Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity

TL;DR

This work demonstrates that Jackiw–Teitelboim gravity coupled to matter reproduces the late-time, ensemble-averaged behavior expected from random-matrix theory and ETH. The ramp and plateau of correlation functions arise from topology-changing processes in Euclidean gravity, notably the emission and absorption of baby universes, and are encoded in a third-quantized Hilbert space formalism. The authors provide explicit ramp- and plateau-related formulas for two-point and four-point functions, and outline generalizations to higher-point OTOCs, connecting gravitational path integrals to ETH-era predictions. The results strengthen the view that JT gravity captures universal features of chaotic quantum systems via bulk wormhole dynamics and D-brane boundary conditions, with potential implications for SYK and higher-dimensional holography.

Abstract

Quantum black holes are described by a large number of macroscopically indistinguishable microstates. Correlation functions of fields outside the horizon at long time separation probe this indistinguishability. The simplest of these, the thermal two-point function, oscillates erratically around a nonperturbatively small average "ramp" and "plateau" after an initial period of decay; these non-decaying averaged features are signatures of the discreteness of the black hole spectrum. For a theory described by an ensemble of Hamiltonians, the two-point function follows this averaged behavior. In this paper we study certain correlation functions in Jackiw-Teitelboim (JT) gravity and find precise agreement with the behavior expected for a theory described by an ensemble of Hamiltonians with random matrix statistics -- the eigenstates obey the Eigenstate Thermalization Hypothesis (ETH) and the energy levels have random matrix level statistics. A central aspect of our analysis is an averaged bulk Hilbert space description of the relevant behavior. The mechanism behind this behavior is topology change due the the emission and absorption of closed "baby universes". These baby universe effects give two complementary pictures of the non-decaying behavior, related by different continuations of a Euclidean geometry. A long Einstein-Rosen bridge can become short by emitting a large baby universe, and baby universes emitted and reabsorbed at points widely separated in space and time creates a "shortcut", allowing particles to leave the interior of the black hole.

Figures (32)

• Figure 1: Above we have pictured the general features in the spectral form factor. The slope transitions to the plateau at a time $T\sim e^{S_0/2}$Cotler:2016fpe, and the ramp transitions to the plateau at a time $T\sim e^{S_0}$. The height of the plateau is smaller than the $T=0$ value of the spectral form factor by a factor of $e^{-S_0}$.
• Figure 2: Here we have pictured an example geometry that contributes to the Hartle-Hawking wavefunction $\psi_{D,\beta/2}(\ell)$. The light blue circle represents the hyperbolic disk, while the dark blue represents the piece of the disk that we take as our Euclidean spacetime. The wiggly asymptotic boundary has renormalized length $\beta/2$, while the geodesic piece of the boundary has renormalized length $\ell$.
• Figure 3: Matrix elements of the time evolution operator $\langle \ell | e^{-i \frac{T}{2} H_{Bulk}}|\ell'\rangle$ are calculated by the path integral over all geometries with the appropriate boundary conditions.
• Figure 4: Here we have represented equation (\ref{['ZAmplitude']}) to leading order in $e^{-S_0}$. On the left, we show a piece of complexified hyperbolic space with the topology of a disk and a boundary of renormalized length $\beta+i T$. On the right we have shown another piece of complexified hyperbolic space with the topology of a disk and a boundary of length $\beta+i T$. However, this geometry is piecewise Euclidean and Lorentzian, with the two half-disks representing the Euclidean geometries which prepare the Hartle-Hawking state, and the vertical, rectangular piece representing the Lorentzian evolution between these two states. These pieces are joined along slices of zero extrinsic curvature of renormalized lengths $\ell$ and $\ell'$.
• Figure 5: Here we have pictured a contribution to the spectral form factor, viewed as the return probability of a pair of Hartle-Hawking states. Contributions such as these couple the two systems, so that $H_{bulk, LR}\neq H_{Bulk, R}- H_{Bulk, L}$.