Negative energies and the breakdown of bulk geometry

Abstract

One central question in quantum gravity is to understand how and why predictions from semiclassical gravity can break down in regimes with low spacetime curvature. One diagnostic of such a breakdown is that states which are orthonormal at the semiclassical level can receive large corrections to their inner products from quantum fluctuations. We study this effect by examining inner products in pure 2D JT gravity. Previous work showed that black hole states with long interiors exhibit a breakdown at length scales of order $e^{S_0}$, where $S_0$ is a parameter analogous to $1/G_N$ in higher dimensions. This breakdown is caused by the discreteness of the spectrum of the dual boundary random matrix theory. We show that the full sum over quantum fluctuations indicates a more dramatic breakdown at parametrically shorter lengths of order $e^{S_0/3}$. In the dual boundary description, these corrections arise from negative energy states appearing in rare members of the random matrix ensemble. These results demonstrate that non-perturbative effects can invalidate the semiclassical description at much smaller length scales than previously expected, providing a new mechanism for the breakdown of effective gravitational theories.

Figures (5)

• Figure 1: Bulk and boundary descriptions of $\ket{\ell}$ states in JT gravity.
• Figure 2: Three versions of the density of states of JT gravity, $\rho(E) = e^{-S_0} \sum_{E_a} \delta(E-E_a)$. Red: discrete spectrum for a given realization of the ensemble at finite $S_0$. Blue:$S_0\to \infty$ limit of the ensemble average $\overline{\rho(E)}$, which matches the bulk density of states $\rho_0(E)$. Orange:$\overline{\rho(E)}$ at finite $S_0$ in the universal "Airy" regime. The key feature of the spectrum at finite $S_0$ underlying the breakdown in \ref{['deviation_result']} and \ref{['variance_result']} is the appearance of negative energies in some members of the ensemble, and ( green) the accompanying large value of $\braket{\ell|E}$ at negative energies.
• Figure 3: The complex energy contour $\Gamma$ for the JT gravity matrix integral is shown in blue. The contour is obtained by following the steepest descent line of $V_{\rm eff}(E)$, starting from its first maximum on the negative real axis at $E=-1/8$, into the upper half plane. The saddle-points $E^{\ast}(k)$ of \ref{['f3']} for various values of $k \equiv \bar{\ell} e^{-S_0}$ from $k=0.001$ to $k=3$ are shown in red. For small negative $k \lesssim 0.047$, $E^{\ast}(k)$ lies at increasingly negative values on the real line as we increase $k$. For $k \gtrsim0.047$, $E^{\ast}(k)$ moves into the upper half plane and its real part becomes less negative, and eventually positive, on increasing $k$ further.
• Figure 4: For $k=1$, we show the saddle-point value of $E$ in the complex plane that gives the dominant contribution to the integral over $\Gamma$ (shown in the blue dashed contour) as the red point, and one example of another saddle-point that does not contribute to the integral in green. The purple lines are lines of constant $\mathop{\rm Im} f(E)$, where $f(E)\equiv -\frac{V_{\rm eff}(E)}{e^{S_0}} + 2k\sqrt{-2E}$ is the exponent in \ref{['f3']}. The contour plot shows the value of $\mathop{\rm Re} [f(E)]$ , which allows us to distinguish the steepest descent contours (along which $\mathop{\rm Re} [f(E)]$ decreases fastest) from the steepest ascent contours (along which $\mathop{\rm Re} [f(E)]$ increases fastest) among the purple lines. The left and right plots respectively show these steepest ascent/descent contours for the red and green points. We see that for the red point, the steepest ascent contour intersects $\Gamma$ while the steepest descent contour asymptotes to $\Gamma$, indicating that it contributes to the integral. For the green saddle-point, the steepest ascent contour does not intersect $\Gamma$ and the steepest descent contour asymptotes in a direction far from $\Gamma$, indicating that it does not contribute.
• Figure 5: In (a), the blue data points show the real part of the exponent of the integrand in \ref{['f3']} evaluated at the saddle-point value of $E$ (found numerically), and the red line shows the analytic estimate $\bar{\ell}(1- \frac{1}{\log(8\pi^2\bar{\ell} e^{-S_0})})$ for this quantity from \ref{['f6']}. (b) shows the numerical evaluation of the full quantity $\braket{\ell|V^{\dagger}V|\ell'}$ on including the imaginary part in the exponent and adding contributions from $\Gamma$ and $\Gamma'$. The resulting value of the overlap shows large oscillations as a function of $\bar{\ell}$ for large $\bar{\ell}$ (we show only up to $\ell e^{-S_0}=10$ as an illustration, as the numerical values rapidly become very large for larger $\ell$).