标题： 代数旋量和狄拉克-赫斯滕斯旋量以及旋量场 显示英文标题

标题： Algebraic and Dirac-Hestenes Spinors and Spinor Fields

摘要： 在物理学（以及数学）教科书中的狄拉克理论的一阶或二阶量子化表述几乎都使用协变狄拉克旋量场。一个例外是海斯蒂内斯最初提出的该理论（一阶量子化）的表述，现在许多作者也使用这一表述。在那里，使用了一个新的旋量场概念（作为非齐次偶多向量场的和）。然而，仔细分析（如下文详细说明）表明，海斯蒂内斯最初的定义不可能是正确的，因为它与费尔茨恒等式的含义相冲突。在本文中，我们开始了一个项目，专门考察海斯蒂内斯所使用的对象的数学和物理基础，以进行全面定义。为了做到这一点，我们给出了闵可夫斯基时空上的代数旋量场（ASF）和狄拉克-海斯蒂内斯旋量场（DHSF）的初步定义，作为一些明确的数学对象对的等价类，其中一对中的一个成员是一个非齐次偶微分形式。我们的定义的必要性通过仔细分析狄拉克理论的可能表述以及费尔茨恒等式集合的含义得到了说明。我们认为，本文澄清了该主题文献中出现的一些误解（过去和最近的）。接下来将有一篇后续论文，在其中将作为称为左旋-克莱夫福德丛的向量丛的适当部分给出 ASF 和 DHSF 的最终定义。本文还包含附录（A-E），其中展示了关于克莱夫福德代数理论的一组真正有用的成果（包括许多“行业技巧”），这对于理解文本是必要的。 显示英文摘要

摘要： Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF) and Dirac-Hestenes spinor fields (DHSF) on Minkowski spacetime as some equivalence classes of well defined pairs of mathematical objects, one of the members of the pair being an even nonhomegeneous differential form. The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections a vector bundle called the left spin-Clifford bundle. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of Clifford algebras (including many `tricks of the trade') necessary for the intelligibility of the text.