Critical Study on Vector Matrices Realization of Hurwitz Algebras

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 nonassociative;
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.