Talk:Representation theory of the Lorentz group/GA1

GA Review
The edit link for this section can be used to add comments to the review.''

Reviewer: SparklingPessimist (talk · contribs) 18:00, 21 April 2017 (UTC)

I will be reviewing this article to make sure it fits within GA standards. Sparkling Pessimist  Scream at me!  18:00, 21 April 2017 (UTC)


 * Thanks for taking this on! There was some discussion of this GA nomination at Wikipedia talk:WikiProject Mathematics/Archive/2016/Dec that may be helpful for your review. —David Eppstein (talk) 23:59, 28 April 2017 (UTC)

Sorry for the delay, it's a really long mathematical article and I'm still trying to get through it. Thank you for your paitence. Sparkling Pessimist  Scream at me!  01:55, 29 April 2017 (UTC)
 * I think you're confusing me for the nominator? I'm just a bystander, interested in improving our mathematics coverage but with too little expertise on this particular subject to dare editing or reviewing it. —David Eppstein (talk) 03:34, 29 April 2017 (UTC)
 * I am the nominator. Please take all the time you need. There's absolutely no hurry. YohanN7 (talk) 07:06, 2 May 2017 (UTC)

Will also review
Thank you, for such a thorough article! I will review the article in the next few days as well. Jakob.scholbach (talk) 02:37, 7 May 2017 (UTC)

Finite-dimensional representations

 * The history section is lacking a reference. (The footnotes given are primary sources.) In the interest of space (this topic will appear again later...) I also suggest using the harvtxt template or similar, such as " essentially completed ...".
 * Rossmann has a few historical tidbits. But I don't know of any suitable dedicated source on the history. YohanN7 (talk) 15:21, 8 May 2017 (UTC)
 * Maybe Curtis' book? Jakob.scholbach (talk) 01:36, 9 May 2017 (UTC)
 * Now secondary references are in place. Though derirable, it is a difficult editing problem to change to harvtxt and preserve the flow. I put it on a low priority. YohanN7 (talk) 11:11, 23 May 2017 (UTC)


 * I find the first paragraph of "Strategy" confusingly written, and also partly too verbose (the readers of this article don't need to explanations such as "falsehood such as 0 = 1", say). I do have a little background in Lie algebras etc., but "One assumes heuristically that all representations that a priori could exist, do exist. " strikes me as unprecise and "Second, one can better understand the representations that do exist." does not convey meaning to me.
 * Not all has been written by me, and I agree. I rewrote it, and tried to substantiate in technical terms what is meant. I wrote it without references at hand, and will put in more references later. YohanN7 (talk) 11:44, 12 May 2017 (UTC)
 * We are moving towards the right direction; however, I am still firmly convinced that the way of writing is suboptimal. Since you cite Hall §7, take a look at how he writes: he opens the chapter with a statement --very much up front--what is the goal and how we accomplish it. In many sections, including the "Step 1", you offer the insight at the very end, which I think is not helpful, especially not with an article of this total length. Jakob.scholbach (talk) 18:08, 12 May 2017 (UTC)
 * I will think about this. (I copied your signature to the above reply. Hope you don't mind.) YohanN7 (talk) 09:46, 15 May 2017 (UTC)
 * The way Hall writes in the intro to Chapter 7 works well in the confines of the book. In it, all concepts used in the intro are introduced and used extensively in earlier chapters. An attempt to do the same thing here results in "blue link hell". I have, however, created a subsection at the end highlighting the result. YohanN7 (talk) 11:31, 15 May 2017 (UTC)
 * Why are there "2 Steps" -- you state that Cartan's theorem involves both steps. I would suggest restructuring like this. §3.2.1 Cartan's theorem on highest weights §3.2.2 Alternative constructions. Jakob.scholbach (talk) 18:08, 12 May 2017 (UTC)
 * The existence statement in Cartan's theorem is proved using one of the first three mentioned methods of constructions. (The last two aren't general.) "The three" can probably be modified to imply Cartan's theorem in full, but this is not how it is approached in my main reference on this (which is Hall's first edition). In his second edition (also referenced here), he refers to the steps (tentative classification and existence respectively) as the "easy part" and the "hard part". Maybe Rossmann could be used for "full" (the irrreps exists and are the only ones) Peter–Weyl and Borel–Weil approaches. At any rate, I have a good reference for the Verma module approach (and Lie algebras in general) that I'll put in. YohanN7 (talk) 09:46, 15 May 2017 (UTC)


 * The first and second paragraph belong, IMO, not to this article, but rather to Finite-dimensional representations of Lie groups (or something like that).
 * It is rewritten, but you may still feel the same way. There are, at least, two problems with your suggestion. First off, we don't have the relevant article. Secondly, this article is purposely written one level down. That is, it assumes that immediate prerequisites (like Cartan subalgebra) are not always available. To make the article readable for a slightly wider audience, a brief summary of notions are provided here, not using blue links to tersely written articles themselves consisting mostly of blue links. These summaries are also referred to elsewhere in the text. That said, prerequisites of prerequisites (like diagonalizable) are not provided (barring the non-technical intro given in a hide box). That would undoubtedly be going too far.
 * This approach (one level down) has been suggested to me because this is a highly technical graduate-level subject. (The whole thing is an experiment after all.) YohanN7 (talk) 12:24, 15 May 2017 (UTC)


