Reconciling a quantum gravity minimal length with lack of photon dispersion

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 push the minimal length scale to distances smaller than the Planck length. This poses a challenge for such minimal distance models. Here we propose a modification of the position and momentum operators, xˆ and pˆ, which lead to a minimal length scale, but preserve the photon energy-momentum relationship E=pc. In this way there is no dispersion of photons with different energies. Additionally, this can be accomplished without modifying the commutation relationship [xˆ,pˆ]=iħ.