Talk:Representation theory of the Lorentz group

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Recent edits are an improvement in clarity. Some comments;


 * I would fix a slight quibble in Properties of the (m, n) representations - there is the text
 * "Since the angular momentum operator is given by $J = A + B$, the highest weight of the rotation subrepresentation will be $m + n$. So for example, the ($1⁄2$, $1⁄2$) representation has spin 1 and spin 0 subspaces..."
 * which exemplifies the subspaces of ($1⁄2$, $1⁄2$) as 0 and 1 case before giving the general sequence of subspaces - it would be clearer to give the general subspaces then the example. (Maschen)
 * Done (YohanN7)


 * Also should the section The group talk a bit more directly about infinitesimal rotations in SO(3)? That's what it seems to infer. (Maschen)
 * By "small" here is meant small enough so that both the inverse function theorem and the Baker-Campbell-Hausdorff formula hold. This may generally be the whole connected component. They hold at least in an open neighborhood U containing the identity. When they hold, the exponential mapping yield a representation. Now, if g is far away from the identity 1, choose a path from 1 to g and write
 * $$1 = g_1^{-1}(g_1g_2^{-1})(g_2g_3^{-1})(g_3g_4^{-1})\cdots(g_{n-1}g^{-1})g,$$
 * where the gi are on the path and close enough to each other so that the factors enclosed by parentheses are in U. Use the exp formula for these. Now just solve for g and take Π of both sides using that Π is a homomormhism for the small element. Finally let that last expression define' Π(g):
 * $$\Pi(g) \equiv \Pi(g_{n-1}g^{-1})^{-1}\cdots\Pi(g_3g_4^{-1})^{-1}\Pi(g_2g_3^{-1})^{-1}\Pi(g_1g_2^{-1})^{-1}\Pi(g_1^{-1})$$ (YohanN7)


 * Notation...

The notations D(Λ) = "representation of the Lorentz group" and (m, n) = "finite dimensional irreducible representations" seem clear enough, however the notation (say) "D(1/2, 0)" (i.e. including the superscript) is confusing... (Maschen)
 * The D(m, n) is always the group rep corresponding to the Lie algebra rep (m, n). Sometimes (m, n) is used for group reps too. (YohanN7)


 * My burning questions are...


 * is D(2m) = (m, 0) for half-integer m, or equivalently D(m) = (m/2, 0) for integer m? Or... (Maschen)
 * is D(m) = (m, 0) for integer or half-integer m? (Maschen)

Just alternative notations/conventions?... (Maschen)
 * I don't know, but see above. (The D should always mean group rep.) (YohanN7)


 * In this paper by Gábor Zsolt Tóth (see appendix A4); letting m and n be integers:
 * D(m) = (m/2, 0) &oplus; (0, m/2),
 * $\overline{D}$(m) = (m/2, m/2),
 * D(m, n) = (m/2, n/2) &oplus; (n/2, m/2),
 * what does the last expression have for equivalent notation:
 * D(m, n) = D(?) &oplus; D(?) ?
 * while... (Maschen)


 * The WP article takes m, n to be half-integers in (m, n), so does that translate to
 * "D(2m, 2n) = (m, n) &oplus; (n, m)" ?
 * and this has what notation:
 * "D(m, n) = D(?) &oplus; D(?) ? (Maschen)


 * In all... is the statement
 * D(m, n) = (m/2, n/2) &oplus; (n/2, m/2) = "(2m  +  1)(2n  +  1)-dimensional irreducible representations of D(Λ)"
 * true? (Maschen)


 * Are any of the differing conventions, where to put the 1/2 factor, or just the choice of what is integer and half-integer, is "the" standard? (Maschen)


 * It looks like Tóth defines properly what he is doing (58),(59) and (60) in terms of our notation. (YohanN7)


 * I have the feeling that mathematicians prefer (m, n) = pair of integers. For su(2) and so(3), the corresponding statement is certainly true. (YohanN7)


 * It would be good to clarify the correspondence between index notations (tensors and spinors) and the (m, n) representations, again indicated in that paper... (Maschen)

Minor comments though, good work! (I might as the maths ref desk btw). M&and;Ŝc2ħεИτlk 08:40, 16 February 2013 (UTC)

Thanks for the comments. I'll in time add a small section on notation and conventions to the article - and perhaps the formula above. YohanN7 (talk) 15:43, 16 February 2013 (UTC)


 * That would definitely help. What you say above:
 * By "small" here is meant small enough so that both the inverse function theorem and the Baker-Campbell-Hausdorff formula hold..."
 * could be stated in those plain words in the group section of the article though. The section does say that, but didn't seem very obvious before, and (with respect to the rewrite) it still doesn't (apologies)...


 * Also, please let's not intersect each other’s posts - else future readers (including ourselves) will not know who wrote what. I relabelled which posts are which above. M&and;Ŝc2ħεИτlk 00:05, 17 February 2013 (UTC)


 * The condition that the theorem based on inverse function theorem holds is named condition (A), and the corresponding condition for Baker-Campbell-Hausdorff formula is condition (B). In the open set $
 * U$ both (A) and (B) hold. I don't know if I did this before or after the last reply (00:05, 17 February), probably after.
 * Intersecting posts without signing, what was I THINKING about? Not about signing apparently... I personally think that "intersecting" in a longer bullet list is (depending on context of course) quite ok provided one signs (and there is mutual concent;). YohanN7 (talk) 11:08, 17 February 2013 (UTC)


 * No worries. M&and;Ŝc2ħεИτlk 00:43, 18 February 2013 (UTC)

Commutative diagram
I'd like to add a commutative diagram or two to the article when it has "settled". How do I make them? Volunteers? YohanN7 (talk) 13:00, 18 February 2013 (UTC)


 * You can't produce them in the current TeX rendering, use xymatrix in LaTeX then export to a pdf or svg file (although this didn't work for me recently, for some reason), see also help:displaying a formula, or just use any graphics program and "draw" it. If you tell me what to produce I can add the diagram. M&and;Ŝc2ħεИτlk 13:16, 18 February 2013 (UTC)


 * Forgot to add; when you're ready, just list all morphisms like so:


 * $$A \xrightarrow{F_1} B, A \xrightarrow{F_2} C, B \xrightarrow{F_3} D, \cdots$$


 * and I could put them together. M&and;Ŝc2ħεИτlk 02:38, 19 February 2013 (UTC)


 * Wonderful! But we better wait a little. I can still see minor notation changes, like $Π->Π_{U}$. YohanN7 (talk) 12:53, 19 February 2013 (UTC)

The covering group + Other edits

 * 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 $p:SL(2;C)→SO(3;1)^{+}$ 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 $SO(3;1)^{+}$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)


 * I added the explicit formula for the $π_{(m,n)}$ representation to the Lie algebra section. I don't really have a reference for this, I reverse engineered it from the formula on component form in Explicit formulas, which I do have a reference for. I am known to screw up on occasion, so it wouldn't be out of place to verify what I have written. At any rate, the section feels utterly incomplete without that formula. The (verifiable) component formula is, by itself, too detailed. YohanN7 (talk) 19:05, 22 February 2013 (UTC)


 * Added remark that, by choice of phase, projective reps (double valued reps) can be made continuous locally but not for the whole group (ref Wigner).


 * Renamed Π to ΠU in formula (G2).


 * Replaced formula (G3) with a simpler version (fewer inverses) and sourced it (ref Hall). YohanN7 (talk) 20:22, 22 February 2013 (UTC)


 * Commutative diagram


 * Plenty of minor tweaks to Full Lorentz group. Fixed an error in the description of the adjoint action of a group representation on its algebra representation. Pseudoscalars defined. Antiunitarity and antilinearity of T included. Got rid of the too prudent if and only if.


 * Charge conjugation parity $C$ is now mentioned in Full Lorentz group because it is not direcly related to Lorentz symmetry. It is really off topic, but since it is the "missing ingredient" that together with $P$ and $T$ make up CPT, I think it is worth mention that it is not related from Lorentz invariance. YohanN7 (talk) 02:59, 23 February 2013 (UTC)


 * Explicit formulas added for Weyl spinor and bispinor (Dirac spinor) Lie algebra reps at the bottom of the article in Weyl spinors and bispinors. YohanN7 (talk) 07:22, 23 February 2013 (UTC)


 * Physics and math project templates YohanN7 (talk) 18:11, 24 February 2013 (UTC)


 * ... and Relativity YohanN7 (talk) 18:54, 24 February 2013 (UTC)

Representation theory of SL2(C)
Several days ago I redirected here. Today I realized that can simply explain only how representations of the Möbius group (PSL2(C)) are included to Lorentzian representation theory, whereas SL2(C) is its covering, not a subgroup. Could somebody explain how projective representations of the Lorentz group become true representations of SL2(C)? The article should not be silent about SL2(C) if only because it’s SL2(C) which explains why some reps are true Lorentzian representations and others are only projective ones. Incnis Mrsi (talk) 15:35, 26 June 2013 (UTC)


 * Good point. But I am not entirely sure that an explanation belongs in this article, since it is a general feature of the theory of covering groups and representations.


