Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume-Dependent Corrections — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner-Eckart reduced matrix elements in infinitely-extended systems, with corrections scaling as inverse volume — a topological/geometric constraint on operator matrix elements. | MATH: Wigner-Eckart theorem: ⟨α′,j′m′|T^k_q|α,jm⟩ = (1/√(2j+1))⟨j m k q|j′m′⟩ ⟨α′j′||T^k||αj⟩. SSB correction: ⟨α′j′||T^k||αj⟩_broken = ⟨α′j′||T^k||αj⟩_unbroken + O(1/V), where V is system volume. The correction term is not simply proportional to the identity on the degenerate ground-state manifold but carries a geometric phase (Berry-like) connection. | CONNECTION: The 1/V scaling is the inverse of the volume — in d dimensions, V ~ L^d, so corrections scale as L^{-d}. This is the same power-law structure as Goldstone mode normalization (∝ 1/√V per mode). The degeneracy manifold of SSB vacua is a coset space G/H — for continuous groups this is a symmetric space (e.g., S^2 for SU(2)→U(1), CP^n for SU(n+1)→U(n)), whose curvature encodes the correction. No d
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026spontaneous,
title = {{Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume-Dependent Corrections — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23526351},
url = {https://zenodo.org/records/23526351},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume-Dependent Corrections — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23526351
UR - https://zenodo.org/records/23526351
AB - FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner-Eckart reduced matrix elements in infinitely-extended systems, with corrections scaling as inverse volume — a topological/geometric constraint on operator matrix elements. | MATH: Wigner-Eckart theorem: ⟨α′,j′m′|T^k_q|α,jm⟩ = (1/√(2j+1))⟨j m k q|j′m′⟩ ⟨α′j′||T^k||αj⟩. SSB correction: ⟨α′j′||T^k||αj⟩_broken = ⟨α′j′||T^k||αj⟩_unbroken + O(1/V), where V is system volume. The correction term is not simply proportional to the identity on the degenerate ground-state manifold but carries a geometric phase (Berry-like) connection. | CONNECTION: The 1/V scaling is the inverse of the volume — in d dimensions, V ~ L^d, so corrections scale as L^{-d}. This is the same power-law structure as Goldstone mode normalization (∝ 1/√V per mode). The degeneracy manifold of SSB vacua is a coset space G/H — for continuous groups this is a symmetric space (e.g., S^2 for SU(2)→U(1), CP^n for SU(n+1)→U(n)), whose curvature encodes the correction. No d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -