  Polarization of radiation is a powerful tool to study cosmic magnetism and
analysis of polarization can be used as a diagnostic tool for large-scale
structures. In this paper, we present a solid theoretical foundation for using
polarized light to investigate large-scale magnetic field structures: the
cosmological polarized radiative transfer (CPRT) formulation. The CPRT
formulation is fully covariant. It accounts for cosmological and relativistic
effects in a self-consistent manner and explicitly treats Faraday rotation, as
well as Faraday conversion, emission, and absorption processes. The formulation
is derived from the first principles of conservation of phase-space volume and
photon number. Without loss of generality, we consider a flat
Friedmann-Robertson-Walker (FRW) space-time metric and construct the
corresponding polarized radiative transfer equations. We propose an all-sky
CPRT calculation algorithm, based on a ray-tracing method, which allows
cosmological simulation results to be incorporated and, thereby, model
templates of polarization maps to be constructed. Such maps will be crucial in
our interpretation of polarized data, such as those to be collected by the
Square Kilometer Array (SKA). We describe several tests which are used for
verifying the code and demonstrate applications in the study of the
polarization signatures in different distributions of electron number density
and magnetic fields. We present a pencil-beam CPRT calculation and an all-sky
calculation, using a simulated galaxy cluster or a model magnetized universe
obtained from GCMHD+ simulations as the respective input structures. The
implications on large-scale magnetic field studies are discussed; remarks on
the standard methods using rotation measure are highlighted.
