Aspects of black hole quantum mechanics and thermodynamics in 2+1 dimensions.

We discuss the quantum mechanics and thermodynamics of the (2+1)dimensional black hole, using both minisuperspace methods and exact results from Chern-Simons theory. In particular, we evaluate the first quantum correction to the black hole entropy. We show that the dynamical variables of the black hole arise from the possibility of a deficit angle at the (Euclidean) horizon, and briefly speculate as to how they may provide a basis for a statistical picture of black hole thermodynamics. email: [email protected] email: [email protected] In the twenty years since Bekenstein’s proposal that black holes have an entropy [1] and Hawking’s discovery that they can evaporate [2], a great deal has been learned about the thermodynamics of black holes. Nevertheless, some key questions remain unanswered: 1. Despite considerable work over the past few years, the “information loss paradox”— the apparently nonunitary transition from pure particle states to thermal Hawking radiation—remains an open problem (see, for example, [3]). 2. Standard approaches to black hole thermodynamics involve semiclassical approximations of one kind or another, and can tell us little about the final stages of the process of evaporation, where the effects of quantum gravity are sure to be important. 3. Although a number of interesting suggestions have been made, we do not yet have a generally accepted model of the microscopic statistical mechanics that should presumably underlie black hole thermodynamics. The recent discovery of black hole solutions in (2+1)-dimensional gravity offers a promising new arena for investigating such problems. In contrast to (3+1)-dimensional general relativity, the (2+1)-dimensional model has only finitely many physical degrees of freedom. As a result, questions about quantum gravity can be explored in considerable detail, and we can be reasonably confident that our conclusions are at least self-consistent. The purpose of this paper is to begin that exploration. The plan of the paper is the following. In section 1, we discuss the geometry of the Euclidean black hole and its relation to hyperbolic three-space IH, the complete Riemannian space of constant negative curvature, with appropriate identifications. We explain how to generalize these identifications to allow for a conical singularity (and a helical twist) at the horizon. The holonomies of this generalized geometry become the dynamical variables of the black hole, and are assigned Poisson brackets from previously available results for the Chern-Simons formulation. In section 2 we discuss the black hole from the Hamiltonian point of view, concentrating on a minisuperspace model. We again find that the parameters describing a singularity at the horizon become the dynamical variables of the black hole, with Poisson brackets equivalent to those derived from the Chern-Simons approach. We show that the partition function arises from a sum over these parameters, and we calculated it in the classical approximation, obtaining the standard black hole entropy. We then evaluate the first quantum correction, again using known results from the Chern-Simons formulation. For large black holes, we find that this correction does not involve h̄, and merely renormalizes the gravitational constant, a typical occurrence in Chern-Simons theory. Finally, we devote section 3 to speculation on the possible microscopic origin of black hole entropy as it emerges from the descriptions obtained in the first two sections. 1. The Euclidean Black Hole We shall investigate black hole thermodynamics in terms of the “Wick-rotated” Euclidean black hole. One may take the point of view that the Euclidean geometry emerges as a complex stationary point of the Lorentzian action (see, for example, [4]), or one may argue that the