标题： 代数与Dirac-Hestenes旋量和旋量场 显示英文标题

标题： Algebraic and Dirac-Hestenes Spinors and Spinor Fields

摘要： 几乎所有物理学（以及数学）教材中关于狄拉克理论的表述，无论是采用第一还是第二量子化形式，都使用了协变的狄拉克旋量场。唯一的例外是由赫斯蒂尼斯最初提出的狄拉克理论（一阶量子化）的表述，现在已被许多作者采用。在那里，引入了一个新的旋量场概念（作为非齐次偶数多矢量场的和）。然而，经过仔细分析（详见下文）表明，赫斯蒂尼斯的原始定义是不正确的，因为它与费兹恒等式的含义相冲突。本文启动了一项计划，致力于研究赫斯蒂尼斯所用对象的数学和物理基础。为了实现这一点，我们在闵可夫斯基时空上初步定义了代数旋量场（ASF）和狄拉克-赫斯蒂尼斯旋量场（DHSF），这些定义是一些明确配对的数学对象的等价类，其中一对中的一个成员是一个偶数非齐次微分形式。通过狄拉克理论可能表述的仔细分析以及费兹恒等式集合的意义，我们展示了这些定义的必要性。我们认为，本文澄清了出现在该领域文献中的某些误解（包括过去的和最近的）。它将由后续论文接续，该论文将在称为左旋-克利福德丛的适当截面上给出ASF和DHSF的最终定义。本文还包含了附录（A-E），这些附录展示了一系列关于克利福德代数理论的真正有用的结果（包括许多“行话技巧”），以使文本易于理解。 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.