Finite-Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner–Eckart (W-E) relations in infinite systems via finite-volume corrections, with Goldstone modes mediating the deviation. | MATH: W-E theorem: ⟨ j' m' | T(k)q | j m ⟩ = 1/(√(2j+1)) ⟨ j' || T(k) || j ⟩ ⟨ j m; k q | j' m' ⟩ (Clebsch-Gordan coefficient). SSB correction: For unbroken group G → H, the leading correction to W-E is **not** a constant reduced matrix element but acquires a **Goldstone pole** — scaling as 1/(p2) in momentum space, giving finite-volume correction ∝ 1/L2 (where L is system size) for the ratio of matrix elements. The paper (arXiv:2007.03539) proves corrections are **non-analytic** in symmetry-breaking parameter, proportional to ⟨ π | Jμ | 0 ⟩ / fπ (Goldstone decay constant). | CONNECTION: The Clebsch-Gordan coefficients are intimately tied to SU(2) root systems — the same algebraic structure as cry
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research
- Spontaneous Symmetry Breaking Alters Wigner–Eckart Relations via Volume-Scaled Goldstone Corrections — E8 Intelligence Research
- Beyond Landau: Symmetry Breaking and Goldstone Corrections to Wigner-Eckart Relations — E8 Intelligence Research
- Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume Corrections — E8 Intelligence Research
- Spontaneous Symmetry Breaking Alters Wigner-Eckart Reduced Matrix Elements with Volume-Dependent Corrections — E8 Intelligence Research
BibTeX
@misc{caldin2026finitevolume,
title = {{Finite-Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658383},
url = {https://zenodo.org/records/23658383},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658383
UR - https://zenodo.org/records/23658383
AB - FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner–Eckart (W-E) relations in infinite systems via finite-volume corrections, with Goldstone modes mediating the deviation. | MATH: W-E theorem: ⟨ j' m' | T^(k)_q | j m ⟩ = 1/(√(2j+1)) ⟨ j' || T^(k) || j ⟩ ⟨ j m; k q | j' m' ⟩ (Clebsch-Gordan coefficient). SSB correction: For unbroken group G → H, the leading correction to W-E is **not** a constant reduced matrix element but acquires a **Goldstone pole** — scaling as 1/(p^2) in momentum space, giving finite-volume correction ∝ 1/L^2 (where L is system size) for the ratio of matrix elements. The paper (arXiv:2007.03539) proves corrections are **non-analytic** in symmetry-breaking parameter, proportional to ⟨ π | J^μ | 0 ⟩ / f_π (Goldstone decay constant). | CONNECTION: The Clebsch-Gordan coefficients are intimately tied to SU(2) root systems — the same algebraic structure as cry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -