The bundles of algebraic and Dirac-Hestenes spinor fields

The main objective of 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) $\mathfrak{M}=$($M,g,\nabla,\tau_{g},\uparrow$) admitting a spin structure, and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) in the case where $\mathfrak{M}$ is a Lorentzian spacetime (the general case when $\mathfrak{M}$ is a RCST will be discussed in another publication). To this aim we introduce the Clifford bundle of multivector fields ($\mathcal{C\ell}(M,g)$) and the left ($C\ell_{\mathrm{Spin}_{1,3}^{e}}^{l}(M)$) and right $\mathcal{C \ell}_{\mathrm{Spin}_{1, 3}^{e}}^{r} (M)$) 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 $\mathcal{C \ell}_{\mathrm{Spin}_{1, 3}^{e}}^{l}(M)$) is clarified. We study in DHSF $\mathbf{\Psi}\in\sec\mathcal{C\ell}_{\mathrm{Spin}_{1,3}^{e}}^{l}(M)$ (denoted DE $\mathcal{C\ell}^{l})$ on a Lorentzian spacetime is found. We also obtain a representation of the DE $\mathcal{C\ell}^{l}$ in the Clifford bundle $\mathcal{C\ell}(M,g)$. It is such equation that we call the DHE and it is satisfied by Clifford fields $\mathit{\psi}_{\Xi}\in\sec\mathcal{C\ell}(M,g)$. This means that to each DHSF $\mathbf{\Psi}\in\sec$ $\mathcal{C\ell}_{\mathrm{Spin}_{1,3}^{e}}^{l}(M)$ and $\Xi\in\sec P_{\mathrm {Spin}_{1, 3}^{e}}(M)$, there is a well defined sum of even multivector fields $\mathit{\psi}_{\Xi}\in \sec \mathcal {C\ell} (M,g)$ (EMFS) associated with $\Psi$. Such a 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 andspinor fields of all kinds. We emphasize that the DE $\mathcal{C\ell}^{l}$ 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.