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

A bstract Asymptotic observables in quantum field theory beyond the familiar S -matrix have recently attracted much interest, for instance in the context of gravity waveforms. Such observables can be understood in terms of Schwinger-Keldysh-type ‘amplitudes’ computed by a set of modified Feynman rules involving cut internal legs and external legs labelled by time-folds. In parallel, a homotopy-algebraic understanding of perturbative quantum field theory has emerged in recent years. In particular, passing through homotopy transfer, the S -matrix of a perturbative quantum field theory can be understood as the minimal model of an associated (quantum) L ∞ -algebra. Here we bring these two developments together. In particular, we show that Schwinger-Keldysh amplitudes are naturally encoded in an L ∞ -algebra, similar to ordinary scattering amplitudes. As before, they are computed via homotopy transfer, but using deformation-retract data that are not canonical (in contrast to the conventional S -matrix). We further show that the L ∞ -algebras encoding Schwinger-Keldysh amplitudes and ordinary amplitudes are quasi-isomorphic (meaning, in a suitable sense, equivalent). This entails a set of recursion relations that enable one to compute Schwinger-Keldysh amplitudes in terms of ordinary amplitudes or vice versa.