Building the E10 lattice with integer octonions

[john baez]

Greg Egan just proved something nice: the E10 lattice, famous in string theory and supergravity, can be described as the lattice of self-adjoint matrices with integral octonions as entries!

I'm not sure this result is new, but I've been wanting it for quite a while and haven't seen it anywhere. So, it's nice to know.

 

[marcus]

Great to see you! Thanks for dropping in with that nice bit of news!

 

[marcus]

Your mentioning (matrices of) octonions reminded me (via matrices of quaternions) of the paper that came out yesterday by Chamseddine Connes Mukhanov. It employs 2-by-2 matrices of quaternions. I can't imagine any connection with E10, but this CCM paper got me curious because it proposes a way to "reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model."

http://arxiv.org/abs/1411.0977
Geometry and the Quantum: Basics
Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov
(Submitted on 4 Nov 2014)
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M₂(H) and M₄(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.
33 pages, 2 figures

 

[john baez]

Yes, Connes and Chamseddine have been using the quaternions to get the weak force from noncommutative geometry for a long time, basically because SU(2) is the unit quaternions. However, they have been doing it in ever more elegant ways as the years go by. Recently they came out with a paper that got the Pati-Salam grand unified theory from seemingly natural assumptions. I hadn't seen this even newer paper.

 

[marcus]

https://plus.google.com/117663015413546257905/posts/D2jALr2L4Wt
Thanks for the "publicly shared" google+ link! It reminds me of the good times when there was "This Week's Finds".

I'm a little put off by the newer social media and structured online communities. So was reluctant to participate in google+

But having publicly shared essays that introduce one to major research that has recently appeared in the TWF manner is SUPER!
So I will be going back there.

Tell me, John Baez, what is the drawback with Connes' approach that prevents wider adoption? Is it too hard? Is there some missing piece to the picture?

It looks like this is the way to go if you want a unified model of dynamic geometry and the standard particle model. Why haven't a whole lot more people jumped on the bandwagon? I'll post the link and abstract to that 2013 paper you discussed in your introductory essay.

http://arxiv.org/abs/1304.8050
Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification
Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom
(Submitted on 30 Apr 2013)
The assumption that space-time is a noncommutative space formed as a product of a continuous four dimensional manifold times a finite space predicts, almost uniquely, the Standard Model with all its fermions, gauge fields, Higgs field and their representations. A strong restriction on the noncommutative space results from the first order condition which came from the requirement that the Dirac operator is a differential operator of order one. Without this restriction, invariance under inner automorphisms requires the inner fluctuations of the Dirac operator to contain a quadratic piece expressed in terms of the linear part. We apply the classification of product noncommutative spaces without the first order condition and show that this leads immediately to a Pati-Salam SU(2)_Rx SU(2)_Lx SU(4) type model which unifies leptons and quarks in four colors. Besides the gauge fields, there are 16 fermions in the (2,1,4)+(1,2,4) representation, fundamental Higgs fields in the (2,2,1), (2,1,4) and (1,1,1+15) representations. Depending on the precise form of the order one condition or not there are additional Higgs fields which are either composite depending on the fundamental Higgs fields listed above, or are fundamental themselves. These additional Higgs fields break spontaneously the Pati-Salam symmetries at high energies to those of the Standard Model.
41 pages, 2 figures.

I've been interested in that young person Walter vSuijlekom ever since
http://arxiv.org/abs/1301.3480
a paper he wrote with Matilde Marcolli that actually tries out a spectral geometry extension of LQG spin networks. WvS is at Nijmegen, where Renate Loll is. To an amateur non-expert like myself it's encouraging to see different approaches to quantum geometry-and-matter coming together. van Suijlekom looks strong, Nijmegen looks like a good place.

In your essay you helped readers by giving a link to an expository essay about grand unifications that you and Heurta wrote. Here's the section on Pati-Salam if anyone here wants to glance at it.
http://math.ucr.edu/~huerta/guts/node18.html
It seems to "intuitize" GUT mathematics to some extent. As Archimedes said "give me a diagram that commutes and I will move the world" or words to that effect.

 

[john baez]

You don't need to join Google+ to read public posts there; you can read all my stuff here. By now there's enough to waste a whole day or two reading it.

I'm not working seriously on quantum gravity or n-categories anymore; I'm doing network theory and math related to climate science and biology. But I still like fundamental physics and pure math. So I post about it, but most of my posts are meant to be readable by ordinary folks - unlike This Week's Finds, which was more serious.

"Tell me, John Baez, what is the drawback with Connes' approach that prevents wider adoption? Is it too hard?"

It's quite hard, but also some quantum field theorists have argued that it's really just another way of formulating ordinary quantum field theory. See Jacques Distler's blog post for a nice example of a well-informed physicist with an unimpressed, "where's the beef" attitude.

