Lorentz boosts and Wigner rotations: self-adjoint complexified quaternions

TL;DR

This paper explores Lorentz boosts and Wigner rotations using self-adjoint complexified quaternions, providing explicit algebraic formulas, analyzing non-associativity, and connecting to the Baker-Campbell-Hausdorff theorem for more efficient computations.

Contribution

It introduces a quaternionic framework for Lorentz transformations that simplifies calculations and offers new insights into the composition and properties of 4-velocities.

Findings

Explicit algebraic formulas for 4-velocity composition

Relation of Wigner rotation to non-associativity of velocity composition

Connection to Baker-Campbell-Hausdorff theorem

Abstract

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.