Hole Area Quantization ∗

It has been argued by several authors that the quantum mechanical spectrum of black hole horizon area must be discrete. This has been confirmed in different formalisms, using different approaches. Here we concentrate on two approaches, the one involving quantization on a reduced phase space of collective coordinates of a Black Hole and the algebraic approach of Bekenstein. We show that for non-rotating, neutral black holes in any spacetime dimension, the approaches are equivalent. We introduce a primary set of operators sufficient for expressing the dynamical variables of both, thus mapping the observables in the two formalisms onto each other. The mapping predicts a Planck size remnant for the black hole. 1. What to Quantize? The central question of obtaining the theory of Quantum Gravity seems to be : what to " quantize ". A perturbative approach has been successfully attempted especially in the elegant work of B. DeWitt and is adequate if one learns to contend with the limitations of a nonrenormalizable theory. However the nonperturbative aspects of the theory would remain inaccessible. The formulation in terms of New Canonical Variables of Ashtekar may be taken to be the minimal consistent nonperturbative approach to Gravity. It is a formulation amenable to solution on loop spaces, originally pioneered by Mandelstam for QCD. Obtaining phenomenologically interesting solutions however remains an unsolved problem. This has spurred a number of other approaches wherein one assumes certain ground states suggested by classical General Relativity as possible vacuua. By fo-cusing on a few collective coordinates one attempts a quantization of these. Several approaches to Quantum Cosmology may be viewed in this light and seem to enjoy success, again when interpreted with caution. In the following we shall discuss such