A Nonperturbative Toolkit for Quantum Gravity

Abstract

We propose a method for demonstrating equivalences beyond the saddlepoint approximation between quantities in quantum gravity that are defined by the Euclidean path integral, without assumptions about holographic duality. The method involves three ingredients: (1) a way of resolving the identity with an overcomplete basis of microstates that is under semiclassical control, (2) a drastic simplification of the sum over topologies in the limit where the basis is infinitely overcomplete, and (3) a way of cutting and splicing geometries to demonstrate equality between two different gravitational path integrals even if neither can be explicitly computed. We illustrate our methods by giving a general argument that the thermal partition function of quantum gravity with two boundaries factorises. One implication of our results is that universes containing a horizon can sometimes be understood as superpositions of horizonless geometries entangled with a closed universe.

Figures (25)

• Figure 1: Cartoon of cutting a gravity path integral boundary condition to define a state.
• Figure 2: Action of $e^{-\beta H}$ on states. ( a) Action of the Hamiltonian. ( b) Example state. ( c) Action of the Hamiltionian on a state. ( d) Survival amplitude after Euclidean time evolution.
• Figure 3: Two distinct ways of preparing states in $\mathcal{H}_{L \cup R}$. ( a) Construction of an example state in $\mathcal{H}_{LR}$. ( b) Construction of an example state in $\mathcal{H}_{\mathcal{B}_L}\otimes \mathcal{H}_{\mathcal{B}_R}$.
• Figure 4: Asymptotic boundary condition for the gravity path integral defining the shell state. ( a) Cut-open Euclidean boundary with topology $\mathbb{I}_{\frac{\beta_L+\beta_R}{2}}\times\mathbb{S}^{d-1}$ for preparation of the shell states. The shell operator $\mathcal{O}_{i}$ is pictured in red. In AdS/CFT we can also perform the path integral in the boundary CFT with insertion of a $\mathbb{S}^{d-1}$ symmetric operator dual to the shell. ( b) Euclidean boundary with the $\mathbb{S}^{d-1}$ suppressed. We adopt this convention for the rest of the paper. Here $\beta_{L,R}/2$ are Euclidean "preparation times".
• Figure 5: Shell asymptotic boundary condition for $\overline{\langle j|i\rangle}$, consisting of the operator insertions $\mathcal{O}_{i}$ and $\mathcal{O}^{\dagger}_{j}$ separated by asymptotic time extent $\beta_{L}$ and $\beta_{R}$ respectively. The red lines represent the shells propagating into the bulk.