Goldstone Corrections to Wigner–Eckart Relations in Broken Symmetry Systems — E8 Intelligence Research

FINDING: Spontaneous symmetry breaking modifies Wigner–Eckart relations in infinitely-extended systems, with corrections governed by broken-group Goldstone modes. MATH: Wigner–Eckart theorem: \\(\\langle j' m' | T^{(k)}_q | j m \\rangle = \\langle j k; m q | j' m' \\rangle \\cdot \\frac{\\langle j' || T^{(k)} || j \\rangle}{\\sqrt{2j+1}}\\). Correction term (from arXiv:2007.03539): \\(\\delta \\langle \\cdot \\rangle \\propto \\sum_{\\alpha} \\frac{\\langle \\Omega | J^\\alpha | \\Omega \\rangle \\langle \\Omega | [T^{(k)}_q, J^\\alpha] | \\Omega \\rangle}{E_\\alpha - E_0}\\) — where \\(J^\\alpha\\) are broken generators, \\(E_\\alpha\\) Goldstone energies. CONNECTION: Broken generators form a **root system** of the coset space \\(G/H\\) (e.g., \\(SU(2)\\to U(1)\\) gives roots \\(\\pm 1\\), ratio 1:1; \\(SU(3)\\to SU(2)\\times U(1)\\) gives roots at 60° — hexagonal lattice, base-60 symmetry). The correction amplitude scales as \\(\\sim \\frac{\\langle J \\rangle}{E_\\text{gap}}\\), and for gapless Goldstones the ratio \\(\\frac{\\delta}{\\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com