Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes

Asymptotic observables in quantum feld theory beyond the familiar S-matrixhave recently attracted much interest, for instance in the context of gravity waveforms. Suchobservables can be understood in terms of Schwinger-Keldysh-type ‘amplitudes’ computedby a set of modifed Feynman rules involving cut internal legs and external legs labelledby time-folds.In parallel, a homotopy-algebraic understanding of perturbative quantum feld theoryhas emerged in recent years. In particular, passing through homotopy transfer, the S-matrixof a perturbative quantum feld theory can be understood as the minimal model of anassociated (quantum) L∞-algebra.Here we bring these two developments together. In particular, we show that SchwingerKeldysh amplitudes are naturally encoded in an L∞-algebra, similar to ordinary scatteringamplitudes. As before, they are computed via homotopy transfer, but using deformationretract data that are not canonical (in contrast to the conventional S-matrix). We furthershow that the L∞-algebras encoding Schwinger-Keldysh amplitudes and ordinary amplitudesare quasi-isomorphic (meaning, in a suitable sense, equivalent). This entails a set of recursionrelations that enable one to compute Schwinger-Keldysh amplitudes in terms of ordinaryamplitudes or vice versa.