A two-way approach to out-of-time-order correlators

Out-of-time-order correlators (OTOCs) are a standard measure of quantum chaos. Of the four operators involved, one pair may be regarded as a source and the other as a probe. A usual approach, applicable to large-$N$ systems such as the SYK model, is to replace the actual source with some mean-field perturbation and solve for the probe correlation function on the double Keldysh contour. We show how to obtain the OTOC by combining two such solutions for perturbations propagating forward and backward in time. These dynamical perturbations, or scrambling modes, are considered on the thermofield double background and decomposed into a coherent and an incoherent part. For the large-$q$ SYK, we obtain the OTOC in a closed form. We also prove a previously conjectured relation between the Lyapunov exponent and high-frequency behavior of the spectral function.