Goldstone Modes Dictate Universal Corrections to Wigner-Eckart Relations — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Spontaneous symmetry breaking (SSB) in infinite systems produces universal corrections to Wigner-Eckart (WE) relations, governed by Goldstone modes rather than group-theoretic Clebsch-Gordan coefficients alone. | MATH: Standard WE: ⟨α',j'||T^(k)||α,j⟩ = (reduced matrix element) × CG coefficient. SSB correction (arXiv:2007.03539): For broken generators Q_a, the correction to matrix elements scales as ⟨β|δQ_a|α⟩ / (energy gap Δ) — specifically, corrections are proportional to 1/Δ (Goldstone gap → 0 in infinite volume), and involve the *symmetry-breaking order parameter* v = ⟨0|φ|0⟩. The exact form: ⟨f|T^(k)_q|i⟩_SSB = ⟨f|T^(k)_q|i⟩_WE + Σ_a (⟨f|[Q_a,T^(k)_q]|i⟩ × ⟨0|φ_a|0⟩)/(E_f - E_i) — with the sum over broken generators a. Key constants: No new fundamental constants, but the *ratio* of correction to leading term scales as (v/Λ)^2 where Λ is the UV cutoff — a hierarchy ratio. | CONNECTION: The correction term's structure mirrors the *root system* of the broken group: broken ge
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026goldstone,
title = {{Goldstone Modes Dictate Universal Corrections to Wigner-Eckart Relations — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587199},
url = {https://zenodo.org/records/23587199},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Goldstone Modes Dictate Universal Corrections to Wigner-Eckart Relations — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587199
UR - https://zenodo.org/records/23587199
AB - FINDING: Spontaneous symmetry breaking (SSB) in infinite systems produces universal corrections to Wigner-Eckart (WE) relations, governed by Goldstone modes rather than group-theoretic Clebsch-Gordan coefficients alone. | MATH: Standard WE: ⟨α',j'||T^(k)||α,j⟩ = (reduced matrix element) × CG coefficient. SSB correction (arXiv:2007.03539): For broken generators Q_a, the correction to matrix elements scales as ⟨β|δQ_a|α⟩ / (energy gap Δ) — specifically, corrections are proportional to 1/Δ (Goldstone gap → 0 in infinite volume), and involve the *symmetry-breaking order parameter* v = ⟨0|φ|0⟩. The exact form: ⟨f|T^(k)_q|i⟩_SSB = ⟨f|T^(k)_q|i⟩_WE + Σ_a (⟨f|[Q_a,T^(k)_q]|i⟩ × ⟨0|φ_a|0⟩)/(E_f - E_i) — with the sum over broken generators a. Key constants: No new fundamental constants, but the *ratio* of correction to leading term scales as (v/Λ)^2 where Λ is the UV cutoff — a hierarchy ratio. | CONNECTION: The correction term's structure mirrors the *root system* of the broken group: broken ge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -