  We calculate the motion of binary mass systems in gravity up to the fourth
post--Newtonian order. We use momentum expansions within an effective field
theory approach based on Feynman amplitudes in harmonic coordinates by applying
dimensional regularization. We construct the canonical transformations to ADM
coordinates and to effective one body theory (EOB) to compare with other
approaches. We show that intermediate poles in the dimensional regularization
parameter $\varepsilon$ vanish in the observables and the classical theory is
not renormalized. The results are illustrated for a series of observables for
which we agree with the literature.
