions#Example: SL(2,C). The second part is summarised there and the details can be given in an elementary way. The proof of the Plancherel theorem for SL(2,C) itself is described in various places. It is a much easier theorem to prove than the real case of SL(2,R). One approach is explained in the Appendix to Chapter VI of Guillemin and Sternberg's book "Geometric Asymptotics"; it applies to all complex semisimple Lie groups and is, according to them, Gelfand's original argument. I will firstly try to add the material on L2(G / K) in a brief form; and then I will try to devise what I consider the "simplest" account for L2(G). I am adding some parallel content to another article (Differential forms on a Riemann surface#Poisson equation), which is how I returned to this topic. Mathsci (talk) 10:42, 14 December 2016 (UTC)
replies: text: Sounds excellent! YohanN7 (talk) 15:19, 14 December 2016 (UTC)
replies: text: Thanks. I don't think your new sections on strategy and steps are the optimal way to present that material. I will think about how that material can be improved. The classification should come after the global description of the irreducible representations, which does not happen at present. Mathsci (talk) 19:05, 14 December 2016 (UTC)
replies: text: Okay. I was mostly following the finite-dimensional case where a "tentative classification" actually precedes construction in some common texts. The tentative classification is then validated by explicit construction. The "steps" 1-4 were from the Tung reference, which is an undergraduate text, and the approach is (implicitly) followed in Harish-Chandra's paper. But any improvement or total rewrite is of course welcome. I'm not all that happy with the present version either. It is certainly not optimal. YohanN7 (talk) 08:29, 15 December 2016 (UTC)
replies: text: The subject matter itself has two main problems in infinite dimensions: firstly it is not undergraduate material; and secondly it requires some effort to locate good sources that give a concise and comprehensible treatment for SL(2,C) (including the full Plancherel theorem). I am still looking. I added Gelfand, Graev and Vilenkin as another source; the treatment there is reasonable, but there are other approaches. Mathsci (talk) 08:48, 15 December 2016 (UTC)
replies: text: In the meanwhile, the present classification resides at the bottom of the infinite-rep section. (But I wouldn't agree that Harish-Chandra's paper is undergraduate material.) YohanN7 (talk) 15:52, 15 December 2016 (UTC)

text: The present classification isn't as poverty-stricken as it might seem as it, as far as I can see, with some work, provides explicit formulas for the non-zero matrix elements in every irreducible representation of the Lie algebra, finite-dimensional or infinite-dimensional. At least for the unitary ones in the latter case. YohanN7 (talk) 12:20, 17 December 2016 (UTC)

text: I am working on the material for the Plancherel theorem in my user space. In its initial form the references were given in the history section. Gelfand, Graev and Vilenkin was missing from the references. At the moment I am working out content related to what would now be called the reduced C* algebra of SL(2,C), i.e. the closure in operator norm of the *-algebra of convolution operators λ(f) for f in Cc(G). Mathsci (talk) 05:18, 20 December 2016 (UTC)
replies: text: Okay. All references are still there in the history section. Can anyone of them be used for inline citations? (Those references in Russian are problematic for some people, and hard to find.) YohanN7 (talk) 08:57, 20 December 2016 (UTC)
replies: text: I am concentrating on producing content at the moment. I don't know who added the cyrillic text: it was not a good idea. Three reasonable references are the book of Knapp, the book of Naimark on the Representations of the Lorentz group and the book of Gelfand, Graev and Vilenkin. The latter two have both been translated into English. Understanding of the underlying structure has advanced significantly since 1947. I think it is possible to convey that in the case of SL(2,C), but it requires some effort. Mathsci (talk) 13:43, 20 December 2016 (UTC)
thread_title:  Infinite-dimensional representations 

message: It seems a bit strange that in such a long article about reps of the Lorentz group there is no mention of the adjoint representation of the Lorentz group...
:That is a point. YohanN7 (talk) 08:21, 9 January 2017 (UTC)
replies: text: Looks like you fixed it adequately already. I don't think it needs very much more. Perhaps it could be mentioned in Induced representations on the Clifford algebra and the Dirac spinor representation that the constructs there are generalizations of the adjoint rep. YohanN7 (talk) 08:32, 9 January 2017 (UTC)
thread_title:  Adjoint Representation? 

