This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.PhysicsWikipedia:WikiProject PhysicsTemplate:WikiProject Physicsphysics articles
This article is within the scope of WikiProject Chemistry, a collaborative effort to improve the coverage of chemistry on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.ChemistryWikipedia:WikiProject ChemistryTemplate:WikiProject ChemistryChemistry articles
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics articles
Why is the transformation of the spinor fields according to the representation of the Lorentz group irrelevant, and why is it irrelevant/dubious that the literature uses Lorentz group representations to construct relativistic wave equations? The paper by T. Jaroszewicz and P.S Kurzepa discusses this, and so does the book by E. Abers (chapter 12). Well, it may be erroneous to say the equations are "derived from" the irreps, but one way or another they are used...
If I remember correctly, the group of spatial rotations SO(3) has one irreducible representation for spin 1/2. And there are two ways to extend it to a representation of the Lorentz group which differ in how they handle boosts. The four-vector representation is a tensor product of these two. JRSpriggs (talk) 14:08, 8 July 2013 (UTC)[reply]
No, you do not. In short:
There are infinite families of irreps;
There are two complex spin-½ irreps of SU(2) (orprojective representations of SO(3)), they differ only in reversing of the complex structure. When you extend SO(3) to SO(3,1) then yes, they also differ in how they handle boosts, but these are distinct irreps over complex numbers already in 3-d.
The article should not put such gross emphasis on the Lorentz group and classification of its irreps. Real irreps certainly are used for description of spin, but they do not constitute a comprehensive theory of spin: it is a quantum-mechanical phenomenon and is too complex to be compressed to this article. Incnis Mrsi (talk) 17:30, 8 July 2013 (UTC)[reply]
JRSpriggs: Yes, the correspondence between SO(3) and SU(2) does seem familiar and is given in many books, but how tensor products fit in is still confusing for me. It'll absorb in someday. (I was being presumptuous in the lead by writing that reps could be reduced into tensor products of irreps and no further, which as user:Mathsci and Incnis Mrsi indicated here that this isn't the case...).
Incnis Mrsi: Well, yes this article shouldn't dwell too much on just the Lorentz group but as you can see there are plenty of empty sections on other groups to fill in, so it wouldn't just be the Lorentz group. This article needs to indicate how and what the irreps of the Lorentz group are with some synopsis of their importance. M∧Ŝc2ħεИτlk12:16, 9 July 2013 (UTC)[reply]
User:R.e.b. and IP User:96.60.16.101 have done a good job at correcting the article after I tried to change from a redirect to an article and failed. However, R.e.b. introduced the concept of "indecomposable" as different to "irreducible". Maybe the term is confused in places (not just me), maybe the terminology is just used differently. Note R.e.b. actually wrote indecomposable without adding citations to support this terminology, and exactly where I added references which do in fact use the term "irreducible representation".
"Indecomposable" refers (in Halls book) to the Lie algebra itself, while irreducible (universally) refers to reps. Technicalities about the terminology: Suppose that there is an invariant subspace U of a rep. Then the orthogonal complement of U may or may not be an invariant subspace itself. In the former case, the rep is reducible (to U and its complement), but not in the latter case. If the space breaks up completely (by repeatedly finding invariant subspaces and their invariant complements), then the rep is completely reducible. If every rep of a group/algebra breaks up this way, it has the complete reducibility property. Semisimple Lie algebras all have the complete reducibility property.
I have quickly glanced the new article, and I suggest that the term indecomposable should be avoided because it isn't standard, and apparently has multiple meanings. YohanN7 (talk) 14:46, 10 November 2013 (UTC)"[reply]
I have just modified one external link on Irreducible representation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).
If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
If you found an error with any archives or the URLs themselves, you can fix them with this tool.
