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Recent edits are an improvement in clarity. Some comments;
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)
Just alternative notations/conventions?... (Maschen)
Minor comments though, good work! (I might as the maths ref desk btw). M∧Ŝ c2ħε Иτlk 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)
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)
Several days ago I redirected Representation theory of SL2(C) ( | talk | history | links | watch | logs) 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)
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)
[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
The commutation relations of the Lie algebra so(3;1) are
In three-dimensional notation, these are
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 [eij), wouldn't it be better to write the commutation relations as:
since we have already defined what J1, J2, J3, K1, K2, K3? Call me nitpicky but it would be much clearer. M∧Ŝ c2ħε Иτlk 06:58, 19 November 2013 (UTC)
[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 + i K)/2 and B = (J − i K)/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.
- ← HERE
In other words, one has the isomorphism...
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)
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 Tαβ − 1/4Tξξ gαβ is not normally zero, whereas gαβ − 1/4δξξgαβ = 0, isn’t it? Incnis Mrsi ( talk) 19:46, 19 November 2013 (UTC)
Made a major addition. Small problems:
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)
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)
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)
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)
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)
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):
Lie groups and Lie algebras |
---|
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)
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)
Can someone verify/correct this:
(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)
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)
Endnote 101 has "both" misspelled. (It says "botyh".)
There was another spelling error I just corrected, but in this case I can't edit endnotes.
I rarely see spelling errors in Wikipedia articles. Yet here I saw two. Please proofread this article.
166.137.101.174 ( talk) 21:49, 20 July 2014 (UTC)Collin237
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)
"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)
{{
cite arXiv}}
: Invalid |ref=harv
(
help)
YohanN7 (
talk)
14:24, 2 December 2016 (UTC)In "The unitarian trick" section:
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)
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)
"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)
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)
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)
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)
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)
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.)
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)
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)
@ 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)
@ 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). – Finnusertop ( talk ⋅ contribs) 13:02, 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:
– Finnusertop ( talk ⋅ contribs) 18:54, 20 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)
Just more remarks on this article in relation to the Good article assesment. This is not an official GA but more my ideas on how to improve the article. I am not a specialist on the field (and reading the article does not really help me either :)
So here my remarks:
The optimal length of an article is around 40.000 bytes wp:size
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)
- 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)
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)
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...
I notice that in the section Concrete realizations, representations are discussed. But these are not the same as the representations. So what is (e.g.) a representation? Is it a representation (of dimension 3) or a 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)
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)
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:
Michael Hardy ( talk) 16:01, 15 May 2017 (UTC)
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)
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)
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)
Lead, end of second paragraph: the link to Plancherel formula does not work. Boris Tsirelson ( talk) 19:13, 7 June 2017 (UTC)
Dashes instead of a math symbol—namely, «D – 2»—linger for two years, and in multiple instances. Worse: originated not from a clueless editor, even not from a stupid script making blind replacements for U+002D. How can such guru as YohanN7 commit this? Not good… guys, you should learn to manage without me, at the end. Incnis Mrsi ( talk) 15:24, 3 May 2019 (UTC)
The section removed
here, with Prerequisites outlined
and instructive passages like These can be thought of, in the passive view, as (instantly!) giving the coordinate system (and with it the observer) a velocity in a chosen direction
, is a clear-cut example of pedagogical material that is inappropriate here per
the NOTTEXTBOOK policy:
Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter. It is not appropriate to create or edit articles that read as textbooks, with leading questions and systematic problem solutions as examples.
Also relevant in the same policy:
Describing to the reader how people or things use or do something is encyclopedic; instructing the reader in the imperative mood about how to use or do something is not
Wikilinks give connections to articles further explaining related topics. It’s not the place of the article to outline prerequisites for learning material; articles neutrally describe facts. — MarkH21 talk 18:42, 5 May 2020 (UTC)
too trivial. Also note that introducing and presenting broader context (in moderation) for accessibility to a topic is fine, but the pedagogical language and presentation style of the removed section is not. — MarkH21 talk 02:11, 6 May 2020 (UTC)
Thank you to all contributes 41.190.245.201 ( talk) 13:25, 30 December 2021 (UTC)
I don't want to dive in and make these edits myself because (a) I'm not sure I'm right, (b) a lot of people have clearly put a great deal of hard work into this page and could have better ideas about how my observations could be used, but firstly there seems to be a reference (in the "Explicit formulas: Weyl spinors and bispinors" section) to equation G1, which doesn't exist. I'm pretty sure it should be a reference to equation A2 instead. Also, I think equation A2 is derived from the formulas A=(J+iK)/2 and B=(J-iK)/2 at the start of the section "The Lie algebra", and maybe it would improve clarity if this were made explicit. Lastly, I wonder if it's worthwhile showing an equivalent calculation to W1 for the (1/2,1/2) case, to show how the vector representation comes out of A2. 1.125.111.33 ( talk) 01:58, 23 April 2023 (UTC)
This is the
talk page for discussing improvements to the
Representation theory of the Lorentz group article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Archives: 1 |
Representation theory of the Lorentz group has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it. | ||||||||||
| ||||||||||
A
fact from this article appeared on Wikipedia's
Main Page in the "
Did you know?" column on
September 25, 2017. The text of the entry was: Did you know ... that while working on the
representations of the Lorentz group, an encounter with
Dirac convinced
Harish-Chandra that he did not have "the mysterious sixth sense which one needs in order to succeed in physics"? |
This article is rated GA-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||
|
Recent edits are an improvement in clarity. Some comments;
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)
Just alternative notations/conventions?... (Maschen)
Minor comments though, good work! (I might as the maths ref desk btw). M∧Ŝ c2ħε Иτlk 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)
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)
Several days ago I redirected Representation theory of SL2(C) ( | talk | history | links | watch | logs) 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)
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)
[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
The commutation relations of the Lie algebra so(3;1) are
In three-dimensional notation, these are
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 [eij), wouldn't it be better to write the commutation relations as:
since we have already defined what J1, J2, J3, K1, K2, K3? Call me nitpicky but it would be much clearer. M∧Ŝ c2ħε Иτlk 06:58, 19 November 2013 (UTC)
[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 + i K)/2 and B = (J − i K)/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.