Nonetheless there are some very good people working on the noncommutative geometry approach, so it's not even clear that a bandwagon would speed progress at this point.

 

[marcus]

You don't need to join Google+ to read public posts there; you can read all my stuff here. By now there's enough to waste a whole day or two reading it.
...most of my posts are meant to be readable by ordinary folks ...

So that's the menu.
https://plus.google.com/117663015413546257905/posts
It's an informative and attractive format, with graphics and lead sentence clues as to content.

Near the top of the menu I see the piece about Greg Egan's E10 discovery. The essay makes the connection with billiard ball bouncing chaos and the (even classical) confusion at the "big bang" start of expansion. So that makes what Egan found about E10 wide-audience interesting. Not so technical.
It reminds me of things I heard Steve Carlip talk about one time: asymptotic silence, BKL, spontaneous dimensional reduction. It also is evocative of a recent paper by Aurelien Barrau and friends about signature change, at high density near start of expansion, with lightcones shrunk down to lines (zero speed of light) spreading out to cones as signature shifts to Lorentz. In case anyone is interested here's the Barrau et al "deafness at extreme density" paper.
http://arxiv.org/abs/1411.0272
I posted about it here:
https://www.physicsforums.com/threa...ven-if-it-is-right.775104/page-2#post-4901880
Maybe it should be described as "disconnectedness at extreme density". Even near neighbors lose causal contact. What is the entropy of a bunch of causally isolated points, no one of which knows anything about the rest of the universe? From the standpoint of those points I would say zero. It knows all there is to know. It's a curious idea. But I want to look at your essay about what Greg Egan found out!

 

[marcus]

You call the essay "Cosmological billiards and space-time crystals"
It's a fascinating title. And you start out with the classical BKL...

When you put time in reverse and shrink it down, even without matter, geometry itself goes crazy.
And this can be modeled by billiard balls bouncing around in odd shaped cells---which intuitizes the chaos for us.
Now how do you bring in the new theorem which Egan just proved?

It is going to have to do with spacetime crystals.
Yes, here's the "n-category cafe" link:
https://golem.ph.utexas.edu/category/2014/11/integral_octonions_part_7.html
I'm new to the format, so I don't see how to get a permanent link for the G+ essay. Maybe there is none but in any case we can go to:
https://plus.google.com/117663015413546257905/posts
and find it in the menu and click on its "read more..."
Ahah! I see how to get the permalink, by clicking the date at the top. It is:
https://plus.google.com/117663015413546257905/posts/aALBNUwnWSn

 

[Jim Kata]

Professor Baez, Egan showed that:
\mathfrak{h}_{2}\left(\mathbf{O}\right) \cong E_{10}
any speculation on
\mathfrak{h}_{3}\left(\mathbf{O}\right) \cong \ ?

 

[john baez]

Check out my last two comments in that discussion, here and here. Right now things are at a very suspenseful stage; I woke up last night and started mentally trying a calculation that would be required to get everything to work in an maximally nice way. I'll post something else when I figure out exactly what's going on.

 

[john baez]

Okay, I proved that \mathfrak{h}_3(\mathbf{O}), the lattice of 3 \times 3 self-adjoint matrices with integral octonions as entries, is the same as \mathrm{D}_{24}^{+++}. This is a lattice in 27 dimensions which is believed to play a role in bosonic string theory analogous to the role played by \mathrm{E}_{11} in superstring theory.

A bit more precisely: just as the 11-dimensional lattice \mathrm{E}_{11} = \mathrm{E}_8^{+++} is connected to M-theory and contains a 10-dimensional lattice \mathrm{E}_{10} = \mathrm{E}_8^{++} which shows up in superstring theory, the 27-dimensional lattice \mathrm{D}_{24}^{+++} contains a 26-dimensional lattice \mathrm{D}_{24}^{++} which shows up in bosonic string theory. Some have suggested that there's a theory waiting to be discovered in 27 dimensions, "bosonic M-theory", which will complete this pattern. However, nobody seems to know!

You can see the details starting here. I explain the {}^{++} and {}^{+++} constructions, which are ways of getting lattices that have 2 or 3 more dimensions than the one you started with.

I'm an agnostic about whether any of these structures are relevant to the physics of our universe, but the math sure is fun!

 

[MTd2]

I am under the impression Tony Smith has proved this many years ago. But I am not sure. I sent him an email.

 

[Jim Kata]

Just rambling, but what would fascinate me is if the 3 generations of particles in the standard model could be explained by those 3 copies of E_8 in the off diagonal terms in h_3(O).

 

[inflector]

To date, my most significant contribution to physics and math is probably sending John the invite to Google Plus, and then hounding him a bit to get him to go there.

Not much I realize, but hey, it's a start.