Clifford Z₂-Grading Realizes Ising Anyon Fusion Rules — E8 Intelligence Research

Abstract

FINDING: Clifford algebra's Z₂-grading (even/odd elements) underpins the fusion rules of Ising anyons, enabling non-Abelian statistics via Majorana zero modes — the Alicea lecture confirms this is a realized physical framework, not just abstract algebra. | MATH: Clifford algebra Cl(p,q) has Z₂-grading: Cl = Cl⁰ ⊕ Cl¹, where Cl⁰ (even) is a subalgebra. For Ising anyons, fusion rules: σ × σ = 1 + ψ (σ = non-Abelian anyon, ψ = fermion, 1 = vacuum); σ × ψ = σ; ψ × ψ = 1. These are the fusion rules of the Ising category, isomorphic to the even subalgebra of Cl(1,1) or the Majorana algebra {γᵢ, γⱼ} = 2δᵢⱼ. Non-Abelian statistics: braiding of σ particles yields unitary matrices U = exp(θ γᵢγⱼ/2) with θ = π/4 for exchange, giving U² = −1 (fermionic sign). | CONNECTION: The Z₂-grading mirrors the even/odd split in root systems of type Dₙ (n = 2m), where the even subalgebra corresponds to the Dₙ weight lattice mod 2 — directly linked to the Ising model's critical point (central charge c = 1/2). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com