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Is the wigner D function a representation of SO(3)?

  1. May 8, 2013 #1
    Hello everyone,

    I'm reading a bit about the Wigner D matrix, defined by
    [itex]\mathscr{D}\left(\hat{n},\phi \right) = \exp[-\frac{i \phi}{\hbar}\vec{J}\cdot \hat{n}][/itex].

    Now I'm wondering : is the map [itex] \pi : \text{SO(3)} \to \text{GL}\left( \mathscr{H} \right) [/itex] given by [itex] R\left(\hat{n},\phi \right) \mapsto \mathscr{D}\left(\hat{n},\phi \right)[/itex] a representation of SO(3) on some Hilbert space [itex]\mathscr{H}[/itex]?

    I would say yes, but I have no clue on how to prove that this map is a homomorphism so that it is indeed a representation. Further, if it is a representation, is it reducible or irreducible?

    Thanks for any help
  2. jcsd
  3. May 8, 2013 #2


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    Through the work of Wigner and Bargmann, the map [itex] \pi [/itex] you mention maps SU(2) (not SO(3)) onto some subset of the unitaries of a complex separable Hilbert space (essentially [itex] L^2 (R^3,C^{2s+1}) [/itex]). This is such a standard result, that I can't think of the best book treating this subject. Barut and Raczka, maybe.
    Last edited: May 8, 2013
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