  The transport equations for polarized radiation transfer in non-Riemannian,
Weyl-Cartan type space-times are derived, with the effects of both torsion and
non-metricity included. To obtain the basic propagation equations we use the
tangent bundle approach. The equations describing the time evolution of the
Stokes parameters, of the photon distribution function and of the total
polarization degree can be formulated as a system of coupled first order
partial differential equations. As an application of our results we consider
the propagation of the cosmological gamma ray bursts in spatially homogeneous
and isotropic spaces with torsion and non-metricity. For this case the exact
general solution of the equation for the polarization degree is obtained, with
the effects of the torsion and non-metricity included. The presence of a
non-Riemannian geometrical background in which the electromagnetic fields
couple to torsion and/or non-metricity affect the polarization of photon beams.
Consequently, we suggest that the observed polarization of prompt cosmological
gamma ray bursts and of their optical afterglows may have a propagation effect
component, due to a torsion/non-metricity induced birefringence of the vacuum.
A cosmological redshift and frequency dependence of the polarization degree of
gamma ray bursts also follows from the model, thus providing a clear
observational signature of the torsional/non-metric effects. On the other hand,
observations of the polarization of the gamma ray bursts can impose strong
constraints on the torsion and non-metricity and discriminate between different
theoretical models.