 * You mention a "first step", but never (explicitly) a "second step". I also suggest using subsections if these steps are clearly delineated. "Then the classification part. " is not complete.
 * I agree about the "second step". I'll simply list three standard methods of realizing representations; Verma modules, the Peter–Weyl theorem and the Borel–Weil theorem, as well as the ad hoc method of "looking for representations where they might be found" (tensor products, Clifford algebras). This method is not to be sneezed at though fallible because when it works, it yields concrete representations. Then there's in the simplest cases the method of "starting from scratch and guessing". This last method is actually working in the present case (SU(2)/SL(2, C)). YohanN7 (talk) 15:21, 8 May 2017 (UTC)
 * Now there is a second step. The Unitarian trick gets a section of its own too. YohanN7 (talk) 11:44, 12 May 2017 (UTC)


 * Here and elsewhere: "one obtains", "one sees" should be avoided (see WP:MOS).
 * Also, "Let ... be" should be avoided. Jakob.scholbach (talk) 01:36, 9 May 2017 (UTC)
 * All "one has", etc are reformulated. YohanN7 (talk) 13:24, 17 May 2017 (UTC)


 * Also here and elsewhere: I would avoid self-referencing when not necessary, e.g. in "a topic which is investigated in some depth".

* The links to equations "(A1)" etc. don't work for me.
 * ✅ YohanN7 (talk) 15:55, 8 May 2017 (UTC)

* In equation A1, some C's should be C.
 * ✅ YohanN7 (talk) 14:33, 12 May 2017 (UTC)


 * I am not convinced that putting the explicit bases of J_i and K_i below is a good idea. What is the motivation here?
 * As an outsider, I'd strongly applaud their inclusion here... The thing is, lots of readers I know would only come here for these matrices... The'd recall such exist, and would desperately try to access them in a hurry. Where else would they go? Cuzkatzimhut (talk) 15:40, 12 May 2017 (UTC)
 * I did not mean to suggest to remove the formula, I rather suggest it to where it is used first in this article Jakob.scholbach (talk) 18:08, 12 May 2017 (UTC)
 * Originally, the section Conventions and Lie algebra bases was meant to serve as an appendix of sorts, in order to keep it non-technical, or at least non-numerical early on. Should I move the Lie algebra basis to its own section early on? YohanN7 (talk) 13:24, 17 May 2017 (UTC)


 * Is the list of "There is a ..."-items just saying that there are equivalences of categories of the said representations?
 * It was explained above the list how it was supposed to be interpreted, At any rate, I have reformulated. You aren't the first to have had complaints about that list. The redundant SL(2, R) is now removed. YohanN7 (talk) 16:04, 8 May 2017 (UTC)
 * I still think it would be beneficial to state it as an equivalence of categories (possibly including a brief explanation of this notion). (Or are they not equivalent?) This makes it clear that coproducts are preserved. An additional statement would be that the tensor structure on the four categories is compatible along these equivalences. Again, all of this would be ideally like so: "According to the general theory of [simply-connected] Lie groups, there is an equivalence of categories of representations of G and of g. Since ... is simply connected, this can be applied to G=...". Jakob.scholbach (talk) 01:36, 9 May 2017 (UTC)
 * You may be (probably are) right that the irreps constitute a category. There are at least three problems. I am not fluent in category theory, and the references do not mention it in this context. Then, a credible brief exposition of the notion of a category might not at all be very brief. (The "one step down" philosophy (see comments elsewhere) requires a credible exposition.) Then it would still remain unclear why we have a category (hence with preserved coproducts). The "why" is (I believe) explained in the paragraph that (as of now) resides between the (now two) lists. Observe here that compactness of $SU(2)$ is an essential ingredient (Peter–Weyl theorem for characters) in establishing the pne-to-one correspondences that would get lost if only simple connectedness is appealed to. I am not convinced about the introduction of category theory. YohanN7 (talk) 13:43, 15 May 2017 (UTC)
 * This basic google query shows immediately results in the direction I mentioned. See, for example "An Introduction to Lie Groups and Lie Algebras" by Kirillov, Theorem 4.3 on p. 53. Your response, btw, also point towards another weakness of the article: it is strongly based on just a few references, which has a potential effect of biasing the presentation towards the point of view of the author in question.
 * Also, a big detour in category theory is not at all necessary: a "credible" explanation could look like this: 1. quote the theorem as in Kirillov, say. 2. "Here, an equivalence of categories is, roughly speaking, a one-to-one correspondence between the objects on both sides: in this case, representations of a Lie algebra g and a simply connected Lie group G. Moreover, this one-to-one correspondence is also compatible with homomorphisms of the objects on both sides. This, in particular, includes a correspondence between irreducible representations: these are characterized by the property that they have no non-zero subobjects, a property which is preserved under any equivalence of categories. [If need be, add references]. Jakob.scholbach (talk) 15:40, 15 May 2017 (UTC)
 * Now Kirillov is in the reference list, and is referenced (once so far). I'll try to add more, and I'll also try to introduce reps of groups and algebras as categories. But alas, and this is a deficiency on my part, I still don't see how category theory adds anything beyond different terminology. I'll have to read up. Also, the new versions (in two places) of the unitarian trick immediately show that cross products and direct sums (presumably called coproducts in the world of categories) can be introduced.


