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Stream: deprecated: mathematics

Topic: octonions and 10d spacetime

John Baez (Nov 30 2020 at 06:53):

I wrote another installment:

• Octonions and the Standard Model (part 9).

Here's the basic idea:

The Riemann sphere $\mathbb{C}\mathrm{P}^1$ leads a double life in physics. On the one hand it's the set of states of a complex qubit. In this guise, physicists call it the 'Bloch sphere'. On the other hand it's the set of directions in which you can look when you're an inhabitant of 4d Minkowski spacetime - as we all are. In this guise, it's called the 'celestial sphere'. But these two roles are deeply connected! In 4d spacetime a Weyl spinor is described by a complex qubit: that is, a unit vector in $\mathbb{C}^2$. A state - that is, a unit vector modulo phase - simply says which way the spinor is spinning. Its spin can point in any direction, and these directions are points in the celestial sphere.

John Baez (Nov 30 2020 at 06:54):

Last time I started explaining how to generalize some of these ideas from $\mathbb{C}\mathrm{P}^1$ to what I'm really interested in, $\mathbb{O}\mathrm{P}^2$. Again this has two roles. On the one hand it's the set of states of an octonionic qutrit. On the other hand it's the heavenly sphere in a 27-dimensional spacetime modeled on the exceptional Jordan algebra. This is a funny spacetime where the lightcone is described not by the usual sort of quadratic equation like

$t^2 - x^2 - y^2 - z^2 = 0$

but instead by a cubic equation.

John Baez (Nov 30 2020 at 06:54):

It would be easy for me to get lost in the pleasures of this geometry. But I have a concrete goal in mind. The symmetries of $\mathbb{O}\mathrm{P}^2$ form the group $\mathrm{E}_6$, and I'm trying to use this to understand a fact about $\mathrm{E}_6$. Namely, this Lie group has four Lie subgroups:

• the double cover of the Lorentz group in 10 dimensions
• translations in right-handed spinor directions
• translations in left-handed spinor directions
• 'dilations' (rescalings)

and these give Lie subalgebras whose direct sum, as vector spaces, is all of $\mathfrak{e}_6$:

$\mathfrak{e}_6 \cong \mathfrak{so}(9,1) \oplus \mathbb{O}^2 \oplus (\mathbb{O}^2)^\ast \oplus \mathbb{R}$

John Baez (Nov 30 2020 at 06:55):

I proved these facts back in Part 7, but now I'm trying to understand them better. The duality between points and lines in projective plane geometry turns out to be the key!

Jalex Stark (Nov 30 2020 at 08:54):

I think if you want a theory of everything, you need to relax the exact representations to approximate representations, where mass-energy excitations are things which increase the error parameter

Jalex Stark (Nov 30 2020 at 08:56):

Then black holes are just decoding issues --- places where the mass is high enough that you can't tell which irrep to snap back to

Jalex Stark (Nov 30 2020 at 08:57):

This should let you use experimental results any the actual schwarzild radius to verify whether you picked the right lie algebra to study representations of

David Michael Roberts (Dec 01 2020 at 06:03):

@John Baez That direct sum of Lie algebras isn't a product of Lie algebras, but I was wondering if there is some nontrivial extension of Lie algebras happening.

John Baez (Dec 01 2020 at 15:50):

Good comment!

There's a nice formula for the Lie bracket of e6 in terms of operations on the four summands I listed: so(9,1), its two spinor representations S+ and S-, and its trivial representation on R.

John Baez (Dec 01 2020 at 15:52):

There are various things going on... for example if we look at just so(9,1) + S+, that'll be a Lie subalgebra of e6, and it's a semidirect product of so(9,1) and its right-handed spinor representation S+, which is an abelian Lie algebra acted on by so(9,1).

John Baez (Dec 01 2020 at 15:54):

You can do the same for so(9,1) $\oplus$ S+ $\oplus$ R.

John Baez (Dec 01 2020 at 15:57):

But the bracket of a guy in S+ and a guy in S- gives you a guy in so(9,1). S- is just the dual of S+, while the adjoint rep so(9,1) is self-dual, so we can take the action

so(9,1) $\otimes$ S+ $\to$ S+

and dualize it to get a linear map

S+ $\otimes$ S- $\to$ so(9,1)

and this is the bracket between guys in S+ and S-, up to a constant factor that I haven't computed, probably $\pm 1$.

John Baez (Dec 01 2020 at 15:59):

All this is most easily understood by analogy to the case of sl(2,C) acting on the Riemann sphere, where you can visualize what's going on. That's why I keep harping on that analogy: it brings everything down to earth.

David Michael Roberts (Dec 02 2020 at 03:59):

Hmm, I was more wondering if the whole e6 Lie algebra was an extension somehow, but this is silly: isn't it meant to be one of the (semi)simple Lie algebras? (though I'm also not quite sure what's happening here over R instead of C)

John Baez (Dec 02 2020 at 05:44):

It's semisimple. I'm working over R and looking at a specific real form of E6, but it's still semisimple. As I tried to explain on the n-Category Cafe, I'm building it by gluing together two semidirect products.