Wigner–Eckart Theorem: Separating Geometry from Dynamics via Projective Lorentz Generalization — E8 Intelligence Research

FINDING: Wigner–Eckart theorem separates geometric coupling (Clebsch–Gordan coefficients) from reduced matrix elements, with a projective-representation generalization for non-compact Lorentz groups. MATH: - Core theorem: \\(\\langle j' m' | T^{(k)}_q | j m \\rangle = \\langle j m; k q | j' m' \\rangle \\cdot \\frac{\\langle j' \\| T^{(k)} \\| j \\rangle}{\\sqrt{2j'+1}}\\) - Projection theorem (special case, \\(k=1\\)): \\(\\langle j m' | \\mathbf{V} | j m \\rangle = \\frac{\\langle j \\| \\mathbf{V} \\| j \\rangle}{j(j+1)} \\langle j m' | \\mathbf{J} | j m \\rangle\\) - Generalization (arXiv:1509.05633): For arbitrary Lie groups (non-compact), the theorem holds via recoupling between finite-dimensional and admissible infinite-dimensional representations — requires projective representations, i.e., cocycles \\(\\omega(g_1,g_2)\\) satisfying \\(\\omega(g_1,g_2)\\omega(g_1g_2,g_3) = \\omega(g_1,g_2g_3)\\omega(g_2,g_3)\\) (2-cocycle condition). - Magnetic point groups: projective reps arise from anti-unitary symmetr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com