 * Question: Is it a good idea to write a (short!) section on equivalence of representations ($G$-maps, $g$-maps, presumably "morphisms")? This would seem like a nice spot (if it is to be done) to introduce categories $Lie$ (our Lie groups), $lie$ (our Lie algebras) and ${Π:G → GL(V)|Π a morphism}$ and ${π:G → gl(V)|π a morphism}$ (our representations, one for each group/Lie algebra). Standard notation is probably totally different from this, but I'm sure you understand what I mean. YohanN7 (talk) 13:16, 16 May 2017 (UTC)
 * (unindent) I personally would phrase several statements (like the unitarian trick, the complexification) like this, but I am not an expert. Jakob.scholbach (talk) 04:43, 18 May 2017 (UTC)


 * I would try to avoid a way of writing which makes implicit assumptions. For example "Now, the representations of sl(2, C) ⊕ sl(2, C), which is the Lie algebra of SL(2, C) × SL(2, C), are supposed to be irreducible" -- by whom? and why are they supposed to be irreducible.
 * Now they are required to be irreducible. It might still be asked "by whom", but I think the context (we are looking for irreducible representations, and the unitarian trick preserves irreducibility) provides the answer. I'll make sure that complete reducibility is mentioned at an earlier point (explaining why irreducible reps only are sought for). YohanN7 (talk) 16:04, 8 May 2017 (UTC)
 * Why not write "The irreducible representations of ... necessarily have [this and this property]."? For my personal taste, the attitude of "requiring" things makes the write-up sound more mechanical. Jakob.scholbach (talk) 01:36, 9 May 2017 (UTC)
 * The "requirement" is now made by appeal to the section Strategy. YohanN7 (talk) 12:34, 15 May 2017 (UTC)


 * if done consistently -- verbose and also not to the point? After all direct sums etc. do have a standard (consistent) definition.
 * Consistently across the list. This belongs here in my opinion. YohanN7 (talk) 16:09, 8 May 2017 (UTC)
 * There is now an additional list, compatible with the entries in (A1), where cross products and direct sums are explicitly (and consistently!) introduced. YohanN7 (talk) 12:34, 15 May 2017 (UTC)


 * The section "Unitarian trick" is quite long. I would suggest working towards a more focussed presentation, by using clear statements about relations of representations of G and its Lie group, and the various types of complexifications considered. This also makes it clear how much of the discussion you do for SL(2,C) carries over to the case SL(2,C) x SL(2,C). Again, I can only underline that this article is way too long. (ThAny opportunities for streamlining the exposition should be used. Jakob.scholbach (talk) 01:36, 9 May 2017 (UTC)
 * It is much shorter now, an in my opinion clear. Generalities outlined in "strategy" and applied to case at hand in main "Unitarian trick" section. YohanN7 (talk) 11:35, 18 May 2017 (UTC)


 * The material on tensor products of representations belongs rather to the background section.
 * I don't agree. Tensor products have two interpretations, and it is important to tell which one is intended. Later in the article, the other interpretation is used and the equation here is referred to. YohanN7 (talk) 16:09, 8 May 2017 (UTC)


 * "The representations for all Lie algebras and groups involved in the unitarian trick can now be obtained. " -- again this way of writing is unencyclopedic. I would prefer presenting the theory not in the way "we do this, then we do that...". Try to rephrase it more concisely, by pointing out what is known, what tools / theorems are used.
 * Phrase removed. (The word "we" occurs nowhere in the article.) YohanN7 (talk) 08:33, 17 May 2017 (UTC)


 * In "Common representations", much of the explanation just doubles the table. Remove those.
 * I still see a lot of repetition there. Jakob.scholbach (talk) 04:43, 18 May 2017 (UTC)
 * ✅ again. YohanN7 (talk) 13:20, 18 May 2017 (UTC)
 * ✅ again. YohanN7 (talk) 13:20, 18 May 2017 (UTC)


