I ask this question motivated by trying to understand vector spherical harmonics and find or come up with an elegant abstract derivation of their form.
Suppose we have a 3D field of 3D vectors: $\textbf{F}(\textbf{r})$. I think of this vector field as a 3D arrow attached to each point of 3D space. There are a few different types of rotations we can implement on this vector field. Suppose we have a rotation matrix $\textbf{R}$.
- Rotation of the vectors themselves: $\textbf{R}\textbf{F}(\textbf{r})$. This might be associated with intrinsic or spin angular momentum $\textbf{S}$.
- Rotation of the field itself while the vectors stay fixed: $\textbf{F}(\textbf{R}^{-1}\textbf{r})$. This might be associated with extrinsic or orbital angular momentum $\textbf{L}$.
- Rotation of both the vectors and the field: $\textbf{R}\textbf{F}(\textbf{R}^{-1}\textbf{r})$. This might be associated with total angular momentum $\textbf{J}$.
I've gotten this sense about rotation of classical vector fields from reading lots of references about angular momentum, but I've never really seen it spelled out clearly in this way. I've also never seen really good notation to indicate whether an operator acts on the vectors or the field.
For example, it might be written then $\textbf{J}=\textbf{L}+\textbf{S}$, but $\textbf{L}$ operates on the coordinates while $\textbf{S}$ operates on the vectors. What kind of operators even are $\textbf{L}, \textbf{S}$, and $\textbf{J}$ in this case and what is the mathematical space on which they act?
Also, I'd appreciate information about this topic which does NOT rely basically at all on quantum mechanics. We shouldn't need quantum mechanics to understand, for example, classical antenna theory. Yes, quantum mechanics might be where we first learned about spherical harmonics and angular momentum operators, but using it as a crutch to understand this stuff just leaves me feeling like I've had a circular or incomplete explanation/motivation.
Unfortunately I'm having a little bit of a hard time even formulating a question about all of this but I'm hoping the sentiment I'm driving at makes sense. I would welcome references which discuss rotations on vector fields in a way sympathetic to this discussion.
