The 2-Sphere of Imaginary Units: Quaternions, Pauli Matrices, and Rolling SU(2) — E8 Intelligence Research

Abstract

FINDING: Quaternions, Pauli matrices, and SU(2) form a unified algebraic structure where the square root of −1 is not unique but a 2-sphere of solutions, and Pfaffian systems on SU(2) encode rolling distributions with constant curvature. MATH: - Quaternion square roots of −1: any unit pure quaternion \( q = ai + bj + ck \), \( a^2+b^2+c^2=1 \), satisfies \( q^2 = -1 \). This is the 2-sphere \( S^2 \subset \mathbb{H} \). - Pauli matrices \( \sigma_1, \sigma_2, \sigma_3 \) satisfy \( \sigma_i \sigma_j = \delta_{ij} I + i \epsilon_{ijk} \sigma_k \), generating \( \mathfrak{su}(2) \). The map \( i \mapsto i\sigma_1, j \mapsto i\sigma_2, k \mapsto i\sigma_3 \) gives an isomorphism \( \mathbb{H} \cong \mathbb{R} \oplus \mathfrak{su}(2) \). - Pfaffian systems: For a \((2,3,5)\)-distribution, the Cartan–Nurowski conformal structure is encoded by a Pfaffian form \( \theta \) with \( d\theta \) restricted to the distribution. The SU(2) Pfaffian system arises from rolling two surfaces of Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com