 * Every Lie group has a simply connected covering space with a canonical smooth structure and (up to isomorphism) a canonical group structure making it a Lie group called the universal covering group. The Lie algebra of this group is isomorphic to the Lie algebra of the group one started with, and hence the representations of the two algebras are in one-to-one correspondence. The representation of the Lie algebra of the universal covering group always lifts to a representation of the universal covering group because the latter is simply connected. (This particular point is made in the article.) The covering map from the universal covering group to the original group is a group homomorphism. This covering map is many to one in case the original group is not simply connected; in the Lorentz group case it is 2:1. Thus if one tries to obtain a representation by first going from the original group to the universal covering group and then to its representation (composition of functions), then one needs to choose one of many (two in the Lorentz case) elements in each fiber of the covering map. The result is, in general, not a group homomorphism, but there will be a phase factor (+/-1 in the Lorentz case). This is roughly how projective representations come about.


 * If you can make sense of the above, please go ahead. I think I can make things more precise if needed, including references. YohanN7 (talk) 13:34, 16 July 2013 (UTC)


 * Also, if the kernel of the covering map is contained in the kernel of a representation of the universal covering group, then the representation will "pass to the quotient" yielding a proper representation. (An easily proved lemma for the first isomorphism theorem.) The kernel of the covering map is, in the Lorentz case, {I,-I}. Thus, for example, if the representation of the universal covering group is faithful, meaning it's kernel is {I}, then we are necessarily looking at a projective representation of the quotient (the Lorentz group).


 * I could (a couple of weeks later) write a detailed account of this, including a very explicit proof that SL2(C) is the universal covering group of SO(3;1)+, if at least a couple of users believe it should go into the article. I'm not sure myself that it belongs here other than as a link. YohanN7 (talk) 13:50, 17 July 2013 (UTC)


 * By the way, I suspect that what I called the $(1⁄2,0)⊕(0,1⁄2)$ representation in Weyl spinors and bispinors actually is the $(0,1⁄2)⊕(1⁄2,0)$ representation. In other words, I might have confused left and right Weyl spinors. YohanN7 (talk) 13:46, 16 July 2013 (UTC)

Induced representations
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)


 * 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)


 * Some anonymous hero did fix it. Thanks. YohanN7 (talk) 19:11, 12 November 2013 (UTC)

Commutation relations in "Explicit formulas" section
In the same way one writes basis vectors as e1, e2, (each a different vector, and the subscripts are not the components of the basis vectors, in which case we may write something like [ei]j), wouldn't it be better to write the commutation relations as:


 * $$[J_i,J_j] = i\epsilon_{ijk}J_k, \quad [J_i,K_j] = i\epsilon_{ijk}K_k, \quad [K_i,K_j] = -i\epsilon_{ijk}J_k.$$

since we have already defined what J1, J2, J3, K1, K2, K3? Call me nitpicky but it would be much clearer. M&and;Ŝc2ħεИτlk 06:58, 19 November 2013 (UTC)


 * Yes, boldface is simply wrong. YohanN7 (talk) 11:28, 19 November 2013 (UTC)


 * Forgot to mention, after saying the commutation relations of A and B in words in the The Lie algebra section, we should actually include them:


 * Nice. YohanN7 (talk) 11:28, 19 November 2013 (UTC)


 * then later after the formula $$[M^{\mu\nu},M^{\rho\sigma}] = i(\eta^{\sigma\mu}M^{\rho\nu} + \eta^{\nu\sigma}M^{\mu\rho} - \eta^{\rho\mu}M^{\sigma\nu} -\eta^{\nu\rho}M^{\mu\sigma}).$$ point back up to the Lie algebra section? For now I'll add them the commutation relations for A and B. Stating the commutation relations at the outset rather than just mentioning the groups would also be clearer what is meant, since anyone with regular QM background will immediately recognize these as the angular momentum commutation relations. M&and;Ŝc2ħεИτlk 07:07, 19 November 2013 (UTC)


 * Don't know if the commutation relations should be stated at the outset. They are a "prerequisite" for the article (i.e. really belong in either Lorentz group or Lorentz Lie algebra (if it existed). Also, they aren't used exlpicitly (unless we beef out completely). I'm mainly thinking of preservation of space here.


 * A couple of stylistic issues:
 * I don't like "visible" references. They take up space, and if given, they should I m o be to a really reputable source.
 * The colon-equation style may "exist" in some sense, but blending in that style with what is standard standard (used in the rest of the article) is hurting the eye a bit (and is a stylistic nono).


 * I edited your edit regarding the four (or two or whatever) indices in an equation. I feel that a full explanation of the indices belongs to the Kronecker product article, which is (I think) linked. YohanN7 (talk) 11:28, 19 November 2013 (UTC)


 * I still don't get the issue with colons for indenting formulae. The colon is to indent almost every displayed formula in WP like this:
 * $${x'}^\alpha = \Lambda^\alpha{}_\beta x^\beta \,, $$
 * nothing more.


 * I'm not talking about indentation.


 * Newtons second law reads
 * $$ F = ma.$$
 * Newtons second law reads:
 * $$ F = ma.$$


 * One of the two sentences above is not proper English. I'm saying that even if the :equation-style exists, it's something I don't like (which is largely irrelevant), and one should not mix two styles in the same article, in Wikipedia or elsewhere, (which is relevant). YohanN7 (talk) 01:01, 20 November 2013 (UTC)


 * Fair point about the refs being to "visible", so I'll trim these. However, the books by Abers and Ohlsson are not "unreliable" in any way - they are proper graduate-level quantum theory books as good as any other and are fine for refs (in fairness IMO Ohlsson's book is almost like Wienberg's vol 1 but much more compressed and easier to follow).


 * I Can't comment on Ohlssons book. It's not the point. The books seem (in the article) to be introduced to motivate the notation, which barely needs a reference at all. YohanN7 (talk) 01:01, 20 November 2013 (UTC)


 * I scanned the contents of Ohlsson's book. The Weinberg and Ohlsson books have very different scopes, so you can't compare them. The Ohlsson book is introductory, while Weinberg's is general, and therefore comparatively abstract. YohanN7 (talk) 01:17, 20 November 2013 (UTC)


 * While brevity is important, a well-written article should establish it's content. It did already, I'm not denying that. If we can talk about abstract groups, why not one extra line for immediate visual impact? The aim is not to clamm up the introductory paragraphs and make them unreadable - I will not add anything beyond the commutators of A and B (also the definitions of A and B are displayed LaTeX to clearly stand out instead of being buried in the text). M&and;Ŝc2ħεИτlk 20:29, 19 November 2013 (UTC)
 * I didn't object to the A and B. I questioned the commutation relations for the Mμν far up. YohanN7 (talk) 01:01, 20 November 2013 (UTC)


 * Deleted the colons and refs. I didn't mean to add the commutation relations for the Mμν high up at all. M&and;Ŝc2ħεИτlk 09:19, 20 November 2013 (UTC)


 * Ok, I misunderstood something. By the way, do you write "proper english" or "proper English"? In the former case, one sentence of mine above must look rather silly;) I have seen both versions, but my editor, which I really don't trust, suggests it's English, not english. YohanN7 (talk) 12:56, 20 November 2013 (UTC)

Apologies if my late-coming ignorance of the entire record above makes my point below meaningless, but... The explicit formulas are great, and a student/reader may well wish to see them right in section 1, coming from the Lorentz group article, with its finite Lorentz transformations, etc... But your conventions are a bit funny, as they stand.... The Js are hermitean, but the K's are antihermitean... So then the Ms are not uniformly hermitean or antihermitean themselves, and worse, the As are not hermitean conjugate to the Bs, as defined...spinors beware. Actually,  the As and the Bs do not commute with each other in terms of the explicit Ks as defined...or do they? All could be fixed by dropping the i's in the definition of the Ks, I think, and would make real boost parameters contrast to real angles that would enter with a relative i to them.... but too many convention chefs spoil the broth, and maybe I should wait until the dust settles to enjoy the final word. I don't want to discourage the fabulous idea to highlight the explicit matrices, a "must" for the beginning of the article. I also suspect that the Jjks were never defined, although their connection to the Ms is evident. Cuzkatzimhut (talk) 01:33, 3 December 2013 (UTC)


