= Wigner D-matrix
{c}
{title2=$D^{(j)}_{mm'}(\phi,\theta,\psi)$}
{wiki}
For active rotations with $U=e^{-i\phi J_3}e^{-i\theta J_2}e^{-i\psi J_3}$, the <irreducible spin representation> <matrix> is $D^{(j)}_{mm'}=e^{-im\phi}d^{(j)}_{mm'}(\theta)e^{-im'\psi}$, where $d^{(j)}_{mm'}=\langle j,m|e^{-i\theta J_2}|j,m'\rangle$. Its spin-one-half middle <matrix> is $\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}$. A $2\pi$ rotation acts by $(-1)^{2j}$; only integer-spin representations descend to the <special orthogonal group> in three <dimensions>.
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