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.

 

[denian]

The D^{l}(\theta) representation of 3D rotations is a mathematical tool used to describe and analyze rotations in three-dimensional space. The "l" in the representation refers to the angular momentum quantum number, which determines the type of rotation being represented.

For example, when l=1, the D^{l}(\theta) representation is known as the "spherical harmonic representation" and is commonly used to describe rotations in quantum mechanics. This representation can also be used to construct the general rotation Euler matrix for 3D space.

When l is any other integer or half-integer, the D^{l}(\theta) representation describes rotations in higher dimensional spaces. For instance, when l=2, the representation describes rotations in four-dimensional space.

In essence, the D^{l}(\theta) representation is a generalization of 3D rotations to higher dimensional vector spaces. It allows for a more comprehensive understanding and analysis of rotations in various contexts, such as in quantum mechanics or higher-dimensional geometry.

I hope this explanation helps to clarify the meaning and significance of the D^{l}(\theta) representation for you. If you have further questions or would like more information, please don't hesitate to ask.