 * There is very likely to be errors somewhere, but I don't think it is possible to have the Ms all hermitean. If that was the case, the group generated by them would be unitary, and the Lorentz group doesn't have any finite-dimensional unitary representations at all. The important thing is that the displayed matrices (don't know where they come from) satisfy the right relations, given in terms of the Ms, or the line below, in 3d notation, which I do know where they come from (the given reference). YohanN7 (talk) 04:15, 3 December 2013 (UTC)
 * Yes, you are right on the Ms. The As and Bs look right now.  Their conjugation relation to each other is now −*, or minus transposition, which is fine. I failed to see your discussion near the top of the article, because the actual matrices were not next to it. I suspect using Ms instead of J jks would be clearer. Cuzkatzimhut (talk) 12:51, 3 December 2013 (UTC)
 * Either Ms or Js should be fine, but not, as it was, both. Now only Js. I have ran most commutation relations in a program of mine, and the matrices seem to deliver what they promise to do.
 * I guess the issue is if we should have explicit matrices near the top of the article. I vote "no", but it does not represent a strong opinion. I'm concerned about space preservation, because there are several additions I plan to make to the article. Also, too much hardware near the top of an already technical article might scare people off because it might look more daunting than it really is. How about a very visible link to where the really explicit stuff can be found? YohanN7 (talk) 17:07, 3 December 2013 (UTC)
 * Sure, a link in the 2nd-3rd line of section 1.1, sending one to the bottom, Appendix like, for explicit 4d rep sounds good. Cuzkatzimhut (talk) 17:46, 3 December 2013 (UTC)
 * Ok, I have done something. It's better than nothing, but still not good. Thank you for pointing out these matters. Now, the link comes as the third sentence (or something like that). What I really want to do is to write an introduction to Representation theory of the Lorentz group as well as a smaller one to Representation theory of the Lorentz group, if you see my point.

“traceless metric tensor”
Wouldn’t this enigmatic thing from be replaced with the (traceless) stress–energy tensor? A symmetric 2-form has 10 components and its representation should be (1, 1)“traceless” ⊕ (0, 0). But for the stress–energy tensor $(−1, 1, 1, 1)$ is not normally zero, whereas $J_{i}$, isn’t it? Incnis Mrsi (talk) 19:46, 19 November 2013 (UTC)
 * My guess is that it originated as “traceless sym metric tensor” but several letters were lost in transmission. Incnis Mrsi (talk) 08:39, 20 November 2013 (UTC)


 * Unfortunately, my browser doesn't display the math expression in "A symmetric 2-form has 10 components and its representation should be (1, 1)“traceless” ⊕ (0, 0).". Could you use other versions (archaic ones please, I'm using XP) of the characters?


 * Anyway, the $K_{i}$-representation should have $A = (J + i K)/2$ dimensions. "Symmetric" should mean 10 independent components (dimensions), and "traceless" would reduce this to 9 dimensions. So, the statement in the article (latest version) seems to make sense. But we should have an example of such a tensor field. A traceless stress–energy tensor would do nicely. Is it (or can it be made) traceless? My memory is fading here, and the linked article doesn't tell. YohanN7 (talk) 12:06, 20 November 2013 (UTC)
 * Because you consume a clumsy “official” CSS (say thanks to guys who purged useful tips from WP:«math», as well as to “CSS masters” who do not do anything useful for math styles for about a year, only break more things). Use a good CSS. You even do not need to have fonts locally: MathJax will download them for you. Incnis Mrsi (talk) 17:36, 21 November 2013 (UTC)


 * No thanks. MathJax is too slow. Only idiots use it. YohanN7 (talk) 00:25, 22 November 2013 (UTC)

The covering group
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)


 * 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&and;Ŝc2ħεИτlk 17:43, 20 November 2013 (UTC)


 * Thanks. YohanN7 (talk) 17:58, 20 November 2013 (UTC)


 * By "main theorem of connectedness", are you referring to Zariski's main theorem or Zariski's connectedness theorem or something else? M&and;Ŝc2ħεИτlk 17:51, 20 November 2013 (UTC)


 * 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)

I should have mentioned this as well; The Rossmann book is brilliant (perhaps the very best of the introductory texts in Lie group theory), but it contains a gazillion of minor errors. The formulas in his book are, for this reason, not identical to the ones in the article. Either he or I have screwed up. YohanN7 (talk) 23:05, 20 November 2013 (UTC)


 * Maybe you could start a stub. Let's ask at WikiProject Mathematics. M&and;Ŝc2ħεИτlk 13:02, 11 December 2013 (UTC)

New History section
So, I wrote a history section. Anyone of major importance forgotten? Somebody unduly there? Years correct? I'll do some digging myself, but any help with original references is much appreciated. One reference per name would be great, I think. YohanN7 (talk) 12:39, 11 December 2013 (UTC)


 * Looks great, good work!
 * Wouldn't Pauli come into this somehow for introducing the Pauli spin matrices, which are a special case of the general spin matrices used in the J operators? Lorentz is not mentioned, did he make any contributions to the group theory (I don't think he did, but could be wrong).
 * Just some thoughts... M&and;Ŝc2ħεИτlk 12:53, 11 December 2013 (UTC)


 * Lorentz should probably be there. He basically told Einstein, "Hey, you are dealing with a Lie group." And, after all, it's Lorentz' own Lie group. Einstein should perhaps be there too, for the (verifiable) joking comment "I don't recognize my own theory any more since the mathematicians got hold of it". I'd vote "no" to Pauli, since he was (originally at least) dealing with SO(3) symmetry in a 3-dimensional rep, not with O(3;1) symmetry. Did he come up with the general so(3) reps? If so, then "yes". YohanN7 (talk) 14:50, 11 December 2013 (UTC)


 * I'm not sure about Pauli at all. But others from Einstein and Lorentz also derived some/all of the Lorentz transformations. Should we ignore them? Probably since they just derived the transformations, without contributing to the group theory. What about Poincaré and the Poincaré group (inhomogeneous Lorentz group?) M&and;Ŝc2ħεИτlk 07:26, 13 December 2013 (UTC)

Three new sections
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)

Lie algebra representations from group representations
New mini-section. Can somebody please fix equation G6? There should be a vertical bar in it (the derivative should be evaluated at t = 0). Don't know the TeX for that. YohanN7 (talk) 23:38, 15 February 2014 (UTC)
 * Did you try simply the vertical bar character from þe olde goode ASCII? Yet one remark: you are scanty on spaces. It rarely is relevant in the &lt;math> mode, but irritates in the math mode. Do you have some problems with the spacebar key? Incnis Mrsi (talk) 08:34, 16 February 2014 (UTC)


 * Relentlessly pressing the spacebar over and over again tends to wear the battery in the keyboard down. I avoid it if I can.
 * What exactly do you have in mind? The math mode incorporates "no line break" if that is what worries you. YohanN7 (talk) 17:21, 27 February 2014 (UTC)

[S]O+(3;1) or [S]O(3;1)+?
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)
 * walked several times through the $B = (J − i K)/2$ formula in Representation theory of the Lorentz 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 $A$ as the isomorphism sign instead of $B$? Incnis Mrsi (talk) 08:53, 16 February 2014 (UTC)


 * 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)

Pictures
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)

The universal covering group and a commutative diagram
I have expanded the text on $T_{αβ} − 1⁄4T_{ξ}^{ξ}g_{αβ}$ 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)


 * Fixed. Unfortunately at the cost of making paths out of text. YohanN7 (talk) 21:35, 16 April 2014 (UTC)

Unverified formula
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 $g_{αβ} − 1⁄4δ_{ξ}^{ξ}g_{αβ} = 0$-representations. YohanN7 (talk) 00:32, 16 April 2014 (UTC)

Properties of the (m, n) representations
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 section. YohanN7 (talk) 05:02, 21 April 2014 (UTC)

Spelling
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)Collin237


 * 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)

A bit unclear
In the lead, "fields in classical field theory, most prominently the electromagnetic field, particles in relativistic quantum mechanics" could be misunderstood: "particles in relativistic quantum mechanics" are not "fields in classical field theory".

