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.
