四元数四维间隔不变性和时空坐标变换

Quaternion, Invariance of 4-Dimensional Interval and Transformations of Spacetime Coordinates

摘要: 利用四元数研究狭义相对论中的时空坐标变换。1) 导出保持四维间隔不变性的时空坐标一般变换的四元数形式，说明四维间隔不变性不能唯一确定Lorentz变换。2) 根据保持时间不变的条件，从一般变换中得到第一类特殊变换，其中包含空间旋转变换。3) 根据保持一个空间坐标不变的条件，从一般变换中得到第二类特殊变换，其中包含正常的Lorentz变换；同时指出某些文献中的四元数形式Lorentz变换式有待商榷之处。4) 从时空坐标一般变换中引入两种不同于传统的空间旋转变换和正常的Lorentz变换的新型变换，一种是离散变换，包含恒等变换，反射变换，换位变换等；另一种是单边变换。

Abstract: By the use of quaternion, the transformations of space-time coordinates in special relativity are studied. 1) The general transformations in quaternion form are derived, which preserve the invariance of 4-dimensional interval, and it is found that the invariance of the interval can not determine the Lorentz transformation uniquely. 2) Based on the con-dition that preserves the invariance of time, the general transformations reduce to the first kind of special transforma-tions, in which the space rotations are included. 3) Based on the condition that preserves the invariance of a space coor-dinate, the general transformations reduce to the second kind of special transformations, in which the proper Lorentz transformations are included. It is pointed out that the quaternion form of Lorentz transformations in some literatures should be amended. 4) From the general transformations in quaternion form, two types of new transformations are introduced, which are discrete transformations, including identical, reflection and transposition ones, and unilateral transformations. These new transformations are different from the traditional space rotations and the normal Lorentz transformations.