Spontaneous Symmetry Breaking Alters Wigner–Eckart Relations via Volume-Scaled Goldstone Corrections — E8 Intelligence Research

FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner–Eckart (WE) relations in infinite systems via volume-scaled corrections, not just perturbative shifts. | MATH: The WE theorem factorizes matrix elements as ⟨α′j′m′|T^(k)_q|αjm⟩ = ⟨jk;mq|j′m′⟩ · (reduced element)/√(2j′+1). The arXiv result (2007.03539) proves that when G is spontaneously broken, corrections to these relations scale as **1/√V** (volume V of the system), arising from Goldstone-mode contributions. Specifically, the correction to the reduced matrix element is proportional to ⟨0|φ_G|0⟩/√V, where φ_G is the Goldstone field, and the leading correction to the WE ratio is **O(V-1/2)**, vanishing in the thermodynamic limit but finite for finite systems. | CONNECTION: The 1/√V scaling is the square root of the inverse volume — echoing the ratio **0.7071** (√1/2) if normalized to a unit cell, but more fundamentally it ties to **spherical symmetry breaking down to a subgroup H ⊂ G**, whose coset space G/H has dimension Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com