Wigner–Eckart Theorem: Decomposing Tensor Matrix Elements via Clebsch–Gordan Coefficients — E8 Intelligence Research

FINDING: The search results are pedagogical videos on the Wigner–Eckart theorem, not the specific arXiv paper (2007.03539) on topological sector finite-size corrections. The theorem itself decomposes matrix elements of tensor operators into a reduced matrix element (geometry-independent) and Clebsch–Gordan coefficients (symmetry-determined). | MATH: ⟨j′m′| T^k_q |jm⟩ = ⟨j′‖T^k‖j⟩ · ⟨j m; k q | j′ m′⟩ / √(2j′+1). The Clebsch–Gordan coefficients arise from SU(2) recoupling; they satisfy orthogonality: Σ_{m,q} ⟨j m; k q|j′ m′⟩⟨j m; k q|j″ m″⟩ = δ_{j′j″} δ_{m′m″}. | CONNECTION: Clebsch–Gordan coefficients are intimately tied to the root system of SU(2) (A₁ Lie algebra), whose Weyl group is Z₂. The coefficients themselves are expressible in terms of 3j-symbols, which are related to the Racah–Wigner 6j-symbols — these encode the tetrahedral symmetry of SU(2) recoupling. The reduced matrix element ⟨j′‖T^k‖j⟩ is independent of magnetic quantum numbers, reflecting the rotational symmetry breaki Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com