 * The material "The Lie correspondence" is out of place in this article. I totally appreciate your energy in explaining these topics, but I think it is harming the purpose of this article to include too much background like this. You are basically copying large parts of a textbook here. Maybe even more to the point, much of this material is, if I am not mistaken?, not at all specific to the Lorentz group: §3.4.2 is true for any Lie group, §3.4.3 is true for any simply connected Lie group. A great service would be done to this article and to Lie correspondence (or some other related article) if this material here would be moved there. In this article, we could expect a brief summary of the general theory, applied to the Lie groups (and universal covers) and algebras relevant here.
 * I agree in full barring one point. Most will go out. At the time it was written, there was no article on the Lie correspondence to put it in.
 * The disagreement is this: There's no copying, just relying on one reference for the proof of the soundness of $exp$ for realizing representations and being detailed. If you really want to see me go detailed, have a look in hide box "Combinatoric details" at the bottom here.
 * The Lie correspondence is actually referenced to Rossmann, (not Hall), the only ref I have seen stating it for actual connected matrix Lie groups/Lie algebras and not equivalence classes of simply connected groups and Lie algebras. I don't appreciate (repeatedly on this page!) being accused of copying textbooks. It makes me look bad. I can live with being accused of OR (as WP defines it) once in a while, and, technically, there is some of it in the article. Not all statements are in all books, and I don't have all books. But I have, as time have passed, managed to either at least get confirmation from actual mathematicians or finally found references for almost all statements.
 * This latter point is what you should concentrate on in my opinion. Unreferenced statements, not the referenced ones. I am officially looking for a good reference on the Riemann P-functions and the Lorentz group action on them. YohanN7 (talk) 08:47, 16 May 2017 (UTC)
 * My review was not intended to make you look bad, neither as the author of this article nor as a person. Nonetheless, there is a fundamental agreement what a WP article should (not) be, expressed by WP:NOTTEXTBOOK. The recent edits make the article move in the right direction of using summary style, as opposed to an extremely detailed exposition. Jakob.scholbach (talk) 04:43, 18 May 2017 (UTC)
 * WP:NOTTEXTBOOK is totally uncontroversial. Totally. Pointing out that something is too text-booky is normal critisism. But this,
 * You are basically copying large parts of a textbook here.
 * is not normal criticism. (Similar phrasings occur elsewhere.) If you take a closer look, I base the proof of the exponential map yielding (projective) group representations (which is not the same thing as the Lie correspondence by the way), with ample references to Hall's four-page proof (or do you mean I copied Rossmann in Lie correspondence?). Firstly, four pages is not a large part of a textbook. Secondly, there is nothing that is copied . It is an outline of proof based on a single reference. I saw no reason to reference other sources for a technical, but standard proof. Please chose your words better. YohanN7 (talk) 13:37, 18 May 2017 (UTC)


 * At any rate, the Lie correspondence is reduced to bare essentials (that are refererenced from elsewhere). Most all on the proof of the exponential map yielding (projective) group representations is gone (but still referenced to from the new replacement text). YohanN7 (talk) 13:37, 18 May 2017 (UTC)


 * In the same vein, I think the section on the fundamental group is out of place here: it belongs better to an article about SO(3, 1) (do we have one?), summarized here briefly.
 * The fundamental group manifests itself in the representation theory, and has profound actual physical importance. This section, as well as most relating to spin and spinors is here mostly for the benefit of physics students (without being text-booky). There's a trinity that needs to be treated together for maximum enlightment; spinors (by def thingies transforming under projective reps), the fundamental group and the covering group. Besides, the section is short and to the point. It should go into Lorentz group as well. YohanN7 (talk) 13:49, 18 May 2017 (UTC)


 * I am not sure I understand the reason of including "A geometric view": it seems to be again the standard construction of the universal covering (together with the fact that this is again a Lie group). Can you clarify why this is included?
 * To illustrate how the covering group should be thought about, namely path homotopy classes. This section is now referenced, and now really needed as opposed to before the removal of details on exponentiation of Lie algebra reps, from the new section Group representations from Lie algebra representations. YohanN7 (talk) 14:02, 18 May 2017 (UTC)


 * The beginning of §3.5.3.1. (Concrete realization) is oddly written. I would just write something like this: the group SL(2, C) acts on the (n+1)-dimensional C-vector space of homogeneous polynomials of degree n by [...]. Using standard properties of the adjoint representation [i.e., eliminate (S3)-(S6)], this gives the following formulas for the sl(2, C).
 * ✅ (But the adjoint representation isn't involved here. ) YohanN7 (talk) 14:33, 12 May 2017 (UTC)


 * It is also a bit odd that the \phi_n for sl(2, C) are introduced only here, but already used above.
 * It is explicitly stated that representation theory of $su(2)$, or equivalently complex linear representations of $sl(2, C)$ are taken for granted as building material for this article. These reps are stated for reference, and also because they happen to be ingredients in the real linar $sl(2, C)$/$so(3, 1)$ reps, I. e. the $m, n)$ reps. At any rate, they have their own section now. YohanN7 (talk) 14:02, 18 May 2017 (UTC)


 * The sentence "The (m, n) representations are irreducible, and they are the only irreducible representations." is the main message of §3. Yet, it is buried in the middle of §3. It should appear at, or near the very beginning of §3.
 * Irreducibility and uniqueness should be clear already from section Strategy and the irreducibility is repeatedly pointed out (or "is required") in intermediate sections. The (m, n) representations (for Lie algebra and group) aren't baptized before section The (m, n)-representations of so(3; 1) and not defined in full until after section Realization of representations of SL(2, C) and sl(2, C) and their Lie algebras. I see no appropriate place to place yet another remark that the representations are irreducible (and all irreducible ones) YohanN7 (talk) 16:00, 22 May 2017 (UTC)


 * It would be good to say exactly at this point what the (m, n) representations are (in case a reader just jumps here).
 * ✅ YohanN7 (talk) 16:23, 22 May 2017 (UTC)


 * I believe the Weyl dimension formula is somehow an overkill for sl(2, C)? In any case, again, I think it is worth trimming the section like so: An explicit inspection of the representation, or alternatively applying the Weyl dimension formula shows dim \pi_m = 2m+1.
 * At least I swapped places between the Weyl dimension formula and simply counting. There's a quirk that the notation of the article introduces that is worth pointing out in relation to the general formula. YohanN7 (talk) 16:23, 22 May 2017 (UTC)

Jakob.scholbach (talk) 06:44, 7 May 2017 (UTC)

