# Talk:Representation theory of the Lorentz group

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## My article

I didn't write this, I just moved material here which was written by Linas, whom I have called upon to write his own article the way he sees fit. I will not contribute any further to this article on representations of the Lorentz group.---CH (talk) 03:12, 17 July 2005 (UTC)

Thank you, Chris. It looks like it turned into a marvelous article over the years. User:Linas (talk) 19:04, 30 November 2013 (UTC)

## a bunch of stuff

the mood struck me, so I added a bunch of stuff to this article. This treatment comes from my memory of a QFT by Ryder, and well, I think it looks pretty sloppy. I guess somewhere we should find the represenation theory of the rotation group, upon which this is predicated. A longer exposition on how to form field equations for a given representation would be nice too. And uh, I guess... since the Lorentz group is noncompact, can I expect more than a discrete set of representations? I dunno. Help me out. -lethe talk + 12:36, 5 March 2006 (UTC)

## Z/2?

In the article it mentions "for each j in Z/2, one has the (2j+1)-dimensional spin-j representation spanned by the spherical harmonics with j as the highest weight." in the section of finding representations. By Z/2 the author means ... -1/2, 0, 1/2, 1, 3/2, 2, .... But isn't Z/2 the group of integers modulo 2? In which case, the author is saying that the possible representations are labelled by the group {0,1}?

128.120.51.98 21:14, 4 January 2007 (UTC)kiwidamien

## Z\2Z

You're thinking of the quotient group Z\2Z. --30apr2008

## Finite-dimensional representations

There was a sentence that " The Lorentz group has no unitary representation of finite dimension, except for the trivial representation (where every group element is represented by 1). " This is simply not true, so I removed it. For example, the mapping that takes -Id to -1, and evertyhing else to 1 is a 1 dimensional representation. It's certainly not faithful, but it's unitary (and irreducible as it's one-dimensional). — Preceding unsigned comment added by Goens (talkcontribs) 14:48, 15 July 2012 (UTC)

This article is an advanced piece that goes directly to infinite-dimensional representations of the Lorentz group. Today I added the reference by Paerl that discusses both finite and infinite dimensional representation. Unless there is a protest, I intend to preface the infinite dimensional material with the finite.Rgdboer (talk) 04:53, 19 January 2010 (UTC)

As the Lorentz group has six real dimensions there are more elementary ideas that are useful. May I suggest the biquaternion#Algebraic properties representation of the Lorentz group.Rgdboer (talk) 00:14, 8 July 2010 (UTC)

## su(2) - sl(2;c)

First, on the above section: " The Lorentz group has no unitary representation of finite dimension, except for the trivial representation (where every group element is represented by 1). "

I recognize that scentence. It's from Weinbergs "The Quantum Theory of Fields". It refers to the identity component (the proper ortochronous transformations) of the Lorentz group. It is a true statement from a reliable source, and should be put back because it is of some importance. Perhaps with the qualification that it is about the identity component.

Then there is this su(2) issue. The algebra su(2) is not the complexification of the algebra of the rotation group, so(3). The algebra su(2) is a real algebra isomorphic so so(3). What is actually used is the complexification of su(2), namely sl(2;C). For this latter issue, see e.g. Brian C. Hall, "Lie Groups, Lie Algebras, and representations; An Elementary Introduction".

One might add that this error in terminology exist (explicitly or implicitly) in pretty much every physics book there is. Typically it is itroduced at the same time the "ladder operators" are defined. One makes a complex change of basis of generators of su(2) and lands in sl(2;C). It is not the same thing as the different conventions regarding the different definitions of a Lie Algebra (a factor of i). YohanN7 (talk) 23:03, 12 September 2012 (UTC)

There is more that is pretty much backwards.