"It enters into general relativity because..." — which "it"? Spin? The classical electromagnetic field? Quantum mechanical wave function? The representation theory? Boris Tsirelson (talk) 07:29, 2 December 2016 (UTC)


 * I tried to fix the first sentence, and then "it = the theory" for GR. Does it work? YohanN7 (talk) 08:21, 2 December 2016 (UTC)


 * Nice. Boris Tsirelson (talk) 10:49, 2 December 2016 (UTC)

"Non-compactness implies that no nontrivial finite-dimensional unitary representations exist." Really? The real line is non-compact, but has nontrivial finite-dimensional unitary representations; some of them are faithful (but reducible); some are irreducible (but not faithful). Boris Tsirelson (talk) 11:38, 2 December 2016 (UTC)


 * The formulation should be A connected simple non-compact Lie group cannot have any nontrivial finite-dimensional unitary irreducible representations. It is detailed in the section non-unitarity. Does it look correct? I'll "complete the hypothesis" in the incorrect statement you found. YohanN7 (talk) 12:18, 2 December 2016 (UTC)


 * I got puzzled. No irreducible? Thus, also no reducible? In finite dimension a reducible representation must have a nontrivial irreducible subrepresentation, right? Boris Tsirelson (talk) 14:12, 2 December 2016 (UTC)


 * Ha, yes, it appears to be the case. The statement in the reference from where I extracted the proof is
 * Finite-dimensional unitary reps of non-compact simple Lie groups: Let U : G → U(n) be a unitary representation of a Lie group G acting on a (real or complex) Hilbert space H of finite dimension n ∈ N. Then U is completely reducible. Moreover, if U is irreducible and if G is a connected simple non-compact Lie group, then U is the trivial representation.
 * Page 4 in * YohanN7 (talk) 14:24, 2 December 2016 (UTC)
 * Added missing "unitary" in the statement above. YohanN7 (talk) 14:33, 2 December 2016 (UTC)


 * I suppose a better statement would be
 * A connected simple non-compact Lie group cannot have any nontrivial finite-dimensional unitary representations.
 * with "irreducible" striked out. The Lorentz group has the property of complete reducibility, meaning all reps decompose into a direct sum of irreducibles (I think all semisimple groups have that property). YohanN7 (talk) 14:42, 2 December 2016 (UTC)

In "The unitarian trick" section:
 * The following are equivalent:
 * There is a representation of SL(2, R) on V
 * There is a representation of SU(2) on V

and so on. Is this "there is" really the existence quantifier? If so, it is rather a property of a natural number, the dimension of V. But I guess, you mean much more, something like "The following objects are in a natural one-to-one correspondence". Though, if it is clear that such a representation (for a given dim(V)) is unique (up to isomorphism), then indeed my remark is pedantic. But in this case a short clarification could be helpful. Boris Tsirelson (talk) 14:35, 3 December 2016 (UTC)


 * I'll think of a reformulation. As I recall, it is (as presently formulated) almost verbatim from Knapp. I no longer have access to the book, but the meaning of it all is, as you guess, the statement "The following objects are in a natural one-to-one correspondence" - at least once isomorphisms have been written down explicitly in equation (A1) or the like. I have not thought about much it, but my guess is that the isomorphisms themselves (between the Lie algebras) aren't always unique. I'll leave out any mention of such uniqueness. Your parenthetical "up to isomormhism" seems to refer to what I call equivalence. Correct?, See below. YohanN7 (talk) 08:17, 5 December 2016 (UTC)


 * I left the list as is, but tried to indicate how the presence (or truth) of one item "propagates" to the others. YohanN7 (talk) 11:25, 5 December 2016 (UTC)

And by the way, our "Equivalent representation" page redirects to "Representation theory", and there the word "equivalent" does not occur; "isomorphic" does. Boris Tsirelson (talk) 14:45, 3 December 2016 (UTC)


 * My (and the articles) notion of a equivalence between representations is a nonzero invertible linear map $$ between representation spaces $(1,1)$ such that


 * $$A \circ \pi(x) = \rho(x) \circ A, \quad x \in \mathfrak g \text{ or } x \in G$$


 * where $(1 + 2*1)(1 + 2*1) = 9$ and $M ≈ … ≈ SO(3;1)$ are representations. This notion is the same for both Lie algebras and Lie groups, and the terminology is, as far as I can tell, standard in the literature. But see the talk page. YohanN7 (talk) 08:17, 5 December 2016 (UTC)


 * I edited Representation theory and simply introduced some alternative and at least fairly common terminology (and blue linked the first occurrence of "equivalent representation" here). YohanN7 (talk) 11:54, 5 December 2016 (UTC)


 * Changed from "equivalent" to "isomorphic" after all. YohanN7 (talk) 15:33, 6 December 2016 (UTC)

"1.2.2.2 so(3,1)": "all its representations, not necessarily irreducible, can be built up as direct sums of the irreducible ones" − I'd delete "not necessarily irreducible" here (since it still will not be unclear, not even a bit). :-)  Boris Tsirelson (talk) 12:18, 7 December 2016 (UTC)


 * ✅ YohanN7 (talk) 12:59, 7 December 2016 (UTC)

The abbreviation "irrep" occurs in "1.3 Common representations" but is explained only in "1.7 Induced ..." Boris Tsirelson (talk) 12:25, 7 December 2016 (UTC)
 * Now avoided in 1.3, but left in 1.7 for local use there (where it is actually appropriate). YohanN7 (talk) 13:10, 7 December 2016 (UTC)

1.4.1 The Lie correspondence: "let $≈$ denote the group generated by $≅$" — One could wonder, isn't $SL(2, C)$ itself a group? Boris Tsirelson (talk) 17:32, 7 December 2016 (UTC)


 * It isn't always a group. There is the $μ,ν$ example in the article where it is shown that $V, W$ misses a conjugacy class. That "hole" gets "filled" by taking products of $(π, V)$ matrices (two suffice in this case, a more general theorem (not in article) shows that finitely many suffice, but I have never seen more than two being needed) that are in the image of $(ρ, W)$. If the image is a group then "generation" is harmless. The image of $Γ(g)$ would be left the way it is. So i think it is correctly formulated. (The reference, Rossmann, is the same as the one mentioned in the MO thread.) YohanN7 (talk) 07:55, 8 December 2016 (UTC)


 * Ah, yes, I see. Why not add a word or two for an impatient reader like me?.. Boris Tsirelson (talk) 11:21, 8 December 2016 (UTC)


 * Added "nb". Does it do the trick? YohanN7 (talk) 11:56, 8 December 2016 (UTC)


 * Yes... but why "one takes all finite products of elements in the image (and repeats if necessary)"? Either products of two elements, and repeats; or all finite products, and no need to repeat, right? Boris Tsirelson (talk) 14:10, 8 December 2016 (UTC)


 * I believe you, wasn't sure myself, hence the "repeat" (just in case). Should I strike it out? YohanN7 (talk) 14:21, 8 December 2016 (UTC)
 * I guess it depends on when one enlarges the original set. What if $exp(g)$ and $exp(g)$ with all $SL(2, ℂ)$ in the original image, and then ... Ah, as of writing the striked out text, now I see the light. One round of finite products will most definitely be enough. Thanks! YohanN7 (talk) 14:28, 8 December 2016 (UTC)

The Lie correspondence, again (now 2.4.1): "linear Lie group (i.e. a group representable as a group of matrices)" — One could wonder (again), isn't every Lie group linear? I tried to find the answer in this article, at no avail (or did I miss it?); but it is found in SL2(R) (regretfully, with no source). Boris Tsirelson (talk) 21:17, 8 December 2016 (UTC)
 * There exist exceptions. The universal covers of the special linear groups SL(n,R) n>=2 don't have a a matrix linear rep and so are technically nonlinear (source: Denis Luminet and Alain Valette, Faithful Uniformly Continuous Representations of Lie Groups, J. London Math. Soc. (1994) 49 (1): 100-108, doi:10.1112/jlms/49.1.100). The metaplectic group doesn't have such a rep. --Mark viking (talk) 23:38, 8 December 2016 (UTC)
 * Wow... Now you can use it both in this article and in SL2(R) article. Boris Tsirelson (talk) 05:04, 9 December 2016 (UTC)


