Algebras and States in JT Gravity

TL;DR

This work analyzes the algebra of boundary observables in Jackiw–Teitelboim gravity, showing that coupling to matter elevates the boundary algebra to a Type II$_\infty$ factor with trivial center, enabling a unique, trace-defined entropy up to an additive constant. The authors connect canonical-quantisation and Euclidean-path-integral approaches, establishing that the boundary entropy matches replica-trick results and that the algebra is background-independent without relying on large-N limits. They also explore wormhole and baby-universe effects, demonstrating that baby-universe operators do not define bona fide Hilbert-space operators in the presence of matter, while wormholes perturb the Hilbert space and algebra in controlled, perturbative ways. A key result is that, for any bulk state (including those with baby universes), a boundary observer can always associate a representative pure state on the boundary Hilbert space, illustrating a boundary-centric description of bulk physics. The paper further extends to scenarios with multiple open universes, concluding that boundary observations cannot reveal the number of connected boundaries, underscoring a profound boundary/bulk equivalence at all orders in the wormhole expansion.

Abstract

We analyze the algebra of boundary observables in canonically quantised JT gravity with or without matter. In the absence of matter, this algebra is commutative, generated by the ADM Hamiltonian. After coupling to a bulk quantum field theory, it becomes a highly noncommutative algebra of Type II$_\infty$ with a trivial center. As a result, density matrices and entropies on the boundary algebra are uniquely defined up to, respectively, a rescaling or shift. We show that this algebraic definition of entropy agrees with the usual replica trick definition computed using Euclidean path integrals. Unlike in previous arguments that focused on $\mathcal{O}(1)$ fluctuations to a black hole of specified mass, this Type II$_\infty$ algebra describes states at all temperatures or energies. We also consider the role of spacetime wormholes. One can try to define operators associated with wormholes that commute with the boundary algebra, but this fails in an instructive way. In a regulated version of the theory, wormholes and topology change can be incorporated perturbatively. The bulk Hilbert space $\mathcal{H}_\mathrm{bulk}$ that includes baby universe states is then much bigger than the space of states $\mathcal{H}_\mathrm{bdry}$ accessible to a boundary observer. However, to a boundary observer, every pure or mixed state on $\mathcal{H}_\mathrm{bulk}$ is equivalent to some pure state in $\mathcal{H}_\mathrm{bdry}$.

Figures (24)

• Figure 1: (a) The path integral on a disc that computes ${\mathrm{Tr}}\,{\sf S}$ with ${\sf S}= e^{-\beta_1 H}\Phi_1 e^{-\beta_2 H}\Phi_2 e^{-\beta_3 H}$. The boundary of the disc is made of three segments with renormalized lengths $\beta_1$, $\beta_2,$ and $\beta_3$. At two junctions of segments, operators $\Phi_1$ and $\Phi_2$ are inserted. At the third junction, the two ends of ${\sf S}$ are joined together. (b) The path integral on a disc that computes ${\mathrm{Tr}}\,{\sf S}_1{\sf S}_2$. The boundary of the disc consists of two segments labeled respectively by ${\sf S}_1$ and by ${\sf S}_2$. There is no intrinsic ordering of the two segments so ${\mathrm{Tr}}\,{\sf S}_1{\sf S}_2={\mathrm{Tr}}\,{\sf S}_2{\sf S}_1$.
• Figure 2: (a) The path integral on a half-disc that computes the map from a string ${\sf S}$ to a Hilbert space state $\Psi_{\sf S}$. The half-disc has an asymptotic boundary labeled by the string ${\sf S}$ and a geodesic boundary $\gamma$. (b) The path integral that computes $\langle {\sf S}',{\sf S}\rangle$ and can be used to demonstrate that the map ${\sf S}\to \Psi_{\sf S}$ from a string to a bulk state preserves inner products.
• Figure 3: Two views of a spacetime $M$ which is half of AdS$_2$ (in Euclidean signature). Viewing AdS$_2$ as a hyperbolic disc, half of AdS$_2$ is the half-disc shown in (a); on the other hand, the AdS$_2$ metric can be put in the static form ${\mathrm d}\sigma^2+\cosh^2\sigma {\mathrm d}\tau^2$, and in this form, half of AdS$_2$ looks like a semi-infinite strip with $\tau\leq 0$, as shown in (b).
• Figure 4: Depicted here is a half-disc $D_0$ with an asymptotic boundary labeled by ${\sf S}_0{\sf S}$ and a geodesic boundary (the horizontal line at the top). The path integral on $D_0$ computes $\Psi_{{\sf S}_0{\sf S}}$. $\gamma$ is a geodesic that connects the endpoints $p,r$ of the boundary segment labeled by ${\sf S}_0$. If $\Psi_{{\sf S}_0}=0$, then the path integral in the region $D_1$ bounded by ${\sf S}_0$ and $\gamma$ vanishes, regardless of the fields on $\gamma$, and therefore $\Psi_{{\sf S}_0{\sf S}}=0$.
• Figure 5: (a) This figure shows the path integral that would be used to compute a matrix element $\langle \Psi'|{\sf S}|\Psi\rangle$ of a Hilbert space operator corresponding to a string ${\sf S}$ between initial and final states $\Psi, \Psi'$ in the bulk Hilbert space ${\mathcal{H}}$. $\Psi$ and $\Psi'$ are inserted on geodesic boundaries that asymptotically meet at a point on the right boundary. (b) In the special case that the initial state is $\Psi=\Psi_{\sf T}$, by gluing onto (a) the path integral preparation of the state $\Psi_{\sf T}$, we get a representation of the matrix element $\langle\Psi'|{\sf S}|\Psi_{\sf T}\rangle$. But this coincides with the path integral representation we would use for the inner product $\langle\Psi'|\Psi_{{\sf S}{\sf T}}\rangle$, showing that the standard interpretation of ${\sf S}$ as a Hilbert space operator is consistent with ${\sf S}\Psi_{\sf T}=\Psi_{{\sf S}{\sf T}}$. Note that this picture can also be read to show that if $\Psi_{\sf T}=0$ then $\Psi_{{\sf S}{\sf T}}=0$.