# Does quaternion multiplication relate to Minkowski space?

A quaternion notated as $a+bi+cj+dk$ can also be written in terms of a scalar and a vector $(a,v)$, where $v$ is the three-vector $(b,c,d)$. In this notation, the real part of the product $(p,q)(r,s)$ equals $pr-q\cdot s$, which looks exactly like the inner product in Minkowski space. Does anything deep and interesting come out of this, or is it not really useful? Since we know that quaternions relate to rotations in three dimensions, i.e., to the geometry of Euclidean space, it's tempting to think that this somehow relates them as well to the geometry of Minkowski space.

This may be relevant, but I can't access it: http://iopscience.iop.org/0143-0807/5/1/007

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The paper you linked to "cheats" by allowing $b, c, d$ to be complex. I think the right question to ask is about the Clifford algebra (en.wikipedia.org/wiki/Clifford_algebra) associated to Minkowski space rather than the quaternions. – Qiaochu Yuan Mar 6 '12 at 4:30

There is a definitely a connection, but how good of a connection depends on your expectations. There is a lot of sexy interplay between the metric geometry using Clifford algebras and the quaternions. Clifford (or "geometric") algebras encode geometric information about the underlying space.

I'd like to mention three such algebras. First of all, $C\ell_{0,2}(\mathbb{R})=\mathbb{H}$.

Secondly, for ordinary Euclidean $\mathbb{R}^3$, the algebra $C\ell_{3,0}(\mathbb{R})$ contains $\mathbb{H}$ as a subalgebra of elements which enact rotations in the traditional quaternion way. (Of course, LOTS of Clifford algebras contain copies $\mathbb{H}$ but this is the one that does it in the "natural way".) These are the so-called rotors in the algebra.

Lastly, moving up to Minkowski space with $C\ell_{1,3}(\mathbb{R})$, the rotors are a bit different than the quaternions. The encompass both spatial rotations and Lorentz boosts and mixtures of the two. I think the phenomenon you observed might be a quaternion-like shadow of these rotors.

In your post it looked like you were reading a paper using biquaternions (complex quaternions), and I can recommend another good paper on that if you haven't found it already: check out Lambek's If Hamilton Had Prevailed: Quaternions in Physics (1995) Math.Intelligencer. He is generally known for good exposition and he's knowledgable about this topic in particular (it was part of his dissertation).

P.S.: If you're wondering about $C\ell_{3,1}(\mathbb{R})$, it's also interesting, but it is nonisomorphic to $C\ell_{1,3}(\mathbb{R})$ as $\mathbb{R}$ algebras. For the purposes of physics, I don't think any evidence has arisen to choose one as "better".

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Yes. This has been done more than once. Although familiar with quaternions, the reader may not have heard of biquaternions.

Let -1, $i$, $j$, and $k$ denote generators of the quaternion group, with relations $(-1)^2 = 1,\ i^2 = j^2 = k^2 = ijk = -1$. When $a$, $b$, $c$, and $d$ are real numbers, $$q = a + bi + cj + dk$$ is called a quaternion. When they are allowed to be complex, it is called a biquaternion.

1. Dirac, P. A. M. Application of quaternions to Lorentz transformations. Proceedings of the Royal Irish Academy (Dublin), vol. 50, sect. A, no. 16, 27 November 1945, pp 261-70. Relates usual Lorentz transforms to transforms on projective five space. The transforms are similar in form to Möbius transforms (so use the ability to divide quaternions) but between biquaternions rather than complex numbers.

2. Lanczos, Cornelius (1949), The Variational Principles of Mechanics, University of Toronto Press, pp. 304–312. Uses a linear subspace of biquaternions, one that is not closed under multiplication so is not a sub-algebra, to represent Minkowski space. This is the simplest to understand. For the basics, see Wikipedia article on biquaternions. The section on "Relation to Lorentz transformations" answers your question.

3. Girard, P. R. The quaternion group and modern physics (1984) Eur. J. Phys. vol 5, p. 25–32. Quaternions are one of the very few Clifford algebras consistent with a definition of division. Since that aspect has been of small help, interest in other Clifford algebras has greatly increased while interest in quaternions themselves has declined. Though dated, this article overviews how quaternions fit in with modern notions, and provides historical references to work done to resolve the question you asked. Among other things, the author uses biquaternions to represent the Lorentz groups, including a treatment of Thomas precession.

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Sorry I didn't see this earlier. Quaternion space is not the same as Minkowski space-time and the relationship between them is subtle.

You can do Lorentz transform with quaternions without a whole lot of trouble, I show one way to do it here. I don't see that bi-quaternions are needed.