Quantum gravity and the Coulomb potential

TL;DR

This paper applies loop quantum gravity techniques to regularize the Coulomb potential in quantum mechanics, analyzing how boundary conditions at the singularity influence the energy spectrum in a polymer quantization framework.

Contribution

It introduces a singularity resolution method from loop quantum gravity to the polymer representation of quantum mechanics with Coulomb potential, highlighting the impact of boundary conditions.

Findings

Antisymmetric sector spectrum converges to Coulomb spectrum as lattice spacing decreases.

Symmetric sector's spectrum is significantly affected by boundary conditions at the singularity.

Boundary conditions influence the spectrum even with very small lattice spacing.

Abstract

We apply a singularity resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on R with the singular -1/|x| potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite difference approximation of the Schrodinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at x=0, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized.