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The SL(2,R) totally constrained model: three quantization approaches

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.