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 mi, 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

Famous quotes containing the word symbol:

    “Dylan is to me the perfect symbol of the anti-artist in our society. He is against everything—the last resort of someone who doesn’t really want to change the world.... Dylan’s songs accept the world as it is.”
    —Ewan MacColl (1915–1989)