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.
Categories: math-ph math.MP