John Carlos Baez on Nostr: In this post, I'll show how to connect the periodic table to a quantum field theory ...

In this post, I'll show how to connect the periodic table to a quantum field theory on the 'Einstein universe' — a universe where space is a 3-sphere.

This is the latest step in hundreds of years of pondering the 'Kepler problem': the problem of particles orbiting in ellipses, attracted by an inverse square force. After Kepler and Newton's pioneering work, people realized this problem had a secret 4-dimensional rotation symmetry! Later Pauli applied this symmetry to the hydrogen atom, which is a *quantum* version of the Kepler problem.

In earlier episodes here, I explained that the hydrogen atom is equivalent to a massless particle in the Einstein universe. This explains the 4d rotation symmetry.

This was already known. But then I showed it still works if you take the spin of the electron in the hydrogen atom into account. Then the hydrogen atom is equivalent to a massless *spin-1/2* particle in the Einstein universe!

Then I ‘second quantized’ both theories. This is a systematic procedure you can do to any quantum theory to get a new one. If your original theory described some thing, the second quantized theory describes *collections* of such things.

If we do this to the massless spin-1/2 particle, we get a quantum field theory! This describes collections of noninteracting massless spin-1/2 particles.

If we second quantize the hydrogen atom, we get a theory of multi-electron atoms. But it's a naive theory where the electrons don’t interact. So in this naive theory we don’t get the periodic table of elements.

But in today's post I'll tweak the theory so we do get the observed periodic table, at least roughly!

https://johncarlosbaez.wordpress.com/2025/10/14/the-kepler-problem-part-12/