Volumetric (CUBIC) Octonion Sigma-Matrices and their Properties

الأصل · EN

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.