Octonions

I came across this fascinating article in Quanta Magazine (July 20th 2018) about octonions, a concept I'd never heard of before. Here is an excerpt:

The suspicion, harbored by many physicists and mathematicians over the decades but rarely actively pursued, is that the peculiar panoply of forces and particles that comprise reality spring logically from the properties of eight-dimensional numbers called “octonions.”

I knew about the Irish mathematician Hamilton's discovery of quaternions in the mid-nineteenth century and it was John Graves, a lawyer friend of Hamilton’s, (who) subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space.

A 39 year old mathematical physicist at the University of Cambridge by the name of Cohl Furey has been making progress recently connecting the octonions with the Standard Model of Physics. She has posted a series of short videos on YouTube, explaining what she is doing. Shown below is the first of the fourteen videos:

Below is a graphic that summarises some of the differences between the real and complex numbers as well as the quaternions and octonians. Double-click to enlarge. There's no point trying to summarise what's in the article or posting copious extracts. It's better to read it in full and then watch the videos. However, I have included a small quote.

FIGURE !: Four Special Number Systems

To reconstruct particle physics, Furey uses the product of the four division algebras, ℝ ⊗ ℂ ⊗ ℍ ⊗ 𝕆 (ℝ for reals, ℂ for complex numbers, ℍ for quaternions and 𝕆 for octonions) — sometimes called the Dixon algebra. 

Whereas Dixon and others proceeded by mixing the division algebras with extra mathematical machinery, Furey restricts herself; in her scheme, the algebras “act on themselves.” Combined as ℝ ⊗ ℂ ⊗ ℍ ⊗ 𝕆, the four number systems form a 64-dimensional abstract space. 

Within this space, in Furey’s model, particles are mathematical “ideals”: elements of a subspace that, when multiplied by other elements, stay in that subspace, allowing particles to stay particles even as they move, rotate, interact and transform. The idea is that these mathematical ideals are the particles of nature, and they manifest the symmetries of ℝ ⊗ ℂ ⊗ ℍ ⊗ 𝕆. 

What's fascinating is the possibility that a number system (the octonions) interacting with other number systems (the reals, complex numbers and quaternions) might be able to describe the existence and behaviour of all the particles and forces in the physical universe.

It will be interesting to follow the progress of Furey's research. Hopefully, she will post more YouTube videos. Her last upload was about nine months ago.

ADDENDUM

Here is a more recent article that relates to octonions titled Ask Ethan: Could Octonions Unlock How Reality Really Works?