Poincar\'e invariance of spinning binary dynamics in the post-Minkowskian Hamiltonian approach

We initiate the construction of the global Poincar\'e algebra generators in the context of the post-Minkowskian Hamiltonian formulation of gravitating binary dynamics in isotropic coordinates that is partly inspired by scattering amplitudes. At the first post-Minkowskian (1PM) order, we write down the Hamiltonian in a form valid in an arbitrary inertial frame. Then we construct the boost generator at the same order which uniquely solves all the equations required by the Poincar\'e algebra. Our results are linear in Newton's constant but exact in velocities and spins, including all spin multiple moments. We also compute the generators of canonical transformations that proves the equivalence between our new generators and the corresponding generators in the ADM coordinates up to the second post-Newtonian (2PN) order.