O ct 1 99 9 Canonical quantization of a minisuperspace model for gravity using self - dual variables ∗

The present article summarizes the work of the papers [1] dealing with the quantization of pure gravity and gravity coupled to a Maxwell field and a cosmological constant in presence of spherical symmetry. Let us stress the following : the motivation for this project was not to quantize a black hole. Rather, we regard the present model as an interesting testing ground for the quantization of full 3+1 gravity, in particular by using Ashtekar's self-dual representation. Throughout we assume that the reader is familiar with the Ashtekar-formulation of gravity ([2]). The conventions are as in [1]. To reduce gravity and our matter content to spherical symmetry, we require that the 3-metric and the Maxwell electric (ǫ a) and magnetic fields (µ a) are Lie annihilated by the generators of the SO(3) Killing group. The result of these Killing-reduction prescriptions is the following : Denoting the angular variables by θ, φ, the radial variable by x , and the standard orthonormal basis on the sphere by {n a } we obtain for the gravitational and Maxwell sector respectively 3 are angle-independent functions of x and t)