Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume Corrections — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner-Eckart reduced matrix elements with inverse-volume scaling corrections in infinite systems. | MATH: Wigner-Eckart theorem: ⟨ j' m' | Tkq | j m ⟩ = ⟨ j k; m q | j' m' ⟩ · ⟨ j' || Tk || j ⟩ / √(2j'+1). SSB correction: ⟨ j' || Tk || j ⟩broken = ⟨ j' || Tk || j ⟩symmetric + O(1/V), where V is system volume. The correction term is proportional to V−1 times a Goldstone-mode overlap integral, scaling as ∼ ∫ ddx ⟨ Ω | [Qa, Tkq] | Ω ⟩ / V. | CONNECTION: The 1/V scaling is the topological inverse-volume factor — in 3D, V ∼ L3, so corrections vanish as L−3, a cubic lattice harmonic. The Goldstone bosons are the zero-momentum modes; their count equals the dimension of the coset space G/H, whose rank relates to root-system degeneracies (e.g., SU(2)→ U(1) gives 2
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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BibTeX
@misc{caldin2026spontaneous,
title = {{Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume Corrections — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490659},
url = {https://zenodo.org/records/23490659},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume Corrections — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490659
UR - https://zenodo.org/records/23490659
AB - FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner-Eckart reduced matrix elements with inverse-volume scaling corrections in infinite systems. | MATH: Wigner-Eckart theorem: ⟨ j' m' | T^k_q | j m ⟩ = ⟨ j k; m q | j' m' ⟩ · ⟨ j' || T^k || j ⟩ / √(2j'+1). SSB correction: ⟨ j' || T^k || j ⟩_broken = ⟨ j' || T^k || j ⟩_symmetric + O(1/V), where V is system volume. The correction term is proportional to V^−1 times a Goldstone-mode overlap integral, scaling as ∼ ∫ d^dx ⟨ Ω | [Q_a, T^k_q] | Ω ⟩ / V. | CONNECTION: The 1/V scaling is the topological inverse-volume factor — in 3D, V ∼ L^3, so corrections vanish as L^−3, a cubic lattice harmonic. The Goldstone bosons are the zero-momentum modes; their count equals the dimension of the coset space G/H, whose rank relates to root-system degeneracies (e.g., SU(2)→ U(1) gives 2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -