Shockwave S-matrix from Schwarzian Quantum Mechanics

TL;DR

This paper demonstrates that the semi-classical limit of the out-of-time-order four-point function in Schwarzian quantum mechanics matches the shockwave S-matrix from 2D gravity, linking quantum chaos and black hole scattering.

Contribution

It establishes a precise correspondence between Schwarzian correlators and bulk shockwave scattering amplitudes, extending understanding of quantum gravity and chaos.

Findings

Exact match between OTO four-point function and shockwave S-matrix.

Two-point function of heavy operators reduces to Schwarzian saddle-point.

Connection between OTO correlators and 2D conformal blocks.

Abstract

Schwarzian quantum mechanics describes the collective IR mode of the SYK model and captures key features of 2D black hole dynamics. Exact results for its correlation functions were obtained in JHEP {\bf 1708}, 136 (2017) [arXiv:1705.08408]. We compare these results with bulk gravity expectations. We find that the semi-classical limit of the out-of-time-order (OTO) four-point function exactly matches with the scattering amplitude obtained from the Dray-'t Hooft shockwave $\mathcal{S}$-matrix. We show that the two point function of heavy operators reduces to the semi-classical saddle-point of the Schwarzian action. We also explain a previously noted match between the OTO four point functions and 2D conformal blocks. Generalizations to higher-point functions, and applications, are discussed.