Shockwave S-matrix from Schwarzian Quantum Mechanics

TL;DR

This work establishes a precise link between the Schwarzian quantum mechanics, describing the IR limit of SYK, and bulk JT gravity by matching exact Schwarzian correlators to the Dray–'t Hooft shockwave S-matrix in the large-C semiclassical regime. By developing a Virasoro-CFT-derived framework with explicit Feynman rules and R-matrices, the authors show that the OTO four-point function encodes gravitational shockwave scattering and that the vacuum Virasoro block on the second sheet reproduces the same result. They further analyze two-point functions in heavy regimes and demonstrate eigenstate thermalization-type behavior, connecting heavy operator insertions to geometric saddles and geodesic interpretations. The results solidify the interpretation of the Schwarzian as a boundary avatar of two-dimensional gravity and illustrate a coherent picture linking CFT techniques, Liouville theory, and bulk shockwave physics. Generalizations to higher-point functions and ETH-like phenomena are discussed as promising directions.

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.

Figures (6)

• Figure 1: The 2-to-2 shockwave scattering matrix has non-zero matrix elements between incoming and outgoing pairs of waves on either side of the black hole horizon. The interaction can produce a time delay (left) or a time advance (right). In the latter case, the particles traverse the wormhole.
• Figure 2: Geometric minimization problem. The gray circle is euclidean AdS$_2$. The blue line is the cut-off boundary of AdS. $X$ and $Y$ correspond to the insertions of the two-point function. We separate the boundary in two arcs of length $L_1$, $L_2$ and enclosing area $A_1$,$A_2$.
• Figure 3: Contour followed in the integral. Deforming to the right gives two ${}_4F_3$ functions or the Wilson function. Deforming to the left is more suitable to deduce that as $M\to \infty$ only 2 poles dominate (colored in red).
• Figure 4: Shockwave interaction between two infalling modes and one outgoing mode.
• Figure 5: Full solution with non-zero temperature, with $\tau_1=0$, $\ell = 2C$, $k_2 = 1$, $k_1 = 3$, with saddle values $\tau_2 \approx 0.236$, $\tau_f \approx 1.588$ and $\beta\approx 3.159$.