Integration of the radiative transfer equation for polarized light: The exponential solution

The radiative transfer equation (RTE) for polarized light accepts a convenient exponential solution when the absorption matrix commutes with its integral. We characterize some of the matrix depth variations which are compatible with the commutation condition. Eventually the vector solution may be diagonalized and one may obtain four independent scalar solutions with four optical depths, complex in general. When the commutation condition is not satisfied, one must resort to a determination of an appropriate evolution operator, which is shown to be well determined mathematically, but whose explicit form is, in general, not easy to apply in a numerical code. However, we propose here an approach to solve a general case not satisfying the commutation condition.