The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) when spacetime is a Lorentzian. To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (Cl_{ Spin_{1,3} ^{e}}^{l}(M)) and right (Cl_{ 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 Cl_{ Spin_{1,3} ^{e}}^{l}(M)) is clarified. We study in details the theory of the covariant derivatives of Clifford and left and right spin-Clifford fields. Moreover, we find a consistent Dirac equation for a DHSF (denoted DECl^{l}) on a Lorentzian spacetime. We succeeded also in obtaining a representation of the DECl^{l} in the Clifford bundle Cl(M,g). It is such equation that we call the DHE and it is satisfied by even multivector fields (EMFS). Of course, such a EMFS is not a spinor field. We provide moreover a consistent theory for the covariant derivatives of Clifford and spinor fields of all kinds. We emphasize that the DECl^{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.
英文摘要：
The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) when spacetime is a Lorentzian. To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (Cl_{Spin_{1,3}^{e}}^{l}(M)) and right (Cl_{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 Cl_{Spin_{1,3}^{e}}^{l}(M)) is clarified. We study in details the theory of the covariant derivatives of Clifford and left and right spin-Clifford fields. Moreover, we find a consistent Dirac equation for a DHSF (denoted DECl^{l}) on a Lorentzian spacetime.
We succeeded also in obtaining a representation of the DECl^{l} in the Clifford bundle Cl(M,g). It is such equation that we call the DHE and it is satisfied by even multivector fields (EMFS). Of course, such a EMFS is not a spinor field. We provide moreover a consistent theory for the covariant derivatives of Clifford and spinor fields of all kinds. We emphasize that the DECl^{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.
The Lie derivative of spinor fields: theory and applications
本文从任意可微映射的李导数的一般概念出发，在主丛Q上的约化G-结构P的框架下，发展了一个系统的李微分理论。证明了这些结构允许拉回向量丛i_P^*（TQ）= P \times_Q TQ在P上的一个正则分解.对于经典的G-结构，即线性标架丛的约化G-子丛，这样的一个分解定义了一个无穷小的正则提升.这一升降机延伸到一个延长伽玛结构P.在这个一般的几何框架的旋量领域的李衍生物的概念进行审查。在专门的情况下，科斯曼电梯，我们恢复科斯曼的原始定义。我们还表明，在一个还原的G-结构的情况下，可以引入一个“还原李衍生物”关于某一类广义无穷小自同构，并作为一个有趣的副产品，证明了一个结果，由于Bourguignon和Gauduchon在一个更一般的方式。其次，我们给出了Killing方程的一个新的刻画和推广，并对“旋量场”的Penrose李导数给出了一个几何上的重新解释。最后，我们提出了旋量场的李导数理论在变分学中的一个重要应用。
In a previous paper we showed that Weyl equation possess superluminal solutions and moreover we showed that those solutions that are eigenstates of the parity operator seem to describe a coupled pair of a monopole anti-monopole system. This result suggests to look for a solution of Maxwell equation \partialF^{\infty} =-gJ with a current J as source and such that the Lorentz force on the current is null. We first identify a solution where J=γ^{5} J_{m} is a spacelike field (even if F is not a superluminal solution of the homogeneous Maxwell equation). More surprisingly we find that there exists…
Integrating Factors for Dirac-Schrodinger Operators: Improving Eigenvalue Estimates and Applications to Charged Positive Mass Theorems Outside Horizon(s)
