JT disk spectral preparation and ensemble boundaries

Use natural units $c=\mathrm{\hslash }=k_B=1$. Thus energy $E$ has units ${\mathrm{s}}^{-1}$, inverse temperature $\beta$ and the positive model parameter $C$ have units $\mathrm{s}$, and $S_0\in \mathbb{R}$ is dimensionless. In this item, JT disk model means only the disk-sector spectral model with the density specified below; the label refers to the cited two-dimensional Jackiw–Teitelboim calculations and adopts no additional field equation or path-integral construction. Genus-zero disk contribution means the single connected disk term, with one boundary and no handles, in the formal expansion by spacetime topology. A nonperturbative completion means additional exact model data that select an exact theory beyond the terms fixed by that formal expansion.

A continuous spectral density here is a nonnegative function ${\rho }_0:(0,\mathrm{\infty })\to [0,\mathrm{\infty })$ whose measure ${\rho }_0(E)dE$ assigns model spectral weight to energy intervals. Adopt

${\rho }_0(E):=e^{S_0}\frac{C}{2{\pi }^2}\sinh \text{\hspace{0.17em}⁣}\left(2\pi \sqrt{2CE}\right),E>0.$

The density has units $\mathrm{s}$, so ${\rho }_0(E)dE$ is dimensionless. This postulate specifies a continuous disk-level input only. It does not specify an individual discrete energy spectrum, sum higher-topology terms, choose a nonperturbative completion, or assert that the model describes nature.

Phil Saad, Stephen H. Shenker, and Douglas Stanford, JT gravity as a matrix integral

Thomas G. Mertens and Gustavo J. Turiaci, Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity

Gábor Sárosi, AdS2 holography and the SYK model

The prior restricted semiclassical AdS entropy dictionary Restricted semiclassical entropy dictionary concerns bulk dimension $d+1$ with $d\ge 2$, and is background context only for this two-dimensional disk model: none of its entropy-dictionary assumptions is adopted here, and it does not derive this disk density. The disk expression remains a separate adopted input; any exact boundary quantum system, higher-genus expansion, or nonperturbative completion is outside this postulate.

Let $C>0$ and $\beta >0$ have units of seconds, let $S_0\in \mathbb{R}$ be dimensionless, and use $c=\mathrm{\hslash }=k_B=1$ units as in the adopted JT disk spectral model. Then $E>0$ has units ${\mathrm{s}}^{-1}$ and the density ${\rho }_0(E)$ from that model has units seconds. Define the disk partition function as the Laplace integral

$Z_0(\beta ):={\int }_0^{\mathrm{\infty }}{\rho }_0(E)e^{-\beta E}dE\mathrm{.}$

When $Z_0(\beta )$ is finite and positive, define the normalized canonical energy distribution by $d{P}_{\beta }(E):=Z_0(\beta {)}^{-1}{\rho }_0(E)e^{-\beta E}dE$ and its mean energy by $\bar{E}(\beta ):={\int }_0^{\mathrm{\infty }}Ed{P}_{\beta }(E)$. Put $T:=1\mathrm{/}\beta$ and, at fixed $C,S_0$, write $C_V\mathrm{/}{k}_B:=d\bar{E}\mathrm{/}dT$ for the dimensionless canonical heat-capacity coefficient in these units. Then the integral is finite and

$Z_0(\beta )=e^{S_0}\frac{C\sqrt{2C}}{2\sqrt{\pi }{\beta }^{3\mathrm{/}2}}\exp \text{\hspace{0.17em}⁣}\left(\frac{2{\pi }^{2}C}{\beta }\right),\bar{E}(\beta )=\frac{3}{2\beta }+\frac{2{\pi }^{2}C}{{\beta }^2},\frac{C_V}{k_B}=\frac{3}{2}+\frac{4{\pi }^{2}C}{\beta }>0.$

These are deductions within the adopted continuous disk model, not claims about an individual microscopic spectrum or nature.

Adopted JT disk spectral model

The Gaussian integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$

Differentiation under an improper multiple integral under an integrable derivative bound

Phil Saad, Stephen H. Shenker, and Douglas Stanford, JT gravity as a matrix integral

Thomas G. Mertens and Gustavo J. Turiaci, Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity

Gábor Sárosi, AdS2 holography and the SYK model

The proof uses the published Gaussian integral supplier, the improper-substitution theorem for the affine, rescaling, and square substitutions, and the dominated-differentiation theorem for both parameter-dependent improper integrals. The JT density itself remains the adopted premise [A1]. No path-integral or ensemble assumption is used in the calculation.