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

TL;DR

This paper demonstrates that out-of-time-order asymptotic observables in quantum field theory can be understood through $L_$-algebras and are quasi-isomorphic to time-ordered amplitudes, enabling recursive computation methods.

Contribution

It establishes a homotopy-algebraic framework linking Schwinger-Keldysh amplitudes with ordinary scattering amplitudes via quasi-isomorphisms.

Findings

Schwinger-Keldysh amplitudes are encoded in an $L_$-algebra.

These $L_$-algebras are quasi-isomorphic to those of ordinary amplitudes.

Recursion relations allow computation of out-of-time-order amplitudes from time-ordered ones.

Abstract

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) $L_\infty$-algebra. Here we bring these two developments together. In particular, we show that Schwinger-Keldysh amplitudes are naturally encoded in an $L_\infty$-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 $L_\infty$-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.