The equations of Dirac and theM2(ℍ)-representation ofCl1,3

In its original form Dirac's equations have been expressed by use of the -matrices , =0, 1, 2, 3. They are elements of the matrix algebra M 4 (). As emphasized by Hestenes several times, the -matrices are merely a (faithful) matrix representation of an orthonormal basis of the orthogonal space 1,3, generating the real Clifford algebra Cl 1,3 . This orthonormal basis is also denoted by , =0, 1, 2, 3. The use of the matrix algebra M 4 () to represent Cl 1,3 has some unsatisfactory aspects. The -matrices contain imaginary numbers as entries whereas Cl 1,3 is real. Moreover, as a matrix algebra Cl 1,3 is M 2 () but only a part of M 4 (). For that reason we investigate in this paper several forms of Dirac's equations in terms of M 2 () instead of M 4 (). 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 M 2 (). 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 Cl 1,3 . Therefore we do not give a description of strong interactions by use of M 2 (). Instead of such an approach we describe these interactions using the space of quadruples of bivector fields in Cl 1,3 . The thus obtained description has remarkable formal resemblance to the original Dirac equations using wave functions with values in the linear space 4