The Nonperturbative Hilbert Space of Quantum Gravity With One Boundary

Abstract

We discuss a basis for the nonperturbative Hilbert space of quantum gravity with one asymptotic boundary. We use this basis to show that the Hilbert space for gravity with two disconnected boundaries factorizes into a product of two copies of the single boundary Hilbert space.

Figures (27)

• Figure 1: Cartoon of cutting a gravity path integral boundary condition along $\mathcal{X}$ to define a state.
• Figure 2: 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 $\mathcal{H}_{\mathcal{B}_L}\otimes \mathcal{H}_{\mathcal{B}_R}$. Note that here we are drawing the boundary conditions of the Euclidean geometry. The bulk geometry between the boundary components $\mathcal{B}_{L,R}$ may be either be connected (like a spatial section of the eternal black hole) or disconnected (like a product of two thermal AdS geometries).
• Figure 3: Resolving the factorization problem. ( a) The two-sided Hilbert space $\mathcal{H}_{LR}$, the tensor product Hilbert space $\mathcal{H}_{\mathcal{B}{_L}}\otimes\mathcal{H}_{\mathcal{B}{_R}}$, and the total two-sided Hilbert space $\mathcal{H}_{L\cup R}$. The shaded regions labeled $\mathcal{H}_{2s}$ and $\mathcal{H}_{L} \otimes \mathcal{H}_R$ represent the spans of a set of two-sided "shell states" and a set of tensor products of single-sided "shell states" respectively, both of which are constructed from the Euclidean path integral. See Sasieta:2022ksuBalasubramanian:2022gmoBalasubramanian:2022lnwAntonini:2023hdh for 2-boundary shell states and Sec. \ref{['sec:1s-span']} for 1-boundary shell states. ( b) We will argue that the two- and single-sided shell states span $\mathcal{H}_{LR}$ and $\mathcal{H}_{\mathcal{B}_{L,R}}$ respectively when we take the number of shell states to infinity. The two-sided shell state span was shown in Toolkit. ( c) We will then show that the two-sided shell states spanning $\mathcal{H}_{LR}$ also span $\mathcal{H}_{\mathcal{B}_{L}} \otimes \mathcal{H}_{\mathcal{B}{_R}}$ and vice versa.
• Figure 4: Cut-open path integral boundary condition defining the single-sided shell states. ( a) Euclidean boundary with topology $\mathbb{R}^{<0}\times\mathbb{S}^{d-1}$ for preparation of the shell states. 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. The shell operator $\mathcal{O}_{i}$ is pictured in red and $\beta_{}/2$ is the Euclidean "preparation time". ( b) Euclidean boundary with the $\mathbb{S}^{d-1}$ suppressed. We adopt this convention for the rest of the paper, and sometimes depict this boundary with a curve or a kink to clarify diagrams.
• Figure 5: Shell-strip asymptotic boundary condition for the overlap $\braket{j|i}$ consisting of the line $\lim_{\alpha \to \infty} [-\alpha, \beta + \alpha]$ on which $\mathcal{O}_{i}$ and $\mathcal{O}^{\dagger}_{j}$ are inserted at $\tau =0$ and $\tau=\beta$ respectively.