八元数空间中的两类不相容不变量
Two incompatible types of invariants in the octonion spaces
摘要：
利用八元数坐标系的旋转变换，研究电磁场和引力场的一些不变量和守恒定律。学者们利用八元数同时分析电磁场和引力场，包括八元数场势、场强、场源、线动量、角动量、力矩和力。当八元数坐标系旋转变换时，一个八元数的矢量部分可能发生变化，而八元数的标量部分保持不变。这个性质允许人们推导出一些不变量，例如八元数半径矢量的标量部分、光速和八元数半径矢量的范数。这些不变量是伽利略变换和洛伦兹变换的基本假设。进一步地，从八元数坐标系的旋转变换出发，可以推导出一个八元数空间$\mathbb{O}$中与引力场有关的质量、能量和功率等不变量。与电荷有关的项会随着坐标系的旋转而变换。在另一个八元数空间$\mathbb{O}_u$中，它能够推导出一些不变量，包括与电磁场有关的电荷。与质量和能量有关的项随八元数坐标系的旋转而变化。所以不变量被分成两组。特别地，质量守恒定律和能量守恒定律可以同时有效。但严格意义上的电荷守恒定律和质量守恒定律不能同时成立。这对进一步理解守恒定律是有益的。
英文摘要：
The paper aims to study some invariants and conservation laws relevant to electromagnetic and gravitational fields, by means of the rotational transformations of octonion coordinate systems. The scholars utilize the octonions to analyze the electromagnetic and gravitational fields simultaneously, including the octonion field potential, field strength, field source, linear momentum, angular momentum, torque and force. When the octonion coordinate system transforms rotationally, the vector part of one octonion may alter, while the scalar part of the octonion will remain unchanged. This property allows one to deduce a few invariants, such as the scalar part of octonion radius vector, speed of light, and norm of octonion radius vector. These invariants are the basic postulates for the Galilean transformation and Lorentz transformation. Further, from the rotation transform of octonion coordinate systems, it is capable of deducing several invariants, including the mass, energy and power relevant to gravitational fields, in one octonion space $\mathbb{O}$. And the term relevant to the electric charge will transform with the rotation of coordinate systems. In another octonion space $\mathbb{O}_u$, it is capable of inferring a few invariants, including the electric charge related to electromagnetic fields. And the terms relevant to the mass and energy will vary with the rotation of octonion coordinate systems. So the invariants are divided into two different groups. In particular, the mass conservation law and energy conservation law can be effective simultaneously. But the charge conservation law and mass conservation law are unable to be valid simultaneously, in the strict sense. It is beneficial to further understand the laws of conservation.
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