Poincare Recurrences and Topological Diversity

TL;DR

This paper investigates the challenge of reproducing Poincare recurrences and quasi-periodic correlators in finite entropy thermal systems using AdS/CFT, highlighting the difficulty of capturing true periodicity through summation over spacetime images.

Contribution

It demonstrates that summing over SL(2,Z) images in the BTZ spacetime fails to produce the expected quasi-periodic behavior, emphasizing the need for exact re-summation methods.

Findings

Large corrections occur at critical times in the sum over images.

The correlator does not exhibit quasi-periodicity after summation.

Exact re-summation may be necessary to recover periodicity.

Abstract

Finite entropy thermal systems undergo Poincare recurrences. In the context of field theory, this implies that at finite temperature, timelike two-point functions will be quasi-periodic. In this note we attempt to reproduce this behavior using the AdS/CFT correspondence by studying the correlator of a massive scalar field in the bulk. We evaluate the correlator by summing over all the SL(2,Z) images of the BTZ spacetime. We show that all the terms in this sum receive large corrections after at certain critical time, and that the result, even if convergent, is not quasi-periodic. We present several arguments indicating that the periodicity will be very difficult to recover without an exact re-summation, and discuss several toy models which illustrate this. Finally, we consider the consequences for the information paradox.