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

[Yoran91]

Hello everyone,

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

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

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

 

[dextercioby]

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