On the non-perturbative bulk Hilbert space of JT gravity

TL;DR

The paper tackles the non-perturbative bulk Hilbert space in JT gravity (with and without matter) by defining a geodesic-based, metric-variable Hilbert space with a highly degenerate inner product and explicit null states. It develops a non-perturbative framework that preserves the perturbative Hamiltonian while revealing intrinsic operator ambiguities and a path-integral interpretation via geodesic sums, applicable to both pure JT and JT with matter. Through two concrete case studies—the length operator and the center-of-mass collision energy operator—it demonstrates universal late-time behavior: wormhole lengths plateau at $\sim e^{S_0}$ and a firewall-like observable exhibits O(1) late-time probability with substantial non-perturbative contributions, including gray-hole states. The discussion connects these results to broader themes such as firewalls, Wheeler-DeWitt constraints, baby universes, and potential realizations in other microscopic models, underscoring the importance of a non-perturbative bulk language for quantum gravity.

Abstract

What is the bulk Hilbert space of quantum gravity? In this paper, we resolve this problem in 2d JT gravity, both with and without matter, providing an explicit definition of a non-perturbative Hilbert space specified in terms of metric variables. The states are wavefunctions of the length and matter state, but with a non-trivial and highly degenerate inner product. We explicitly identify the null states, and discuss their importance for defining operators non-perturbatively. To highlight the power of the formalism we developed, we study the non-perturbative effects for two bulk linear operators that may serve as proxies for the experience of an observer falling into a two-sided black hole: one captures the length of an Einstein-Rosen bridge and the other captures the center-of-mass collision energy between two particles falling from opposite sides. We track the behavior of these operators up to times of order $e^{S_\text{BH}}$, at which point the wavefunction spreads to the complete set of eigenstates of these operators. If these observables are indeed good proxies for the experience of an infalling observer, our results indicate an O(1) probability of detecting a firewall at late times that is self-averaging and universal.

Figures (16)

• Figure 1: Two-sided near-extremal black hole setup in which we shall study the collision energy between an observer and a particle, infalling from opposite sides. In the left-most diagram, an infalling $\color{red} W$ particle encounters a collision almost immediately after crossing the horizon. By applying a boost to the middle figure, we get the right-most diagram. We see that ${\color{red} W}$ does not experience any collision with $\color{blue} V$ until ${\color{red} W}$ gets extremely close to the future inner horizon (which is expected to be singular in any realistic near-extremal black hole.) In other words, while there is still a high-energy collision in the growing wormhole, the $W$ particle enjoys a potentially large amount of proper time in the interior of the black hole before the collision. In this sense, a growing wormhole is safer than a shrinking wormhole.
• Figure 2: On the left, we show a sample contribution to the disk partition function. It includes a handle disk. By stripping off the half-disk wavefunctions, we extract a contribution to the inner product $\braket{\ell}{\ell'}$.
• Figure 3: Spectrum of the JT Hamiltonian for one draw of the ensemble. The black curve is $\rho \propto \sinh (2 \pi \sqrt{2E})$, with the proportionality constant determined by requiring that there are 300 eigenvalues in the window.
• Figure 4: Matrix elements $|\mel{\ell_i}{H}{\ell_j}|$ of the Hamiltonian in the eigenbasis of the non-perturbative length operator $\widehat{\ell}_\Delta$. Note that the colors are on a log scale. This plot was generated for a single draw of the ensemble.
• Figure 5: Numerical demonstration of the $\tau$ scaling limit. We plot the average length as a function of $t/N \sim \tau = e^{-S_0} t$. We show two draws of the ensemble with different values of $e^{S_0}$. After an appropriate rescaling, the two curves are quite similar. We also show the slope predicted by the disk. In Appendix \ref{['app:Numerics']}, we show a version of this diagram where we vary more parameters, see Figure \ref{['fig:length_expectation_tau_complete']}.