Abstract
We continue the study of real-time replica wormholes initiated in [1]. Previously, we had discussed the general principles and had outlined a variational principle for obtaining stationary points of the real-time gravitational path integral. In the current work we present several explicit examples in low-dimensional gravitational theories where the dynamics is amenable to analytic computation. We demonstrate the computation of Rényi entropies in the cases of JT gravity and for holographic two-dimensional CFTs (using the dual gravitational dynamics). In particular, we explain how to obtain the large central charge result for subregions comprising of disjoint intervals directly from the real-time path integral.
Highlights
Real-time computation of correlation functions, both time-ordered and out-of-time-order, as well as density operator matrix elements and their moments, in any quantum system either with or without dynamical gravity, requires the use of a suitable timefolded contour, with segments of forward and backward evolution
One often eschews the use of such contours, relying instead on computations in the Euclidean domain, and analytically continuing the answers obtained into the real-time domain, a strategy that works well when the quantum evolution is not subject to nonanalytic sources. While this is strategy is efficient in extracting information about the non-perturbative aspects of the theory, it does not lend insight into the physical dynamical evolution directly. These issues have been well appreciated in the context of quantum field theory for many decades, but have come to fore with recent analyses of new semiclassical configurations that address the black hole information problem
Inspired by the Euclidean path integral arguments [5,6,7,8] that helped derive the static holographic entanglement entropy formula [9] and its quantum generalization [10], recent investigations in low-dimensional gravity theories have argued for the contribution of replica wormhole saddles [11, 12] in the gravitational path integral
Summary
Real-time computation of correlation functions, both time-ordered and out-of-time-order, as well as density operator matrix elements and their moments, in any quantum system either with or without dynamical gravity, requires the use of a suitable timefolded contour, with segments of forward and backward evolution. This example has been well studied both in field theory and gravity and we again use it to provide an illustration of the geometry of the real-time gravitational solution. Appendix C is a quick overview of the Schottky construction of the covering space geometry (both on the boundary and in the bulk) for the computation of second Rényi entropy for 2 disjoint intervals For this case we present an explicit evaluation of the Euclidean action from the bulk solution in appendix D (as far as we are aware this computation has not hitherto been reported in the literature). Appendix E summarizes some familiar sign conventions and useful identities that we employ in the course of our calculation
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