Timeline for What is the relation between irreducible representation of $SO(3)$ and spherical harmonics?
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| Oct 14 at 3:56 | comment | added | ZeroTheHero | Also if you are really really really mathy, then there's a refinement in the jargon, with the use of "modules". In physics, the carrier space is occasionally confounded with the representations (v.g. the quarks are in the representation $\textbf{3}$ of su(3)). The rep should technically be the matrices of the group elements (or the elements in the algebra). | |
| Oct 14 at 3:39 | comment | added | ZeroTheHero | Basically you think of $Y_\ell^m(\theta,\varphi)$ "in the physicist's way" as $\langle \theta\varphi\vert \ell m\rangle$ and the action of $R(\Omega)$ is via $\langle \theta\varphi\vert \left[R(\Omega)\vert \ell m\rangle\right]$. | |
| Oct 14 at 3:37 | comment | added | ZeroTheHero | yes the vector space spanned by the basis is (sometimes) called the "carrier space". I should add that the spherical harmonics $\{Y_\ell^m(\theta,\varphi) ,m=-\ell,\ldots,\ell\}$ for fixed $(\theta,\varphi)$ form a basis, i.e. a rotation parametrized by $\Omega$ (3 angles) can still act on the spherical harmonics, which are parametrized by $2$ angles: $R(\Omega)Y_{\ell}^m(\theta,\varphi)=\sum_{m'} Y_{\ell}^{m'}(\theta,\varphi) D^\ell_{m'm}(\Omega)$. | |
| Oct 14 at 2:35 | comment | added | JEB | @ZeroTheHero thanks for that. I've always had trouble with the mathy part where what's a group, an algebra, a representation, or new-to-me: what carries an irrep. | |
| Oct 13 at 22:44 | comment | added | ZeroTheHero | actually the spherical harmonics are basis functions that span the carrier space of a $2\ell+1$ dimensional irrep, aka they carry the irreps or transform by the irrep, rather than being the irrep itself. The irrep would be the rotation matrices, as maps from the group elements to matrices so that the multiplication rule of elements is preserved. | |
| Oct 13 at 5:46 | vote | accept | MakiseKurisu | ||
| Oct 3 at 14:08 | history | edited | JEB | CC BY-SA 4.0 |
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| Oct 1 at 19:11 | history | edited | JEB | CC BY-SA 4.0 |
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| Oct 1 at 17:50 | history | answered | JEB | CC BY-SA 4.0 |