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) 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 and the left and right spin-Clifford bundles on the spin manifold The relation between left ideal algebraic spinor fields (LIASF) and Dirac–Hestenes spinor fields (both fields are sections of ) 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 (denoted on a Lorentzian spacetime is found. We also obtain a representation of the in the Clifford bundle It is such equation that we call the DHE and it is satisfied by Clifford fields This means that to each DHSF and spin frame there is a well-defined sum of even multivector fields (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 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.
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Research Article| June 14 2004
The bundles of algebraic and Dirac–Hestenes spinor fields
Ricardo A. Mosna;
Ricardo A. Mosna, Waldyr A. Rodrigues; The bundles of algebraic and Dirac–Hestenes spinor fields. J. Math. Phys. 1 July 2004; 45 (7): 2945–2966. https://doi.org/10.1063/1.1757038
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