Testing the master constraint programme for loop quantum gravity: III. SL(2\,\,\protect\bb{R}) models
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 analyse 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,{\bb R}) gauge symmetry and the complications arise because noncompact 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 planck^{2} which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to planck normal ordering constants). The physical Hilbert space can then be obtained after subtracting this normal ordering correction.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 February 2006
 DOI:
 10.1088/02649381/23/4/003
 arXiv:
 arXiv:grqc/0411140
 Bibcode:
 2006CQGra..23.1089D
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 33 pages, no figures