Understanding the D^{l}(\theta) Representation of 3D Rotations

[Phymath]

I'm having difficulty with the $$D^{l}(\theta)$$ representation of 3D rotations what do the mean i suppose one you construct it for l = 1 you get the general rotation Euler matrix for 3-d Space, but what do the l = other integers or half integers mean physically? is the D matrices the generalization of 3-D rotations to different vector spaces? such as a 2 dimensional space for l = 1/2? any explanation would help thanks.

 

[ismaili]

I'm having difficulty with the $$D^{l}(\theta)$$ representation of 3D rotations what do the mean i suppose one you construct it for l = 1 you get the general rotation Euler matrix for 3-d Space, but what do the l = other integers or half integers mean physically? is the D matrices the generalization of 3-D rotations to different vector spaces? such as a 2 dimensional space for l = 1/2? any explanation would help thanks.

Hmm...roughly speaking, in QM, the description of spin is by the the representation of the group SU(2), which is locally isomorphic to the group SO(3), they can share similar representations.
To obtain the representations of group SU(2), we can start from the algebra, i.e. $$[J_i,J_j]=i\epsilon_{ijk}J_k$$, where $$J_i$$ are generators. By the standard procedure which is shown by almost all QM book(e.g. Sakurai), you can see that the representations are labeled by an integer or half integer j, or your l. For the case of j = 1, you could actually think of it as the three dimensional vector representation. For the case of j = 1/2, this is a representation of the intrinsic spin of, say, electrons. For the group SO(3), it turns out that j could be only integers, 0,1,2,...
Representations can be direct product to form a larger representation, this is physically interpreted as the addition of spins.
Any supplement or corrections are welcome.