The assignment of J and K as pseudovectors and vectors respectively looks suspect.
The representations of the algebra sl(2;c) (and hence those of su(2)) do not stand in one-to-one correspondence with representation of the rotation group SO(3)
The section "Full Lorentz group" seems to get the meaning of irreducibility backwards in places. If a representation of the restricted group happens to be irreducible, then it is certainly irreducible under the full group. On the contrary, a representation may be ireducible under the full group, but not irreducable when restricted.
In particular, the (m,n) representation is in general not irreducible (under the restricted Lorentz group). A process entirely analogous to the Clebsch-Gordan decomposition can be applied to the ones that aren't irreducible.

I am probably going to attempt an edit ragarding these points (if nobody objects), a minimal rewrite + addition of the fact that the finite dimensional representations are never unitary. YohanN7 (talk) 11:03, 14 September 2012 (UTC)

So the striked out text above isn't entirely correct either. It should, of course, be this:
In particular, the (m,n) representation is in general not irreducible under the subgroup SO(3). A Clebsch-Gordan decomposition can be applied to the ones that aren't irreducible.
This results in an (m,n)-representation having SO(3)-invariant subspaces of sizes m+n, m+n-1, ..., |m-n| where each occurs exactly once. These subspaces don't mix under rotations but they mix under boosts. An example is given by the vector representation (1/2,1/2) which splits into J=0 (1-dimensional, e.g. time-component of EM vector potential A) and J=1 (3-dimensional, e.g. space components of A).
The assignment of J and K as a pseudovector and vector is pointless here. (Full O(3)) or full Lorentx group is needed for that, the adjoint action of SO(3) will not tell.) YohanN7 (talk) 10:11, 17 September 2012 (UTC)

## Definition of representation

I wonder if this article should mention that some of the representations are projective representations and don't necessarily meet the full definition of a representation, i.e. not always${\displaystyle D(A)D(B)=D(AB)}$ for ${\displaystyle A,B}$ elements of the group, ${\displaystyle D}$ the representation: there could be multiplication by a phase factor. Or would this be obvious to somebody with the necessary background to read this article? Count Truthstein (talk) 21:49, 17 January 2013 (UTC)

What is actually presented (in the finite dimensional case) is representations of the Lie algebra, not of the group itself. They are representations in the true sense. YohanN7 (talk) 15:18, 19 January 2013 (UTC)
Reps "of the Lie algebra, not of the group itself" are of little use for applications, because we need to transform quantities across reference frames. Count Truthstein is right, the concept of projective representation has some physical implications: see Spin-½#Complex_Phase. Incnis Mrsi (talk) 16:06, 19 January 2013 (UTC)
Projective representations and representations with phase factors don't seem to mean exactly the same things in math and physics. The former is a representation (in math), and the latter is a "lift" (in math) of the former where phase factors are introduced. These phase factors need to be such that the associative law still holds if I understand this correctly. Near the identity, this will work automatically by exponentiation of the Lie algebra reps (all phase factors are 1). Either way, the article should absolutely somehow address these issues. YohanN7 (talk) 17:03, 20 January 2013 (UTC)

## 2013-02-13 Edit

Hi!

I changed quite a bit in "Finding representations".

• Corrected main formula. Formerly it said so(3,1) = su(2) + su(2) which is just plain wrong.
• The main thrust used to be to use representations of SO(3) as a basic building block, or at least present things that way. It doesn't work, because there are more reps of su(2) than come from SO(3).
• I emphasized a bit the distinction between groups and algebras (so that rotation group (SO(3)), su(2), and sl(2;C)) are different things. [From previous version: "su(2) is the complexification of the rotation algebra" - just hideous]
• Mention of how to get group reps (as opposed to algebra), and that this can result in projective representations,

I did retain one thing though. The example of spherical harmonics is still there as a representation of SU(2). To make this work logically, one has first to get an su(2) rep, and then proceed from there. I don't really like it. YohanN7 (talk) 17:11, 13 February 2013 (UTC)

• I removed references to spherical harmonics too. Reason: In part because it was like going over the bridge for water, and in part because it wasn't correct, at least not with only the classical spherical harmonics which work only for integer spin, (which is the only case covered in the Wikipedia linked article).
• Removed too the fact that SU(2) is simply connected. True but irrelevant.
• Relevant fact not yet in article: sl(2,C) is the universal covering group of the Lorentz group.
• Reinstated an old remark that the irreps are never unitary.

