In a recent publication we developed a canonical quantization program describing the gravitational collapse of a spherical dust cloud in $2+1$ dimensions with a negative cosmological constant $-\Lambda \equiv -l^{-2}<0$. In this paper we address the quantization of the Banados-Teitelboim-Zanelli (BTZ) black hole. We show that the mass function describing the black hole is made of two pieces, a constant nonvanishing boundary contribution and a discrete spectrum of the form ${\mu }_n=\frac{\hslash }{l}(n+\frac{1}{2})$. The discrete spectrum is obtained by applying the Wheeler-DeWitt equation with a particular choice of factor ordering and interpreted as giving the energy levels of the collapsed matter shells that form the black hole. Treating a black hole microstate as a particular distribution of shells among the levels, we determine the canonical entropy of the BTZ black hole. Comparison with the Bekenstein-Hawking entropy shows that the boundary energy is related to the central charge of the Virasoro algebra that generates the asymptotic symmetry group of the three-dimensional anti-de Sitter space ${\mathrm{AdS}}_3$. This gives a connection between the Wheeler-DeWitt approach and the conformal field theory approach.

• Received 15 December 2007

DOI:https://doi.org/10.1103/PhysRevD.77.064021