Abstract
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 non-associativity of the composition of three 4-velocities, and a necessary and sufficient 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 4×4 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.
Highlights
The use of Hamilton’s quaternions [1,2,3,4,5,6] as applied to special relativity has a very long, complicated, and rather fraught history—largely due to a significant number of rather sub-optimal notational choices being made in the early literature [7,8], which was compounded by the introduction of multiple mutually disjoint ways of representing the Lorentz transformations [7,8,9,10]
The ordinary classical quaternions are generalizations of the complex numbers that can be written in the form [1,2,3,4,5,6,34]: q = a + b i + c j + d k
Dimensional, the norm of an ordinary quaternion q = a0 + a1 i + a2 j + a3 k is given by
Summary
The use of Hamilton’s quaternions [1,2,3,4,5,6] as applied to special relativity has a very long, complicated, and rather fraught history—largely due to a significant number of rather sub-optimal notational choices being made in the early literature [7,8], which was compounded by the introduction of multiple mutually disjoint ways of representing the Lorentz transformations [7,8,9,10]. Formulations were heavily influenced by Minkowski’s x4 = ict notation, with its imaginary time component, essentially using anti-self-adjoint complexified quaternions, with the result that there were a lot of extra and superfluous factors of (complex) i floating around. The authors re-phrase things in terms of self-adjoint complex quaternionic 4-velocities, arguing for a number of simple compact formulae relating Lorentz transformations and the Wigner angle. While certainly elegant, most results based on the BCH expansion seem to not always be computationally useful
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