This thesis is a focused study of the polarization characteristics of radiative transfer in a strong magnetic field. The main process examined here is magnetized Compton scattering in a non-relativistic regime (i.e.\ magnetized Thomson scattering), and we focus on applying this study to predict polarization properties of the X-ray emission from magnetars. Magnetars are a highly magnetic sub-class of neutron stars, characterized by their extremely high surface magnetic fields, comparable to or exceeding the quantum critical field (Bcr≃4.41×1013 Gauss) at which an electron's cyclotron energy and rest mass energy are equal. There are 29 known/candidate magnetars at this time, and they commonly exhibit persistent quasi-thermal surface emission in soft X-rays with flat tails extending into the hard X-rays up to around 150 keV, as well as transient bursting activity in hard X-rays attributed to magnetospheric flares.

Magnetized Thomson scattering refers to electron-photon scattering in a background magnetic field. The field introduces anisotropy to the problem, giving it a more complicated angular dependence. It also produces a strong frequency dependence to the cross section: it is resonant at the cyclotron frequency ωB=eB/mc. Additionally, electron motion perpendicular to the field becomes increasingly suppressed at higher field strengths, leading to a reduction in the cross section for certain incoming photon angles when ω≪ωB. There are complicated polarization characteristics for the process as well, with the differential cross section depending on the initial and final polarization state of the photon. An important distinction occurs between photons that have a component of linear polarization parallel to the field and those that are fully orthogonal to it.

We explore this process in detail using a Monte Carlo simulation model, treating the transfer primarily in slab geometries, a common simplification. This allows a direct comparison with previous work and is an important step towards achieving more complicated geometries and scattering regimes. We fully map the frequency and angular dependence of this process in the optically thick regime, capturing both resonant and non-resonant properties of the scattering. We present results for a model of magnetar persistent surface emission, as well as a simple magnetar flare model. In both cases the transfer is purely due to magnetic Thomson scattering, and we superimpose the emission from regions of different temperature, density, and magnetic field. For magnetar surface emission, we see a phase-dependent linear polarization, forming either a single- or double-peaked pattern with a maximum level of roughly ∼25%, depending on the angle between the observer and the spin axis. This could have important implications for polarimetric determination of the effect of vacuum birefringence as polarized X-rays transfer through the magnetosphere to infinity. For the flare model we see much stronger polarization signals as the emission is coming from more localized regions, and it is highly dependent on viewing geometry and frequency.

A secondary process is also examined due to its importance in magnetized plasmas: the so-called generalized Faraday effect. This is analogous to the ordinary Faraday effect, where the phase lag caused by birefringence of circular eigenmodes of electromagnetic waves produces a constant rotation of the plane of linear polarization for a propagating wave. The generalized effect occurs when the eigenmodes are no longer circular, and it produces a very similar rotation when viewed in terms of the Poincar'e sphere. When this effect is assumed to be prolific, the transfer can be reformulated in terms of the normal modes in what is called the normal mode description. We explore the parameter space in which this description is valid, and develop a method to handle transfer in certain regimes where it is invalid. This prepares the way for incorporating such nuances in future developments of the magnetized radiative transfer problem.