论文标题
使用四元素的非结晶3速度的相对论组合
Relativistic combination of non-collinear 3-velocities using quaternions
论文作者
论文摘要
四局部与特殊相对论具有（一个百年历史的）广泛而相当复杂的互动。由于四局本质上是四维的，并且在处理三维旋转方面做得很好，因此希望季节的使用将简化Lorentz转换的某些代数。本文中，我们报告了非共线3速度的相对论组合的相对好结果。如果我们使用$ v = {2W \ over1+w^2} $定义的相对论的半速度$ w $，并使用$ \ mathbf {w} = w \; \ Mathbf {\ hat n} $，其中$ \ Mathbf {\ hat n} $是单位Quaternion，那么我们将显示\ [\ Mathbf {w} _ {1 \ oplus2} = \ \ \ \ \ \ \ m马理{W} =（1- \ Mathbf {w} _1 \ Mathbf {w} _2）^{ - 1}（\ MathBf {w} _1 +\ \ \ \ \ \ m athbf {w} _2 _2）=（\ Mathbf {w} _1 _1 _1 _1 _1 +\ Mathbf {W} _2）（1- \ Mathbf {W} _2 \ Mathbf {W} _1）^{ - 1}。 \]现在，与$ \ Mathbf {w} _1 $的Quaternion乘法使用$ \ Mathbf {w} _2 $的$ \ Mathbf {w} _1 $的Quaternion乘法现在编码非共线3速度的相对论组合的所有复杂角度依赖性。此结果还可以扩展以获得相关的wigner角度的优雅和紧凑的公式：\ [\ mathrm {e}^{\mathbfΩ} = \ m马理{e}^{ω\; \ Mathbf {\hatΩ}} =（1- \ Mathbf {W} _1 \ MathBf {W} _2）_2）^{ - 1}（1- \ Mathbf {W} _2 \ Mathbf {W} {\ Mathbf {\ hat {n}}} _ {1 \ oplus2} = \ m athrm {e}^{\mathbfΩ/2} \; \; \; {\ Mathbf {W} _1+\ Mathbf {w} _2 \ over | \ Mathbf {w} _1+\ \ \ \ Mathbf {w} _2 |}; \ qquad \ qquad {\ mathbf {\ hat {n}}} _ {2 \ oplus1} = \ mathrm {e}^{ - \mathbfΩ/2} \; \; \; {\ Mathbf {W} _1+\ Mathbf {w} _2 \ over | \ Mathbf {w} _1+\ \ \ \ \ \ Mathbf {w} _2 |}。 \]因此，我们要说的是，许多关键结果最终是由于非共线增强的非交换性而引起的，可以轻松地根据四季度的代数来改写。
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 relatively nice result for the relativistic combination of non-collinear 3-velocities. If we work with the relativistic half-velocities $w$ defined by $v={2w\over1+w^2}$, and promote them to quaternions using $\mathbf{w} = w \; \mathbf{\hat n}$, where $\mathbf{\hat n}$ is a unit quaternion, then we shall show \[ \mathbf{w}_{1\oplus2} = \mathbf{w}_1 \oplus \mathbf{w}_2 =(1-\mathbf{w}_1\mathbf{w}_2)^{-1} (\mathbf{w}_1 +\mathbf{w}_2) = (\mathbf{w}_1 +\mathbf{w}_2)(1-\mathbf{w}_2\mathbf{w}_1)^{-1}. \] 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 an elegant and compact formula for the associated Wigner angle: \[ \mathrm{e}^{\mathbfΩ} = \mathrm{e}^{Ω\; \mathbf{\hatΩ} } = (1-\mathbf{w}_1\mathbf{w}_2)^{-1} (1-\mathbf{w}_2\mathbf{w}_1), \] in terms of which \[ {\mathbf{\hat{n}}}_{1\oplus2} = \mathrm{e}^{\mathbfΩ/2} \;\; {\mathbf{w}_1+\mathbf{w}_2\over |\mathbf{w}_1+\mathbf{w}_2|}; \qquad\qquad {\mathbf{\hat{n}}}_{2\oplus1} = \mathrm{e}^{-\mathbfΩ/2} \;\; {\mathbf{w}_1+\mathbf{w}_2\over |\mathbf{w}_1+\mathbf{w}_2|}. \] 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 algebra of quaternions.
