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Three Generations in E7
It’s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made this pattern seem natural. That’s what my new paper is about.
It’s my third paper about exceptional algebraic structures and the Standard Model. When you classify famous gadgets in algebra, beautiful gadgets with fancy names like ‘simple Lie algebras’ and ‘Euclidean Jordan algebras’ and ‘positive hermitian Jordan pairs’, you tend to get infinite series of them—together with a few exceptions that can be built using the octonions. This is a bit spooky, so I’ve been interested in this for a long time.
A few physicists have hoped that these exceptions are good for something. For example, maybe the quirky features of our best theory of particle physics, the Standard Model, aren’t accidental. Perhaps they fall out naturally from some exceptional algebraic structure.
It’s a long shot, but we’ve been stuck on figuring out new fundamental laws of particle physics for so long—roughly since the early 1980s—that it’s worth a try.
In 2018, Michel Dubois-Violette and Ivan Todorov noticed that the gauge group of the Standard Model falls out as symmetries of the so-called ‘exceptional Jordan algebra’ together with some ordinary Jordan algebras sitting inside it. I tried to clarify that here, with a huge amount of help from an excellent young mathematician:
• John Baez and Paul Schwahn, The Standard Model gauge group from the exceptional Jordan algebra. (Blog article here.)
It’s very nice, because the Jordan algebras in question arise naturally when you try to axiomatize the foundations of quantum physics. It would be so cool if something about quantum physics made the Standard Model seem mathematically natural!
But really this result only concerns the gauge bosons in the Standard Model: the photon, gluons, and the W and Z bosons. It says nothing about the fermions—that is, the quarks and leptons. And it seems quite hard to get those into the picture.
In 2020, Latham Boyle tried to solve this problem by tensoring the exceptional Jordan algebra with the complex numbers. This made one generation of fermions appear quite naturally! But the connection to the foundations of quantum physics seemed lost: tensoring the exceptional Jordan algebra with the complex numbers seems at first like it might be just a formal trick.
This spring, Latham and his student Endre Bokor and I showed the connection to quantum physics is not lost: