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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?