Post-Minkowskian Hamiltonians in general relativity

We describe the computation of post-Minkowskian Hamiltonians in general relativity from scattering amplitudes. Using a relativistic Lippmann-Schwinger equation, we relate perturbative amplitudes of massive scalars coupled to gravity to the post-Minkowskian Hamiltonians of classical general relativity to any order in Newton's constant. We illustrate this by deriving a Hamiltonian for binary black holes without spin up to second order in the post-Minkowskian expansion and explicitly demonstrate the equivalence with the recently proposed method based on an effective field theory matching.