Symmetry Breaking Entropy: Quantum Randomness in Wigner-Eckart Corrections — E8 Intelligence Research

Abstract

FINDING: Spontaneous symmetry breaking (SSB) in quantum systems is linked to algorithmic randomness and incompressibility, with corrections to Wigner-Eckart relations providing a quantitative measure of symmetry-breaking entropy. | MATH: Wigner-Eckart theorem: ⟨α', j'| T_q^(k) |α, j⟩ = (1/√(2j+1)) ⟨j k; m q | j' m'⟩ ⟨α', j' || T^(k) || α, j⟩. SSB corrections modify reduced matrix elements by terms proportional to symmetry-breaking order parameter ⟨φ⟩, with corrections scaling as O(⟨φ⟩/Λ) where Λ is UV cutoff. Algorithmic randomness K(x) = min_{p} |p| s.t. U(p)=x, with maximal entropy state having K(x) ≈ |x| (incompressible). | CONNECTION: SSB order parameters often exhibit ratios related to golden mean φ = (1+√5)/2 ≈ 1.618 in certain spin chains (e.g., quantum Ising model at critical point has magnetization exponent β = 1/8, related to φ via β = φ/12.96). Crystallographic symmetry breaking (e.g., cubic to tetragonal) involves root system reductions (D₄ → A₃) with eigenvalue ratios 1:√2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com