Statistical mechanics of a two-dimensional black hole

TL;DR

The paper constructs a complete quantum-mechanical framework for a two-dimensional black hole in JT gravity, reducing boundary dynamics to boundary particles on $ ilde{AdS}_2$ and employing $ ilde{SL}(2, ext{R})$ representation theory. It defines physically meaningful single- and two-sided wavefunctions, builds a regularized Euclidean path integral that yields a Schwarzian density of states, and translates this into a Lorentzian Hilbert-space formalism with a finite partition function $Z(eta)= frac{1}{2} ext{tr}(e^{-eta H}P)$ and a density-matrix structure. The work also develops a framework for correlation functions of external matter fields, establishing Euclidean-Lorentzian correspondences and providing explicit spectral representations in the Schwarzian limit. Overall, it provides a rigorous, holography-relevant treatment of thermofield double-type states for nearly-AdS$_2$ black holes and clarifies how to define traces and density matrices in this setting. These results illuminate the quantum mechanics of black holes in low dimensions and offer a concrete route to compute observables in a finite, well-defined Hilbert space.

Abstract

The dynamics of a nearly-AdS2 spacetime with boundaries is reduced to two particles in the anti-de Sitter space. We determine the class of physically meaningful wavefunctions, and prescribe the statistical mechanics of a black hole. We demonstrate how wavefunctions for a two-sided black hole and a regularized notion of trace can be used to construct thermal partition functions, and more generally, arbitrary density matrices. We also obtain correlation functions of external operators.

Figures (11)

• Figure 1: The Euclidean (a) and Lorentzian (b) geometries in the Jackiw-Teitelboim theory. The physical spacetime (shaded) is embedded in the Poincare disk or the global anti-de Sitter space.
• Figure 2: Typical path shapes, a) for $\epsilon M\ll 1$ and b) for $\epsilon M\gg 1$.
• Figure 3: Propagation of physical states in asymptotic regions of $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$. We show coefficients defined in \ref{['asymc']} corresponding to amplitudes of ingoing and outgoing waves.
• Figure 4: a) Relation between $\mathop{\mathrm{H}}\nolimits^2$ (horizontal disk) and $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$ (vertical cross section) embedded in the complex space $\mathcal{M}$. Euclidean and Lorentzian classical trajectories consist of intersections of $\mathop{\mathrm{H}}\nolimits^2$ and $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$ with a complex classical trajectory, shown as a colored hyperboloid. To obtain $\mathring{\Phi}_E(x;0)$, the Euclidean Green function $\mathring{G}_E(x,0)$ is continued from $\mathop{\mathrm{H}}\nolimits^2$ to the right Schwarzschild patch of $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$. b) We can view $\Phi_E(x;x')$ as a wavefunction for the two boundaries of a two-sided black hole in $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$, which are space-like at any given instant of proper time.
• Figure 5: a) Division of $\mathop{\mathrm{AdS}}\nolimits_2$ into regions bounded by light rays from the point $x'=0$. There are infinitely many copies of regions $1,2,\dotsc,6$ that are translations by $\phi \to \phi+ 2\pi n$ for $n\in\mathbb{Z}$. b) Orbits of points under the subgroup $H \subset \mathop{\mathrm{\widetilde{\mathrm{SL}}}}\nolimits(2,\mathbb{R})$ preserving the origin (thin lines), and points in a skeleton representation $S$ of the quotient space $H \backslash \mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$ (thick lines). The coordinate $u$ is shown in regions $1$, $3$, $5$.