Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of the algebra of the Lorentz transformations. Herein we report a new and relatively nice result for the relativistic combination of non-collinear 3-velocities. We work with the relativistic half-velocities w defined by $v=\frac{2w}{1+w^2}$, so that $w=\frac{v}{1+\sqrt{1-v^2}}=\frac{v}{2}+\mathcal{O}\left(v^3\right)$, and promote them to quaternions using $\mathbf{w}=w\widehat{\mathbf{n}}$, where $\widehat{\mathbf{n}}$ is a unit quaternion. We shall first show that the composition of relativistic half-velocities is given by ${\mathbf{w}}_{1\oplus 2}\equiv {\mathbf{w}}_1\oplus {\mathbf{w}}_2\equiv {(1-{\mathbf{w}}_1{\mathbf{w}}_2)}^{-1}({\mathbf{w}}_1+{\mathbf{w}}_2)$, and then show that this is also equivalent to ${\mathbf{w}}_{1\oplus 2}=({\mathbf{w}}_1+{\mathbf{w}}_2){(1-{\mathbf{w}}_2{\mathbf{w}}_1)}^{-1}$. Here as usual we adopt units where the speed of light is set to unity. Note that all of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of ${\mathbf{w}}_1$ with ${\mathbf{w}}_2$. This result can furthermore be extended to obtain novel elegant and compact formulae for both the associated Wigner angle $\Omega$ and the direction of the combined velocities: ${\mathrm{e}}^{\mathbf{\Omega }}={\mathrm{e}}^{\Omega \widehat{\mathbf{\Omega }}}={(1-{\mathbf{w}}_1{\mathbf{w}}_2)}^{-1}(1-{\mathbf{w}}_2{\mathbf{w}}_1)$, and ${\widehat{\mathbf{w}}}_{1\oplus 2}={\mathrm{e}}^{\mathbf{\Omega }/2}\frac{{\mathbf{w}}_1+{\mathbf{w}}_2}{|{\mathbf{w}}_1+{\mathbf{w}}_2|}$. Finally, we use this formalism to investigate the conditions under which the relativistic composition of 3-velocities is associative. Thus, we would argue, many key results that are ultimately due to the non-commutativity of non-collinear boosts can be easily rephrased in terms of the non-commutative algebra of quaternions.

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四元数的非直线3速度的相对论组合

四元数具有（相对一个世纪以上的）广泛且非常复杂的相互作用，具有相对论。由于四元数本质上是4维的，并且在处理3维旋转方面做得如此出色，因此一直希望使用四元数可以简化Lorentz变换的某些代数。在这里，我们报告了非共线3速度相对论组合的一个新的且相对较好的结果。我们使用由定义的相对论半速度w$v=\frac{2w}{\mathrm{1个}+w^2}$， 以便 $w=\frac{v}{\mathrm{1个}+\sqrt{\mathrm{1个}-v^2}}=\frac{v}{2}+\mathcal{Ø}（v^3）$，并使用它们将其提升为四元数 $\mathbf{w}=w\widehat{\mathbf{ñ}}$，在哪里 $\widehat{\mathbf{ñ}}$是单位四元数。我们将首先证明相对论半速度的组成是${\mathbf{w}}_{\mathrm{1个}\oplus 2}\equiv {\mathbf{w}}_{\mathrm{1个}}\oplus {\mathbf{w}}_2\equiv {（\mathrm{1个}-{\mathbf{w}}_{\mathrm{1个}}{\mathbf{w}}_2）}^{-\mathrm{1个}}（{\mathbf{w}}_{\mathrm{1个}}+{\mathbf{w}}_2）$，然后表明这也等同于 ${\mathbf{w}}_{\mathrm{1个}\oplus 2}=（{\mathbf{w}}_{\mathrm{1个}}+{\mathbf{w}}_2）{（\mathrm{1个}-{\mathbf{w}}_2{\mathbf{w}}_{\mathrm{1个}}）}^{-\mathrm{1个}}$。像往常一样，这里我们采用光速设为统一的单位。请注意，非共线3速度相对论性组合的所有复杂角度相关性现在都以的四元数乘法编码${\mathbf{w}}_{\mathrm{1个}}$ 与 ${\mathbf{w}}_2$。此结果可以进一步扩展，以获得相关的维格纳角的新颖优雅且紧凑的公式$\Omega$ 以及组合速度的方向： ${\mathrm{Ë}}^{\mathbf{\Omega }}={\mathrm{Ë}}^{\Omega \widehat{\mathbf{\Omega }}}={（\mathrm{1个}-{\mathbf{w}}_{\mathrm{1个}}{\mathbf{w}}_2）}^{-\mathrm{1个}}（\mathrm{1个}-{\mathbf{w}}_2{\mathbf{w}}_{\mathrm{1个}}）$和 ${\widehat{\mathbf{w}}}_{\mathrm{1个}\oplus 2}={\mathrm{Ë}}^{\mathbf{\Omega }/2}\frac{{\mathbf{w}}_{\mathrm{1个}}+{\mathbf{w}}_2}{|{\mathbf{w}}_{\mathrm{1个}}+{\mathbf{w}}_2|}$。最后，我们使用这种形式主义来研究3速度的相对论组成是相联的条件。因此，我们认为，许多最终归因于非共线性增强的非交换性的结果可以很容易地用四元数的非交换代数来表述。