@User:Jakob.scholbach:Thank you for reviewing. Do you mind if I intersect your bullet list above when I comment? YohanN7 (talk) 08:05, 8 May 2017 (UTC)


 * Sure, go ahead. Jakob.scholbach (talk) 13:42, 8 May 2017 (UTC)


 * I have made a few edits, but it strikes me that it might be good to await the comments of SparklingPessimist. The article isn't supposed to move much while being evaluated. YohanN7 (talk) 14:51, 8 May 2017 (UTC)
 * I'll not be able to do more editing until Wednesday. YohanN7 (talk) 16:41, 8 May 2017 (UTC)
 * Sure. I will also need time to review the entire article. I don't know the exact process of the GAN, but I believe in this case it makes sense to put the nomination on hold. At least if I were the first reviewer, I would suggest this, based on the fact that massive changes to the article are very likely necessary to make it conform to the GA (and general WP) guidelines. (I should emphasize: this is not to say the article is bad!) Jakob.scholbach (talk) 01:36, 9 May 2017 (UTC)

Prerequisites

 * The example of the dihedral group is, IMO, not relevant enough to this article to be included.
 * Skipped. The non-tech intro now assumes the corresponding knowledge on part of the reader. YohanN7 (talk) 14:18, 22 May 2017 (UTC)


 * Symmetry of space and time: this largely reads like a (nice!) story, but not like a WP article (somehow not neutral). The section also has some typos.
 * It is now edited. YohanN7 (talk) 14:18, 22 May 2017 (UTC)


 * Lorentz transformations: remarkably, the article (and in particular this section) fails to simply define what the Lorentz group is.
 * Lorentz group now mathematically defined. (Odd that you this time want here what is (this time) clearly defined in the linked main article. Besides, the Lorentz group was defined in non-technical terms already. YohanN7 (talk) 14:18, 22 May 2017 (UTC)


 * I don't think Cayley tables are in any way relevant to the article? (Where are they used here again? -- The "multiplication table" mentioned later is not really helpful, is it? The closest we can get to a "table" are the generators of the Lie algebra and their relations, I believe?)
 * The multiplication table is the group, abstractly speaking. Preservation of the multiplication table is the defining condition of a faithful representation of any concretely realized group. YohanN7 (talk) 14:18, 22 May 2017 (UTC)


 * Next section: again, unencyclopedic writing ("the expected thing happens", "Ordinary Lorentz transformations matrices do not suffice")
 * Edited YohanN7 (talk) 14:18, 22 May 2017 (UTC)


 * Finite-dimensional representations by matrices: in §1, this is the one closest to a final shape, IMO. (Again, though, the writing should be more sober.)
 * Poetry reduced. YohanN7 (talk) 14:25, 22 May 2017 (UTC)


 * I would move "Infinite-dimensional representations by action on vector spaces of functions" to the section on infinite-dimensional reps later on. (And merge; use encyclopedic writing).
 * I trimmed away stuff from the section on infinite-dimensional reps that is in the non-tech intro. YohanN7 (talk) 13:39, 24 May 2017 (UTC)


 * Next section: "It should be emphasized that in the infinite-dimensional case, one is rarely concerned with these matrices. " -- this testifies how little relevant this section is. I would remove the section completely, possibly leaving a short (1-2 lines) explanation why matrices are less relevant in the infinite-dimensional context.
 * I believe it is highly illuminating, and in the interest of uniformity, to exhibit these representations as matrices. It is particularly relevant for the non-technical introduction. Besides, the matrix elements are of undisputed interest. (In fact, most calculations in QFT are concerned with finding matrix elements of infinite-dimensional representations of various operators, in particular of generators of the Lorentz transformations.) A perfectly valid question is the matrix elements of which matrix?


 * Lie algebra: the section is in general quite vague. I don't see the merit of having vague statements ("often") which can and will be made more precise later. Since this article is about reps of the Lorentz group, let's focus on what is specific about the Lorentz group. The last paragraph (on the metric signature etc.) is somewhat out of place: its relevance is not clear at this point. Jakob.scholbach (talk) 04:43, 18 May 2017 (UTC)
 * You forget that this is a non-technical introduction to representation theory. It would be very odd to leave out the Lie algebra (and the blue links where the reader can look further) and the tremendous simplification it offers. The previous sections talk about the group. The main body of the article talks initially about the Lie algebra. This little section glues it together. YohanN7 (talk) 14:57, 22 May 2017 (UTC)


 * A general reply first. This section is here because it was suggested to me. It was not present before GA nomination was discussed (see talk page). The rationale is to give the reader that is not a graduate student in math/physics at least a chance to understand what representation theory of groups is about, enabling him/her to read at least as far as the application section. It is aimed at a first/second year undergraduate student or maybe a bright high school kid. More than "one level down" in other words. So the assumptions I adopted is that the concept of "group" is (at least vaguely) familiar and likewise for "symmetry" but nothing more. These concepts are knit together, and it goes on from there.
 * With this background, you might see why the section is written the way it is, and why it is in a hide box. That said, without that background, I'd wholeheartedly agree with every single bullet in your list.
 * I think we should ask for input from others whether this section belongs at all. YohanN7 (talk) 07:38, 18 May 2017 (UTC)
 * Awaiting your general comment on my general comment, I'll edit the section for the obvious bullets; typos, "unencyclopedic writing", and the like. YohanN7 (talk) 10:14, 18 May 2017 (UTC)

