  Finite dimensional models that mimic the constraint structure of Einstein's
General Relativity are quantized in the framework of BRST and Dirac's canonical
formalisms. The first system to be studied is one featuring a constraint
quadratic in the momenta (the "super-Hamiltonian") and a set of constraints
linear in the momenta (the "supermomentum" constraints). The starting point is
to realize that the ghost contributions to the supermomentum constraint
operators can be read in terms of the natural volume induced by the constraints
in the orbits. This volume plays a fundamental role in the construction of the
quadratic sector of the nilpotent BRST charge. It is shown that the quantum
theory is invariant under scaling of the super-Hamiltonian. As long as the
system has an intrinsic time, this property translates in a contribution of the
potential to the kinetic term. In this aspect, the results substantially differ
from other works where the scaling invariance is forced by introducing a
coupling to the curvature. The contribution of the potential, far from being
unnatural, is beautifully justified in the light of the Jacobi's principle.
Then, it is shown that the obtained results can be extended to systems with
extrinsic time. In this case, if the metric has a conformal temporal Killing
vector and the potential exhibits a suitable behavior with respect to it, the
role played by the potential in the case of intrinsic time is now played by the
norm of the Killing vector. Finally, the results for the previous cases are
extended to a system featuring two super-Hamiltonian constraints. This step is
extremely important due to the fact that General Relativity features an
infinite number of such constraints satisfying a non trivial algebra among
themselves.