 * There are, at least, two respects in which a would-be-category of matrix Lie groups fails. One is, as mentioned, taking universal covers. The other is taking quotients by normal subgroups. Hall (frequently used here) prove (at least for quotients) examples of both. As I recall he uses the SL(n,R) example in the one case, and the quotient of the Heisenberg group with its center in the other. Also, Ado's theorem can be used to prove that every compact Lie group is a linear group. YohanN7 (talk) 07:28, 9 December 2016 (UTC)
 * Wow again. Boris Tsirelson (talk) 07:47, 9 December 2016 (UTC)
 * According to the article Peter-Weyl theorem, it can, at least in the case of Lie groups, be used to prove the same thing. I don't have my references at hand at the moment, and I may remember wrong, so edits on my part will have to wait. YohanN7 (talk) 08:02, 9 December 2016 (UTC)
 * Really? I do not see this in "Peter-Weyl theorem". Boris Tsirelson (talk) 11:52, 9 December 2016 (UTC)
 * Second to last paragraph in Peter-Weyl theorem (last sentence). YohanN7 (talk) 12:00, 9 December 2016 (UTC)
 * Do you mean "Conversely, it is a consequence of the theorem that any compact Lie group is isomorphic to a matrix group"? Does it give any non-matrix group? Boris Tsirelson (talk) 12:08, 9 December 2016 (UTC)
 * I mean that sentence. Then no, but this is just the point. I think we misunderstand each other here. By the way, I think (but do not know) that "linear group", "matrix group" and "group that has a finite-dimensional faithful" representation always are interpreted to mean the same thing. YohanN7 (talk) 12:41, 9 December 2016 (UTC)

2.2 Strategy: "A subtlety arises due to the doubly connected nature of SO(3, 1)+" — Doubly connected? The link points (via disambig) to "Simply connected space", but "doubly" does not appear there. The article "n-connected" is about a different notion (and "2-connected" is not the "doubly connected"). On the other hand, there is a chapter "Doubly Connected Regions" in a book. Boris Tsirelson (talk) 10:44, 12 December 2016 (UTC)


 * Thanks for pointing this out. I'll write an "nb", or I'll put in in the notion in simply connected. Weinberg vol I gives a very nice purely topological argument. YohanN7 (talk) 13:02, 12 December 2016 (UTC)


 * Now addressed (locally) in fundamental group. YohanN7 (talk) 14:49, 15 December 2016 (UTC)


 * Yes. Probably you mean that a doubly connected space is a space whose fundamental group (or should I say, first homology group?) is of order 2; or maybe, that all elements of this group are of order 2. I wonder, how standard is this terminology. For the "doubly connected regions" in the book mentioned above the fundamental group is infinite cyclic. Boris Tsirelson (talk) 15:11, 16 December 2016 (UTC)


 * Yes, I mean space whose fundamental group (or the first homotopy group) is of order 2, the elements of it being equivalence classes of loops (based at a point). (The simpler, but related, Homology groups are rarely occurring in this context.) Doubly connected certainly occurs, especially in the physics literature, but the terms should not be seen as having a fixed mathematical meaning outside the scope where it is mentioned. It is not vital for the article to have the term, but it feels convoluted to use the more precise "the fundamental group being isomorphic to a two-element group".


 * A standard abuse of terminology, b t w, is to speak of the fundamental group. It is really one for each base point, which sometimes (but rarely) is of importance. YohanN7 (talk) 15:43, 16 December 2016 (UTC)


 * Sure, no problem with "the fundamental group" (since we really mean up to group isomorphisms, and the space is path-connected). And yes, the term "doubly connected" helps. "Should not be seen as having a fixed mathematical meaning outside the scope" − yes, but this is written on the talk page; probably we should give a hint to the reader that he/she should not seek the formal mathematical definition in topology textbooks. Boris Tsirelson (talk) 16:25, 16 December 2016 (UTC)
 * I see, you did it nicely. Boris Tsirelson (talk) 12:51, 17 December 2016 (UTC)

Preparation for GA-nomination
First off, thank you to the editors having responded to my Request For Comments about whether this article could be nominated for GA-status. The request has resulted in encouragement, hands on help, and loads of material to read. It has also resulted in the following list by Mark viking.

We have agreed to placing any replies inside the list, indenting and signing appropriately to keep things in one place. (Therefore I stole Mark's signature for each item on the list below, apologies.)


 * The lead in a GA article is primarily a summary of the content of the article The current lead is mostly about why these reps are important, especially to physics. Probably the introductory/significance/applications material in the lead could be moved into a intro section of the article and the lead rewritten to be mostly a summary of the rest of the article. --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * How about, roughly, taking out the second paragraph and most of the content of the "nb:s", and based on the removed material creating a section "Utility"/"Applications" or whatever we choose to call it? I feel "Introduction" should be reserved for the new section discussed below. The remaining content of the lead can easily be tweaked into a more summary-like style. YohanN7 (talk) 15:55, 7 December 2016 (UTC)


 * The lead has now been rewritten, primarily by taking out the second paragraph and from it creating a new section Applications. I expect the lead and the new section to contain bugs. I'll go through thing more carefully in the coming days.YohanN7 (talk) 10:26, 14 December 2016 (UTC)


 * For highly technical articles, writing in clear prose accessible to a wide audience is an impossible task. Typically for GA articles, a compromise is made, per WP:TECHNICAL, in that the lead and intro sections should be "written one level down." I consider this graduate level stuff, so perhaps write the intro sections for an undergraduate. In this case that would a mean brief description what you mean by a rep (matrices in the finite dimensional case, basis functions in the infinite case) and a quick review of the Lorentz group (as rotations, boosts, space inversion and time reversal, etc) and algebra. This will give the causal reader some basic idea of what the article is about; that may be all they want. --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * Nice idea. It should have occurred to me, but it didn't. I am now drafting a proposal for such a section. I'll try to begin with the mathematical notion of group and the physical notion of symmetry, and argue that they go hand in hand joining in the notion symmetry group. From there, keeping the multiplication table of the group at the center of the discussion all the time, I try to go as fast as possible via matrices to infinite-dimensional representations on function spaces (with vector space structure). (These too will be exposed as (infinite-dimensional) matrices by choice of basis.)


 * This is not easy, and will be very hard to source properly. YohanN7 (talk) 13:52, 7 December 2016 (UTC)


 * An attempt at it is in place. It is for now in a hide box, because it occupies two screens full. What go be taken out? It is obviously too big... YohanN7 (talk) 13:41, 8 December 2016 (UTC)


 * I have an alternative idea that I employed in Lie algebra extension (which is probably even more technical than this article). There, background material makes up the last 40% of the article, with links to it dispersed in the main body of the text. YohanN7 (talk) 14:01, 8 December 2016 (UTC)


 * Where are the nonlinear reps? The answer may be Wigner's theorem. --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * There is admittedly plenty that the article doesn't cover, but it covers a lot. It covers the building blocks of all linear representations (and "linear" is a technical prerequisite for "representation"). A GA article need not have full coverage. With my ignorance as an excuse, the non-linear reps (or rather actions) will have to wait until later unless someone else decides to write a section. YohanN7 (talk) 13:17, 16 December 2016 (UTC)


 * There is some inconsistency in the notation of the article from use of different fonts. For a lie algebra, mathfrak is used for displayed equations and in the text, either ordinary bolded letters or math bolded letters are used. It made me think as to whether these were are meant to be the same objects or not. --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * This is a major problem due to the size of the article. The background is Wikipedia's historical unwillingness to provide a classy math rendering scheme. Until recently, the default for readers not logged in (i.e. for the vast majority of readers) was PNG rendering of Tex. This worked acceptably only for displayed Tex. Inline Tex was on most devices (actual computer screens, not phones) displayed it the wrong size (way, way to big) and text wasn't even aligned. The only acceptable compromise working decently on, as far as I know, most devices was to use Tex for displayed math only and to use math templates for inline math. Default has changed, and is now, I believe, MathML. It could all be converted to Tex, but I really don't look forward to it. YohanN7 (talk) 14:02, 7 December 2016 (UTC)


 * To see what the situation was (for years, even decades), go to Preferences->Appearance->Math and chose PNG. Do this with a big screen. Then have a look at e.g. Lie group–Lie algebra correspondence, a nice article, except that it is (for me) totally unreadable (using PNG) due to the font issue. It is all I see, I can concentrate on the content. Then go to this article and compare. I'd say it still looks decent in this respect.


