  We compare different treatments of the constraints in canonical quantum
gravity. The standard approach on the superspace of 3--geometries treats the
constraints as the sole carriers of the dynamic content of the theory, thus
rendering the traditional dynamical equations obsolete. Quantization of the
constraints in both the Dirac and ADM square root Hamiltonian approaches leads
to the well known problems of time evolution. These problems of time are of
both an interpretational and technical nature. In contrast, the geometrodynamic
quantization procedure on the superspace of the true dynamical variables
separates the issues of quantization from the enforcement of the constraints.
The resulting theory takes into account states that are off-shell with respect
to the constraints, and thus avoids the problems of time. We develop, for the
first time, the geometrodynamic quantization formalism in a general setting and
show that it retains all essential features previously illustrated in the
context of homogeneous cosmologies.