On hold?
@SparklingPessimist, @Jakob.scholbach: I suggest we put this on hold for a week. I'll have plenty more time then to edit. YohanN7 (talk) 08:41, 11 May 2017 (UTC)


 * I agree. I have not yet read the other sections in detail, but I imagine some of my comments above could be extrapolated to these sections as well (moving material to related articles where sensible, provide brief summaries here; cite general facts where possible; trim details of proof unless these are crucial to the understanding of the rest of the article). Jakob.scholbach (talk) 14:06, 11 May 2017 (UTC)


 * A general comment: I'll certainly trim out a number of things e.g. much from "Lie correspondence". It, by the way, was written in a time when we had no article on it. However, the fact that some material ought to be found in the main articles, does not mean that it doesn't belong here. It is a virtue of this article that it avoids the all too common blue link hell. Concepts that are short enough to be explained here are explained here. If an important concept can be illuminated simply (from a different point of view), it is illuminated here. This is the vein in which the article is written, and it was as such I nominated it for GA. I am not interested in minimalism, but I am willing to trim. YohanN7 (talk) 11:55, 12 May 2017 (UTC)


 * I think I would agree with both here. As a theoretical physicist sending students here (WP, not this article) and having them demand explication of the articles which they find too scattered and compressed (in hyper-elegant sparse mathematese), I see the tangible demerits of blue ink hell. I think the wikilcnks should be for deeper, more detailed briefing on the points explained to a curious, perceptive reader here... if she/he wished to digress. But I assume this article is for the reader who wants to get the "big picture" and reads it in one sitting. So, brief summaries of all the pieces are actually welcome. My daffy rule of thumb might be that excessive sub sectioning could be avoided by a coherent summary in fewer supersections that string a memorable summary together. But, of course, I'm not doing this, so I'd be reluctant to proffer "unfunded mandates"! Cuzkatzimhut (talk) 13:48, 12 May 2017 (UTC)

Size considerations
I think our above discussion shows a fundamental disagreement: I am not raising these points to say the article is bad etc., but I just don't see it fits within the WP format right now, and I don't see it will fit after the kinds of small edits here and there discussed above. has done some tremendously detailed and careful work here, but unless these points are adressed in a much more fundamental way, I don't expect to support this GA nomination. I am expressing my opinion at this point since I believe closing the GA nomination later for this reason, after a lot of copyediting on various sections (which is also still necessary) might just cause a lot of frustration. I will be seeking more input from WT:WPM and Jakob.scholbach (talk) 15:17, 15 May 2017 (UTC)
 * there is a standard guideline, WP:TOOBIG, which states that an article > 100 KB "Almost certainly should be divided". This tool, which does not even count all relevant text, indicates a prose size of 171 KB (or, a printed pdf version is (including references, though), 35 pages). This article is, simply put, violating this guideline. While I appreciate the intention of making mathematical content available in down to earth terms, I don't think this goal is (currently) being achieved, effectively.
 * on a more content-based ground, I still have the impression that the article (currently) fails to guide the reader quickly to the most relevant pieces of information. Also, the article is, in parts, a rather faithful copy of a textbook, again conflicting with a guideline (WP:NOTTEXTBOOK).

I have also posted on the physics project. Jakob.scholbach (talk)


 * I think it is too early to give up. The big edit will be to take out most of "Lie correspondence" and the "exponential mapping". The latter is probably what you refer to as "a rather faithful copy of a textbook". It is detailed (on occasion more detailed than the reference used) and sticks to one reference, but that doesn't make it a copy. In any case, I'll continue editing on Wednesday. YohanN7 (talk) 16:01, 15 May 2017 (UTC)


 * Sure, I did not intend to discourage you from continuing. Take your time! I only wanted to avoid a foreseeable frustration on your part if you work on lots of little details throughout the article, when there is (IMO) a big structural issue with this article conflicting with the WP guidelines. Jakob.scholbach (talk) 16:34, 15 May 2017 (UTC)

The relevant number is (I ran the tool) I think this is acceptable for a technical article on a broad subject. Cutting out more topics will either violate "Broad coverage" or degrade the article quality. Not all sections are for everyone, but all sections are for someone. YohanN7 (talk) 09:25, 24 May 2017 (UTC)
 * Prose size (text only): 72 kB (14939 words) "readable prose size"

Pinpointing a paragraph in WP:Size
Here is a quote from WP:SIZE highlighting the issue:
 * Articles that cover particularly technical subjects should, in general, be shorter than articles on less technical subjects. While expert readers of such articles may accept complexity and length provided the article is well written, the general reader requires clarity and conciseness. There are times when a long or very long article is unavoidable, though its complexity should be minimized. Readability is a key criterion.
 * (extra emphasis mine)

Well, it was stated even before the GA nomination that it is an experiment. Can a technical article on a broad subject become GA, and actually be good in the real world sense? If the paragraph is read literally, the answer is, in my opinion, blatantly no.

