Non linear quantum electrodynamics, longitudinal photon state and cosmological distances

In this work, I study the non-linear QED effects to the Maxwell equations due to the interaction of the electromagnetic waves with an external magnetic field with arbitrary direction with respect to the direction of propagation of the electromagnetic waves. I show that in the case when the external magnetic field has a longitudinal component with respect to the electromagnetic wave direction of propagation, a longitudinal photon state is generated in a magnetized vacuum. The mixing of transverse and longitudinal photon states causes a decrease in the group velocity with respect to the velocity of electromagnetic waves in a vacuum. In a cosmological context, as far as a cosmic magnetic field is concerned, the decrease of the group velocity of any form of electromagnetic radiation would cause changes in several distance measures in cosmology. The most affected of these distance measures turn out to be the comoving distance and the luminosity distance. The absolute change in distance measures are usually large but the relative changes are very small.