Talk:Particle physics and representation theory

message: This article is very vague about a number of points and I would recommend that it be put on status to require "clean up" by an expert.  The most pressing need is the need for a few equations to eliminate some of the hand waving.  There are also factual errors in the text.  For example, SU(3) is the gauge of the strong force, but SU(2) (left) is the gauge of the weak force.  They are distinct gauge groups.  SU(2) (left) is weakly violated (and hence is only an approximate symmetry; SU(3) is an exact symmetry.
:I don't understand those "factual errors". Yes, SU(2) (left) is the gauge of the weak force, but does the article say (or imply) otherwise? The article does say that color SU(3) is an example of an exact symmetry, which is true. So exactly which sentence(s) of the article are you disputing? Steve
General picture: the reference to Weinberg's book seems to be more appropriate in Wigner's theorem. Poincaré group is believed to be a subgroup of G only in a flat space (which is not the case for our Universe). Also, representations of Poincaré group are characterized by mass squared and some other quantum number (spin, helicity) which depends on the sign of m^2. In particular, there are massless representations with continuous "spin". It also seems appropriate to change SU(2) to SO(3) in the "hypothetical example" section. Perhaps, we should merge Eightfold_way_(physics) with this article? --Terminus0 (talk) 01:29, 8 November 2009 (UTC)
replies: text: I tried to make those corrections to the article, they're all good points. Thanks! I'm skeptical about the merge, I think of this article as a general physical idea, and Eightfold way, Georgi-Glashow model, Wigner's classification, isospin, etc. as offshoots of it. But I don't feel too strongly, it could work either way. --Steve (talk) 05:29, 8 November 2009 (UTC)

text: I added a few words to suggest a "coordinate-free" interpretation. Do you think that helps? --Steve (talk) 19:03, 17 May 2010 (UTC)
thread_title:  sloppy 

message: From the first paragraph, "This postulate states that each particle is an irreducible representation of the symmetry group of the universe."
To which postulate is this referring? Mathchem271828 05:35, 6 January 2007 (UTC)
thread_title:  which postulate 

message: There appears to be some sort of error, giving the message "Failed to parse (<math_output_error>): \{g|p_0\rangle|g\in G\}." in the general picture section. I don't know what its supposed to say so can't fix it. Sheepe2004 (talk) 12:57, 20 May 2009 (UTC)
replies: text: Seems to be fixed, see discussion at Wikipedia:Village pump (technical)/Archive 134#Strange math bug. --Steve (talk) 22:29, 20 May 2009 (UTC)
thread_title: error in maths

message: I think the "General picture" paragraph should be rewritten (I would have done it myself hadn't my English sucked so much). It should be stressed that the space of states of a quantum theory is a projective Hilbert space PH. That is not a minor issue, since it follows that symmetries of the system, which form a group PU(H), are, in general, automorphisms of PH (i.e. bijections of PH that preserve |<v/|v|,w/|w|>|). 
Now, there is a theorem, let's call it Wigner's theorem, which tells us that all automorphisms of PH are induced by unitary or antiunitary tranformations of H. So, given a Lie group G of symmetries (say Poincare or Galileo's group for example) what we want is to classify projective representations of G (that is continuous homomorphisms from G to PU(H)). The point is that doing that may give us a clue about what the space H should be. 
It would be very nice if one could lift the projective representation to a unitary (G → U(H)) one, since those are much more well studied. There are two kind of obstructions that may arise. The first is that G may not be simply connected so that it is not even sure if there is a continuous map from G to U(H) associated to each representation. This problem is solved by passing to the universal over of G. The second problem in lifting projective representations is a cohomological one, which may be stated for the Lie algebra of G: every projective rapresentation of G determines an element a of the second cohomology (with real coefficients) H2(g,R) of the Lie algebra g of G, and this representation can be lifted to a unitary one iff a = 0. This is called Bergmann's theorem. This second obstruction is very important since: for the Poincare group we have  H2(p,R) = 0 so that all projective representations of the (double cover of the) Poincare group can be lifted to unitary ones, which is not the case for the Galilean group for example.
About the unitary representations of the (double cover of the) Poincare group, today it is done using the so called Mackey machine. The treatment in the first book about QFT of Weinberg uses a kind of ad hoc method which, if memory doesn't fail me, has the problem of not giving a practical description of H (which for irreducible rep., using the Mackey machine, one directly sees is a space of square integrable functions on orbits in momentum space) making it more difficult to link this description to the the usual one with relativistic wave functions in position space (which is a kind of Fourier transform with some issues to deal with).
Good references on theese subjects are Simms Lie groups and quantum mechanics; Varadarajan  Geometry of Quantum Theory (last 5 chapters); Varadarajan "Introduction to supersymmetry for mathematicians" (the very nice outline in the first chapter); and last but not last the beautiful "Quantum field theory, a tourist guide for mathematicians" by Folland.  Skyhc (talk) 12:58, 17 May 2010 (UTC)
replies: text: I tried to start correcting some of these errors you pointed out, but I can't get very far because I don't know enough and don't have time to learn it. Of course if you try editing it yourself I will try to improve your English (which actually sounds fine to me). I will also be attempting to keep it comprehensible for physics undergraduates who don't know much math beyond basic linear algebra and maybe group theory, who are I think much of the audience for this article. --Steve (talk) 19:02, 17 May 2010 (UTC)

text: I strongly second Steve above. If you're knowledgeable, which sounds to me like you certainly are, you should not hesitate to edit. Mct mht (talk) 02:01, 21 January 2011 (UTC)
thread_title:  Remarks