The logic is this: The known irreps of su(2) give all of the irreps of sl(2;C). These, in turn, give all those of so(3;1)C which then finally restrict to so(3;1). All reps (irreducible or not) are then direct sums of the irreps. Group representations, possibly projective, may be obtained by exponentiation.
The former article (as per 2013-02-12) gave the impression that spherical harmonics and SO(3) are sufficient building material for the so(3;1) irreps. This is just not the case. The spherical harmonics are a beautiful illustration of the Lie Group SO(3). In addition, it is true that, given a representation of the Lorentz group, one may restrict it to SO(3). It just doesn't cover all the cases when one tries to go the other way. YohanN7 (talk) 19:48, 13 February 2013 (UTC)

I rewrote the section Full Lorentz group completely.

• The previous wording was a bit awkward. "...not only is this not an irreducible representation, it is not a representation at all,..."
• More importantly, it was not entirely correct as it stood. The (m,n)+(n,m) representations do not automatically include parity, it must be specified separately what constitutes the parity inversion representative.
• Time reversal is now discussed along the same lines.
• The terminology "vector" and "pseudo-vector" is now introduced alongside the equations that motivates it. (I removed it from Finding representations) YohanN7 (talk) 00:13, 14 February 2013 (UTC)

Inline citations now in place. Main math source is Hall (ref section), an easy introductory text on Lie groups, algebras and reps. Main physics source is Weinberg (ref section), a not-quite-so-easy text on QFT that covers a lot of the more advanced concepts when it comes to projective representations (Section 2.7 + Appendix B). YohanN7 (talk) 08:45, 14 February 2013 (UTC)

## 2013-02-13 Planned Edits

So, I have already done quite a bit. I'll wait a while before I proceed with more, but here is a list (in order of priority) of what I plan to include. Comments, both "yay" and "nay" are appreciated.

• Explicit so(3;1) matrices in the standard representation.
• The Lie algebra so(3;1) (i.e. the commutation relations among the above matrices).
• A clear description of the connection between finite dimensional representations of a Lie group and that of its Lie algebra. (The exponential mapping)
• The topic of simple connectedness, and therefore the failure of the exponential mapping to yield a proper representation.
• How a projective representation is still obtained, and why it is useful
• A nontrivial example of how this works out using Clifford algebras. YohanN7 (talk) 00:38, 14 February 2013 (UTC)
• Another slight problem: The electromagnetic vector potential, which is a 1-form, lives in this rep. In the literature, depending on how precise the presentation wants to be, the EM potential either is or is not a 4-vector. In contexts where they want to be precise, they point out that the EM potential is not a 4-vector precisely because it does not transform under the (1/2,1/2) representation. The transformation rule is actually A -> ΛA + the gradient of an arbitrary function. Therefore, I think the example is a particularly bad choice. YohanN7 (talk) 11:16, 14 February 2013 (UTC)
Yes, I agree about the vector potential. From the theoretical point of view, it is the connection in a U(1) gauge theory, not a (co)vector field at all. Incnis Mrsi (talk) 07:40, 16 February 2013 (UTC)
So what would you say is a good example? I can see nothing wrong with simply using the coordinates xμ of events in spacetime, except that it wouldn't be as "nifty" the other examples. YohanN7 (talk) 17:42, 16 February 2013 (UTC)
The four-momentum is the best we could invent. The wavefunction of a massive vector boson is another reasonable choice, though. Incnis Mrsi (talk) 19:11, 16 February 2013 (UTC)
Done. Besides, when I gathered enough courage I'll move the examples section so that it comes directly after "Finding representations". Done. YohanN7 (talk) 13:04, 19 February 2013 (UTC) Examples ought to appear early. YohanN7 (talk) 20:13, 16 February 2013 (UTC)
• So, I added a section Induced representations. This is really general representation theory, but the induced transformations ΠAΠ-1 on End(V) are a bit special for the Lorentz group since it is doubly connected: Projective reps on V become reps proper on End(V). Besides, it partly anwers the question "What are projective (here double valued) reps good for?" YohanN7 (talk) 18:51, 16 February 2013 (UTC)
• Go all the way in defining group reps, i.e. show the formula for "defining Π along a path" Done. In the process I removed this: "This can be compared to the situation with SO(3), so(3), su(2), and SU(2). The irreducible representation of the latter three all stand in one-to-one-correspondence with each other, because su(2) and so(3) are isomorphic and SU(2) is simply connected, but only those representations of so(3) coming from representation of su(2) with odd dimension (integer spin) lift via the exponential mapping to an actual representation of SO(3)." I don't think the analogy is bad, but it is very marginally simpler than the example at hand. This means saved space. I'd like to use it for the following:
• The map exp:(so(3;1)->SO(3;1)+ is certainly not one-to-one, it is for most g∈SO(3;1)+ many-to-one. Is it onto? For pure rotations it is onto, and I believe it's onto for pure boosts as well. Every proper orthocronous LT can be written as a pure boost times a pure rotation. This doesn't immediately answer the question because ei(θ · J + ζ · K) ≠ ei(θ · J)ei(ζ · K) because J and K do not commute. I'd appreciate help here. YohanN7 (talk) 23:14, 16 February 2013 (UTC)

