The debate and controversy concerning the momentum of light in a dielectric medium (Abraham vs Minkowski) is well-known and still not fully resolved. In this paper, we investigate the origin of both momenta in the frame of special relativity by considering photons in media as relativistic quasiparticles. We demonstrate for the first time to the best of our knowledge that the Minkowski form of the photon mass, momentum, and energy follows directly from the relativistic energy conservation law. We introduce a new expression for the momentum of light in a dispersive medium, consistent with the experimentally observed propagation of photons at the group velocity. Finally, the effect of light-induced optical stretching is discussed, which can be used for experimental verification of the existing expressions for the photon momentum.