Abstract
Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of the algebra of the Lorentz transformations. Herein we report a new and relatively nice result for the relativistic combination of non-collinear 3-velocities. We work with the relativistic half-velocities w defined by v=2w1+w2, so that w=v1+1−v2=v2+O(v3), and promote them to quaternions using w=wn^, where n^ is a unit quaternion. We shall first show that the composition of relativistic half-velocities is given by w1⊕2≡w1⊕w2≡(1−w1w2)−1(w1+w2), and then show that this is also equivalent to w1⊕2=(w1+w2)(1−w2w1)−1. Here as usual we adopt units where the speed of light is set to unity. Note that all of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of w1 with w2. This result can furthermore be extended to obtain novel elegant and compact formulae for both the associated Wigner angle Ω and the direction of the combined velocities: eΩ=eΩΩ^=(1−w1w2)−1(1−w2w1), and w^1⊕2=eΩ/2w1+w2|w1+w2|. Finally, we use this formalism to investigate the conditions under which the relativistic composition of 3-velocities is associative. Thus, we would argue, many key results that are ultimately due to the non-commutativity of non-collinear boosts can be easily rephrased in terms of the non-commutative algebra of quaternions.
Highlights
Hamilton first described the quaternions in the mid-1800s, primarily with a view to finding algebraically simple ways to handle 3-dimensional rotations
The anti-hermitian 2 × 2 matrices, essentially −1 × (Pauli matrices). (The factor of −1 is important.) this does not mean that replacing quaternions by Pauli matrices in any way simplifies our results below; it just complicates the formalism
For current purposes we focus our attention on pure quaternions
Summary
Hamilton first described the quaternions in the mid-1800s, primarily with a view to finding algebraically simple ways to handle 3-dimensional rotations. Perhaps one of the reasons for this is that there are a number of sub-optimal notational choices in Silberstein’s original work [1,2,3], and the fact that there is no generally accepted way of using quaternions to represent Lorentz transformations, with many different authors employing their own quite distinct methods [1,2,3,4,5,6,7,8,9]. (The factor of −1 is important.) this does not mean that replacing quaternions by Pauli matrices in any way simplifies our results below; it just complicates the formalism. Neither does this mean that any of our results below are at all “well-known” in this alternate notation. There is much more than pedagogy going on—the results reported in our article are (apart from a consistency check or two) both novel and interesting (see [13].)
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