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 \(j_1 \otimes j_2 = \oplus_j (j)\), the coefficients \(\langle j_1 m_1 j_2 m_2 | J M\rangle\) satisfy orthogonality: \(\sum_{m_1,m_2} \langle j_1 m_1 j_2 m_2 | J M\rangle \langle j_1 m_1 j_2 m_2 | J' M'\rangle = \delta_{JJ'}\delta_{MM'}\). Recursion via ladder operators: \(\sqrt{(J\mp M)(J\pm M+1)}\langle j_1 m_1 j_2 m_2 | J, M\pm 1\rangle = \sqrt{(j_1\pm m_1)(j_1\mp m_1+1)}\langle j_1, m_1\mp 1, j_2 m_2 | J M\rangle + \sqrt{(j_2\pm m_2)(j_2\mp m_2+1)}\langle j_1 m_1, j_2, m_2\mp 1 | J M\rangle\). Racah formula gives closed form as sum over integers \(z\): \(\langle j_1 m_1 j_2 m_2 | J M\rangle = \delta_{M,m_1+m_2} \sqrt{(2J+1)\frac{(J+j_1-j_2)!(J-j_1+j_2)!(j_1+j_2-J)!}{(J+j_1+j_2+1)!}} \times \sqrt{(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