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Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra
International Journal of Geometric Methods in Modern Physics ( IF 2.1 ) Pub Date : 2021-10-05 , DOI: 10.1142/s0219887821502261
Joseph Wilson 1 , Matt Visser 1
Affiliation  

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations eσi in the spin representation (a.k.a. Lorentz rotors) in terms of their generators σi: ln(eσ1eσ2) =tanh1 tanh σ1 +tanh σ2 + 1 2[tanh σ1,tanh σ2] 1 + 1 2{tanh σ1,tanh σ2}. This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension 4, naturally generalizing Rodrigues’ formula for rotations in 3. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex 2 × 2 matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

中文翻译:

基于几何代数的时空显式贝克-坎贝尔-豪斯多夫-戴金公式

我们提出了一个紧凑的 Baker-Campbell-Hausdorff-Dynkin 公式,用于洛伦兹变换的组成eσ一世在自旋表示(又名洛伦兹转子)的发电机方面σ一世 ln(eσ1eσ2) =-1 σ1 + σ2 + 1 2[ σ1, σ2] 1 + 1 2{ σ1, σ2}. 这个公式对于几何代数是通用的 (又名实数克利福德代数)的维数 4, 自然地推广 Rodrigues 的旋转公式 3. 特别是,它适用于 Hestenes 时空代数框架内的洛伦兹转子,并为构成洛伦兹发电机提供了一种有效的方法。复杂的计算机实现是可能的2 × 2由泡利自旋矩阵实现的矩阵表示。该公式应用于相对论的组成3-速度产生的推力和伴随的维格纳角的简单表达式。
更新日期:2021-10-05
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