Relativistic Runge-Lenz vector: from ${\cal N}=4$ SYM to SO(4) scalar field theory

TL;DR

This paper extends the classical Runge-Lenz vector to a relativistic context within an SO(4) symmetric scalar field theory derived from ${ m extbf{N}=4}$ SYM, enabling spectrum calculations of the relativistic hydrogen atom.

Contribution

It introduces a relativistic Runge-Lenz vector, generalizes the Kustaanheimo-Stiefel transformation, and connects these to the spectrum of the relativistic hydrogen atom.

Findings

Relativistic Runge-Lenz vector generates SO(4) algebra.

Calculated relativistic hydrogen atom spectrum using SO(4) symmetry.

Related results to the relativistic oscillator and cusp anomalous dimension.

Abstract

Starting from ${\cal N}=4$ SYM and using an appropriate Higgs mechanism we reconsider the construction of a scalar field theory non-minimally coupled to a Coulomb potential with a relativistic SO(4) symmetry and check for scalar field consistency conditions. This scalar field theory can also be obtained from a relativistic particle Lagrangian with a proper implementation of the non-minimal coupling. We provide the generalization of the non-relativistic construction of the Runge-Lenz vector to the relativistic case and show explicitly that this new vector generates the SO(4) algebra. Using the power of the SO(4) symmetry, we calculate the relativistic hydrogen atom spectrum. We provide a generalization of the Kustaanheimo-Stiefel transformation to the relativistic case and relate our results with the corresponding relativistic oscillator. Finally, in the light of these results, we reconsider the calculation of the hydrogen atom spectrum from the cusp anomalous dimension given in [2].