Mass spectrum and statistical entropy of the BTZ black hole from canonical quantum gravity

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 -Λ≡-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 μ_n=(ℏ)/(l)(n+(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 AdS₃. This gives a connection between the Wheeler-DeWitt approach and the conformal field theory approach.