Wigner 3j Symbols as Tetrahedral Recoupling of Clebsch-Gordan Coefficients — E8 Intelligence Research

FINDING: Wigner 3j symbols are a recoupling form of Clebsch-Gordan coefficients, with closed-form expressions linked to tetrahedral symmetry via the Racah sum rule and the 6j symbol's geometric interpretation as tetrahedral edge lengths. MATH: - 3j symbol: \({pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 {pmatrix}\) related to Clebsch-Gordan coefficient \( j_1 m_1 j_2 m_2 | j_3 m_3 \) by a phase factor. - 6j symbol: \(\{j_1 j_2 j_3; j_4 j_5 j_6\}\) satisfies the Racah sum rule, which is isomorphic to the Biedenharn-Elliott identity for tetrahedral symmetry. - Closed-form for tetrahedral group: The 6j symbol for equal angular momenta \(j\) yields values proportional to \({1}{√2j+1}\) times a tetrahedral angle factor, with exact rational values for \(j = 1/2, 1, 3/2, ...\) given by the Racah formula: \(\{j j j; j j j\} = {(-1)³ʲ}{(2j+1)} √(3j)!/(j!)^3 (2j+1)!\) for integer \(j\). - Key constant: For \(j=1\), the 6j symbo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com