Spontaneous Symmetry Breaking Alters Wigner–Eckart Relations via Volume-Scaled Goldstone Corrections — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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
Related
- Beyond Landau: Symmetry Breaking and Goldstone Corrections to Wigner-Eckart Relations — E8 Intelligence Research
- Finite-Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research
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- Goldstone Modes Dictate Universal Corrections to Wigner-Eckart Relations — E8 Intelligence Research
- Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume Corrections — E8 Intelligence Research
BibTeX
@misc{caldin2026spontaneous,
title = {{Spontaneous Symmetry Breaking Alters Wigner–Eckart Relations via Volume-Scaled Goldstone Corrections — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490794},
url = {https://zenodo.org/records/23490794},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spontaneous Symmetry Breaking Alters Wigner–Eckart Relations via Volume-Scaled Goldstone Corrections — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490794
UR - https://zenodo.org/records/23490794
AB - 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
ER -