message: This edit may have introduced an error. See also Adjoint Representation? on this talk page. The adjoint representation of a simple Lie algebra is irreducible. (A non-trivial invariant subspace would be an ideal. Also, inspection of the commutation relations explicitly confirms that there certainly aren't any 3-dimensional invariant subspaces of .) YohanN7 (talk) 14:20, 15 May 2017 (UTC)
thread_title:  Error? 

message: Some parts of this seem to have been done according to the erroneous by seemingly widespread view that lower-case Greek letters should not be italicized in non-TeX mathematical notation. But TeX italicizes them. It is only capital Greek letters that should not be italicized:
:  \Phi(\alpha,\beta,\gamma,\delta), \Psi(\varepsilon,\zeta,\eta,\theta) 
Michael Hardy (talk) 16:01, 15 May 2017 (UTC)
replies: text: I am aware of it, and have earlier today caught some such errors. (Will continue to edit on Wednesday.) YohanN7 (talk) 16:04, 15 May 2017 (UTC)
thread_title:  Italics 

message: Recent edits have completely changed the typesetting conventions in the article to use mathfrak for Lie algebras and blackboard bold for fields, apparently in violation of WP:MSM.  My own preference would be to refer to the older styling, for the reasons laid out at WP:MSM.  Is there consensus for that?  Sławomir Biały (talk) 10:29, 25 May 2017 (UTC)
replies: text: You are right. I feel a little bit bad about this because Latex-yow did post on my talk page suggesting this, and I approved. The long term plan ought perhaps to be to convert all math to Latex. If so, this has at present (on my part) low priority. YohanN7 (talk) 07:19, 26 May 2017 (UTC)
replies: text: I would point out two things: (1) the previous convention is very uncommon, lower case mathfrak is widely understood to be the character set for Lie algebras in mathematical publishing, and (2) the article was also inconsistent in its use, for example in many block display LaTeX formulas you had mathfrak with the text that immediately followed using math html bold. Finally the pictured diagrams are all using mathfrak. At any rate the conversion will be finished before Monday.Latex-yow (talk) 09:56, 26 May 2017 (UTC)

text: Conversion to mathfrak (and mathbb) complete. I would support a full migration to latex formulas as there are many spacing issues with math html.Latex-yow (talk) 22:23, 26 May 2017 (UTC)
replies: text: ..., just as there are sizing issues with inline Latex. But, I'll also support full migration to Latex because things aren't quite as bad as they used to be. At any rate, uniformity is a must. YohanN7 (talk) 09:30, 27 May 2017 (UTC)
thread_title: Conversion to mathfrak

message: Lead, end of second paragraph: the link to Plancherel formula does not work. Boris Tsirelson (talk) 19:13, 7 June 2017 (UTC)
thread_title:  Small remark(s) 

message: Dashes instead of a math symbol—namely, «»—linger for two years, and in multiple instances. Worse: originated not from a clueless editor, even not from a stupid script making blind replacements for U+002D. How can such guru as  commit this? Not good… guys, you should learn to manage without me, at the end. Incnis Mrsi (talk) 15:24, 3 May 2019 (UTC)
thread_title:  Typography 

message: Thank you to all contributes 41.190.245.201 (talk) 13:25, 30 December 2021 (UTC)
thread_title:  Amazing article ! 

message: I don't want to dive in and make these edits myself because (a) I'm not sure I'm right, (b) a lot of people have clearly put a great deal of hard work into this page and could have better ideas about how my observations could be used, but firstly there seems to be a reference (in the "Explicit formulas: Weyl spinors and bispinors" section) to equation G1, which doesn't exist. I'm pretty sure it should be a reference to equation A2 instead. Also, I think equation A2 is derived from the formulas A=(J+iK)/2 and B=(J-iK)/2 at the start of the section "The Lie algebra", and maybe it would improve clarity if this were made explicit. Lastly, I wonder if it's worthwhile showing an equivalent calculation to W1 for the (1/2,1/2) case, to show how the vector representation comes out of A2. 1.125.111.33 (talk) 01:58, 23 April 2023 (UTC)
thread_title:  Possible misreferenced equation, and suggested clarifications