The bundles of algebraic and Dirac–Hestenes spinor fields

Abstract

Our main objective in this paper is to clarify the ontology of Dirac–Hestenes spinor fields (DHSF) and its relationship with even multivector fields, on a Riemann–Cartan spacetime (RCST) M=(M,g,∇,τg,↑) admitting a spin structure, and to give a mathematically rigorous derivation of the so-called Dirac–Hestenes equation (DHE) in the case where 𝔐 is a Lorentzian spacetime (the general case when 𝔐 is a RCST will be discussed in another publication). To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (ClSpin1,3el(M)) and right (ClSpin1,3er(M)) spin-Clifford bundles on the spin manifold (M,g). The relation between left ideal algebraic spinor fields (LIASF) and Dirac–Hestenes spinor fields (both fields are sections of ClSpin1,3el(M)) is clarified. We study in detail the theory of covariant derivatives of Clifford fields as well as that of left and right spin-Clifford fields. A consistent Dirac equation for a DHSF Ψ∈sec ClSpin1,3el(M) (denoted DECll) on a Lorentzian spacetime is found. We also obtain a representation of the DECll in the Clifford bundle Cl(M,g). It is such equation that we call the DHE and it is satisfied by Clifford fields ψⅪ∈sec Cl(M,g). This means that to each DHSF Ψ∈sec ClSpin1,3el(M) and spin frame Ⅺ∈sec PSpin1,3e(M), there is a well-defined sum of even multivector fields ψⅪ∈sec Cl(M,g) (EMFS) associated with Ψ. Such an EMFS is called a representative of the DHSF on the given spin frame. And, of course, such a EMFS (the representative of the DHSF) is not a spinor field. With this crucial distinction between a DHSF and its representatives on the Clifford bundle, we provide a consistent theory for the covariant derivatives of Clifford and spinor fields of all kinds. We emphasize that the DECll and the DHE, although related, are equations of different mathematical natures. We study also the local Lorentz invariance and the electromagnetic gauge invariance and show that only for the DHE such transformations are of the same mathematical nature, thus suggesting a possible link between them.