## Feedback

Recent edits are an improvement in clarity. Some comments;

"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
${\displaystyle 1=g_{1}^{-1}(g_{1}g_{2}^{-1})(g_{2}g_{3}^{-1})(g_{3}g_{4}^{-1})\cdots (g_{n-1}g^{-1})g,}$
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):
${\displaystyle \Pi (g)\equiv \Pi (g_{n-1}g^{-1})^{-1}\cdots \Pi (g_{3}g_{4}^{-1})^{-1}\Pi (g_{2}g_{3}^{-1})^{-1}\Pi (g_{1}g_{2}^{-1})^{-1}\Pi (g_{1}^{-1})}$ (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)
D(m) = (m/2, 0) ⊕ (0, m/2),
D(m) = (m/2, m/2),
D(m, n) = (m/2, n/2) ⊕ (n/2, m/2),
what does the last expression have for equivalent notation:
D(m, n) = D(?)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) ⊕ (n, m)" ?
and this has what notation:
"D(m, n) = D(?)D(?) ? (Maschen)
• In all... is the statement
D(m, n) = (m/2, n/2) ⊕ (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∧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∧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 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∧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∧Ŝc2ħεИτlk 13:16, 18 February 2013 (UTC)
Forgot to add; when you're ready, just list all morphisms like so:
${\displaystyle A{\xrightarrow {F_{1}}}B,A{\xrightarrow {F_{2}}}C,B{\xrightarrow {F_{3}}}D,\cdots }$
and I could put them together. M∧Ŝc2ħεИτlk 02:38, 19 February 2013 (UTC)
Wonderful! But we better wait a little. I can still see minor notation changes, like Π->ΠU. 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 Representation theory of SL2(C)  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

[references excluded]

The metric signature is (−1, 1, 1, 1) and the physics convention for Lie algebras is used in this article. The Lie algebra of so(3;1) is in the standard representation given by