- ← HERE
In other words, one has the isomorphism...
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)
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 Tαβ − 1/4Tξξ gαβ is not normally zero, whereas gαβ − 1/4δξξgαβ = 0, isn’t it? Incnis Mrsi ( talk) 19:46, 19 November 2013 (UTC)
Made a major addition. Small problems:
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)
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)
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)
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)
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)
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):
Lie groups and Lie algebras |
---|
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)
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)
Can someone verify/correct this:
(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)
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)
Endnote 101 has "both" misspelled. (It says "botyh".)
There was another spelling error I just corrected, but in this case I can't edit endnotes.
I rarely see spelling errors in Wikipedia articles. Yet here I saw two. Please proofread this article.
166.137.101.174 ( talk) 21:49, 20 July 2014 (UTC)Collin237
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)
"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)
{{
cite arXiv}}
: Invalid |ref=harv
(
help)
YohanN7 (
talk)
14:24, 2 December 2016 (UTC)In "The unitarian trick" section:
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)
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)
"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)
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)
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)
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)
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)
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.)
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)
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)
@ 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)
@ 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). – Finnusertop ( talk ⋅ contribs) 13:02, 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:
– Finnusertop ( talk ⋅ contribs) 18:54, 20 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)
Just more remarks on this article in relation to the Good article assesment. This is not an official GA but more my ideas on how to improve the article. I am not a specialist on the field (and reading the article does not really help me either :)
So here my remarks:
The optimal length of an article is around 40.000 bytes wp:size
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)
- 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)
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)
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...
I notice that in the section Concrete realizations, representations are discussed. But these are not the same as the representations. So what is (e.g.) a representation? Is it a representation (of dimension 3) or a 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)
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)
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:
Michael Hardy ( talk) 16:01, 15 May 2017 (UTC)
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)
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)
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)
Lead, end of second paragraph: the link to Plancherel formula does not work. Boris Tsirelson ( talk) 19:13, 7 June 2017 (UTC)
Dashes instead of a math symbol—namely, «D – 2»—linger for two years, and in multiple instances. Worse: originated not from a clueless editor, even not from a stupid script making blind replacements for U+002D. How can such guru as YohanN7 commit this? Not good… guys, you should learn to manage without me, at the end. Incnis Mrsi ( talk) 15:24, 3 May 2019 (UTC)
The section removed
here, with Prerequisites outlined
and instructive passages like These can be thought of, in the passive view, as (instantly!) giving the coordinate system (and with it the observer) a velocity in a chosen direction
, is a clear-cut example of pedagogical material that is inappropriate here per
the NOTTEXTBOOK policy:
Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter. It is not appropriate to create or edit articles that read as textbooks, with leading questions and systematic problem solutions as examples.
Also relevant in the same policy:
Describing to the reader how people or things use or do something is encyclopedic; instructing the reader in the imperative mood about how to use or do something is not
Wikilinks give connections to articles further explaining related topics. It’s not the place of the article to outline prerequisites for learning material; articles neutrally describe facts. — MarkH21 talk 18:42, 5 May 2020 (UTC)
too trivial. Also note that introducing and presenting broader context (in moderation) for accessibility to a topic is fine, but the pedagogical language and presentation style of the removed section is not. — MarkH21 talk 02:11, 6 May 2020 (UTC)
Thank you to all contributes 41.190.245.201 ( talk) 13:25, 30 December 2021 (UTC)
I don't want to dive in and make these edits myself because (a) I'm not sure I'm right, (b) a lot of people have clearly put a great deal of hard work into this page and could have better ideas about how my observations could be used, but firstly there seems to be a reference (in the "Explicit formulas: Weyl spinors and bispinors" section) to equation G1, which doesn't exist. I'm pretty sure it should be a reference to equation A2 instead. Also, I think equation A2 is derived from the formulas A=(J+iK)/2 and B=(J-iK)/2 at the start of the section "The Lie algebra", and maybe it would improve clarity if this were made explicit. Lastly, I wonder if it's worthwhile showing an equivalent calculation to W1 for the (1/2,1/2) case, to show how the vector representation comes out of A2. 1.125.111.33 ( talk) 01:58, 23 April 2023 (UTC)