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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.

  • Part 1: a quick overview of Kepler’s work on atoms and the solar system, and more modern developments.

  • Part 2: why the eccentricity vector is conserved for a particle in an inverse square force, and what it means.

  • Part 3: why the momentum of a particle in an inverse square force moves around in a circle.

  • Part 4: why the 4d rotation group SO(4)\text{SO}(4) acts on bound states of a particle in an attractive inverse square force.

  • Part 5: quantizing the bound states of a particle in an attractive inverse square force, and getting the Hilbert space L 2(S 3)L^2(S^3) for bound states of a hydrogen atom, neglecting the electron’s spin.

  • Part 6: how the Duflo isomorphism explains quantum corrections to the hydrogen atom Hamiltonian.

  • Part 7: why the Hilbert space of bound states for a hydrogen atom including the electron’s spin is L 2(S 3)⊗ℂ 2.L^2(S^3) \otimes \mathbb{C}^2.

  • Part 8: why L 2(S 3)⊗ℂ 2L^2(S^3) \otimes \mathbb{C}^2 is also the Hilbert space for a massless spin-1/2 particle in the Einstein universe.

  • Part 9: a quaternionic description of the hydrogen atom’s bound states (a digression not needed for later parts).

  • Part 10: changing the complex structure on L 2(S 3)⊗ℂ 2L^2(S^3) \otimes \mathbb{C}^2 to eliminate negative-energy states of the massless spin-1/2 particle, as often done.

  • Part 11: second quantizing the massless spin-1/2 particle and getting a quantum field theory on the Einstein universe, or alternatively a theory of collections of electrons orbiting a nucleus.

  • Part 12: obtaining the periodic table of elements from a quantum field theory on the Einstein universe.

I explain how the Hamiltonian of this quantum field theory has to be tweaked a bit to give the ‘Madelung rules’ for the order in which electrons get added as we go along the periodic table:

Posted at November 3, 2025 8:51 AM UTC

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