{\displaystyle {\begin{aligned}J_{1}&=J^{23}=-J^{32}=i{\biggl (}{\begin{smallmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{smallmatrix}}{\biggr )},\\J_{2}&=J^{31}=-J^{13}=i{\biggl (}{\begin{smallmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\\\end{smallmatrix}}{\biggr )},\\J_{3}&=J^{12}=-J^{21}=i{\biggl (}{\begin{smallmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\\\end{smallmatrix}}{\biggr )},\\K_{1}&=J^{01}=J^{10}=i{\biggl (}{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\\\end{smallmatrix}}{\biggr )},\\K_{2}&=J^{02}=J^{20}=i{\biggl (}{\begin{smallmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\\\end{smallmatrix}}{\biggr )},\\K_{3}&=J^{03}=J^{30}=i{\biggl (}{\begin{smallmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\\\end{smallmatrix}}{\biggr )}.\end{aligned}}}

The commutation relations of the Lie algebra so(3;1) are

${\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=i(\eta ^{\sigma \mu }M^{\rho \nu }+\eta ^{\nu \sigma }M^{\mu \rho }-\eta ^{\rho \mu }M^{\sigma \nu }-\eta ^{\nu \rho }M^{\mu \sigma }).}$

In three-dimensional notation, these are

${\displaystyle [{\mathbf {J} }_{i},{\mathbf {J} }_{j}]=i\epsilon _{ijk}{\mathbf {J} }_{k},\quad [{\mathbf {J} }_{i},{\mathbf {K} }_{j}]=i\epsilon _{ijk}{\mathbf {K} }_{k},\quad [{\mathbf {K} }_{i},{\mathbf {K} }_{j}]=-i\epsilon _{ijk}{\mathbf {J} }_{k}.}$

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:

${\displaystyle [J_{i},J_{j}]=i\epsilon _{ijk}J_{k},\quad [J_{i},K_{j}]=i\epsilon _{ijk}K_{k},\quad [K_{i},K_{j}]=-i\epsilon _{ijk}J_{k}.}$

since we have already defined what J1, J2, J3, K1, K2, K3? Call me nitpicky but it would be much clearer. M∧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:

[refs excluded]

According to the general representation theory of Lie groups, one first looks for the representations of the complexification, so(3;1)C of the Lie algebra so(3;1) of the Lorentz group. A convenient basis for so(3;1) is given by the three generators Ji of rotations and the three generators Ki of boosts. First complexify the Lie algebra, and then change basis to the components of A = (J + iK)/2 and B = (JiK)/2. In this new basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra su(2) and moreover that they commute with each other.

${\displaystyle \left[A_{i},A_{j}\right]=\varepsilon _{ijk}A_{k}\,,\quad \left[B_{i},B_{j}\right]=\varepsilon _{ijk}B_{k}\,,\quad \left[A_{i},B_{j}\right]=0\,,}$ ← HERE

In other words, one has the isomorphism...

Nice. YohanN7 (talk) 11:28, 19 November 2013 (UTC)
then later after the formula ${\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=i(\eta ^{\sigma \mu }M^{\rho \nu }+\eta ^{\nu \sigma }M^{\mu \rho }-\eta ^{\rho \mu }M^{\sigma \nu }-\eta ^{\nu \rho }M^{\mu \sigma }).}$ 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∧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:
${\displaystyle {x'}^{\alpha }=\Lambda ^{\alpha }{}_{\beta }x^{\beta }\,,}$
nothing more.
${\displaystyle F=ma.}$
${\displaystyle F=ma.}$
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∧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∧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#Finite-dimensional representations as well as a smaller one to Representation theory of the Lorentz group#Finite-dimensional representations#The Lie algebra, if you see my point.

