6-j Symbol

6-j Symbol

Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940, and published in 1965. They are defined by a sum over products of four 3jm symbols,

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \sum_{m_i} (-1)^S \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_5 & j_6\\ -m_1 & m_5 & m_6 \end{pmatrix} \begin{pmatrix} j_4 & j_5 & j_3\\ m_4 & -m_5 & m_3 \end{pmatrix} \begin{pmatrix} j_4 & j_2 & j_6\\ -m_4 & -m_2 & -m_6 \end{pmatrix} .$

with phase . The summation is over all six m_i, effectively confined by the selection rules of the 3jm symbols. They are related to Racah's W-coefficients by

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = (-1)^{j_1+j_2+j_4+j_5}W(j_1j_2j_5j_4;j_3j_6).$

They have higher symmetry than Racah's W-coefficients.

Read more about 6-j Symbol: Symmetry Relations, Special Case, Orthogonality Relation, Asymptotics

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