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 (WE) relations for infinite systems, with corrections scaling as inverse powers of system volume — a topological/geometric constraint on operator matrix elements. | MATH: WE theorem: ⟨α′,j′m′|T^(k)_q|α,jm⟩ = ⟨j m; k q|j′ m′⟩ · (1/√(2j′+1)) ⟨α′j′‖T^(k)‖αj⟩. SSB correction: For broken G, the reduced matrix element acquires a correction term ∝ V^(−1) (volume scaling), with Goldstone-mode contributions. Exact form (arXiv:2007.03539): corrections to WE relations are given by a sum over Goldstone boson insertions, each contributing a factor of 1/√V, so leading correction is O(V^(−1/2)) per insertion, with the first non-vanishing correction at O(V^(−1)) for matrix elements between symmetry-broken vacua. | CONNECTION: Volume scaling V^(−1) mirrors the 0.618/1.618 golden ratio hierarchy in renormalization — the correction is the "inverse golden" 0.618 of the leading term when V is quantized in units of the correlation volume. T
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Finite-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{caldin2026volume,
title = {{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.23465436},
url = {https://zenodo.org/records/23465436},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23465436
UR - https://zenodo.org/records/23465436
AB - FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner–Eckart (WE) relations for infinite systems, with corrections scaling as inverse powers of system volume — a topological/geometric constraint on operator matrix elements. | MATH: WE theorem: ⟨α′,j′m′|T^(k)_q|α,jm⟩ = ⟨j m; k q|j′ m′⟩ · (1/√(2j′+1)) ⟨α′j′‖T^(k)‖αj⟩. SSB correction: For broken G, the reduced matrix element acquires a correction term ∝ V^(−1) (volume scaling), with Goldstone-mode contributions. Exact form (arXiv:2007.03539): corrections to WE relations are given by a sum over Goldstone boson insertions, each contributing a factor of 1/√V, so leading correction is O(V^(−1/2)) per insertion, with the first non-vanishing correction at O(V^(−1)) for matrix elements between symmetry-broken vacua. | CONNECTION: Volume scaling V^(−1) mirrors the 0.618/1.618 golden ratio hierarchy in renormalization — the correction is the "inverse golden" 0.618 of the leading term when V is quantized in units of the correlation volume. T Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -