The equations of Dirac and theM 2(?)-representation ofCl 1,3

Abstract

In its original form Dirac's equations have been expressed by use of the γ-matrices γμ, μ=0, 1, 2, 3. They are elements of the matrix algebra M4(ℂ). As emphasized by Hestenes several times, the γ-matrices are merely a (faithful) matrix representation of an orthonormal basis of the orthogonal spaceℝ1,3, generating the real Clifford algebra Cl1,3. This orthonormal basis is also denoted by γμ, μ=0, 1, 2, 3. The use of the matrix algebra M4(ℂ) to represent Cl1,3 has some unsatisfactory aspects. The γ-matrices contain imaginary numbers as entries whereas Cl1,3 is real. Moreover, as a matrix algebra Cl1,3 is M2(ℍ) but only a part of M4(ℂ). For that reason we investigate in this paper several forms of Dirac's equations in terms of M2(ℍ) instead of M4(ℂ). In Section1 we survey Dirac's equations describing the interaction of matter with electromagnetic, electroweak, and strong fields. Section2 deals with electromagnetic/weak interactions employing M2(ℍ). Finally, in Section3 we deal with Dirac's equations for strong interactions between quarks. In contrast to su(2) ⊕ u(1), the Lie algebra su(3) is not isomorphic to any subalgebra of Cl1,3. Therefore we do not give a description of strong interactions by use of M2(ℍ). Instead of such an approach we describe these interactions using the space of quadruples of bivector fields in Cl1,3. The thus obtained description has remarkable formal resemblance to the original Dirac equations using wave functions with values in the linear spaceℂ4.