Volumetric (CUBIC) Octonion Sigma-Matrices and their Properties

Abstract

This work rests upon the certainty that only fields of real and complex numbers, quaternions and octonions have algebras of all four arithmetical operations. Also quaternions are good to represent 3-dimensional Euclid space and 4-dimensional Minkowski space. Moreover algebra of Pauli classical sigma-matrices isomorphic to a split quaternion algebra. In this article author supposes that the hyperspace can be described with octonions and if 4-dimensional Minkowski space is a projection of the hyperspace, consequently equations for 4-dimensional space must be projections of hyperspace equations. Therefore he obtains a new object as volumetric analog of Pauli sigma-matrices for 8 (or maybe 24)-dimensional hyperspace. Also the author represents general form of a volumetric Dirac equation solution. As is well known nonassociativity is the main distinctive feature of octonions. But square matrices cannot be nonassociative. Therefore the author had to introduce cubic sigma-matrices. From another hand a volumetric object has 3 indices, as a result of it one cannot introduce multiplicative operator for cubic matrices, their projections only can be objects of the multiplicative operator. Meanwhile an uncertainty appears because of a rest index. To overcome the uncertainty one need to consider all the 3 projections of an octonion equation.