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 ${\sigma }_i$: $$\ln (e^{{\sigma }_1}{e}^{{\sigma }_2})={\tanh }^{-1}\left(\frac{\tanh {\sigma }_1+\tanh {\sigma }_2+\frac{1}{2}[\tanh {\sigma }_1,\tanh {\sigma }_2]}{1+\frac{1}{2}\{\tanh {\sigma }_1,\tanh {\sigma }_2\}}\right)$$ This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension $\le 4$, 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 $2\times 2$ 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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• Spacetime algebra as a powerful tool for electromagnetism