关于洛伦兹和遥平行de Sitter时空结构中守恒定律、物质运动方程和粒子场的注记
Notes on Conservation Laws, Equations of Motion of Matter and Particle Fields in Lorentzian and Teleparallel de Sitter Spacetime Structures
摘要：
我们讨论了相互作用张量场和生活在$M=\mathrm{S0}(1,4)/\mathrm{S0} (1,3)\simeq\mathbb{R}\times S^{3}$（$\mathring{M}=(\mathbb{R}^{5},\boldsymbol{\mathring{g}})$的子流形）中的粒子的物理学，其中$\boldsymbol{\mathring {g}}$具有签名$(1,4)$。结构$(M,\boldsymbol{g})$其中$(\boldsymbol{g=i}^{\ast}\boldsymbol{\mathring{g}})$是洛伦兹流形。结构$(M,\boldsymbol{g,}τ_{\boldsymbol{g}},\uparrow)$主要用于研究$M$中物理场（和粒子）系统的能量动量守恒定律，并得到相应的运动方程。我们构造了两种不同的de Sitter时空结构$M^{dSL}=(M,\boldsymbol{g,D},τ_{\boldsymbol{g}},\uparrow)$和$M^{dSTP}=(M,\boldsymbol{g,\nabla},τ_{\boldsymbol{g} },\uparrow)$。这两种（度量兼容）连接仅用作数学设备。特别是$M^{dSL}$不应该是（\textbf{GRT}）中任何引力场的模型。我们澄清了文献中出现的一些误解。我们使用的克利福德和自旋克利福德束形式主义，并给出了一个周到的介绍的概念，科马尔目前$\mathcal{J}_{A}$（在GRT}）与任何向量场$\mathbf{A}$。给出了科马尔电流的计算公式及其物理意义。我们还证明了$F=dA$如何在Clifford丛中满足一个编码了Einstein方程内容的类Maxwell方程。我们表明，在GRT有无穷多个守恒电流，独立的事实，即洛伦兹时空拥有或不Killing矢量场，即使当适当的Killing矢量场存在不存在一个保守的能量动量共矢量（不是一个共矢量场）在SRT。
英文摘要：
We discuss the physics of interacting tensor fields and particles living in $M=\mathrm{S0}(1,4)/\mathrm{S0} (1,3)\simeq\mathbb{R}\times S^{3}$ a submanifold of $\mathring{M}=(\mathbb{R}^{5},\boldsymbol{\mathring{g}})$, where $\boldsymbol{\mathring {g}}$ has signature $(1,4)$. Structure $(M,\boldsymbol{g})$ where $(\boldsymbol{g=i}^{\ast}\boldsymbol{\mathring{g}})$ is a Lorentzian manifold. Structure $(M,\boldsymbol{g,}τ_{\boldsymbol{g}},\uparrow)$ is primely used to study the energy-momentum conservation law (for a system of physical fields (and particles) living in $M$ and to get the respective equations of motion. We construct two different de Sitter spacetime structures $M^{dSL}=(M,\boldsymbol{g,D},τ_{\boldsymbol{g}},\uparrow)$ and $M^{dSTP}=(M,\boldsymbol{g,\nabla},τ_{\boldsymbol{g} },\uparrow)$. Both (metrical compatible) connections are used only as mathematical devices. In particular $M^{dSL}$ is not supposed to be the model of any gravitational field in the(\textbf{GRT}). We clarify some misconceptions appearing in the literature. We use the Clifford and spin-Clifford bundles formalism and gives a thoughtful presentation of the concept of a Komar current $\mathcal{J}_{A}$ (in GRT}) associated to any vector field $\mathbf{A}$. A formula for the Komar current and its physical meaning are given. We show also how $F=dA$ satisfy in the Clifford bundle a Maxwell like equation encoding the contents of Einstein equation. We show that in GRT there are infinitely many conserved currents,independently of the fact that the Lorentzian spacetime possess or not Killing vector fields and that even when the appropriate Killing vector fields exist there does not exist a conserved energy-momentum covector (not a covector field) as in SRT.
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