## “traceless metric tensor”

Wouldn’t this enigmatic thing from Representation theory of the Lorentz group#Common representations 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 Tαβ1/4Tξξgαβ is not normally zero, whereas gαβ1/4δξξgαβ = 0, isn’t it? Incnis Mrsi (talk) 19:46, 19 November 2013 (UTC)

My guess is that it originated as “traceless symmetric 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 (1,1)-representation should have (1 + 2*1)(1 + 2*1) = 9 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

• 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∧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∧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∧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∧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∧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 [itex] 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)

I walked several times through the M ≈ … ≈ SO(3;1) formula in Representation theory of the Lorentz group #The Möbius group, and only now noticed that the orthochronous sign is missing. There may be more such eggs in the whole article. By the way, is there some reason to use as the isomorphism sign instead of ? Incnis Mrsi (talk) 08:53, 16 February 2014 (UTC)
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 SL(2, C) 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:

{\displaystyle {\begin{aligned}\phi _{\mu \nu }(X)P&={\frac {\partial P}{\partial z_{1}}}{\frac {dz_{1}}{dt}}|_{t=0}+{\frac {\partial P}{\partial z_{2}}}{\frac {dz_{2}}{dt}}|_{t=0}+{\frac {\partial P}{\partial {\overline {z_{1}}}}}{\frac {d{\overline {z_{1}}}}{dt}}|_{t=0}+{\frac {\partial P}{\partial {\overline {z_{2}}}}}{\frac {d{\overline {z_{2}}}}{dt}}|_{t=0}\\&=-{\frac {\partial P}{\partial z_{1}}}(X_{11}z_{1}+X_{12}z_{2})-{\frac {\partial P}{\partial z_{2}}}(X_{21}z_{1}+X_{22}z_{2})-{\frac {\partial P}{\partial {\overline {z_{1}}}}}({\overline {X_{11}}}{\overline {z_{1}}}+{\overline {X_{12}}}{\overline {z_{2}}})-{\frac {\partial P}{\partial {\overline {z_{2}}}}}({\overline {X_{21}}}{\overline {z_{1}}}+{\overline {X_{22}}}{\overline {z_{2}}})\end{aligned}}.}

(S7)

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 μ,ν-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#Applications 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.

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 *Bekaert, X.; Boulanger, N. (2006). "The unitary representations of the Poincare group in any spacetime dimension". arXiv:. 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 A:VW between representation spaces V, W such that
${\displaystyle A\circ \pi (x)=\rho (x)\circ A,\quad x\in {\mathfrak {g}}{\text{ or }}x\in G}$
where (π, V) and (ρ, W) 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#Equivariant maps and isomorphisms 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)

Done 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 Γ(g) denote the group generated by exp(g)" — One could wonder, isn't exp(g) itself a group? [1] Boris Tsirelson (talk) 17:32, 7 December 2016 (UTC)

It isn't always a group. There is the SL(2, ℂ) example in the article where it is shown that exp misses a conjugacy class. That "hole" gets "filled" by taking products of SL(2, ℂ) 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 exp. If the image is a group then "generation" is harmless. The image of exp 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 A = g1g2... and B = g14g99... with all g 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)#Topology and universal cover (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#Matrix coefficients (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. g vs ${\displaystyle {\mathfrak {g}}}$ 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#Example: SL(2,C). The second part is summarised there and the details can be given in an elementary way. The proof of the Plancherel theorem for SL(2,C) itself is described in various places. It is a much easier theorem to prove than the real case of SL(2,R). One approach is explained in the Appendix to Chapter VI of Guillemin and Sternberg's book "Geometric Asymptotics"; it applies to all complex semisimple Lie groups and is, according to them, Gelfand's original argument. I will firstly try to add the material on L2(G / K) in a brief form; and then I will try to devise what I consider the "simplest" account for L2(G). I am adding some parallel content to another article (Differential forms on a Riemann surface#Poisson equation), which is how I returned to this topic. Mathsci (talk) 10:42, 14 December 2016 (UTC)

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 2j0 ↔ |k| (corresponding to the usual difference in labeling of SU(2)-representations between mathematics and physics). For the complementary series, one needs ν + 1 ↔ t, 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

@YohanN7: 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 <ref>{{harvnb|Author|Year|loc=Location}}</ref> or <ref>{{harvnb|Author|Year}} Location</ref>. The output is slightly different (a comma is missing in the latter case). (talkcontribs) 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)
@Mathsci: 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 (talkcontribs) 13:50, 20 December 2016 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────@YohanN7: 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:

