Derivation of the radiative transfer equation for line polarization studies in the presence of magnetic field in astrophysics

In this paper, we derive in a coherent manner, starting from the basic equations of evolution of a quantum mechanical system, the transfer equation for a polarized pencil of radiation interacting with a bath of atoms, in the presence of a weak magnetic field (in the domain of sensitivity to the Hanle effect: 0.1 ≲ ω Lτ ≲ 10). This paper has been inspired by astrophysical purposes: the interpretation of line polarization induced by anisotropic excitation of the levels, eventually modified by the local magnetic field (the Hanle effect); the polarization can be due to scattering of the incident anisotropic radiation, as in solar prominences, or to impact polarization, as in solar flares. The physical conditions are then those of astrophysical media: any direction of polarization and magnetic field, two-level atom approximation not valid, weak radiation field (so that the bare atom description is convenient), weak density of perturbers (so that the impact approximation is valid). In the preceding paper (paper I), the master equation for the atomic density matrix has been derived in the framework of the impact approximation. In the present paper, we obtain the spontaneous emission 4-vector and absorption and stimulated emission 4 x 4 matrices entering the transfer equation for polarized radiation, as functions of the absorbed and emitted radiation polarization tensors, themselves functions of the Zeeman coherences of the atomic density matrix. In the present work line profiles have been ignored: line profile studies are rejected to future work