One of the important identity for angular momentum theory is the Clebsch-Gordon series, that involved Wigner D-matrix.

The series is deduced from evaluate the follow quantity in two ways

$\langle j_1 m_1 j_2 m_2 | U(R) |j m \rangle$

If acting the rotation operator to the $|jm\rangle$, we insert

$\displaystyle \sum_{M} |jM\rangle \langle | jM| = 1$

$\displaystyle \sum_{M} \langle j_1 m_1 j_2 m_2|jM\rangle \langle jM| U(R) |jm\rangle = \sum_{M} C_{j_1m_1j_2m_2}^{jM} D_{Mm}^{j}$

If acting the rotation operator to the $\langle j_1 m_1 j_2 m_2|$, we insert

$\displaystyle \sum_{N_1 N_2 } |j_1 N_1 j_2 N_2\rangle \langle j_1 N_1 j_2 N_2| = 1$

$\displaystyle \sum_{N_1 N_2} \langle j_1 m_1 j_2 m_2|U(R) | j_1 N_1 j_2 N_2\rangle \langle j_1 N_1 j_2 N_2| jm\rangle$

$\displaystyle = \sum_{N_1N_2} C_{j_1N_1j_2N_2}^{jm} D_{m_1N_1}^{j_1} D_{m_2 N_2}^{j_2}$

Thus,

$\displaystyle \sum_{N_1N_2} C_{j_1N_1j_2N_2}^{jm} D_{m_1N_1}^{j_1} D_{m_2 N_2}^{j_2} = \sum_{M} C_{j_1m_1j_2m_2}^{jM} D_{Mm}^{j}$

We can multiply both side by $C_{j_1 N_1 j_2 N_2}^{jm}$, then sum the $j, m$

using

$\displaystyle \sum_{jm} C_{j_1 N_1 j_2 N_2}^{jm} C_{j_1N_1j_2N_2}^{jm} = 1$

$\displaystyle D_{m_1N_1}^{j_1} D_{m_2 N_2}^{j_2} = \sum_{jm} \sum_{M} C_{j_1m_1j_2m_2}^{jM} C_{j_1N_1j_2N_2}^{jm} D_{Mm}^{j}$