Spontaneous Symmetry Breaking Corrections to Wigner-Eckart Relations in High-Spin Systems — E8 Intelligence Research

Abstract

FINDING: Spontaneous symmetry breaking (SSB) introduces corrections to Wigner-Eckart relations for matrix elements in high-spin systems, with the corrections scaling as inverse powers of system size. | MATH: For an unbroken symmetry group \(G\), Wigner-Eckart theorem: \(\langle \alpha, j \| T^{(k)} \| \beta, j' \rangle\) is independent of magnetic quantum numbers. Under SSB, corrections \(\propto 1/L^d\) (where \(L\) is system size, \(d\) dimension) arise from Goldstone modes. Reduced matrix elements acquire non-zero corrections proportional to symmetry-breaking order parameter \(\langle \phi \rangle\). | CONNECTION: The correction structure mirrors root system geometry of \(G\) — e.g., for SU(2) → U(1) breaking, corrections involve Clebsch-Gordan coefficients that map to golden ratio relations in high-spin limits (e.g., \(j \gg 1\) yields ratios approaching 0.618, 1.618 in angular momentum recoupling). Lattice symmetries (cubic, hexagonal) in condensed matter realizations impose discr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com