Lippmann-Schwinger equation and Scattering amplitude with spin How does the Lippmann-Schwinger equation and the scattering amplitude generalize when we include spin? Also, how does the scattering amplitude up to the first Born approximation change if we include spin?  
EDIT: If somebody could include the scattering amplitude with spin, it would be ideal. Also, I know that the form of these will be the same if we include spin(the state ket now being a tensor product of the spatial ket and the spin ket), I am just confused on putting the spin ket for the states corresponding to before and after the scattering
 A: As mentioned already, the Lippmann-Schwinger equation have the same form whether you include spin or not:
$$
|\Psi_\alpha^\pm\rangle=|\Phi_\alpha\rangle+(E_\alpha+H_0\pm i\epsilon)^{-1}V|\Psi^\pm_\alpha\rangle
$$
or, in integrated form (using a complete set of free states),
$$
|\Psi_\alpha^\pm\rangle=|\Phi_\alpha\rangle+\int\mathrm d\beta\ \frac{\langle\Phi_\beta|V|\Psi_\alpha^\pm\rangle|\Phi_\beta\rangle}{E_\alpha-E_\beta\pm i\epsilon}
$$
Here, $\alpha$ is a collection of parameters that label the states. These may or may not include spin, depending on the context.
The Born approximation corresponds to taking $|\Psi^\pm_\alpha\rangle\approx |\Phi_\alpha\rangle$ inside the integral, so that
$$
|\Psi_\alpha^\pm\rangle=|\Phi_\alpha\rangle+\int\mathrm d\beta\ \frac{\langle\Phi_\beta|V|\Phi_\alpha^\pm\rangle|\Phi_\beta\rangle}{E_\alpha-E_\beta\pm i\epsilon}+\mathcal O(V^2)
$$
The first correction thus corresponds to the matrix element
$$
\langle\Phi_\beta|V|\Phi_\alpha^\pm\rangle
$$
If we consider the one-particle problem, where the labels are the position and spin, we can use the basis $|\boldsymbol x,\pm\rangle$, so that
$$
\langle\Phi_\beta|V|\Phi_\alpha^\pm\rangle=\sum_{s,s'}\int\mathrm d\boldsymbol x\,\mathrm d\boldsymbol y\ \langle\Phi_\beta|\boldsymbol x,s\rangle\langle \boldsymbol x,s|V|\boldsymbol y,s'\rangle\langle \boldsymbol y,s'|\Phi_\alpha^\pm\rangle
$$
and I guess you can take it from here.
For more details, see Weinberg's book, Vol.I, chapter 3.
A: The Lippman-Schwinger equation can be used to perturbatively expand the scattering solutions of a potential scattering problem. The Born approximation is the truncation of the expansion at linear order in the potential.
So you have a particle scattering off a potential. If you include spin nothing changes at all. Physically your potential might be spin dependent and change, but that is all.
In quantum mechanics, which is the context of the Lippman-Schwinger equation spin enters through (anti-)symmetrization of the multi-particle wavefunction under particle exchange. Note that the Lippman-Schwinger equation is explicitly constructed for potential scattering and may only be used to deal with two-body scattering. For three particle scattering there are the Faddeev equations.
