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تسجيل مستخدم جديدExplicit Baker-Campbell-Hausdorff-Dynkin formula for Spacetime via Geometric Algebra
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Joseph Wilson
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We present a compact Baker-Campbell-Hausdorff-Dynkin formula for the composition of Lorentz transformations $e^{sigma_i}$ in the spin representation (a.k.a. Lorentz rotors) in terms of their generators $sigma_i$: $$ ln(e^{sigma_1}e^{sigma_2}) = tanh^{-1}left(frac{ tanh sigma_1 + tanh sigma_2 + frac12[tanh sigma_1, tanh sigma_2] }{ 1 + frac12{tanh sigma_1, tanh sigma_2} }right) $$ This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension $leq 4$, naturally generalising Rodrigues formula for rotations in $mathbb{R}^3$. 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 $2times2$ matrix representation realised 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.
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