THE ISSUE OF TIME EVOLUTION IN QUANTUM-GRAVITY

We discuss the relation between the concept. of time and the dynamic s tructure of quantum gravity. We briefly review the problems of time as sociated with the standard procedures of gravity quantization. By expl icitly utilizing York's analysis of the geometrodynamic degrees of fre edom, and imposing the constraints as expectation value equations, we describe a new procedure of gravity quantization. In particular, this ''minimally constrained canonical'' quantization procedure leads to a linear Schrodinger equation augmented by the super-Hamiltonian and sup ermomentum constraints imposed on expectation values. This approach su pplies a description of time evolution in quantum geometrodynamics fre e from the standard problems of time associated with canonical approac hes. Furthermore, the theory is applicable to the full theory of gener al relativity, without the need to impose symmetries or reduce dimensi onality. Using this method we arrive at an analytic expression for the quantum evolution of the Kasner cosmology, as well as a numerically g enerated solution of the quantized Taub model. The key new feature of this approach is that we impose the constraints weakly - instead of th e quantum-mechanical wave function satisfying the constraints exactly, it need only satisfy them on average. Thus the constraints only const rain the observables.