Abstract
Hamieh and Abbas [1] propose using a 3-dimensional real algebra in a solution of the Dirac equation. We show that this algebra, denoted by , belongs to a large class of quadratic Jordan algebras with subalgebras isomorphic to the complex numbers and that the spinor matrices associated with the solution of the Dirac equation generate a six-dimensional real noncommutative Jordan algebra.
Highlights
IntroductionSome of the better known algebras are: 1) Alternative algebras
Let A be an algebra over a field F not of characteristic two
Hamieh and Abbas [1] propose using a 3-dimensional real algebra in a solution of the Dirac equation. We show that this algebra, denoted by G, belongs to a large class of quadratic Jordan algebras with subalgebras isomorphic to the complex numbers and that the spinor matrices associated with the solution of the Dirac equation generate a six-dimensional real noncommutative Jordan algebra
Summary
Some of the better known algebras are: 1) Alternative algebras. In this variety of algebras, all elements x and y satisfy x, x, y x, y, y 0 for all elements x and y. The octonion division ring is an alternative algebra. These are commutative algebras in which all x and y satisfy x, y, x2 0. The book by Zhevlakov, Slin’ko, Shestakov and Shirshov [10] provides a detailed analysis of the alternative and Jordan rings. In any power associative algebra A with unit element we can introduce the series ex. Any quadratic algebra is power associative and any flexible, quadratic algebra is a noncommutative Jordan algebra. The octonion division ring is a quadratic algebras. Domokos and Kövesi-Domokos [12] propose a quadratic algebra, the “algebra of color” as a candidate for the algebra obeyed by a quantized field describing quarks and leptons (see Wene [13,14], and Schafer [15])
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