But, as for me, I think the paragraph gets everything inside out, backwards, and upside down – in every little detail except for the last sentence: Readability is a key criterion. Yes it is. But what is readability?
 * Is "readability" referring to the readers ability to understand the majority of individual words (and even sentences) in a short article?
 * If it is (and this is my interpretation of the message), then the conclusion that the expert will appreciate a long article and the layman a short article downright laughable. It is exactly the opposite of the real world (NOWIKI) truth.
 * If it is not, then readability only means that the reader will perhaps not get very tired if actually reading (but not understanding) the short article. The reader is expected to
 * Not get mad and leave because he doesn't understand a word.
 * Patiently follow every blue link to other (small) articles, presumably with the intention of following its blue links.

Then apply the same reasoning to the weasel-word "clarity". I'll not repeat it all, but does "clarity" mean that to the reader unknown words are blue linked? Or does it mean that they are explained in place? The expert will certainly appreciate having terms and context explained elsewhere. He has seen it all. but the layman reader, etc...

I think the quoted paragraph is all nonsense. The prospective readers of this article are the junior and senior undergraduate students in physics or engineering. It should be written for them in mind, while still avoiding textbook style. If it isn't, it is for no-one. Slavishly following guidelines (excellent, good, bad, and horrible ones) may (even will) sooner or later imply GA (WIKI!), but will, i m o, never yield actual Good Article Status (real life/NOWIKI). Note: I am not naive enough to expect hoards of readers, but I'd expect ten happy readers per year (and many unhappy) as opposed to one happy reader (an expert) per year saying to himself "what a concisely written and stylistically impeccable article"

If cutting down size by using a chainsaw, the solution is not blue link hell. It is recognition of the fact that this subject is broad (in theory and application) and important (to its practitioners because it is to a large extent physics itself). There could be Representation theory of the Lorentz group (finite dimensions) and Representation theory of the Lorentz group (infinite dimensions). (In fact, the original author of the infinite-dimensional part had plans on expanding it not long ago, so it is not unreasonable to anticipate further growth.) YohanN7 (talk) 09:05, 26 May 2017 (UTC)


 * Oh, "ten happy readers per year (and many unhappy)" is quite feasible goal, since among thousands of students that learn this matter this year, every (not very bad) textbook will satisfy more than ten. The problem (as I see it) is that no textbook will satisfy majority. Why a problem? Since the article will never stabilize; consensus will never be reached; the satisfied minority will always be overruled by the unsatisfied majority. This is why textbooks must be a lot. Tastes differ; students differ. This argument does not depend on the (good or bad) WP policy about "nottextbook". This argument depends rather on the "no content forking" policy. A kind of no-go theorem: a portal without content forking cannot satisfy students. Boris Tsirelson (talk) 18:14, 26 May 2017 (UTC)

Discontinue reviewing
I will discontinue reviewing this article. The discussion above has gotten a little more heated than maybe helpful for the purpose of a review. At any rate, it is probable that a fresh mind (maybe ?) can review this article better than I can. Good luck to everyone. Jakob.scholbach (talk) 14:13, 18 May 2017 (UTC)
 * , I agree and I will be looking at this article this weekend, I just wanted to see your opinion before starting the review. Thank you for your opinion it is very helpful. Sparkling Pessimist   Scream at me!  21:26, 18 May 2017 (UTC)
 * I think my opinion about individual parts of the article is relatively clear from the above comments. (I have not reviewed in detail the applications section and the infinite-dimensional representation part.) To summarize, I think the article currently passes the GA criteria 1b, 2a-2d, 3a (as far as I can tell, I am not an expert in representation theory), 4 and 6a-b. I think it currently fails the criteria 1a (main and basic points such as the definition of a representation, the definition of the Lorentz group are missing, the prose is at times very wordy, little spelling errors also), 3b (this one to a large extent; probably the article size, which IMO is unwarranted, is the clearest violation of WP guidelines) and 5 (in view of the necessary changes, which YohanN7 has recently begun to undertake). In summary (i.e. part 7) I think the article currently fails the GA criteria. Jakob.scholbach (talk) 03:14, 19 May 2017 (UTC)
 * Late reply. I didn't see this post before. Yes, the concepts representation and Lorentz group aren't defined. The article is purposefully written "one level down". But these concepts were deemed by me to be two levels down (expected to be known). But they are to be found, mostly in non-technical terms, in the non-technical hidden intro. It is interesting that you continue reviewing and fail the article after discontinuing. YohanN7 (talk) 13:52, 24 May 2017 (UTC)
 * I believe it is good etiquette on a FA or GA nomination not to second-guess the intentions of the reviewer(s). I have spent much time (and much pain) with reading this article, yet you seem to have little appreciation for this effort.
 * That said, the only reason I have given an informal assessment (which is not to same as to fail the nomination) is because of the specific request of above. Jakob.scholbach (talk) 03:01, 25 May 2017 (UTC)
 * Which request?
 * Your effort is appreciated.
 * Your preemptive (second time, see above thread) failing the GA nomination, effectively not giving me a chance to make the edits, is not. (And you know by now what else has not been appreciated...) The WP criteria, by the way, are mutually inconsistent. E.g. "Broad coverage" (an explicit GA criterion) conflicts with "Size". You follow the paragraph of your liking here, not the relevant one. About the "Not a textbook", we are in agreement. Edits have been made made, removing much of detail and changing the language (one sees..., but never we see...) to totally impersonal (even artificial) language, but same thing here – no waiting for actual edits to be made. You should not have taken on this assessment if you never intended to let the nominee make edits before judgement day. YohanN7 (talk) 07:54, 26 May 2017 (UTC)
 * Well, I clearly stated that the article "currently" (as of writing the comment) did not meet some of the GA criteria (in my opinion). This was merely an assessment of my current opinion about the article, and I repeat that I did not fail the article. This accusation in simply unfounded. If you are really of this opinion, you are seriously misunderstanding my posts, to a point which is not helpful in such a process. Since Sparklingpessimist was asking for my opinion, I posted it here. Openly sharing one's opinion, and collaboration by means of that, is a basic principle of WP. I seriously don't understand why you seem to be so upset about it. In (what you might see as) the worst case, if the nomination actually fails, you can simply renominate it, possibly with some improvements in the meantime.
 * In any case, I think both your and my time is better invested in the article namespace, be it this article or any other. As I said, I wish you well with this article. Jakob.scholbach (talk) 15:08, 26 May 2017 (UTC)