• Dixmier, J.; Malliavin, P. (1978)
• Gelfand, I. M.; Graev, M. I.; Vilenkin, N. Ya. (1966)
• Naimark, M.A. (1964)
• Stein, Elias M. (1970)

(talkcontribs) 18:54, 20 December 2016 (UTC)

I see, Mathsci. My comment was all about polishing. You can disregard it if this is not the right time. (talkcontribs) 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 article is to long (198.000 bytes)

The optimal length of an article is around 40.000 bytes wp:size

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

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 (μ,ν)?

I notice that in the section Concrete realizations, ${\displaystyle (\mu ,\nu )}$ representations are discussed. But these are not the same as the ${\displaystyle (m,n)}$ representations. So what is (e.g.) a ${\displaystyle (1,0)}$ representation? Is it a ${\displaystyle m=1,n=0}$ representation (of dimension 3) or a ${\displaystyle \mu =1,\nu =0}$ 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 μ and ν is fairly easy to miss. Sławomir Biały (talk) 00:29, 6 January 2017 (UTC)

You have a point. The (μ, ν) are for (at least) real-linear sl(2, ℂ)-reps and the (m, n) are for so(3; 1). All numerically mentioned reps refer to so(3; 1). It is standard in the literature to use half-integers for so(3; 1), and (as far as I know) also standard to use integers for sl(2, ℂ). 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 SO(3; 1) 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)
Done

## GA 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. SparklingPessimist 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#GA nomination 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. SparklingPessimist 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)
Rate Attribute Review Comment
1. Well written:
1a. the prose is clear and concise, and the spelling and grammar are correct.
1b. it complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation.
2. Verifiable with no original research:
2a. it contains a list of all references (sources of information), presented in accordance with the layout style guideline.
2b. all in-line citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines.
2c. it contains no original research.
2d. it contains no copyright violations nor plagiarism.
3a. it addresses the main aspects of the topic.
3b. it stays focused on the topic without going into unnecessary detail (see summary style).
4. Neutral: it represents viewpoints fairly and without editorial bias, giving due weight to each.
5. Stable: it does not change significantly from day to day because of an ongoing edit war or content dispute.
6. Illustrated, if possible, by images:
6a. images are tagged with their copyright status, and valid fair use rationales are provided for non-free content.
6b. images are relevant to the topic, and have suitable captions.
7. Overall assessment.

### Will also review

Thank you, YohanN7 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 "Killing (1888) 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? [2] 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)
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)
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.

Done YohanN7 (talk) 15:55, 8 May 2017 (UTC)

* In equation A1, some C's should be C.

Done 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)
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:Ggl(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.
Done
I still see a lot of repetition there. Jakob.scholbach (talk) 04:43, 18 May 2017 (UTC)
Done 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).
Done (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).
Done 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:

• 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 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. YohanN7 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 [[WT:Wiki, in the hope that another math-experienced editor might weigh in. Jakob.scholbach (talk) 15:17, 15 May 2017 (UTC)

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)

• Prose size (text only): 72 kB (14939 words) "readable prose size"

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)

#### 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 SparklingPessimist?) 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. SparklingPessimist 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 SparklingPessimist above. Jakob.scholbach (talk) 03:01, 25 May 2017 (UTC)
Which request?
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#Why not a textbook. 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 Wikipedia:Copyright violations and Wikipedia: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 Wikipedia:Copyright violations and Wikipedia: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)

## 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 ad.) 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:

${\displaystyle \Phi (\alpha ,\beta ,\gamma ,\delta ),\Psi (\varepsilon ,\zeta ,\eta ,\theta )}$

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 exp may or may not be surjective. It belongs in the article that it is, resp. is not for SO(3, 1) and SL(2, C), 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 SO(3, 1) 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)
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 ${\displaystyle \mathrm {SO(8)} }$ 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)