Two Dimensional Representation of the Dirac Equation in Non-Associative Algebra

Abstract

In this note a simple extension of the complex algebra to higher dimension is proposed. Using the proposed algebra a two dimensional Dirac equation is formulated and its solution is calculated. It is found that there is a sub-algebra where the associative nature can be recovered.

On the Extended Lorentz Transformation Model and Its Application to Superluminal Neutrinos

“…For example when calculating cross-section for some physical process in standard model one has to take into account for SP the Application of this formalism to interacting fields and calculation of cross section for some physical process will be left for future investigation. In next section, as we are interested in new physics that might govern the dynamics of SP, we take this opportunity to see the smallest representation of the new DE in the newly proposed algebra [13]. This algebra is called the Generalized-C (GC) extend quantum theory to new class of theories based on the non associative algebra.…”

“…The model will be applied to superluminal neutrino to obtain an estimation of neutrino mass. Moreover, as our main concern here is to probe new physics, we have considered the dynamics of superluminal particles not only within the framework of quantum mechanics but also in the generalized quantum mechanics [13].…”

Algebras of Hamieh and Abbas Used in the Dirac Equation

“…Hamieh and Abbas [1] pass to a representation of the point    G of the algebra in spherical coordinates, …”

“…We construct a large family of quadratic Jordan algebras that contains the three-dimensional real algebra, the so called G algebra, the generalized complex numbers, of Hamieh and Abbas [1], and show that the spinor matrices that arises from using the in a formulation of the Dirac equation generate a six-dimensional noncommutative quadratic Jordan algebra.…”

“…Hamieh and Abbas [1] present a "description of an algebra which can be used in a possible extension of local complex quantum field theories". We further expand their description and show that these algebras are Type D Jordan algebras (see Jacobson [8]).We construct a large family of quadratic Jordan algebras that contains the three-dimensional real algebra, the so called G algebra, the generalized complex numbers, of Hamieh and Abbas [1], and show that the spinor matrices that arises from using the in a formulation of the Dirac equation generate a six-dimensional noncommutative quadratic Jordan algebra. …”