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Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also the case that these matrices (for all positive $j$ multiple of $1/2$) are a representation of the rotation group $SU(2)$, which is a double cover of $SO(3)$.

Now, I can see this for the specific case $j=1$ (3x3 matrices representing rotations in 3-dimensional Euclidean space), but what does it mean for other values of $j$? Say for $j=3/2$, does that represent the subgroup of 3-dimensional rotations given some parameter, in a 4-dimensional ambient geometry?


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Spherical harmonics form a basis of solutions of Laplace's equation, and (three-dimmensional in this case) Laplacian is invariant under rotations, so if f(x,y,z) is a solution to this equation, then g(x,y,z) = f( R(x,y,z : a,b,c) from SO(3)) So, if you have a soution (or a certain quantum-mechanical wave function of some spherically symmetric system) in the basis of spherical harmonics, you can calculate, how its changes in rotated system (using Wigner D-matrix, which is exactly the representation of SO(3) in the basis of spherical harmonics). This is for SO(3) and integer j values (related to the angular momentum of the system).

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