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The free photon dispersion relation is a reference quantity for high precision tests of Lorentz Invariance. We first outline theoretical approaches to a conceivable Lorentz Invariance Violation (LIV). Next we address phenomenological tests based on the propagation of cosmic rays, in particular in Gamma Ray Bursts (GRBs). As a specific concept, which could imply LIV, we then focus on field theory in a non‐commutative (NC) space, and we present non‐perturbative results for the dispersion relation of the NC photon.

Could the photon dispersion relation be non‐linear?