Spontaneous Symmetry Breaking Alters Wigner–Eckart Relations via Goldstone Modes — E8 Intelligence Research

FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner–Eckart (WE) relations in infinite systems, with corrections governed by Goldstone mode structure rather than vanishing in the thermodynamic limit. | MATH: WE theorem: ⟨α′,j′m′|T^(k)_q|α,jm⟩ = ⟨j m; k q|j′m′⟩ · ⟨α′j′‖T^(k)‖αj⟩/√(2j′+1). SSB correction: For broken G, the reduced matrix element acquires a correction term proportional to ⟨0|[Q_a, T^(k)_q]|0⟩/E_Goldstone, where Q_a is the broken charge generator and E_Goldstone → 0 as volume → ∞, but the ratio ⟨0|[Q_a,T^(k)_q]|0⟩/E_Goldstone remains finite — yielding a non-vanishing modification to the WE ratio. The correction scales as 1/√V in intermediate steps but cancels in physical matrix elements via Goldstone pole dominance (arXiv:2007.03539). | CONNECTION: The correction structure mirrors root system geometry: broken generators Q_a correspond to positive roots of G/H coset space; the Goldstone modes are the crystallographic simple roots. The ratio of WE correction to unbro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com