Clebsch–Gordan Coefficients as SU(2) Tensor Product Unitary Transformations — E8 Intelligence Research

FINDING: Clebsch–Gordan coefficients are the unitary transformation matrix elements between uncoupled and coupled angular momentum bases, directly encoding SU(2) tensor product decomposition into irreducible representations. | MATH: For \(j_1 ⊗ j_2 = ⊕_j (j)\), the coefficients \( j_1 m_1 j_2 m_2 | J M\) satisfy orthogonality: \(∑m_1,m_2 j_1 m_1 j_2 m_2 | J M j_1 m_1 j_2 m_2 | J' M' = δJJ'δMM'\). Recursion via ladder operators: \(√(J∓ M)(J± M+1) j_1 m_1 j_2 m_2 | J, M± 1 = √(j_1± m_1)(j_1∓ m_1+1) j_1, m_1∓ 1, j_2 m_2 | J M + √(j_2± m_2)(j_2∓ m_2+1) j_1 m_1, j_2, m_2∓ 1 | J M\). Racah formula gives closed form as sum over integers \(z\): \( j_1 m_1 j_2 m_2 | J M = δM,m_1+m_2 √(2J+1)(J+j_1-j_2)!(J-j_1+j_2)!(j_1+j_2-J)!/(J+j_1+j_2+1)! × √(J+M)!(J-M)!(j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)! Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com