 * The particular problem mentioned, i.e. $exp$ vs $$\mathfrak g$$ exists. I acknowledge that. But what to do? (B t w, until very recently, MathML failed to display all equations in this article corectly.) YohanN7 (talk) 14:15, 7 December 2016 (UTC)


 * There is a lot of good detail in this article, not only about Lorentz group reps, but also Lie correspondence, CBH, relations to other groups, algebraic vs geometric POVs, etc. All good stuff, but a non expert might get lost in the details. What might help is (a) a brief description description of the plan of attack: first concentrate on the restricted Lorentz group, get at group reps from the algebra reps and the Lie correspondence, then add back in partiy and time reversal components, and (b) highlight the main result: a list of the Lorentz group reps. I guess that the Properties of the (m, n) representations section is closest to a main results section in the finite case. --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * Part (a) tentatively implemented in the finite-dimensional case with section strategy. YohanN7 (talk) 10:14, 12 December 2016 (UTC)


 * Part (a) tentatively implemented in the infinite-dimensional case as well, together with a section classification giving an outline of the classification itself. YohanN7 (talk) 17:50, 12 December 2016 (UTC)


 * The Common representations section seems like it should be in the Lorentz Lie algebra section. --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * Now there together with matrix generating formula. YohanN7 (talk) 13:17, 16 December 2016 (UTC)


 * With regard to citations, GA requires a certain citation density. For technical articles WP:SCICITE is often used. In practice that means definitely a citation in each section, and probably a cite in each nontrivial paragraph. The idea is not necessarily to verify controversial statements, but to show the reader where the material is drawn from. Against that custom, this article is pretty well referenced, but there are sections like the infinite dimensional history section that need more sourcing. E.g., who says that infinite reps were first studied in 1947? --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * In respect to the last remark, I simply stroke out "first". It is blatantly incorrect as e.g. Dirac published before that as is mentioned in the article. But the three mentioned 1947 publications were first to classify all reps. Will fix this. YohanN7 (talk) 12:55, 9 December 2016 (UTC)


 * B t w, does anyone know a reference for the representation on the Riemann P-functions? YohanN7 (talk) 12:55, 9 December 2016 (UTC)


 * There are other GA criteria, such as making sure images are legal, described in WP:GACR --Mark viking (talk) 09:56, 7 December 2016 (UTC)


 * Only Gelfand in Plancherel theorem seems problematic. Fair use? YohanN7 (talk) 15:30, 8 December 2016 (UTC)

Again note that it is preferable for anyone itching to comment to do so inside the list with proper indentation though it technically may be breach of etiquette (you'd be editing inside my post that I stole from Mark), it is practical. YohanN7 (talk) 13:52, 7 December 2016 (UTC)

Infinite-dimensional representations
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. 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), which is how I returned to this topic. Mathsci (talk) 10:42, 14 December 2016 (UTC)


 * Sounds excellent! YohanN7 (talk) 15:19, 14 December 2016 (UTC)
 * 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)
 * 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)
 * 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)
 * 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)
 * 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)

@Mathsci. There is a discrepancy between the present classification section and the representations given. For the principal series, I suspect one has $SL(2, ℂ)$ (corresponding to the usual difference in labeling of $exp$-representations between mathematics and physics). For the complementary series, one needs $exp$, but I don't see exactly how this comes about, and how it should be explained in the article (if the present classification stays much longer). YohanN7 (talk) 16:28, 19 December 2016 (UTC)

Also, we need at least one inline citation for the formulas in the Plancherel theorem section. YohanN7 (talk) 17:37, 19 December 2016 (UTC)
 * 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)


 * 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)
 * 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)

Referencing errors
I've marked every sfn link that doesn't match with a long citation with Incomplete short citation. Chances are most of these are typos (Gaev, Graev), the wrong year (Weinberg 2003?), or one author missing (Greiner 1996). You can verify if a sfn is formatted correctly by clicking on it; if it won't take you to a long citation, something is wrong with it.

If you want to be super pedantic, you should use either or. The output is slightly different (a comma is missing in the latter case). – Finnusertop (talk ⋅ contribs) 13:02, 20 December 2016 (UTC)


 * Many thanks!
 * Will fix, and thanks for the hint about the comma thing (have still to figure out what it means). I suspected the Gelfand picture wouldn't pass... YohanN7 (talk) 13:08, 20 December 2016 (UTC)
 * Just as a point of information, the photographs of many mathematicians have been made available by the MFO in Oberwolfach and released under a CC attribution-share-alike license, so can be used on wikipedia. There are two photos of Gelfand, of which this one is the best. So if you want a photo, you can upload that on Commons. Mathsci (talk) 13:24, 20 December 2016 (UTC)
 * Unfortunately neither photo ~reads "Copyright: MFO", which the disclaimer says denotes photos under CC BY-SA. Here is an unrelated photo with the text as an example: Eckes, Christophe – Finnusertop (talk ⋅ contribs) 13:50, 20 December 2016 (UTC)

thank you for fixing. I also note that there are a couple of full citations that don't have any short citations pointing to them. If these are unused, they should be moved to Further reading or removed. These are: – Finnusertop (talk ⋅ contribs) 18:54, 20 December 2016 (UTC)
 * Dixmier, J.; Malliavin, P. (1978)
 * Gelfand, I. M.; Graev, M. I.; Vilenkin, N. Ya. (1966)
 * Naimark, M.A. (1964)
 * Stein, Elias M. (1970)


 * I don't think that these suggestions are helpful. You are discussing the part of the article concerned with the unitary representation theory of SL(2,C) and the Lorentz group in infinite dimensions and in particular the Plancherel theorem for SL(2,C). You have just earmarked for possible removal the two main books on the subject, both regarded as classics. Please read this page more carefully. I have stated quite clearly and unambiguously that I am in the process of adding more material on that subject (various approaches to the Plancherel theorem) and am currently preparing that in my user space. As mentioned on this page, some of it is already on wikipedia in Plancherel theorem for spherical functions. The Stein article will be used for material on intertwining operators, which is an important aspect of the subject. YohanN7 wants this article to become a good article. Unfortunately that is not simply a question of formatting. A large amount of content is missing from the article and I am trying to add it. Your comments therefore seem to be at cross-purposes. The article is not being polished; it is being expanded.  Mathsci (talk) 09:20, 21 December 2016 (UTC)
 * I see, . My comment was all about polishing. You can disregard it if this is not the right time. – Finnusertop (talk ⋅ contribs) 10:31, 21 December 2016 (UTC)

I think organizing the references for readability wouldn't hurt much. Some of the references (like MTW) are, while cited, on separate topics. Some papers are purely historical, etc. This needs, if it is to be done, some thought. One could argue for a subdivision of pure math and physics references. YohanN7 (talk) 11:12, 21 December 2016 (UTC)
 * Please wait until content has been added before trying to assess the references. There is no point in putting the cart before the horse. Mathsci (talk) 12:41, 21 December 2016 (UTC)

More comments for GA nomination
Just more remarks on this article in relation to the Good article assesment. This is not an official GA but more my ideas on how to improve the article. I am not a specialist on the field (and reading the article does not really help me either :)

So here my remarks: The optimal length of an article is around 40.000 bytes wp:size
 * the article is to long (198.000 bytes)

So this would mean the article should be cut in around 5 articles I am not sure how to cut it up but was thinking:

---Introduction material--- There is quite a lot of introduction material on other subjects. I think only a tiny bit on Lorentz groups and representation theory should be in this article. the other parts could go to the article on the subject itself (lie groups lie algebra's and so on) and this article can just link to them. WillemienH (talk) 19:16, 21 December 2016 (UTC)

- the lead is to long and to technical I guess that is related to the above. maybe start with writing a pre-lead WillemienH (talk) 19:16, 21 December 2016 (UTC)

Other general remarks

- The current main image --File:Einstein_en_Lorentz.jpg -- is not really good ( replace it with something that represents an Lorentz group /transformation or something similar )WillemienH (talk) 19:16, 21 December 2016 (UTC)


 * Without stretching the truth very much, one could write


 * $$\text{special relativity}\Leftrightarrow\text{representation theory of the Lorentz group.}$$


 * I can't think of a picture more appropriate than the present one. But I do agree that some illustration of the group would be appropriate. The nearest thing describing the group can be found in Fundamental group. That illustration is still too high dimensional to be faithfully rendered. But I'll think about it. YohanN7 (talk) 10:13, 22 December 2016 (UTC)


 * You will need something 2 or 3 dimensional (like the lillustration like File:Lorentz_boost_electric_charge.svg )WillemienH (talk) 13:49, 29 December 2016 (UTC)


 * No, you need something 6-dimensional - like the group. YohanN7 (talk) 10:19, 30 December 2016 (UTC)

- the article could do with more images but images showing a Lorentz group not the people discovering it.WillemienH (talk) 19:16, 21 December 2016 (UTC)

- the examples used all are different could there be not one main example used troughout the article and then an section how to link other examples to this main example WillemienH (talk) 19:16, 21 December 2016 (UTC)

- there is quite a lot of history in the article (not just under history also at other places, maybe better to split the history to a sepertate page) and only keep the history from the fist use. WillemienH (talk) 19:16, 21 December 2016 (UTC)

