Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models

TL;DR

This paper tests the Master Constraint Programme in Loop Quantum Gravity using complex SL(2,R) models, revealing spectral challenges and proposing a normalization correction to obtain the physical Hilbert space.

Contribution

It extends the Master Constraint Programme analysis to SL(2,R) models, addressing issues with non-compact groups and spectral properties, and introduces a normalization correction for the physical Hilbert space.

Findings

Spectrum of the Master Constraint does not include zero.

Minimum of the spectrum is of order , indicating a normal ordering constant.

Normal ordering correction allows construction of the physical Hilbert space.

Abstract

This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an $SL(2,\Rl)$ gauge symmetry and the complications arise because non -- compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. However, the minimum of the spectrum is of order $\hbar^2$ which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to $\hbar$ normal ordering constants). The physical Hilbert space can then be be obtained after subtracting this normal ordering correction.