# Lippmann-Schwinger Equation in Dirac theory

Consider a scattering process of some particle which is described by a Dirac Equation. We use the Lippmann-Schwinger Equation for the total scattering wave function in representation-free form up to second order

$$\lvert \Psi \rangle = \lvert \Psi_0 \rangle +G_0 V \lvert\Psi_0\rangle+G_0 V G_0 V \lvert\Psi_0\rangle,\qquad\qquad\qquad\qquad(1)$$

where $\lvert\Psi_0\rangle$ is the free wave function before the scattering event, $G_0$ is the free Green's function defined by

$$(E-H_0)G_0 = \mathbb{1}$$

and $V$ is the potential the particle scatters off. The Lippmann-Schwinger Equation above corresponds to the second order Born approximation.

I would like to know what equation $(1)$ looks like if I know what the Hamiltonian, the scattering potential (which are matrices in this case) and the wavefunctions (which are spinors) are in momentum space. So, in order to compute the scattering amplitude of the scattering process I would like to use a represenation of equation $(1)$ with integration over the momentum. Unfortunately, I don't know how to find that representation. I think my problem is that the quantities I'm dealing with now are matrices and vectors unlike in the case of non-relativistic scattering where Schroedinger's Equation is used. Hopefully somebody can help me find a representation which uses the relevant quantities in momentum space.