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Stream: deprecated: mathematics

Topic: a 27-dimensional exceptional spacetime

view this post on Zulip John Baez (Dec 20 2020 at 07:34):

We can think of the exceptional Jordan algebra as a funny sort of spacetime. This spacetime is 27-dimensional, with light rays through the origin moving on a lightcone given by a cubic equation instead of the usual

t2−x2−y2−z2=0 t^2 - x^2 - y^2 - z^2 = 0

in 4-dimensional Minkowski spacetime. But removing this lightcone still chops spacetime into 3 connected components: the past, the future, and the regions you can't reach from the origin without exceeding the speed of light. The future is still a convex cone, and so is the past. So causality still makes sense like it does in special relativity.

At some point I got interested in seeing what physics would be like in this funny spacetime. In 2015, Greg Egan and John Huerta joined me in figuring out the very basics of what quantum field theory would be like in this world. Namely, we figured out a bit about what kinds of particles are possible.

One difference is that we must replace the usual Lorentz group with the 78-dimensional group E6\mathrm{E}_6. But an even bigger difference is this. In 4d Minkowski space, every point in your field of view acts essentially like every other, if you turn your head. But in our 27-dimensional spacetime, the analogous fact fails! There is a 'sky within the sky': some particles moving at the speed of light can only be seen in certain directions. Thus, the classification of particles that move at the speed of light is much more baroque.

This is a big digression from my main quest these days: explaining how people have tried to relate the octonions to the Standard Model. But it would be a shame not to make our results public, and now is a good time.

To explain why this isn't a completely wacky thing to do, I should say that people have completely classified vector spaces with 'self-dual homogeneous convex cones'. The most famous example is the future cone

t2−x2−y2−z2>0,t>0 t^2 - x^2 - y^2 - z^2 \gt 0, \qquad t \gt 0

in 4d Minkowski spacetime. The possibilities are very limited: they correspond to formally real Jordan algebras! There are four infinite families and one exception: the exceptional Jordan agebra h3(O)\mathfrak{h}_3(\mathbb{O}). So, just in the name of exploring possibilities, it seems worth taking a look at physics with h3(O)\mathfrak{h}_3(\mathbb{O}) playing the role of spacetime. I’m not claiming it’s good for real-world physics!

I explain it here:

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