Split-Quaternions: Thomas Rotation

This page begins our journey into poorly charted algebras, by extending the split-complex numbers with an additional split root using the Cayley-Dickson construction, and immediately we encounter a naming crisis.

This crisis stems from the fact that the algebra described here is isomorphic to the algebra James Cockle named the coquaternions. Some suggest that they should then have the same name.

I draw the line thus. If you construct the algebra with mixed roots, it should be called the coquaternions. If you construct the algebra from two split roots it should be called the split-quaternions. Then it is slightly less obvious that the two should be identical from their definitions. Mainly this allows me to use the term split-quaternions when I use them, while respecting those who prefer the older name.

This algebra is also isomorphic to the algebra of 2×2 matrices, but that representation clearly represents a different mental picture, and the mapping is far less trivial than sign conventions and permutations of coordinates.

From the Cayley-Dickson construction we find that the fourth root must be circular. Further, we have the following multiplication table.

$\begin{array}{cccc}1&j_x&j_y&i_{\theta}\\j_x&1&-i_{\theta}&-j_y\\j_y&i_{\theta}&1&j_x\\i_{\theta}&j_y&-j_x&-1\end{array}$

Multiplying by $r=\cos(\theta)-\sin(\theta)i_{\theta}$ works for rotations, but using $b=\cosh(\varphi)-\sinh(\varphi)j_x$ mingles the $y$ and $\theta$ components.

The cure is to find a boost with half the rapidity and use a modified conjugate, as follows.

$r=\cos\frac{\theta}{2}-\sin\frac{\theta}{2}i_{\theta}$
$b=\cosh\frac{\varphi}{2}-\sinh\frac{\varphi}{2}j_x$
$\overline{(a+bj_x+cj_y+di_{\theta})}=(a+bj_x+cj_y-di_{\theta})$

It is straightforward to show that $\overline{q}$ satisfies the conjugation properties listed in the section on the Cayley-Dickson construction. It is also straightforward to show that $x'=qx\overline{q}$ produces the proper rotation or boost.

Consider a boost of rapidity $\tanh\varphi=\frac{10\sqrt{2}}{49}$ in the $x$ direction, followed by a similar boost in the $x$ direction.

$X''=b_y(b_xX\overline{b_x})\overline{b_y}$

Since this algebra is associative, this gives us:

$X''=(b_yb_x)X(\overline{b_x}\,\overline{b_y})=(b_yb_x)X\overline{(b_yb_x)}$
$b_x=\frac{1}{7}(5\sqrt(2)-j_x)$
$b_y=\frac{1}{7}(5\sqrt{2}-j_y)$
$b_yb_x=\frac{1}{49}(50-5\sqrt{2}j_x-5\sqrt{2}j_y+i_{\theta})$

But this provides a rotation. Observing that $rr^*=1$ for $\frac{1}{\sqrt{2501}}(50-i_{\theta})$,

$b_yb_x=\frac{1}{49}(50-5\sqrt{2}j_x-5\sqrt{2}j_y+i_{\theta})\frac{1}{\sqrt{2501}}(50-i_{\theta})\frac{1}{\sqrt{2501}}(50+i_{\theta})$
$b_yb_x=\frac{1}{49\sqrt{2501}}(2501-245\sqrt{2}j_x-255\sqrt{2}j_y)\frac{1}{\sqrt{2501}}(50+i_{\theta})$

Remembering to double our angles and boosts, shows that this is the same as a rotation about an angle with $\tan\theta=\frac{-100}{2499}$, followed by a boost of $\tanh{\varphi}=\frac{50020\sqrt{2501}}{6505101}$ at an angle of $\tan\theta=\frac{51}{49}$ towards the y-axis relative to the x-axis.

This is exactly the Thomas rotation.

The above can be done by computer using 4 component split-quaternion, not the three component space-time and 9 component boost matrices. Further, extracting the rotation angle is far simpler in the split-quaterion formulation than in the matrix formulation.

This could provide faster computations. Further, matrices are notoriously difficult to keep numerically stable.