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.

1. Introduction

Let be an algebra over a field F not of characteristic two. The associator is a trilinear mapping

of into that measures the lack of associativity in.

One scheme of classifying nonassociative algebras involves placing conditions on the associator of certain sets of elements. Some of the better known algebras are:

1) Alternative algebras. In this variety of algebras, all elements x and y satisfy

for all elements x and y. The octonion division ring is an alternative algebra. An interesting variation is psuedooctonion algebra (Okubo [5,9]).

2) Jordan algebras. These are commutative algebras in which all x and y satisfy

A Type D Jordan algebra is the Jordan algebra of the symmetric bilinear form q on a vector space. Albert [3] has shown that any algebra of Type D has a basis with multiplication given by

The algebra will be semisimple if for all.

3) Noncommutative Jordan algebras. A generalization of the alternative and Jordan algebras that requires all x and y satisfy a generalization of the commutative law

that is, the algebras are flexible, and

The book by Zhevlakov, Slin’ko, Shestakov and Shirshov [10] provides a detailed analysis of the alternative and Jordan rings.

The above algebras are all power associative since each element a generates an associative subalgebra; equivalently, for positive integers. In any power associative algebra with unit element we can introduce the series

for ignoring the question of convergence.

An algebra over a field F is called quadratic if, for every x in

where are in F and e is the identity of. The quantities and are called the trace and norm of the element x, respectively. The trace is a linear functional on see Schafer [7]. The norm defines a symmetric bilinear form on via

Say is nondegenerate if is. Any quadratic algebra is power associative and any flexible, quadratic algebra is a noncommutative Jordan algebra.

A quadratic algebra is flexible if and only if the trace is associative; that is, for all in. If is flexible then the mapping is an involution in (see Braun and Koecher [11], p. 216).

Lemma 1. The Hamiltonian division ring is a quadratic algebra.

Proof. Let be an element of the Hamitonian division ring. Direct computation shows that

.

Example 1. The octonion division ring is a quadratic algebras.

Example 2. 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 also Wene [13,14], and Schafer [15]).

2. Construction of the Algebras

The elements of the algebra are the elements of the real vector space with basis. The addition is the vector space addition and multiplication is defined by, , is the identity and the distributive laws. We note that the algebra is commutative and has divisors of zero.

An immediate generalization of this algebra has a basis, over the field of real numbers and multiplication defined by where is the identity. For want of a better name called these the Abbas algebras. As noted above, these algebras are Type D Jordan algebras. Note that the algebra is the construction for; the results for the Abbas algebras apply to the. Each Abbas algebra contains a copy of the complex numbers.

Lemma 2. The Abbas algebras are quadratic algebras.

Proof. Let H denote a Abbas algebra. Then if, , Einstein summation convention where Then

Adding both sides gives

and we see that and.

A commutative quadratic algebra will be a Jordan algebra.

Since the algebra is commutative the trace is associative; the norm is symmetric.

Lemma 3. The norm of a Abbas algebra is nondegenerate.

Proof. Let H denote a Abbas algebra. Then if, is arbitrary and is fixed, then

3. Special Algebras

Hamieh and Abbas [1] pass to a representation of the point of the algebra in spherical coordinates, and . The subalgebras, called special algebras and denoted by are the subalgebras spanned by all elements in which the “azimutal phase angle is constant”. Each of these subalgebras is (isomorphic to) the complex numbers.

Lemma 4. The algebra is isomorphic to an algebra of two by two matrices

under the usual matrix operation of addition and multiplication.

Proof. The straight forward verification that the mapping is an isomorphism is left to the reader.

Lemma 5. Each of the algebras is isomorphic to the complex numbers.

Proof. We note that if then

if. If or, then and the subalgebra is (isomorphic to) the complex numbers. Otherwise, or for some. Let, then

The multiplication, using the basis will be given by

4. The Spinor Matrices

The classical reference on spinors and wave equations is the book by Corson [16].

The associator spinor matrices are

, , ,

where. Denoting the 2 by 2 identity matrix by, these matrices satisfy

The spinor matrices generate a 6-dimensional real algebra with elements

that contains the matrix representation of the algebra. Denote this algebra by.

Lemma 6. The algebra is a quadratic algebra.

Proof. If is an element of, then

Adding the left and right sides gives

Lemma 7. The algebra is flexible.

Proof. Because of the trilinearlty of the associator, we can write the elements x and y of the associator

as and. Then

Theorem 1. The algebra is a quadratic noncommutative Jordan algebra.

5. The Dirac Equation

We proceed as in Hamieh and Abbas [1]. The Dirac equation over the complex numbers is often written as

utilizing the Einstein summation convention for . A more general form is, setting

where.

Upon substituting the matrices for and simplifying we get

In dimensions x and t, the solution is given by

p and are respect the momentum and energy. N is a normalization factor.

6. Conclusion

We have shown that the algebra belongs to a large class of Jordan algebras and have examined a few of the algebraic properties of these algebras and, like the Jordan algebra and the algebra of color, there is a very rich mathematical structure to further explore.

REFERENCES

NOTES

^*This paper is in final form and no version of it will be submitted for publication elsewhere.