Scalar field quantization on the (2+1)-dimensional black hole background.

The quantization of a massless conformally coupled scalar eld on the 2+1 dimensional Anti de Sitter black hole background is presented. The Green's function is calculated, using the fact that the black hole is Anti de Sitter space with points identiied, and taking into account the fact that the black hole spacetime is not globally hyperbolic. It is shown that the Green's function calculated in this way is the Hartle-Hawking Green's function. The Green's function is used to compute hT i, which is regular on the black hole horizon, and diverges at the singularity. A particle detector response function outside the horizon is also calculated and shown to be a fermi type distribution. The back-reaction from hT i is calculated exactly and is shown to give rise to a curvature singularity at r = 0 and to shift the horizon outwards. For M = 0 a horizon develops, shielding the singularity. Some speculations about the endpoint of evaporation are discussed.