On the reconstruction map in JT gravity

TL;DR

This work addresses how to reconstruct semiclassical bulk operators in AdS/CFT within JT gravity by constructing a reconstruction map R^* that is compatible with Iliesiu’s holographic map V. By introducing action-angle variables in the effective and fundamental theories, the authors define R^* so that P→N and e^{iX}→e^{iθ}, with a Fourier-based prescription that truncates phase space to retain semiclassical accuracy at short times. They explicitly reconstruct the wormhole length operator, derive its matrix elements, and use quantum ergodicity and random-matrix theory to predict non-perturbative wormhole dynamics, including a non-monotonic mean length, suppressed fluctuations until the Heisenberg time, and a heavy-tailed late-time length distribution, with numerical simulations supporting the qualitative picture. The results shed light on the interplay between holographic reconstruction, non-perturbative gravity, and notions of complexity, while highlighting robustness but potential limitations when matter is introduced. Overall, the paper provides a concrete, principled framework for connecting effective JT gravity to its non-perturbative dual, yielding testable predictions about wormhole dynamics in quantum gravity regimes.

Abstract

An open question in AdS/CFT is how to reconstruct semiclassical bulk operators precisely enough that non-perturbative quantum effects can be computed. We propose a set of physically-motivated requirements for such a reconstruction map, and explicitly construct a map satisfying these requirements in Jackiw-Teitelboim (JT) gravity. Our map is found by canonically quantizing "action-angle" variables for JT gravity, which are chosen to ensure that the spectrum of the fundamental quantum theory matches known results from the gravitational path integral. We then study unitary quantum dynamics in this theory, and obtain analytical predictions for the dynamics of the wormhole length, including its quantum fluctuations, leveraging techniques from quantum ergodicity theory. Level repulsion in the non-perturbative JT spectrum implies that the average wormhole length is non-monotonic in time, that fluctuations in wormhole length are non-perturbatively suppressed until nearly the Heisenberg time, and that the late-time-evolved Hartle-Hawking state has a heavy-tailed distribution of lengths. We discuss the implications of our results for the "complexity = volume" conjecture.

Figures (10)

• Figure 1: Numerical illustration of the dynamics of the wormhole length $x$ as a function of the time $t$ in a microcanonical state with minimum energy $E_{\min}$, associated with a Heisenberg time $t_{{\rm H}}(E_{\min}) \sim \exp(S(E_{\min}))$. The slow spread of wavepackets up to late times and non-monotonicity in the expectation value of length are clear hallmarks of ergodic dynamics due to random matrix statistics in the fundamental theory. See Sec. \ref{['sec:wormholedynamics_numerics']} for numerical details.
• Figure 2: A cartoon of the modification of phase space suggested by our proposed reconstruction map, which truncates the range of $X$ from $(-\infty,\infty)$ to $[-\pi,\pi)$. In $(x,p)$ coordinates, this deletes the red region, which consists of wormholes long compared to $\mathrm{e}^{S(E)}$. The boundary conditions we place at these new walls can be understood as gluing the remaining phase space in the way depicted, making the paths cyclic.
• Figure 3: Wormhole dynamics as a function of time in the Hartle-Hawking state with $\beta = 9$. Compared to the microcanonical state in Fig. \ref{['fig:microcanonicalnumerics']}, $p(x)$ shows the emergence of a long-tailed distribution due to the thermal spread of velocities, as well as a more spread out bump (see Appendix \ref{['app:HH']}).
• Figure 4: Bump in $\langle x_{\mathrm{alt}}(t)\rangle$, given \ref{['eq:Xmatrixelements2']}, which is qualitatively similar to the behavior of $\langle x(t)\rangle$ shown in Figures \ref{['fig:microcanonicalnumerics']} and \ref{['fig:HHnumerics']}.
• Figure 5: Probability disributions $\lvert \langle \theta'\vert \tilde{\theta}\rangle\rvert^2$ for the wavepackets in \ref{['eq:quasianglewavepackets']} for different values of $-\pi \leq \theta \leq \pi$ in $\theta' \in (-\pi,\pi]$, with $d=60$. These wavepackets become increasingly localized as $d\to\infty$, and always contain a complete orthonormal basis for any finite $d$. The $N_0$-dependent oscillations in the phase of the wavepackets are not visible in this plot of probability density, but will be visible in the real and imaginary parts.