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Lethe asked for volunteers (rep theory)

  1. Jun 13, 2003 #1


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    Lethe posted this in the group theory thread:

    " you know, this thread is turning into a pretty nice lie group/lie algebra thread. there is the differential forms thread. now all we need if for someone to start a representation theory thread, and we ll have all the maths we need to do modern particle physics.

    who wants to volunteer?"
    Last edited: Jun 13, 2003
  2. jcsd
  3. Jun 13, 2003 #2


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    what should be an easy question

    I had a question in that area
    that I posted in "theoretical" forum:

    "Let φ SU(2) ---> SO(3) be the double covering

    Any irreducible representation of SO(3) pulls back by φ
    to provide an irreducible representation of SU(2) on the
    same finite dimensional Hilbert space.

    this seems clear, almost not worth saying:
    the pullback is obviously irred. and has the same dimension.

    I have a question about the other direction----suppose an
    irred. rep. of SU(2) factors thru φ

    then it obviously gives a representation of SO(3), same space
    same dimension and all, but

    under what circumstances is the representation irreducible? "

    this may be a dumb question for all I know.

    BTW to me, at the Lie algebra level, they look the same
    To me at least, so(3) looks like su(2).
    I have the feeling that the representation of SO(3)
    that you get ought to be irreducible under fairly general
    circumstances, but this should be part of time-honored lore
    that other people here have. So I ask.

    I have a strong preference for online textbooks not only
    because I can get them free and quickly but also because
    we all share online stuff and can be "on the same page" easily
    I would like to know an online "rep theory for dummies" textbook
    Last edited: Jun 13, 2003
  4. Jun 13, 2003 #3
    well we should start with some prerequisites.

    we need just a little bit of group theory: definition of a group, subgroup, group homomorphism, group isomorphism, normal subgroups too, maybe? but perhaps this is already in/should go in the group theory for dummies thread.

    then a little bit of linear algebra and matrix groups. what is an eigenvalue, eigenvector, what are SU(n), SO(n), GL(n) etc. determinants, transposes, hermitian adjoints. inner product spaces. that ought to do it? maybe all our linear algebra needs (e.g. for this thread and for the differential forms thread) should be consolidated into on intro to linear algebra thread? what do you think?

    if you ve already got the prerequisites, then you re ready to see the definition of a representation, which should be the natural starting point, eh?

    A representation ρ of a group is a group homomorphism from the the group G to GL(V), the group of linear transformations of a vector space V. this means that


    so ρ(g) and ρ(h) are vector space isomorphisms of V. the dimension of V is the dimension of the representation. if ρ is an isomorphism (that is, one to one and onto), then we call it a faithful representation.

  5. Jun 13, 2003 #4
    perhaps i am being a bit hasty. we should agree on notations, starting points, etc. before we jump right in, eh?

    i m going to think about your question marcus. my knee-jerk response is that i think it will always be irreducble. but i haven t thought it through.

    but for right now, it s friday night, and i have a need to get sloshy, so that s all you ll get from me tonight.
  6. Jun 13, 2003 #5


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    this is the best kind of reaction!
    you share your intuition!
    I also have the feeling that it should always be irreducible
    (and also a terrible sense of my own naivete, because
    in a good subject like this there must be surprises)
    get good and sloshy
    perhaps will see you tomorrow
  7. Jun 14, 2003 #6


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    I got an answer to that question!

    The induced rep on SO(3) is ALWAYS irreducible.

    I will get back in a minute and post in more detail, but
    you for sure know what I mean since this was your intuition
    about what happened (and mine too) it was almost self-evident
    but needed a little time to be sure about.
  8. Jun 14, 2003 #7


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    a useful theorem

    I found a useful theorem in Brian Hall's book about groups and reps:

    Let G be a connected matrix Lie group with Lie algebra {g}

    I cant type the usual GOTHIC style lowercase letter g, so just say {g}.

    Let Π be a representation of G acting on a space V, and let π be the associated Lie algebra representation. Then a subspace W of V is invariant for π if and only if it is invariant for Π.
    And in particular, π is irreducible if and only if Π is irreducible.


    I think we both had the sense of that in mind, tip of the tongue, but I at least wanted reassurance. It is not a deep or hard theorem---he makes the proof a homework problem---but summarizes one of those things which tho obvious should IMHO always be remembered. What a nice subject this is!
  9. Jun 14, 2003 #8


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    the trouble with unicodes on PF

    Unicodes just produce squares for some people

    When I want to type a psi, I say & psi;

    without the space, and that makes ψ

    For integral sign:

    & int;


    When you use those number codes ("unicodes"?)
    my browser just writes a box
    Maybe a mentor will appear and tell me how to change
    the settings. But there is no FAQ that I could find here at PF
    about it. And probably there are some other people in the
    same boat.

