Unitary S matrices with long-range correlations and the quantum black hole

Abstract

We propose an S matrix approach to the quantum black hole in which causality, unitarity and their interrelation play a prominent role. Assuming the 't Hooft S matrix ansatz for a gravitating region surrounded by an asymptotically flat space-time we find a non-local transformation which changes the standard causality requirement but is a symmetry of the unitarity condition of the S matrix. This new S matrix then implies correlations between the in and out states of the theory with the involvement of a third entity which in the case of a quantum black hole, we argue is the horizon S matrix. Such correlations are thus linked to preserving the unitarity of the S matrix and to the fact that entangling unitary operators are nonlocal. The analysis is performed within the Bogoliubov S matrix framework by considering a spacetime consisting of causal complements with a boundary in between. No particular metric or lagrangian dynamics need be invoked even to obtain an evolution equation for the full S matrix. Constraints imposed by the new causality requirement and implications for the effectiveness of field theoretical descriptions and for complementarity are also discussed. We find that the tension between information preservation and complementarity may be resolved provided the full quantum gravity theory either through symmetries or fine tuning forbids the occurrence of closed time like curves of information flow. Then, even if causality is violated near the horizon at any intermediate stage, a standard causal ordering may be preserved for the observer away from the horizon. In the context of the black hole, the novelty of our formulation is that it appears well suited to understand unitarity at any intermediate stage of black hole evaporation. Moreover, it is applicable generally to all theories with long range correlations including the final state projection models.