This is just a first impression of me and maybe i am a bit convictive (I nominated an article and it resulted in a degrading of the article) WillemienH (talk) 13:58, 21 December 2016 (UTC)


 * I'll get back tomorrow and try to respond more thoroughly. Suggestion: Could you edit your post and make it a bullet list, signing each bullet? That way we could keep the discussion organized without scattering it (meaning I could respond per bullet in the list). YohanN7 (talk) 15:14, 21 December 2016 (UTC)


 * I would rather love the photo of Einstein & Lorentz: They are both at the stage in life when they would be thinking about the subject, and look like real people. OK, now, more seriously: I have experienced not a few students who failed to derive maximum benefits from WP, where I sent them, simply because the articles lack sufficient background info to make what is presented meaningful. WP is not meant to be read off a smart phone. Any reader may skip over dull material, by the sweep of a cursor. But clicking on side tabs to find one's bearings has proven very dangerous and unworkable for students, as I indicated. I, for one, strongly applaud a longer, more complete, and thorough coverage of an unapologetically technical  subject such as this. The short attention span reader maybe should not be on this article! Cuzkatzimhut (talk) 22:19, 21 December 2016 (UTC)


 * it is a bit do you want to go historical or technical, compare it with chemistry and hyperbolic geometry, in most articles on chemistry you will not find a mention of Lavoisier while he started the whole subject. Most readers would find references to him strange and mentioning him would make many articles difficult to read.
 * But on the other hand I still have not found a book on hyperbolic geometry that does not mention Lobachevski or Bolyai
 * I guess it has to do whith the age (or maturity?) of the subject. WillemienH (talk) 13:49, 29 December 2016 (UTC)


 * Or the age and maturity of you? You are clearly not constructive here. Leave. YohanN7 (talk) 10:22, 30 December 2016 (UTC)

Just a notice that I'm unlikely to do more edits the coming holidays (and i doubt a reviewer will be in place until after them). The edit first in line is to move the section on the matrix exponential elsewhere. YohanN7 (talk) 11:45, 29 December 2016 (UTC)

The articles related to this article: Particle physics and, representation theory , Poincaré group representations,   Galilean group representations,Lie groups in physics Representation theory Lie group representation Lie algebra representation are maybe better to be developed to the level of this group (or maybe some of the present article can be moved to there) so that this article only needs to mention the differences. (I took these links from --Template:Lie_groups -- ) maybe better to make more templates to which the article belongs. WillemienH (talk) 13:49, 29 December 2016 (UTC)


 * So you are suggesting that I, after years of work, actually should have written an other article, preferably about something else? Or preferably not at all? YohanN7 (talk) 10:12, 30 December 2016 (UTC)


 * Uhhhh: le mieux est l'ennemi du bien. As I indicated already, there is a very strong advantage in keeping all the relevant stuff under one roof--I am,personally, delighted at the grand synthesis achieved here... Might as well treat it as a WP experiment, comparable to the legendary classic articles of the Encyclopedia Brittanica of past centuries...
 * I have dealt with hours of WP linkage dysfunction trying to get to the right stuff bouncing from link to pestiferous link... And I know from experience that interested students I send to WP ultimately give up, on account of the hit-or-miss nature of wikilink-pinball... victims of WP's encouragement of the short (and brainless) attention span.  There is no problem in vertical skittering up and down the article to access the stuff, back and forth. Out of sight/site it would be out of mind. Only hyper-organized browser tab mavens could possibly do that on several pages in parallel.
 * Please don't scatter the stuff around, just to achieve the common denominator of catering to the outsider dictated by the culture... This is a technical article for people eager to be here to actually learn the stuff, as opposed to those eager for a broad impression of what is there, so as to go to a good book... Cuzkatzimhut (talk) 15:17, 30 December 2016 (UTC)

Adjoint Representation?
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)


 * 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)

(m,n) or (&mu;,&nu;)?
I notice that in the section Concrete realizations, $$(\mu,\nu)$$ representations are discussed. But these are not the same as the $$(m,n)$$ representations. So what is (e.g.) a $$(1,0)$$ representation? Is it a $$m=1,n=0$$ representation (of dimension 3) or a $$\mu=1,\nu=0$$ representation (of dimension 2)? It seems like the article should settle on one notation for the representations, because there is a genuine possibility of confusion, especially since the definition of m and n in terms of &mu; and &nu; is fairly easy to miss. Sławomir Biały (talk) 00:29, 6 January 2017 (UTC)


 * You have a point. The $A = g_{1}g_{2}...$ are for (at least) real-linear $B = g_{14}g_{99}...$-reps and the $g$ are for $g$. All numerically mentioned reps refer to $2j_{0} ↔ |k|$. It is standard in the literature to use half-integers for $SU(2)$, and (as far as I know) also standard to use integers for $ν + 1 ↔ t$. But yes, I can see the problem. One solution might be to have a sentence in the lead last thing, alerting the reader to this. At any rate, some confusion would remain even if labeling was the same. Do we mean a $(μ, ν)$ or a $sl(2, ℂ)$ rep? Same matrices, but different properties as reps (as far as faithfulness and "projectiveness" goes). YohanN7 (talk) 08:16, 9 January 2017 (UTC)



Error?
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 $(m, n)$.) YohanN7 (talk) 14:20, 15 May 2017 (UTC)

Italics
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)


 * 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)

Conversion to mathfrak
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)


 * 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)


 * 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)


 * 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)


 * ..., 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)

Redundancy in Material
Editing I noticed that there are several proofs and passages that cover material not specific to the topic. For example we have proofs showing that the kernel of a group homomorphism is a normal subgroup, that is a general group theory fact, the proof should not appear on this page. Another example is showing that exp is not surjective, again this is a fact in the broader theory of Lie algebras and Lie groups, does not need to be repeated here. I propose that we remove these two and other similar material to shorten this article. Latex-yow (talk) 21:19, 1 June 2017 (UTC)
 * For your first example, yes, it should be removed. For the second example, no, because $so(3; 1)$ may or may not be surjective. It belongs in the article that it is, resp. is not for $so(3; 1)$ and $so(3; 1)$, because it is of practical importance. YohanN7 (talk) 07:04, 2 June 2017 (UTC)
 * The onto-not onto sections have, by the way, already be shortened to outlines of proof and not actual proof. (The "onto" in the $sl(2, ℂ)$ case is not easily found in the literature. It seems to be taken for granted in physics.) Can you be more specific about "other similar material"? YohanN7 (talk) 09:27, 2 June 2017 (UTC)


 * Others include: strategy, step one, step two; the bit about Lie algebra reps from group reps and vice versa and the adjoint representation. I mean this is the article for the representation theory of a particular Lie group. That means the reader ought to know (1) group theory, (2) representation theory, (3) Lie groups, (4) lie algebras and (5) representation theory of Lie groups and Lie algebras. If you don't know one of these you should not be reading this article, period. Latex-yow (talk) 05:20, 3 June 2017 (UTC)


 * I can understand that point of view, though it is a bit extreme. Take for instance understanding of Lie groups. This, by itself, comes with a considerable amount of prerequisites, like solid foundations in abstract algebra and smooth manifolds. In turn, smooth manifolds, by itself, requires grounding in, for instance, topology. Within a university curriculum for graduate students in mathematics, this can be arranged for. But the intended audience for this article is not only grad students in math. It includes undergrad students in physics and engineering. These do not have the required prerequisites available. If you look in a physics book, everything about group theory (including rep theory) is somehow "pulled out of the hat". This article attempts to make a connection to the underlying actual mathematics for those readers.


 * Even if a prerequisite like representation theory is available, there are enough odd features about the Lorentz group (non-compactness, non-simple connectedness) that warrants the discussion (strategy, step one, step two, Group reps from Lie algebra reps respectively) because they are usually ignored even in introductory graduate level mathematics texts. These texts focus almost invariably on compact groups and never on projective representations. The latter is heavily used in applications and has undoubtedly confused several generations of students. (Feynman: If we can't explain spin to our students, do we understand it?)


 * The article seeks to demonstrate how the general theory applies in this particular case. As I said, I understand your point of view, but it isn't the only one. Several sections have actually been proposed to me (on this page). Among these are the non-technical introduction and the strategy section. Explicitly, the article is written "one level down". It could be written as you suggests. This would reduce accessibility to a selected few, but quite possibly it would formally become an impeccable WP article. But in my view, it would have little value. YohanN7 (talk) 08:54, 5 June 2017 (UTC)


 * Not convinced at all by your reasoning here. Yes Lie groups requires a lot of things, that is because it is a very advanced subject in mathematics and a graduate level topic in many universities. I find the idea that there are no good texts on representations of non-compact Lie groups to be baffling (have you looked at Knapp's 800 page book??) also the Feynman quote doesn't apply here, there he is saying we really don't know what spin is, here we know all the math the problem is what is appropriate for this page. I would like to disabuse of the notion you have that assuming familiarity with the topics will lower the number of people using the page, nobody will learn about Lie groups and representation from the contents of this page alone. That is one of the points of the Wikipedia linking to its other pages.


