Elimination of the spin supplementary condition in the effective field theory approach to the post-Newtonian approximation

TL;DR

This work tackles the elimination of the covariant spin supplementary condition (SSC) within the EFT-based post-Newtonian framework for spinning binaries. It develops and compares two reduction strategies—Dirac brackets and an action-principle approach—to produce fully reduced canonical Hamiltonians depending only on physical degrees of freedom, in particular promoting spin variables to Newton-Wigner-type canonical forms. By transforming existing non-reduced next-to-leading order (NLO) spin-orbit and spin-spin potentials (Levi, Porto, and Porto–Rothstein) to canonical form, the authors demonstrate agreement with ADM Hamiltonians up to infinitesimal canonical transformations, and provide explicit generators for these transformations. The results streamline the EFT formulation, improve Feynman rules, and pave the way for NNLO extensions and integration with effective-one-body formalisms for gravitational-wave astrophysics.

Abstract

The present paper addresses open questions regarding the handling of the spin supplementary condition within the effective field theory approach to the post-Newtonian approximation. In particular it is shown how the covariant spin supplementary condition can be eliminated at the level of the potential (which is subtle in various respects) and how the dynamics can be cast into a fully reduced Hamiltonian form. Two different methods are used and compared, one based on the well-known Dirac bracket and the other based on an action principle. It is discussed how the latter approach can be used to improve the Feynman rules by formulating them in terms of reduced canonical spin variables.