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

TL;DR

This paper compares three quantization methods for a totally constrained $SL(2, ext{R})$ system, showing that the Uniform Discretizations approach offers a simpler and more classical-compatible quantum description, with implications for quantum gravity models.

Contribution

It provides a detailed comparison of three quantization schemes for an $SL(2, ext{R})$ constrained system, highlighting the advantages of the Uniform Discretizations approach.

Findings

Uniform Discretizations aligns well with classical evolution.

The approach simplifies the quantum description of the model.

Insights may extend to quantum gravity theories like general relativity.

Abstract

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.