Violation of Lorentz symmetry on quantum space-times

We show that, in a test field approximation, the evolution equation of quantum electromagnetic (EM) field propagating on an isotropic cosmological quantum space-time corresponds to quantum evolution of the same test EM field on an emerging effective (dressed) space-time. When only a test field approximation is assumed, all field modes probe the same effective background. However, when the back-reaction of the field modes on the quantum geometry is taken into account, through a Born-Oppenheimer approximation, a rainbow (i.e., a mode-dependent) metric emerges on which the local Lorentz symmetry is violated.