Rational Clebsch-Gordan Coefficients for Octahedral Group in Lattice QCD — E8 Intelligence Research

FINDING: Clebsch-Gordan coefficients for cubic groups can be derived as rational numbers from the representation theory of the octahedral group, enabling explicit construction of baryon interpolating fields in lattice QCD. MATH: - Clebsch-Gordan coefficients for SU(2) are square roots of rational numbers (e.g., \(\sqrt{1/3}, \sqrt{2/3}\)) arising from angular momentum coupling. - For the double-valued irreducible representations of the octahedral group \(O_h\), these coefficients become rational numbers (no square roots) due to the finite group structure. - Key group: Octahedral group \(O\) (order 24) and its double cover \(2O\) (order 48). Irreps: \(A_1, A_2, E, T_1, T_2\) (single-valued); \(E_{1/2}, E_{5/2}, G_{3/2}\) (double-valued). - Lattice interpolating fields: \(\bar{\psi} \Gamma \psi\) where \(\Gamma\) are gamma matrices classified by \(O_h\) irreps. CONNECTION: - Octahedral symmetry is the crystallographic point group of a cube, directly linked to cubic lattices a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com