Classical Observables from Causal Response Functions

We revisit the calculation of classical observables from causal response functions, following up on recent work by Caron-Huot at al. [JHEP 01 (2024) 139]. We derive a formula to compute asymptotic in-in observables from a particular soft limit of five-point amputated response functions. Using such formula, we re-derive the formulas by Kosower, Maybee and O'Connell (KMOC) for the linear impulse and radiated linear momentum of particles undergoing scattering, and we present an unambiguous calculation of the radiated angular momentum at leading order. Then, we explore the consequences of manifestly causal Feynman rules in the calculation of classical observables by employing the causal (Keldysh) basis in the in-in formalism. We compute the linear impulse, radiated waveform and its variance at leading and/or next-to-leading order in the causal basis, and find that all terms singular in the $\hbar \to 0$ limit cancel manifestly at the integrand level. We also find that the calculations simplify considerably and classical properties such as factorization of six-point amplitudes are more transparent in the causal basis.