Abstract
Recent developments on black holes have shown that a unitarity-compatible Page curve can be obtained from an ensemble-averaged semi-classical approximation. In this paper, we emphasize (1) that this peculiar manifestation of unitarity is not specific to black holes, and (2) that it can emerge from a single realization of an underlying unitary theory. To make things explicit, we consider a hard sphere gas leaking slowly from a small box into a bigger box. This is a quantum chaotic system in which we expect to see the Page curve in the full unitary description, while semi-classically, eigenstates are expected to behave as though they live in Berry’s ensemble. We reproduce the unitarity-compatible Page curve of this system, semi-classically. The computation has structural parallels to replica wormholes, relies crucially on ensemble averaging at each epoch, and reveals the interplay between the multiple time-scales in the problem. Working with the ensemble averaged state rather than the entanglement entropy, we can also engineer an information “paradox”. Our system provides a concrete example in which the ensemble underlying the semi-classical Page curve is an ergodic proxy for a time average, and not an explicit average over many theories. The questions we address here are logically independent of the existence of horizons, so we expect that semi-classical gravity should also be viewed in a similar light.
Highlights
Is a non-unitary theory), one can explicitly demonstrate the emergence of the Page curve by evaluating the average over the underlying ensemble [7, 9]
Recent developments on black holes have shown that a unitarity-compatible Page curve can be obtained from an ensemble-averaged semi-classical approximation
We emphasize (1) that this peculiar manifestation of unitarity is not specific to black holes, and (2) that it can emerge from a single realization of an underlying unitary theory
Summary
Let us start by considering a collection of N hard spheres, each with a radius a, enclosed in a cubic box of length L. ∂B1 ∪∂B2, but only on ∂B1 ∪∂B2 −H where H is the part of the domain which corresponds to the location of the hole The region within this vanishing condition of the wave function is our true domain, and we denote it by D. Let NS and NL denote the number of particles in the smaller and larger box at a particular epoch. YNL) can loosely be thought of as denoting the position vectors of the particles in the smaller and larger boxes respectively.. YNL) can loosely be thought of as denoting the position vectors of the particles in the smaller and larger boxes respectively.5 In terms of these coordinates, we can define the wavefunctions as.
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