# 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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