Search "textbook" and you'll find what this supposed heat is about. I hope the original reviewer will concentrate on the Good article criteria, such as ""well written", ""broad coverage", "no OR" etc (actual criteria are found in previous thread) and not repeatedly accuse me of coping textbooks. I have been accused of OR before, but never "copying large parts of textbooks". YohanN7 (talk) 14:31, 18 May 2017 (UTC)

Textbook or not textbook?
My opinion: User:Tsirel. Boris Tsirelson (talk) 06:50, 19 May 2017 (UTC)


 * The dispute is not over WP:NOTTEXTBOOK. I fully agree that a Wikipedia article shouldn't be a textbook. I am no WP spring chicken.


 * It is a dispute about accusations of me having "copied large parts of a textbook". Very different indeed. It, if true, would go under Copyright violations and Plagiarism. But I reserve my rights to protest when someone sweepingly, without even being precise about which large parts in which textbook, I allegedly have copied. Not a single sentence has been given as example. YohanN7 (talk) 07:34, 19 May 2017 (UTC)


 * I see now that Copyright violations and Plagiarism are exactly criterion 2.d. in the GA criteria. YohanN7 (talk) 07:40, 19 May 2017 (UTC)


 * Sure, but this unfortunate and imprudent phrase of Jacob is already in the past; he wrote on your talk page "I did not mean to say you plagiarize a text-book, but that the exposition is like in a textbook (concerning of the level of detail)." Do you need more official apologies from him?
 * And now the dispute is (should be) not about copyvio but about being text-bookish (or not). Boris Tsirelson (talk) 08:22, 19 May 2017 (UTC)


 * , my apologies for not immediately accepting your apology. This unfortunate little incident is now in the past on my part. Tsirel is right. It should have been in the past even days ago. Maybe we both did wrong, and should have sorted it out quickly on the user talk pages instead. YohanN7 (talk) 10:44, 19 May 2017 (UTC)

My opinion: WP:NOTTEXTBOOK is not very enlightening about the properties making up a textbook or not. There is just this hint of not allowing for leading questions and systematic problem solutions as examples as establishing a textbook quality. Even examples, specifically those intended to inform are deemed appropriate within WP. I do not follow the terse ideal Jakob.scholbach seems to strive for, but I am convinced that any editor, producing high quality output for any article should be allowed to add meat to the dry bones to an extent that suites his paradigm of a GA, the size of an article does not matter very much, when its structure is elaborated. The no-textbook look and feel is grosso modo a weasely killer phrase, prohibiting acessible and easily readable WP articles. ... even when considered a WP spring chicken. Purgy (talk) 10:48, 24 May 2017 (UTC)

Short break
I'll not edit this weekend. YohanN7 (talk) 12:50, 19 May 2017 (UTC)

No more planned big edits
@

I have now made the changes I seem fit to meet Jacob's criticism in the bullet lists, and in his non-review review, sometimes half-ways, sometimes more and sometimes less. I suggest you create a new bullet list for points remaining in your opinion. Please, be as precise as you can with what you mean and (if applicable) location in article for every bullet. "There are spelling errors", "Write more encyclopedic" or "I find it amazing that..." wouldn't be helpful (for different reasons). I say this because Jacob's list, sometimes with contradicting requirements, was not very easy to work with. YohanN7 (talk) 11:59, 24 May 2017 (UTC)
 * I was going to step in and close this since everything's clearly done, but my only criticism is that the article does seem overly long. Are we sure that there are some parts that aren't overly detailed, or couldn't be in the main Lorentz group article instead? Wizardman  13:29, 20 August 2017 (UTC)
 * SparklingPessimist has recently returned to editing on Wikipedia, and they said they were going to stop by here to resume work on the review but haven't done so as yet. I think we've arrived at a "now or never" point, especially as regards Wizardman's queries. BlueMoonset (talk) 05:20, 21 August 2017 (UTC)
 * BlueMoonset, I have returned to editing and I see no reason why this shouldn't be closed as a pass given all of the work that has been done during my absence Sparkling Pessimist   Scream at me!  05:58, 21 August 2017 (UTC)
 * Wizardman, leaving this to your best judgment. BlueMoonset (talk) 04:43, 27 August 2017 (UTC)

No one else seems to have an issue with the length, though I still think it's a little overkill, so I'll pass this. Wizardman 13:56, 1 September 2017 (UTC)