Simple model of pointlike spacetime defects and implications for photon propagation

A model in which pointlike defects are randomly embedded in Minkowski spacetime is considered. The distribution of spacetime defects is constructed to be Lorentz-invariant. It does not introduce a preferred reference frame, because it is based on a sprinkling process. A field-theoretic action for the photon and a fermion is set up, in which the photon is assumed not to couple to the defects directly, but via a scalar field. We are interested in signs for Lorentz violation caused by the spacetime defects, which are expected to reveal themselves in the photon sector. A modification of the photon dispersion relation may result as a quantum effect and we compute it at leading order perturbation theory. The outcome of the calculation is that the photon dispersion law remains conventional, if the defect distribution is dense, homogeneous, and isotropic. This result sheds some new light on Lorentz violation in the framework of a small-scale structure of spacetime. It shows that Lorentz invariance can be preserved even in the presence of a spacetime structure that is supposed to emerge at the Planck scale. This conclusion has already been drawn on general grounds in other publications, where the current paper delivers a demonstration by a direct computation in a simple model.