The existence of the eikonal is proven for conservative dynamics, and this work addresses inclusion of radiative effects. Note that the eikonal may not be single-valued. Well-definedness of its derivatives suffices, since the eikonal itself is not an observable quantity; only though its difference the eikonal represents observable quantities, even for the quantities that cannot be computed from the scattering generator equation [Eq. (1)] such as the time delay [22, 23] and polarization rotation [24, 25].

The positive frequency expansion $f(x)={\int }_k{e}^{-ikx}f(k)$ and outgoing convention (reading coefficients of $f(-k)$ as vertex rules) are employed to ensure consistency with the Feynman rules given in Ref. [44].

The d’Alembertian operator $\square \iff -k^2$ has an extra minus sign in momentum space.

There is a distinction made between the $i{0}^{+}$ prescription and the causality prescription. The $i{0}^{+}$ prescription can be retarded, advanced, or Feynman; the causality prescription can only be retarded or advanced.

For spinning systems, ${\sum }_i\{{\chi }_{(1)},S_i^{\mu \nu }\}\frac{\partial }{\partial S_i^{\mu \nu }}$ implements precession by the LO spin kick, consistent with the conventional picture of saddle-point shifts [40]. In other words, the ${\mathcal{D}}_{SL}$ operator introduced in Ref. [63] can be viewed as compensation terms for neglecting nested brackets in the scattering generator equation [Eq. (1)] and “noncommutativity” of the impact parameter space [Eq. (17)]; see, e.g., the scattering observable computations of Ref. [64] based on Dirac brackets. Note that Poisson brackets can be used in place of Dirac brackets for variables manifestly satisfying constraint conditions [29].

This relation follows from translation invariance of LO radiation eikonal ${\chi }_{(1.5)}^{H^1}[H(x)]$ in position space, where $x^{\mu }$ is dual to $k^{\mu }$ in Eq. (8).