  The wave-function in quantum gravity is supposed to obey the Wheeler-DeWitt
(WDW) equation, however there is neither a satisfactory probability
interpretation nor a successful solution to the problem of time in the WDW
framework. To gain some insight on these issues we compare quantization of
ordinary systems, first in the usual way having the Schr\"{o}dinger equation
and second by promoting them as parametrized theories by introducing embedding
coordinate fields, which yields first class constraints and the WDW equation.
We observe that the time evolution in the WDW framework can be described with
respect to the embedding coordinates, where the WDW equation becomes
Schr\"{o}dinger like, i.e. it involves first order timelike functional
derivatives. Moreover, the equivalence with the ordinary quantization procedure
determines a suitable Hilbert space with a viable probability interpretation.
We then apply the same construction to general relativity by adding embedding
fields without any prior coordinate choice. The reparametrized general
relativity has two different types of diffeomorphism invariance, which arises
from world-volume and target-space reparametrizations. As in the case of
ordinary systems, the time evolution can be described with respect to the
embedding fields and the WDW equation becomes Schr\"{o}dinger like; the
construction is almost identical to an ordinary parametrized field theory in
terms of time evolution and Hilbert space structure. However, this time, the
constraint algebra enforces the wave-function to be in a subspace of states
annihilated by an operator that can be identified as the Hamiltonian. The
implications of these results for the canonical quantization program, and in
particular for the minisuperspace quantum cosmology, are discussed.
