Classical Observables from Causal Response Functions

TL;DR

This paper develops a new formula for calculating classical observables from causal response functions, simplifies computations using the causal basis, and clarifies the classical limit in scattering processes.

Contribution

It introduces a formula linking soft limits of response functions to classical observables and demonstrates advantages of the causal basis in simplifying calculations and ensuring manifest causality.

Findings

Derived a formula for asymptotic in-in observables from soft limits.

Re-derivation of KMOC formulas for impulse and momentum.

Simplification and cancellation of singular terms in the classical limit.

Abstract

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.