Vector Matrices Realization of Hurwitz Algebras

TL;DR

This paper introduces a matrix-based realization of Hurwitz algebras that preserves geometric and algebraic structures relevant to physics, extending classical and quantum frameworks with associative and alternative algebra properties.

Contribution

It presents a novel matrix realization of Hurwitz algebras that maintains key algebraic properties and links to physical theories, extending to higher dimensions.

Findings

Real and complex numbers are commutative and associative in this realization.

Real quaternions maintain associativity.

Real octonions form an alternative algebra.

Abstract

We present the realization of Hurwitz algebras in terms of 2x2 vector matrices, which maintain the correspondence between the geometry of the vector spaces used in the classical physics and the underlined algebraic foundation of the quantum theory. The multiplication rule used is modification of the one originally introduced by M.Zorn. We demonstrate that our multiplication is not intrinsically non-associative; the realization of the real and complex numbers is commutative and associative, the real quaternions maintain associativity and the real octonion matrices form an alternative algebra. The extension to the calculus of the matrices (with Hurwitz algebra valued matrix elements) of the arbitrary dimensions is straightforward. We discuss briefly the applications of the obtained results to the extensions of the standard Hilbert space formulation of the quantum physics and to the alternative wave mechanical formulation of the classical field theory.