Octonions and the Standard Model

I avoid talking about fundamental physics or pure math here—I do that on the n-Category Café. I also avoid talking about category theory, except for its applications to electrical circuits, chemical reaction networks and the like. I discuss more ‘pure’ aspects of category theory on the n-Category Café and the Category Theory Community Server.

I’ve been fascinated by the octonions for a long time now: they’re an enigmatic link between many ‘exceptional’ structures in geometry and group theory.

• John Baez, The octonions.

There have been various attempts to use the octonions in physics. While they play a clear role in superstring theory, which is mathematically beautiful but distant from what we observe in nature, there are also some hopes that they could explain the quirky patterns in the forces and particles we actually see. I’m not extremely optimistic about these hopes, but there are some tantalizing facts here and there, so I’ve decided to write some blog articles explaining them.

I should emphasize that I’m not proposing or even advocating any theory of physics here! Instead, I’m just collecting and explaining some interesting relations between octonionic mathematics and the Standard Model. I thought about this stuff for a long time, so I wanted to write it up before I forget it all—especially some work I did with Greg Egan and John Huerta back in November 2015.

Here are the posts so far:

• Octonions and the Standard Model 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under SU(3).

• Octonions and the Standard Model 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.

• Octonions and the Standard Model 3. How a lepton and a quark fit together into an octonion – at least if we only consider them as representations of SU(3), the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group SU(3).

• Octonions and the Standard Model 4. Introducing the exceptional Jordan algebra: the 3×3 self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the 2×2 adjoint octonionic matrices form precisely the Standard Model gauge group.

I didn’t give the proof of that result.   Instead I moved in a different direction, which should eventually loop back:

• Octonions and the Standard Model 5. How to think of the 2×2 self-adjoint octonionic matrices as 10-dimensional Minkowski space, and pairs of octonions as left- or right-handed Majorana-Weyl spinors in 10 dimensional spacetime.

• Octonions and the Standard Model 6.  The linear transformations of the exceptional Jordan algebra that preserve the determinant form the exceptional Lie group E6. How to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and left-handed spinors in 10d Minkowski spacetime.

• Octonions and the Standard Model 7. How to describe the Lie group E6 using 10-dimensional spacetime geometry. This group is built from the double cover of the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.

• Octonions and the Standard Model 8  A geometrical way to see how E6 is connected to 10d spacetime, based on the octonionic projective plane.

• Octonions and the Standard Model 9.  Duality in projective plane geometry, and how it lets us break the Lie group E6 into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.

• Octonions and the Standard Model 10.  Jordan algebras, their symmetry groups, their invariant structures — and how they connect quantum mechanics, special relativity and projective geometry.

• Octonions and the Standard Model 11.  Particle physics on the spacetime given by the exceptional Jordan algebra: a summary of work with Greg Egan and John Huerta.

As usual, once I start writing about something I get more interested in it. There’s a lot more left to say, and it’s a lot of fun, so there will be more posts.