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arXiv 1502.05685math-phmath.MP
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de Sitter流形中自旋1/2费米子波动方程的Clifford丛方法

A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold

W. A. Rodrigues, S. A. Wainer, M. Rivera-Tapia, E. A. Notte-Cuello, I. Kondrashuk

备注：
This version contains a new appendix and improves the presentation of section 5
发表信息：
Advances in Applied Clifford Algebras 26 (2016) 253-277
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摘要：

本文用Clifford丛方法研究了de Sitter流形中自由自旋$1/2$费米子的波动方程，该费米子是一个生活在体时空$\mathbb{R}^{4,1}=(\mathring{M}=\mathbb{R}^{5},\boldsymbol{\mathring{g}})$中的具有拓扑结构$M=\mathrm{S0}(4,1)/\mathrm{S0}(3,1)$的膜，它具有度规场$\boldsymbol{g:=-i}^{\ast}\boldsymbol{\mathring{g}%}$，其中$\boldsymbol{i}:M\rightarrow\mathring{M}$是包含映射.为了得到Dirac方程在闵可夫斯基时空中的类似结构$\mathring{M}$，我们利用$C_{2}$线性化中给出的约束条件，对de Sitter群的李代数的两个Casimir不变量$C_{1}$和$C_{2}$进行适当的因式分解，将$C_{1}$线性化.通过这种方式，我们得到一个方程，我们称之为\textbf{DHESS1,}，在以前的研究中，其他作者只是假设。接下来我们推导出一个波动方程（称为\textbf{DHESS2}）的自由自旋$1/2$费米子在德西特流形使用启发式的论点，这是一个明显的推广启发式的论点（详见附录D）允许导出闵可夫斯基时空中的狄拉克方程，并表明这一著名方程只不过表达了自由粒子的动量是常数的事实给定速度场的类时积分曲线上的向量场。值得注意的是，\textbf{DHESS1}和\textbf{DHESS2} \重合。在我们的文件中的主要成分之一是使用的概念的Dirac-Hestenes旋量领域。附录B和C回顾了这个概念及其与物理学家通常使用的协变狄拉克旋量场的关系。

英文摘要：

In this paper we give a Clifford bundle motivated approach to the wave equation of a free spin $1/2$ fermion in the de Sitter manifold, a brane with topology $M=\mathrm{S0}(4,1)/\mathrm{S0}(3,1)$ living in the bulk spacetime $\mathbb{R}^{4,1}=(\mathring{M}=\mathbb{R}^{5},\boldsymbol{\mathring{g}})$ and equipped with a metric field $\boldsymbol{g:=-i}^{\ast}\boldsymbol{\mathring{g}%}$ with $\boldsymbol{i}:M\rightarrow\mathring{M}$ being the inclusion map. To obtain the analog of Dirac equation in Minkowski spacetime in the structure $\mathring{M}$ we appropriately factorize the two Casimir invariants $C_{1}$ and $C_{2}$ of the Lie algebra of the de Sitter group using the constraint given in the linearization of $C_{2}$ as input to linearize $C_{1}$. In this way we obtain an equation that we called \textbf{DHESS1,}which in previous studies by other authors was simply postulated.$.$Next we derive a wave equation (called \textbf{DHESS2}) for a free spin $1/2$ fermion in the de Sitter manifold using a heuristic argument which is an obvious generalization of a heuristic argument (described in detail in Appendix D) permitting a derivation of the Dirac equation in Minkowski spacetime and which shows that such famous equation express nothing more than the fact that the momentum of a free particle is a constant vector field over timelike \ integral curves of a given velocity field. It is a remarkable fact that \textbf{DHESS1}and \textbf{DHESS2}\ coincide. One of the main ingredients in our paper is the use of the concept of Dirac-Hestenes spinor fields. Appendices B and C recall this concept and its relation with covariant Dirac spinor fields usualy used by physicists.

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