From Momentum Expansions to Post-Minkowskian Hamiltonians by Computer Algebra Algorithms

TL;DR

The paper demonstrates that complete post-Minkowskian Hamiltonians for binary gravitating systems can be recovered algorithmically from velocity expansions within an EFT framework, by deriving and solving first-order factorizable recurrences for the momentum-expansion coefficients using guessing and difference-field methods. It first validates the approach in the equal-mass case up to 3PM, obtaining explicit closed-form V1–V3, and then extends to unequal masses by introducing a rational-ρ variable that preserves structure and yields analytic V1–V3 expressions. The work highlights a practical, automatable pipeline (GSAGE, Sigma, and related tools) for reconstructing PM potentials, with careful handling of coordinate choices and functional forms (including algebraic and special-function terms). The results reinforce the viability of bridging EFT velocity expansions and PM Hamiltonians, potentially streamlining high-order gravitational scattering calculations and future amplitude-based analyses.

Abstract

The post-Newtonian and post-Minkowskian solutions for the motion of binary mass systems in gravity can be derived in terms of momentum expansions within effective field theory approaches. In the post-Minkowskian approach the expansion is performed in the ratio $G_N/r$, retaining all velocity terms completely, while in the post-Newtonian approach only those velocity terms are accounted for which are of the same order as the potential terms due to the virial theorem. We show that it is possible to obtain the complete post-Minkowskian expressions completely algorithmically, under most general purely mathematical conditions from a finite number of velocity terms and illustrate this up to the third post-Minkowskian order given in \cite{Bern:2019crd}.