Explicit Baker-Campbell-Hausdorff-Dynkin formula for Spacetime via Geometric Algebra

Abstract

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 $2\times2$ 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.