 * The way you should think about it is this, your view means everything I objected to should be repeated in every Wikipedia page on a specific Lie group. That includes the exceptional Lie groups or simply SO(8) (which has its own page and it is not simply connected).


 * At any rate I won't be doing anything about this beyond making my point here (so don't worry I am not going to delete anything).Latex-yow (talk) 21:19, 5 June 2017 (UTC)


 * "We call a situation hopeless when the obvious way out of it does not suit us at all."


 * I know that "go to Wikiversity" may sound here like "go to hell" or at least "go away". At the risk of insulting the author (certainly worthy of praise) I want to say that (a version of) this article, probably, could be the best article on Wikiversity ("a small honor" you reply?), and, more importantly, it could really help to many students; and, still more, this is the only way, since here it will not survive for long (yes, I am a pessimist, really, I know). Because it is "Representation theory of the Lorentz group for physics undergraduates" rather than just "Representation theory of the Lorentz group". Wikiversity, in contrast to Wikipedia, admits content forking. And, as far as I understand, it is OK to write in a WP article something like "Wikivesity has an article on this matter" (with a link, of course). Boris Tsirelson (talk) 22:51, 5 June 2017 (UTC)
 * I am experimenting on Wikiversity with a copy of this article. Hope there is no evil in doing so (otherwise please let me know). Boris Tsirelson (talk) 19:56, 6 June 2017 (UTC)
 * Needless to say, YohanN7 is more than welcome to do it better than me. Boris Tsirelson (talk) 19:29, 7 June 2017 (UTC)


 * I want to assure you (both) that I will listen to what you say. I'll not be infinitely stubborn. Please read the following with that assurance of mine in mind. I'll need to do some thinking


 * (re Latex-yow) Yes, I have had a look in Knapp's book. (It is referenced in the article, especially in the context of the unitarian trick.) Just saying that most introductory books focus on the mathematically straightforward, i.e. the complex semisimple Lie algebras and their compact real forms. Thus real non-compact Lie algebras are usually ignored in introductory texts. I say nowhere that no good books on non-compact groups don't exist. They are invariably just more advanced. (Knapp's introductory treatment, which I like, is by no means the easiest approach, and it does rely on Lie machinery more than e.g. Hall's or Rossmann's treatments).


 * The differences between an article on the Lorentz group and a hypothetical article on $$\mathrm{SO(8)}$$ is that the former group is notable and the latter is not. The Lorentz group has as important an application that a mathematician could ever dream of. (To the extent that mathematician's like application. Some take pride in their math being utterly useless (outside math) = "pure mathematics". G. H. Hardy was one such, and would probably be shocked that applications have been found to some of his "pure math".) Whether the mathematician likes this or not, the article (any article on the subject) is also the concern of the physicist and students. Since spin exists (in nature), and since it is a big topic, it deserves treatment, though it could be tossed off on mathematical grounds, because spin reps aren't technically representations (only group actions on projective Hilbert spaces).


 * It is a misconception that (I would believe that) this article aims to actually teach students. It aims to be understandable in such a way that individual sentences should at least explicitly hint at what the sentence actually means for someone not an expert. It, by courtesy, offers brief descriptions of concepts defined in linked articles. I believe that this is a virtue. When I was working as a programmer (C++), I had a reputation of writing code that seemed to do absolutely nothing, except calling other code (usually templates, so that the compiler generated the code actually doing something). So when a newbie was faced with my code, it wasn't exactly the ideal way for him of learning programming or even understanding it. I wouldn't like WP articles to be built the way I built my C++ libraries, or applications on top of them, though they could be written in 10 lines of code where ordinary code using ordinary libraries would use 100. (The WP counterpart of my code would be a short article compost almost exclusively of blue links, often self-referential.)


 * (re Boris) I'll need to think about what you wrote, and have a look at your experiment before commenting.


 * (Finally) It is not surprising that it is 100% (assuming I am correct about Latex-yow being a mathematician) accomplished mature mathematicians that take the "pure view". I have seen other views from other people, not necessarily as extreme as mine, but still not taking the "pure view" of the accomplished mathematician. But these are absent now. I don't know if this represents a change of their minds, or just expresses the fact that physicists are less engaged than mathematicians. (This has happened in pure physics articles as well.) YohanN7 (talk) 08:52, 7 June 2017 (UTC)

I linked the Wikiversity version. At present, I don't have any plans on devoting much time to it, though I see plenty of stuff that could go in, e.g. related branching rules, CG decomposition, more detailed derivation of the infinite-dimensional reps (and inclusion of non-unitary ones; the finite-dimensional ones arise as special cases of these), more applications, perhaps discussion about central extensions, etc. (Editing this monster is more than enough.) YohanN7 (talk) 07:30, 14 June 2017 (UTC)

I have put a couple of sections in hide boxes. It is noteworthy that every topic in hide boxes (except for one subsection) have, one way or another, found its way into the article as a result of other editor's comments/viewpoints, either here on this talk page, or (rarely) by other means of communication. What else from the stuff that is in plain view ought not be in plain view? YohanN7 (talk) 07:30, 14 June 2017 (UTC)

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

Typography
Dashes instead of a math symbol—namely, «$SO(3; 1)$»—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)

Textbook-like background and prerequisites section
The section removed, with Prerequisites outlined and instructive passages like These can be thought of, in the passive view, as (instantly!) giving the coordinate system (and with it the observer) a velocity in a chosen direction, is a clear-cut example of pedagogical material that is inappropriate here per the NOTTEXTBOOK policy: Also relevant in the same policy: Wikilinks give connections to articles further explaining related topics. It’s not the place of the article to outline prerequisites for learning material; articles neutrally describe facts. — MarkH21talk 18:42, 5 May 2020 (UTC)
 * I see your points in principle, but 95% of the bitter complaints about WP one gets in PSE and places where students congregate are about the imperious, elliptical, click-tag game WP is subjecting them to.  This particular GA article, whose development I watched  with admiration,  has already done yeomanly service for years, reminding physics students of basic facts and jargon, with all the material accessible bypassing the need for out-clicking.
 * It used to be a go-to reference. Coming in as a mathematician, (and not a physicist, the principal target audience of this article), and declaring that something is too trivial and pedagogical, even if hidden !, ensures the disappointed students shake their fists and vent more.  I could not disagree with them, as I saw a fine article made less helpful to them in real time, on the basis of highly arguable principle.  Readers and students will move on and away.  Cuzkatzimhut (talk) 00:42, 6 May 2020 (UTC)
 * It doesn't matter if there are complaints at StackExchange, and my background doesn't matter. The fact of the matter is that, by Wikipedia policy, the Wikipedia is not a textbook and the purpose of Wikipedia is not to serve as a substitute for textbooks and other learning resources. If you disagree with the policy, then take it up at Wikipedia talk:What Wikipedia is not or Village pump (policy).By the way, article content should never be hidden by MOS:DONTHIDE. I have also never declared any content here as too trivial. Also note that introducing and presenting broader context (in moderation) for accessibility to a topic is fine, but the pedagogical language and presentation style of the removed section is not. — MarkH21talk 02:11, 6 May 2020 (UTC)
 * The baby should not be summarily thrown away with the bathwater. If you felt you could actually improve the presentation style, you are of course more than welcome to do this, beyond trashing clarifications that make the article more accessible to a general audience, including physicists, the major consumers of this article, also a policy of WP. Proposing an improved section here would allow the other editors to assess its merits. Cuzkatzimhut (talk) 14:01, 6 May 2020 (UTC)
 * I think the other sections are fine and sufficient though. Context can be given in the other sections where needed, without a general introduction/prerequisite section. — MarkH21talk 15:04, 6 May 2020 (UTC)
 * I suppose students could be steered to the Wikiversity link on the left column of the page. Cuzkatzimhut (talk) 16:56, 6 May 2020 (UTC)
 * Yes, Wikiversity and Wikibooks are very suitable place to direct students and StackExchange questions. — MarkH21talk 17:12, 6 May 2020 (UTC)

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

Possible misreferenced equation, and suggested clarifications
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)