For scattering theory in quantum mechanics, one can use the Dyson Equation which states that the Green's function which is a solution to the equation
$$ (E - H_0 - V)G = 1$$
is given by
$$ G = G_0 + G_0 V G, $$
where $H_0$ is the free Hamiltonian, $V$ is a potential and $G_0$ is the solution to $(E-H_0)G_0=1$. I'm wondering how to compute the scattering amplitude from the Dyson Equation when the Green's function is a matrix, as in Dirac theory. In this case, the wave function would be a spinor. Can anybody tell me how to get the scattering amplitude then?