Quantization of the Linearized Kepler Problem

TL;DR

This paper explores the quantization of the linearized Kepler problem via group theoretical methods, revealing its symmetry group as SU(2,2) and representing energies through harmonic oscillators with specific constraints.

Contribution

It introduces a novel group-theoretical quantization approach for the linearized Kepler problem, including negative, positive, and zero energy cases, based on the Kustaanheimo-Stiefel transformation.

Findings

Symmetry group identified as SU(2,2)

Negative energy case as four harmonic oscillators with constraints

Positive energy case as four repulsive oscillators with constraints

Abstract

The linearized Kepler problem is considered, as obtained from the Kustaanheimo-Stiefel (K-S)transformation, both for negative and positive energies. The symmetry group for the Kepler problem turns out to be SU(2,2). For negative energies, the Hamiltonian of Kepler problem can be realized as the sum of the energies of four harmonic oscillator with the same frequency, with a certain constrain. For positive energies, it can be realized as the sum of the energies of four repulsive oscillator with the same (imaginary) frequency, with the same constrain. The quantization for the two cases, negative and positive energies is considered, using group theoretical techniques and constrains. The case of zero energy is also discussed.