Split-Octonions: Helicity

Adding a third spatial dimension using the Cayley-Dickson construction leads us to the split-octonions. This adds a third spatial direction to the axial vector, and increases the angle to a three component polar vector. We also have a new component, whose nature will be deferred.

q=e+\vec{J}+\vec{I}+k

Generalized Conjugation

There are also three generalized conjugations that are useful, related to the similarly named concepts from Clifford algebras. Unlike Clifford algebras, there is no matrix formulation, as the split-octonions are not associative.

Conjugate: q^*=q=e-\vec{J}-\vec{I}-k
Grade Inversion: q^{\dagger}=q=e-\vec{J}+\vec{I}-k
Reversion: \overline{q}=e+\vec{J}-\vec{I}+k

It is useful to note that the composition of any two distinct generalized conjugations is the third. qq^*=q^*q returns a scalar norm for q. qq^{\dagger}=(q^{\dagger}q)^* returns a quaternion norm of q. We also have the following multiplication rules.

(xy)^*=y^*x^*
(xy)^{\dagger}=x^{\dagger}y^{\dagger}
\overline{(xy)}=(xy)^{\dagger*}=\overline{y}\,\overline{x}

Unlike Clifford algebras, the split-octonions are not associative. This means that there is no matrix representation.

When (\vec{J}\cdot\vec{I})^2=(ek)^2 for p and q, however, we have p(qx\overline{q})\overline{p}=(pq)x\overline{(pq)}. This will become important when we discuss rotations and boosts.

Split-Octonion Representations

Often it is useful to express the axial and polar vectors in either Cartesian coordinates, or using magnitude-direction.

q=e+\vec{A}j+\vec{P}i+k

q=e+Aj_{\hat{v}}+Pi_{\hat{u}}+k

Further, there are several possible sign conventions available to us. The following sign convention is somewhat non-standard, representing a grade inversion of most presentations in literature, but allows the equations of rotations and boosts to be the same form as split-quaternions and quaternions.

For calculations by hand it is useful to express the multiplication table in terms of the standard dot and cross products of vectors.

\begin{array}{cccc}1&\vec{J}&\vec{I}&k\\\vec{J}&\begin{array}{c}\cdot: +1\\ \times: -\vec{I}\end{array}&\begin{array}{c}\cdot: -k\\ \times: +\vec{j}\end{array}&-\vec{I}\\ \vec{I}&\begin{array}{c}\cdot: +k\\ \times: +\vec{j}\end{array}&\begin{array}{c}\cdot: -1\\ \times:-\vec{I}\end{array}&-\vec{J}\\k&\vec{I}&\vec{J}&1\end{array}

For computer algorithms it is useful to express this explicitly for each component.

\begin{array}{cccccccc}1&j_x&j_y&j_z&i_x&i_y&i_z&k\\j_x&1&-i_z&i_y&-k&j_z&-j_y&-i_x\\j_y&i_z&1&-i_x&-j_z&-k&j_x&-i_y\\j_z&-i_y&i_x&1&j_y&-j_x&-k&-i_z\\i_x&k&j_z&-j_y&-1&-i_z&i_y&-j_x\\i_y&-j_z&k&j_x&i_z&-1&-i_x&-j_y\\i_z&j_y&-j_x&k&-i_y&i_x&-1&-j_z\\k&i_x&i_y&i_z&j_x&j_y&j_z&1\end{array}

This allows successive boosts and rotations to be nested, and expressed as qx\overline{q}, where q is any unit norm split-octonion such that \left(\vec{J}\cdot\vec{I}\right)^2=(ek)^2. This limits the sub-algebra to six degrees of freedom, as required to be isomorphic with the restricted Lorentz group. In particular, the equations for a pure rotation or boost can be expressed as follows.

r=\cos\frac{\theta}{2}-\sin\frac{\theta}{2}i_{\hat{u}}
b=\cosh\frac{\varphi}{2}-\sinh\frac{\varphi}{2}j_{\hat{v}}

[Note: these rotations were derived using an incorrect version of the split-octonion multiplication table. Conjugating using these values will not cause the rotations as envisioned when this post was originally written, and certainly will not form the restricted Lorentz group.]

Helicity

But what is this new quantity named k? This representation is in homage to the representation of the coquaternions of James Cockle.

The first thing worth noting is that the rotation and boost formula are not broken if we swap the sign convention on k, changing the sign of the elements in the rows and columns headed by k and all appearances of k in the tables.

Next, we note that it is a dot product of a polar and axial vector. This naturally leads to the identification of this quantity as a pseudo-scalar.

The classical representation of a pseudo-scalar is helicity. If we have a helix and rotate it, it appears to move in a certain direction. Likewise, it we translate it about its axis it will appear to be rotating.

Further, if we examine the reflection the helicity of the reflection will be opposite that of the original, unlike scalars which are unchanged by reflections.

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