Hyperquaternions and physics

Abstract

The paper develops, within a new representation of Clifford algebras in terms of tensor products of quaternions called hyperquaternions, several applications. The first application is a quaternion 2D representation in contradistinction to the frequently used 3D one. The second one is a new representation of the conformal group in (1+2) space (signature $+--$) within the Dirac algebra $C_(2,3) \simeq $$\mathbb{C\otimes H\otimes H}$ subalgebra of $\mathbb{H\otimes H\otimes H}$. A numerical example and a canonical decomposition into simple planes are given. The third application is a classification of all hyperquaternion algebras into four types, providing the general formulas of the signatures and relating them to the symmetry groups of physics.