    Greg has a sticky about symbols at the top of most forums.
    Would prefer if we could just use that vocabulary
    so no special numbers need be memorized and so on,
    but also could adapt if told how to do it.
  10. Jun 14, 2003 #9


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    The online Lie Group text I've been using

    "An Elementary Introduction to Groups and Representations"

    is at arXiv:math-ph/0005032

    However, its main focus is on the matrix lie groups/algebras and not on the general case. I'm not sure how much of a drawback that would be.
  11. Jun 14, 2003 #10


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    That is Brian Hall's online textbook! I like it a lot.
    If there is some sort of ongoing discussion of people
    helping each other learn groups and representations
    that would be an excellent choice to give something to focus

    It is 128 pages, takes a while to print off. But whole book seems

    Still worried that my browser doesnt see "unicodes" if that is the word for it. In Lethe's post I see some little squares.
  12. Jun 15, 2003 #11
    i don t think those codes are unicode. they are ancient html codes. using the token (e.g. psi) and using the number should give the exact same results. so there is no need to memorize numbers.

    if your browser can t render the symbol, it will just make a box. which symbols are you unable to see?

    a more complete list than what is in the sticky thread can be found here. go there and see how your browser stacks up. i would like to continue using those codes, but if it is a problem, we could look at other solutions (one solution might be for you to use a a different browser!).

    how does your browser render the differential forms thread? i used a lot of symbols in that one.
  13. Jun 15, 2003 #12


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    I see you are suggesting I switch to Mozilla Firebird.
    You asked about the diff forms thread----I dont see any
    boxes there! Very strange.
    Am used to this browser and tend to be reluctant
    to change things about the computer.
    I have no trouble reading other people's posts on PF
    probably because they use the symbols listed on the sticky.
    for some reason those symbols work for my browser.
    Dont know if other people on PF are in the same boat
    Am wondering how this will work out.

    There may be a setting or a preference I can change
    on my browser.

    Or some other kind of code I can type to get symbols.
    Greg may have suggestions.

    Last edited: Jun 15, 2003
  14. Jun 16, 2003 #13


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    Group theory really is at the heart of physics and is needed to describe the relations between symmetry and conservation laws, the geometry of spacetime, and the classification of particles along with their interactions.

    When you guys have learned enough group theory, I'd like to help out teaching how all that mathematical technology is used in high energy and elementary particle theory.

    I can discuss quantum field theory, string theory, LQG, or virtually any other theory people here want to learn about and in as much depth as they can stand.

    Who else around here wants to help out with this stuff?
  15. Nov 19, 2003 #14
    Lie groups text

  16. Nov 19, 2003 #15


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    Re: Lie groups text

    Hi Dr. Hall,

    Welcome to physicsforums! Thank you for the update, your book looks very informative indeed.

    - Warren
    Last edited: Nov 19, 2003
  17. Nov 19, 2003 #16


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    rebirth anybody?

    Hey let's get the thread going again. Now we have Latex to talk about instead of HTML codes, and two choices for a common text (I vote for the online one). Let's dig into it!
  18. Nov 27, 2003 #17
    I'm in once my workload diminishes (about a week's time from now).
  19. Nov 28, 2003 #18


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    here's an area of confusion I have---relating to reps

    in highbrow papers something like this is
    sometimes said in an offhand manner (presuming that everyone

    "quantizing a classical theory is nothing but finding
    representations in hilbertspace of the algebra of observables"

    well they never say it quite that simply, but I get the impression
    that some people EQUATE quantum theories with hilbertspace
    representations of algebras, or groups since groups and algebras are associated so closely

    can someone clarify? to what extent or under what limitations are quantum theories subsumed under the heading of representations?

    if one takes that attitude, what does having the right classical limit mean--and what role does the inner product of the hilbertspace play?

    (perhaps I know some of this but it might be good to get the relation between QM and representation theory established at the getgo)
  20. Nov 29, 2003 #19
    yeah, i do, for example.

    well, i would say that the act of creating a quantum theory is completely subsumed under the idea of finding a representation on a Hilbert space of the classical Poisson algebra.

    i think the inner product on the Hilbert space contains the symplectic form of the classical theory
  21. Feb 19, 2004 #20

    I'm in. The questions I have are about how the decomposition of a continous group or algebra into its irreducible representations is related to the physical structures seen in particle physics. I've been reading a lot on representation theory and Lie groups and algebras, learning to multiply reps using Young tableaux. - The book sitting in front of me right now is Georgi's Lie Algebras in Particle Physics. I was wondering if anyone thought it was possible to use genetic algorithms to breed a rep that would solve the puzzle of the universe.
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