Second Quantization for the Kepler Problem

TL;DR

This paper explores the quantum Kepler problem for a spin-1/2 particle, revealing deep symmetries and connecting bound states to solutions of the Weyl equation on a spherical spacetime, with implications for atomic structure modeling.

Contribution

It establishes a unitary equivalence between the bound state Hilbert space of the quantum Kepler problem and the Weyl equation solutions on 3, linking quantum mechanics, symmetry, and field theory.

Findings

Hilbert space of bound states is equivalent to Weyl equation solutions on 3

Fermionic Fock space corresponds to a massless spin-1 field on 3

Modified Hamiltonian reproduces Madelung rules for electron filling

Abstract

The Kepler problem concerns a point particle in an attractive inverse square force. After a brief review of the classical and quantum versions of this problem, focused on their hidden $\text{SU}(2) \times \text{SU}(2)$ symmetry, we discuss the quantum Kepler problem for a spin-$\frac{1}{2}$ particle. We show that the Hilbert space $\mathcal{H}$ of bound states for this problem is unitarily equivalent, as a representation of $\text{SU}(2) \times \text{SU}(2)$, to the Hilbert space of solutions of the Weyl equation on the spacetime $\mathbb{R} \times S^3$. This equation describes a massless left-handed spin-$\frac{1}{2}$ particle. We then form the fermionic Fock space on $\mathcal{H}$ and show this is unitarily equivalent to the Hilbert space of a massless left-handed spin-$\frac{1}{2}$ free quantum field on $\mathbb{R} \times S^3$, again as representations of $\text{SU}(2) \times \text{SU}(2)$. By modifying the Hamiltonian of this free field theory, we obtain the well-known "Madelung rules". These give a reasonable approximation to the observed filling of subshells as we consider elements with more and more electrons, and match the rough overall structure of the periodic table.