Wigner-Eckart Theorem: Separating Geometry from Physics in Tensor Operators — E8 Intelligence Research

FINDING: The Wigner-Eckart theorem separates matrix elements of tensor operators into a geometric (Clebsch-Gordan) part and a physical (reduced matrix element) part, with applications extending to finite magnetic groups. | MATH: ⟨α′j′m′|T^k_q|αjm⟩ = ⟨j m; k q|j′ m′⟩ × (1/√(2j′+1)) ⟨α′j′‖T^k‖αj⟩ — the reduced matrix element ⟨α′j′‖T^k‖αj⟩ is independent of m, m′, q; Clebsch-Gordan coefficients ⟨j m; k q|j′ m′⟩ carry all magnetic quantum number dependence; for finite magnetic groups, the theorem generalizes via irreducible representations of the magnetic point group (including time-reversal symmetry). | CONNECTION: Clebsch-Gordan coefficients are directly related to 3-j symbols and Racah coefficients, which encode the SU(2) root system — the same SU(2) structure underlies the golden ratio φ = (1+√5)/2 in the Fibonacci-like degeneracy sequences of angular momentum addition (e.g., j ⊗ j decompositions yield multiplicities following Fibonacci numbers for large j, with ratios approaching φ). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com