Lattice U(1) Gauge Theory and Weyl Group B4: Bridging Discrete to Continuum — E8 Intelligence Research

FINDING: U(1) lattice gauge theory recovers continuum electromagnetism via Wilson loops; Weyl group B4 (hyperoctahedral) governs root system symmetries in Lie theory, linking discrete lattice to continuous gauge structure. | MATH: U(1) gauge group on lattice: \(S = -β ∑plaquettes Re[U_μ(x)U_ν(x+μ̂)U_μ^(x+ν̂)U_ν^(x)]\), with \(U_μ(x) = eiθ_μ(x)\), \(β = 1/g^2\). Continuum limit: \(S → 1/4g^2∫ FμνFμνd^4x\). Weyl group B4: order \(2^4 · 4! = 384\), generated by reflections \(s_i\) with Cartan matrix \(Aᵢⱼ = 2 α_i,α_j/ α_i,α_i\) for simple roots \(α_i\) (i=1..4), lengths squared ratio 1:2 (short:long). | CONNECTION: B4 root system has 64 roots — 32 short (length²=1) and 32 long (length²=2). The ratio of squared lengths is 2:1, which is the square of \(√2 ≈ 1.414\), not directly golden. However, the Weyl group order 38 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com