Wormholes from Averaging over States

Abstract

An important question about black holes is to what extent a typical pure state differs from the ensemble average. We show that this question can be answered within semi-classical gravity. We focus on the quantum deviation, which measures the fluctuations in the expectation value of an operator in an ensemble of pure states. For a large class of ensembles and observables, these fluctuations are calculated by a correlation function in the eternal black hole background, which can be reliably calculated within semi-classical gravity. This implements the idea of [arXiv:2002.02971] that wormholes can arise from averages over states rather than theories. As an application, we calculate the size of the long-time correlation function $\langle A(t) A(0)\rangle$.

Figures (2)

• Figure 1: Statistical ensemble corresponding to $\ket{\text{\small MCTFD}}$. The function $g$ selects a window of energy states in an auxiliary Boltzmann distribution. We pick three windows (shaded areas), corresponding to the regimes explained in the text. From left to right: $\beta\ll S'(E_{0})$ , $(S'(E_{0})-\beta)\Delta E \ll 1$ and $\beta \gg S'(E_{0})$. The dashed areas correspond to the states that dominate the ensemble in the different regimes. The energy uncertainty is determined by these states only.
• Figure 2: The two contractions that contribute to the connected correlation function. The contraction (a) is time independent, the contraction (b) decays exponentially in time.