Reconciling a quantum gravity minimal length with lack of photon dispersion

TL;DR

This paper proposes a modification to quantum operators that introduces a minimal length scale without causing photon dispersion, reconciling quantum gravity theories with observational constraints.

Contribution

It introduces a new operator modification that maintains photon dispersion relations while incorporating a minimal length scale, addressing previous observational challenges.

Findings

Photon dispersion is avoided despite the minimal length scale.

The modification preserves the standard energy-momentum relationship for photons.

The approach aligns quantum gravity models with gamma-ray burst observations.

Abstract

Generic arguments lead to the idea that quantum gravity has a minimal length scale. A possible observational signal of such a minimal length scale is that photons should exhibit dispersion. In 2009 the observation of a short gamma ray burst seemed to bound the minimal length scale to distances smaller than the Planck length, implying that spacetime appeared continuous to distances below the Planck length. This poses a challenge for such minimal distance models. Here we propose a modification of the position and momentum operators, ${\hat x}$ and ${\hat p}$, which lead to a minimal length scale, but preserve the photon energy-momentum relationship $E = p c$. In this way there is no dispersion of photons with different energies. This can be accomplished without modifying the commutation relationship $[{\hat x}, {\hat p}] = i \hbar$.