We develop a framework for the multiple scattering of a polarized wave. We consider particles with spin propagating in a medium filled with scatterers. We write the amplitudes of each spin eigenstate in a local mobile frame. One of the axes is in the direction of propagation of the particle. We use this representation to define a directional Green's operator of the homogeneous medium and also to write the spin-dependent scattering amplitudes. We show that this representation reveals a Berry phase. We establish a generalized Green-Dyson equation for the multiple-scattering problem in this framework. We show that the generalized Green-Dyson equation can be solved by linear algebra if one uses a representation of the rotations based on Wigner D matrices. The properties of light scattering are retrieved if we use spin 1 particles. Our theory allows to take into account several kinds of anisotropies such as circular or linear dichroism and birefringence, Faraday effects, and Mie scattering within the same formalism. Several anisotropies can be present at the same time.
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