Lorentz提升和Wigner旋转：自伴复杂化四元数
Lorentz boosts and Wigner rotations: self-adjoint complexified quaternions
摘要：
在这里，我们将从（复杂化的）四元数观点考虑洛伦兹助推和维格纳旋转。我们将证明，对于一个适当定义的自伴复四元数4-速度，纯洛伦兹助推可以用连接两个惯性系的相对4-速度的四元数平方根来表述。简单的计算，然后导致相当明确和相对简单的代数公式的组成4速度和维格纳角。随后，我们将维格纳旋转的三个4-速度的合成的一般非结合性，并制定了一个必要和充分条件的结合性举行。最后，我们将4-速度的组成与Baker-Campbell-Hausdorff定理的具体实现联系起来。与普通的4 × 4洛伦兹变换相比，自伴复化四元数的使用从计算的角度来看可以节省存储和更快速的计算，从教学的角度来看可以得到相对简单和明确的公式。
英文摘要：
Herein we shall consider Lorentz boosts and Wigner rotations from a (complexified) quaternionic point of view. We shall demonstrate that for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. We subsequently relate the Wigner rotation to the generic non-associativity of the composition of three 4-velocities, and develop a necessary and sufficient condition for associativity to hold. Finally, we relate the composition of 4-velocities to a specific implementation of the Baker-Campbell-Hausdorff theorem. As compared to ordinary 4x4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.
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