triality in GA

[Fred Lunnon]

For some while, I have been intrigued but baffled by this mysterious topic.
One obstruction is that standard treatments all insist on dragging in the
notion of "spinors", which even Michael Atiyah admits to difficulty in
understanding: see
https://www.youtube.com/watch?v=SBdW978Ii_E .
Another is that they gleefully wave hands over all manner of beautiful
properties, while steadfastly avoiding any discussion of how they might
algorithmically be realised.

An isolated honourable exception is provided by
Pertti Lounesto "Octonions and Triality"
Advances in Applied Clifford Algebras vol. 11 no.2, 191--213 (2001)
http://deferentialgeometry.org/papers/loun112.pdf .
However my earlier attempt to implement his Cl(8) formulation failed,
through misinterpreting his notation. This difficulty has now been resolved,
and the latest upload at
https://github.com/FredLunnon/ClifFred/
includes a demo of octonions and triality in Spin(8) . [Inter alia, I have
at last succeeded in discovering an application for GA with grade exceeding
6 .]

All later demos there in GA_scripts.py obviously require an accompanying
gloss summarising their motivation and methods: if somebody cares to
complain sufficiently vociferously, they might eventually receive one.
A further
complication is my inability to appreciate why spinors (and representations in
general) might be useful or interesting at all: in particular I utterly fail
to grasp the purported connection between them and bosons and fermions of the
standard model in particle physics (of which my ignorance remains lamentably
extensive).

Any available enlightenment would be appreciated!

Fred Lunnon

[Fred Lunnon]

Correction: I became confused by loose wording or reading, eg. of sect. 3.3 in
https://empg.maths.ed.ac.uk/Activities/Spin/Lecture3.pdf
For triality "spinors" just means (as expected) members of the Spin(8) group,
or equivalently normalised even versors in Cl(8) under usual
Clifford product. Now a connection with particle physics begins to
make more sense,
as geometry in spaces with different metrics.

Fred Lunnon

[Fred Lunnon]

My triality tester has discovered a (minuscule) error: Lounesto
sect. 5 claims ---

<< A non-linear automorphism of Spin(8) might also interchange -1 with
either of (+/-)e12...8. Such an automorphism of Spin(8), of order 2,
is said to be a `swap automorphism', denoted by swap(u) for u in
Spin(8). >>

But though swap S1 exchanges -1,+J as expected, the demo program
reveals that S2 fixes both +1,-1 , and exchanges +J,-J ; instead the
`companion' C (which he ignores) exchanges -1,-J .

OK, so it's hardly Nobel prize material ... but it made me happy! WFL

[John Gonsowski]

Would be really nice if Lounesto was still alive.  To get the spinors used for fermions in physics, you can add the dual 24-cell to the 24 cell of Spin(8)'s root system to get the F4 root system.  This adds a vector-half spinor-half spinor triality to the Spin(8)/Cl(8) bivectors. To see spinor fermions vs bosons/position-momentum operators for Cl(8) you can use a Hodge Star map:

http://www.tony5m17h.net/hstar.html  

This is from Tony Smith who hung out at Clifford algebra conferences with Lounesto.  The odd grades are for spinor fermions and the even grades are for bosons/position-momentum.  The diagonal with grade 0 then 14 of grade 4 then grade 8 is the primitive idempotents for Cl(8) mentioned by Lounesto.  I also via Smith use an elementary cellular automata rules space partitioning table to picture the grading of Cl(8) for physics:

https://prespacetime.com/index.php/pst/article/view/1348/1397