  We provide a detailed comparison of the different approaches available for
the quantization of a totally constrained system with a constraint algebra
generating the non-compact $SL(2,\mathbb{R})$ group. In particular, we consider
three schemes: the Refined Algebraic Quantization, the Master Constraint
Programme and the Uniform Discretizations approach. For the latter, we provide
a quantum description where we identify semiclassical sectors of the
kinematical Hilbert space. We study the quantum dynamics of the system in order
to show that it is compatible with the classical continuum evolution. Among
these quantization approaches, the Uniform Discretizations provides the simpler
description in agreement with the classical theory of this particular model,
and it is expected to give new insights about the quantum dynamics of more
realistic totally constrained models such as canonical general relativity.

General Relativity and Quantum Cosmology