Semiclassical Quantum States for Black Holes

I discuss the semiclassical approximation for the Wheeler–DeWitt equation when applied to the CGHS model and spherically symmetric gravity. Special attention is devoted to the issues of Hawking radiation, decoherence of semiclassical states, and black hole entropy. Dedicated to Tullio Regge on his sixty–fifth birthday. To appear in the Proceedings of the Second Conference on Constrained Dynamics and Quantum Gravity, Santa Margherita, Italy, September 1996, edited by V. de Alfaro et al. (Nucl. Phys. B, Suppl., 1997). One of the most important unsolved problems in theoretical physics is to understand the full non–perturbative evolution of a black hole. Since a consistent quantum theory of gravity does not seem available yet, such an understanding remains elusive. A quantum theory of gravity is needed in particular to provide a derivation of black hole entropy from quantum statistical mechanics. As long as the full theory is unknown, research is focused on approaches which are either candidates for such a theory, or probably provide interesting models within which some of the interesting questions can be addressed. One of these approaches is canonical quantum gravity. Since the full Wheeler– DeWitt equation cannot be tackled in general, a restriction to special models and approximation schemes is necessary. In the following I shall briefly discuss some results which can be obtained from semiclassical approximations. I shall assume that there are no anomalies present. Otherwise the scheme has to be modified [1]. Most results are in accordance with results found by other methods, but it is illuminating to view them from a different perspective. I shall restrict attention in the following to basically two aspects: First, the treatment of Hawking radiation in the functional Schrödinger equation and its decohering influence on black hole superpositions and, second, the notion of black hole entropy for semiclassical quantum states. Most of the material was developed in collaboration with Jean-Guy Demers [2] and Thorsten Brotz [3], and I refer to these papers for more details. The semiclassical approximation to canonical quantum gravity, in its simplest version, is performed through an expansion with respect to the gravitational constant G [4]. Starting from the full quantum equations (WheelerDeWitt equation and diffeomorphism constraints) Ĥ⊥Ψ = 0, ĤiΨ = 0 (1) for a wave functional Ψ depending on three–geometries (as represented by three–metrics) and matter fields, an ansatz of the form