Posts with this logo use 
November 30, 2025
Octonions and the Standard Model (Part 12)
Posted by John Baez
Having spent a lot of time pondering the octonionic projective plane and its possible role in the Standard Model of particle physics, I’m now getting interested in the ‘bioctonionic plane’, which is based on the bioctonions rather than the octonions .
The bioctonionic plane also has intriguing mathematically connections to the Standard Model. But it’s not a projective plane in the axiomatic sense — and it can’t be constructed by straightforwardly copying the way you build a projective plane over a division algebra, since unlike the octonions, the bioctonions are not a division algebra. Nonetheless we can define points and lines in the bioctonionic plane. The twist is that now some pairs of distinct lines intersect in more than one point — and dually, some pairs of distinct points lie on more than one line. It obeys some subtler axioms, so people call it a Hjelmslev plane.
I am not ready to give a really good explanation of the bioctonionic plane! Instead, I just want to lay out some basic facts about how it fits into mathematics — and possibly physics.
November 22, 2025
Beyond the Geometry of Music
Posted by John Baez
Yesterday I had a great conversation with Dmitri Tymoczko about groupoids in music theory. But at this Higgs Centre Colloquium, he preferred to downplay groupoids and talk in a way physicists would enjoy more. Click here to watch his talk!
November 5, 2025
The Inverse Cube Force Law
Posted by John Baez
Here’s a draft of my next column for the Notices of the American Mathematical Society. It’s about the inverse cube force law in classical mechanics.
November 3, 2025
Second Quantization and the Kepler Problem
Posted by John Baez
The poet Blake wrote that you can see a world in a grain of sand. But even better, you can see a universe in an atom!
Bound states of hydrogen atom correspond to states of a massless quantum particle moving at the speed of light around the Einstein universe — a closed, static universe where space is a 3-sphere. We need to use a spin-½ particle to account for the spin of the electron. The states of the massless spin-½ particle where it forms a standing wave then correspond to the orbitals of the hydrogen atom. This explains the secret 4-dimensional rotation symmetry of the hydrogen atom.
In fact, you can develop this idea to the point of getting the periodic table of elements from a quantum field theory on the Einstein universe! I worked that out here:
but you can see a more gentle explanation in the following series of blog articles.
November 2, 2025
Dynamics in Jordan Algebras
Posted by John Baez
In ordinary quantum mechanics, in the special case where observables are described as self-adjoint complex matrices, we can describe time evolution of an observable using Heisenberg’s equation
where is a fixed self-adjoint matrix called the Hamiltonian. This framework is great when we want to focus on observables rather than states. But Heisenberg’s equation doesn’t make sense in a general Jordan algebra. In this stripped-down framework, all we can do is raise observables to powers and take real linear combinations of them. This lets us define a ‘Jordan product’ of observables:
but not commutators and not multiplication by . What do we do then?