Beyond Landau: Symmetry Breaking and Goldstone Corrections to Wigner-Eckart Relations — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Phase transitions may occur without hidden symmetry breaking, challenging the Landau paradigm; spontaneous symmetry breaking (SSB) modifies Wigner-Eckart relations in infinite systems via Goldstone-mode corrections. | MATH: Landau free-energy functional F[ϕ] = ∫ ddx (|∇ϕ|2 + rϕ2 + uϕ4); SSB correction to Wigner-Eckart: ⟨ j'||Tk||j⟩broken = ⟨ j'||Tk||j⟩sym + ∑n (⟨ j'||V||n⟩⟨ n||Tk||j⟩)/(Ej − En + iϵ) (Goldstone pole dominates as q→ 0); unbroken group G, broken subgroup H, coset dim = # Goldstone modes = dim(G/H). | CONNECTION: Goldstone modes correspond to generators of G/H — root system weights of Lie algebras (e.g., SU(2)→ U(1) gives ratio 1:2 in magnon dispersion); crystallographic point groups (e.g., D6 for graphene) constrain SSB patterns; no explicit golden ratios emerge, but the coset structure echoes root lattice geometry
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026beyond,
title = {{Beyond Landau: Symmetry Breaking and Goldstone Corrections to Wigner-Eckart Relations — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553800},
url = {https://zenodo.org/records/23553800},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond Landau: Symmetry Breaking and Goldstone Corrections to Wigner-Eckart Relations — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553800
UR - https://zenodo.org/records/23553800
AB - FINDING: Phase transitions may occur without hidden symmetry breaking, challenging the Landau paradigm; spontaneous symmetry breaking (SSB) modifies Wigner-Eckart relations in infinite systems via Goldstone-mode corrections. | MATH: Landau free-energy functional F[ϕ] = ∫ d^dx (|∇ϕ|^2 + rϕ^2 + uϕ^4); SSB correction to Wigner-Eckart: ⟨ j'||T^k||j⟩_broken = ⟨ j'||T^k||j⟩_sym + ∑_n (⟨ j'||V||n⟩⟨ n||T^k||j⟩)/(E_j − E_n + iϵ) (Goldstone pole dominates as q→ 0); unbroken group G, broken subgroup H, coset dim = # Goldstone modes = dim(G/H). | CONNECTION: Goldstone modes correspond to generators of G/H — root system weights of Lie algebras (e.g., SU(2)→ U(1) gives ratio 1:2 in magnon dispersion); crystallographic point groups (e.g., D_6 for graphene) constrain SSB patterns; no explicit golden ratios emerge, but the coset structure echoes root lattice geometry Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -