On Post-Minkowskian Hamiltonians in General Relativity

TL;DR

The paper develops a framework to obtain post-Minkowskian Hamiltonians for two-body General Relativity directly from relativistic scattering amplitudes using a Lippmann-Schwinger approach. It derives explicit 1PM and 2PM potentials for non-spinning binaries and demonstrates their equivalence to results obtained via EFT matching, while also computing the corresponding classical scattering angle and showing cancellations of unphysical imaginary parts. The method provides a direct amplitude-based route to high-order two-body dynamics without relying on low-velocity expansions, and it suggests a natural path to higher-loop corrections and connections to eikonal approaches. Overall, it offers a unifying, relativistic framework that links quantum scattering data to classical gravitational dynamics.

Abstract

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 an Hamiltonian for binary black holes without spin up to 2nd order in the post-Minkowskian expansion and demonstrate explicitly the equivalence with the recently proposed method based on an effective field theory matching.