Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

• Main Authors: Thomas Berry, Matt Visser
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Physics
• Subjects: special relativity; quaternions; Lorentz boosts; composition of velocities; Wigner angle
• Online Access: https://www.mdpi.com/2624-8174/3/2/24

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated 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. The Wigner rotation is subsequently related to the generic <i>non-associativity</i> of the composition of three 4-velocities, and a <i>necessary and sufficient</i> condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mo>×</mo><mn>4</mn></mrow></semantics></math></inline-formula> 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.