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The groups O(3), SO(3) and SU(2)

  1. Jan 1, 2010 #1
    1. The problem statement, all variables and given/known data

    How can irreducible representations of O(3) and SO(3) be determined from the irreducible representations of SU(2)?

    3. The attempt at a solution

    I believe there is a two-one homomorphic mapping from SU(2) to SO(3); is that enough for some shared representations? If I had an idea of *why* irreducible reps. can determined for O(3) and SO(3) from SU(2), I might have a better notion of *how* to go about proving it.

    Cheers!
     
  2. jcsd
  3. Jan 1, 2010 #2

    George Jones

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    Mathematically, what is a representation of a group G?
     
  4. Jan 1, 2010 #3
    Typically a matrix, I believe, for which [tex]\Gamma(T_{1}T_{2}) = \Gamma(T_{1})\Gamma(T_{2})[/tex] holds, where [tex]T_{1}, T_{2}[/tex] belong to [tex]G[/tex]
     
  5. Jan 2, 2010 #4

    dextercioby

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    There are 2 isomorphisms you need to use:

    [tex] \mbox{SO(3)}\simeq\frac{\mbox{SU(2)}}{\mathbb{Z}_{2}} [/tex]

    and

    [tex] \mbox{O(3)} = \mbox{SO(3)} \times \{-1_{3\times 3}, 1_{3\times 3} \}[/tex]

    There are at least 50 books or so discussing the connection b/w SO(3) and SU(2), however there are many less computing all representations of O(3) starting from the ones of SO(3) deducted from the ones of SU(2).
     
    Last edited: Jan 2, 2010
  6. Jan 2, 2010 #5

    Thank you very much for your answer. I understand the second statement, but what exactly are you doing in the first - simply slicing off the complex aspect by mathematical fiat?

    Cheers :)
     
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