Restoring Unitarity in BTZ Black Hole

Whether or not system is unitary can be seen from the way it, if perturbed, relaxes back to equilibrium. The relaxation of semiclassical black hole can be described in terms of correlation function which exponentially decays with time. In the momentum space it is represented by infinite set of complex poles to be identified with the quasi-normal modes. This behavior is in sharp contrast to the relaxation in unitary theory in finite volume: correlation function of the perturbation in this case is quasi-periodic function of time and, in general, is expected to show the Poincar\'e recurrences. In this paper we demonstrate how restore unitarity in the BTZ black hole, the simplest example of eternal black hole in finite volume. We start with reviewing the relaxation in the semiclassical BTZ black hole and how this relaxation is mirrored in the boundary conformal field theory as suggested by the AdS/CFT correspondence. We analyze the sum over $SL(2,{\bf Z})$ images of the BTZ space-time and suggest that it does not produce a quasi-periodic relaxation, as one might have hoped, but results in correlation function which decays by power law. We develop our earlier suggestion and consider a non-semiclassical deformation of the BTZ space-time that has structure of wormhole connecting two asymptotic regions semiclassically separated by horizon. The small deformation parameter $\lambda$ is supposed to have non-perturbative origin to capture the finite N behavior of the boundary theory. The discrete spectrum of perturbation in the modified space-time is computed and is shown to determine the expected unitary behavior: the corresponding time evolution is quasi-periodic with hierarchy of large time scales $\ln 1/\lambda$ and $1/\lambda$ interpreted respectively as the Heisenberg and Poincar\'e time scales in the system.