Reformulating polarized radiative transfer for astrophysical applications (I). A formalism allowing non-local Magnus solutions

Abstract: The solar atmosphere is diagnosed by solving the polarized radiative transfer problem for plasmas in Non-Local Thermodynamic Equilibrium (NLTE). A key challenge in multidimensional NLTE diagnosis is to integrate efficiently the radiative transfer equation (RTE), but current methods are local, i.e. limited to constant propagation matrices. This paper introduces a formalism for non-local integration of the RTE using the Magnus expansion. We begin by framing the problem in terms of rotations within the Lorentz / Poincare group (Stokes formalism), motivating the use of the Magnus expansion. By combining the latter with a highly detailed algebraic characterization of the propagation matrix, we derive a compact analytical evolution operator that supports arbitrary variations of the propagation matrix and allows to increasingly consider any order in the Magnus expansion. Additionally, we also reformulate the inhomogeneous part of the RTE, again using the Magnus expansion, and introducing the new concept of inhomogeneous evolution operator. This provides the first consistent, general, and non-local formal solution to the RTE that is furthermore efficient and separates integration from the algebraic formal solution. Our framework is verified analytically and computationally, leading to a new family of numerical radiative transfer methods and potential applications such as accelerating NLTE calculations. With minor adjustments, our results apply to other universal physical problems sharing the Lorentz / Poincare algebra in special relativity and electromagnetism.