Multivector Dirac Equation and Z2-gradings of Clifford Algebras

As soon as Dirac proposed the equation bearing his name, alternative multivector formulations of it have been proposed, with either physical or mathematical motivations. In general, these approaches seek a better understanding of the geometric foundations underlying both the Dirac theory and the concept of spinor fields, besides new applications. As a distinguished example, the Hestenes’s formulation [2, 3] makes extensive use of the spacetime algebra (the real Clifford algebra C1,3(R) of spacetime) to give spinors a geometrical interpretation. In this work, we generalize certain aspects of such an approach in order to obtain multivector Dirac equations associated to a large class of representations of the gamma matrices. This is done by replacing the usual even/odd decomposition of the spacetime algebra with more general Z2gradings. Some examples are given and the chiral case, which is not addressed by the usual approach, is considered in detail. A Lagrangian formulation is briefly discussed. A relationship between this work and certain quaternionic models of (the usual) quantum mechanics is obtained. Finally, we discuss under what conditions the Hestenes’s form can be recovered and we suggest a geometrical interpretation for the corresponding situation. References