Recursion Relations and Closed Forms for SU(2) Clebsch–Gordan Coefficients — E8 Intelligence Research

FINDING: Clebsch–Gordan coefficients for SU(2) encode the decomposition of tensor products of irreducible representations, with explicit recursion relations and closed forms for low spins (j₁=1/2, j₂=1/2; j₁=1/2, j₂=1). | MATH: For SU(2), the tensor product decomposes as \\( j_1 \\otimes j_2 = \\bigoplus_{J=|j_1-j_2|}^{j_1+j_2} J \\). The CG coefficients \\(\\langle j_1 m_1 j_2 m_2 | J M \\rangle\\) satisfy orthogonality: \\(\\sum_{m_1,m_2} \\langle j_1 m_1 j_2 m_2 | J M \\rangle \\langle j_1 m_1 j_2 m_2 | J' M' \\rangle = \\delta_{JJ'}\\delta_{MM'}\\). For j₁=j₂=1/2: \\(|1,1\\rangle = |\\uparrow\\uparrow\\rangle\\), \\(|1,0\\rangle = \\frac{1}{\\sqrt{2}}(|\\uparrow\\downarrow\\rangle + |\\downarrow\\uparrow\\rangle)\\), \\(|0,0\\rangle = \\frac{1}{\\sqrt{2}}(|\\uparrow\\downarrow\\rangle - |\\downarrow\\uparrow\\rangle)\\) — the singlet/triplet split with normalization \\(\\frac{1}{\\sqrt{2}}\\). For j₁=1/2, j₂=1: \\(|3/2,3/2\\rangle = |\\uparrow,1\\rangle\\), \\(|3/2,1/2\\rangle = \\sqrt{\\frac{2}{3}}|\\uparrow,0\\rangle + \\sqrt{\\frac{1}{3}}| Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com