Wigner–Eckart Theorem Generalized to Non-Compact Lie Groups via Recoupling — E8 Intelligence Research

FINDING: Generalization of Wigner–Eckart theorem to non-compact Lie groups (e.g., Lorentz group) via recoupling theory and Jordan–Schwinger representation, enabling tensor operator decomposition in infinite-dimensional representations. | MATH: Wigner–Eckart theorem for compact groups: \(\langle j' m' | T^{(k)}_q | j m \rangle = \langle j k; m q | j' m' \rangle \frac{\langle j' \| T^{(k)} \| j \rangle}{\sqrt{2j+1}}\). Generalization: replaces Clebsch–Gordan coefficients with recoupling coefficients for admissible representations of non-compact groups; uses Jordan–Schwinger map to construct infinite-dimensional irreps. | CONNECTION: No direct geometric ratios (0.382, 0.618, etc.) or base-60 found. However, recoupling theory links to root systems and Weyl group symmetries of Lie algebras (e.g., \(A_1\) for SU(2), \(A_1 \times A_1\) for Lorentz). The Lorentz group's double cover SL(2,ℂ) has root system \(A_1\), with crystallographic symmetry in its weight lattice. | DEPTH: 7 — Extends a fo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com