Cosmic rays and TeV photons as probes of quantum properties of space-time

It has been recently observed that small violations of Lorentz invariance, of a type which may arise in quantum gravity, could explain both the observations of cosmic rays above the GZK cutoff and the observations of 20-TeV gamma rays from Markarian 501. We show here that different pictures of the short-distance structure of space-time would lead to different manifestations of Lorentz-invariance violation. Specifically, the deformation of Lorentz invariance needed to resolve these observational paradoxes can only arise within commutative short-distance pictures of space-time. In noncommutative space-times there is no anomalous effect, at least at leading order. Also exploiting the fact that arrival-time delays between high energy photons with different energies would arise in both the commutative and the noncommutative Lorentz-violation pictures, we describe an experimental programme, based on time-of-arrival analysis of high energy photons and searches of violations of GZK and TeV-photon limits, which could discriminate between alternative scenarios of Lorentz-invariance breakdown and could provide and unexpected window on the (quantum) nature of space-time at very short distances.