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.

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

High precision astrometry, space missions and certain tests of General Relativity, require the knowledge of the metric tensor of the solar system, or more generally, of a gravitational system of N extended bodies. Presently, the metric of arbitrarily shaped, rotating, oscillating and arbitrarily moving N bodies of finite extension is only known for the case of slowly moving bodies in the post-Newtonian approximation, while the post-Minkowskian metric for arbitrarily moving celestial objects is known only for pointlike bodies with mass-monopoles and spin-dipoles. As one more step towards the aim of a global metric for a system of N arbitrarily shaped and arbitrarily moving massive bodies in post-Minkowskian approximation, two central issues are on the scope of our investigation. (i) We first consider one extended body with full multipole structure in uniform motion in some suitably chosen global reference system. For this problem a co-moving inertial system of coordinates can be introduced where the metric, outside the body, admits an expansion in terms of Damour–Iyer moments. A Poincare transformation then yields the corresponding metric tensor in the global system in post-Minkowskian approximation. (ii) It will be argued why the global metric, exact to post-Minkowskian order, can be obtained by means of an instantaneous Poincare transformation for the case of pointlike mass-monopoles and spin-dipoles in arbitrary motion.

Gravitational field of one uniformly moving extended body and N arbitrarily moving pointlike bodies in post-Minkowskian approximation

High precision astrometry, space missions and certain tests of General Relativity, require the knowledge of the metric tensor of the solar system, or more generally, of a gravitational system of N extended bodies. Presently, the metric of arbitrarily shaped, rotating, oscillating and arbitrarily moving N bodies of finite extension is only known for the case of slowly moving bodies in the post-Newtonian approximation, while the post-Minkowskian metric for arbitrarily moving celestial objects is known only for pointlike bodies with mass-monopoles and spin-dipoles. As one more step towards the aim of a global metric for a system of N arbitrarily shaped and arbitrarily moving massive bodies in post-Minkowskian approximation, two central issues are on the scope of our investigation: 
(i) We first consider one extended body with full multipole structure in uniform motion in some suitably chosen global reference system. For this problem a co-moving inertial system of coordinates can be introduced where the metric, outside the body, admits an expansion in terms of Damour-Iyer moments. A Poincare transformation then yields the corresponding metric tensor in the global system in post-Minkowskian approximation. 
(ii) It will be argued why the global metric, exact to post-Minkowskian order, can be obtained by means of an instantaneous Poincare transformation for the case of pointlike mass-monopoles and spin-dipoles in arbitrary motion.

Time evolution of relativistic particle systems can be accomplished by means of a Schrodinger wave equation, provided the Hamiltonian of the N particle system satisfies some commutator equations involving the other generators of a suitable representation of the Poincare group on the initial data space of the Schrodinger equation. This set of operator equations is solved in the N free particle case. Further, the structure of N interacting particle Hamiltonians is worked out to include second‐order relativistic corrections, i.e., (1/c2)‐terms. It is shown that Breit and Barker–O’Connell Hamiltonians (nonzero spin particles) and the Bažanski Hamiltonian (zero‐spin particles) are particular cases of this more general Hamiltonian.

Approximate relativistic quantum Hamiltonians for N interacting particle systems

Within the framework of the previous paper, we complete the set of equations of motion by including the spin propagation equation at first order. We check this equation with the known result on the precession, which is obtained by means of slow motion approximation of our result. A new scheme of expanding equations of motion is also introduced. It will be useful to undertake higher-order calculations.

J. Ibáñez

Papers

Gravitational interaction of two spinning particles in general relativity