Second quantization for the Kepler problem

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 SU (2) × SU (2) symmetry, we discuss the quantum Kepler problem for a spin- 1 2 particle. We show that the Hilbert space H of bound states for this problem is unitarily equivalent, as a representation of SU (2) × SU (2), to the Hilbert space of solutions of the Weyl equation on the spacetime R × S 3 . This equation describes a massless chiral spin- 1 2 particle. We then form the fermionic Fock space on H and show this is unitarily equivalent to the Hilbert space of a massless chiral spin- 1 2 free quantum field on R × S 3 , again as representations of SU (2) × 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.