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A basic assumption in classical GR is that the metric field is nowheredegenerate in spacetime. In particular the induced metric on Cauchy surfacesmust be nowhere degenerate. It is only under this assumption that one canderive the hypersurace deformation algebra between the initial valueconstraints which is absolutely transparent from the fact that the {\itinverse} of the induced metric is needed to close the algebra. This statementis independent of the density weight that one may want to equip the spatialmetric with. Accordingly, the very definition of a non-anomalous representationof the hypersurface defomation algebra in quantum gravity has to address theissue of non-degenracy of the induced metric that is needed in the classicaltheory. In the Hilbert space representation employed in Loop Quantum Gravity(LQG) most emphasis has been layed to define an inverse metric operator on thedense domain of spin network states although they represent induced quantumgeometries which are degenerate almost everywhere. It is no surprise thatdemonstration of closure of the constraint algebra on this domain meetsdifficulties because it is a sector of the quantum theory which is classicallyforbidden and which lies outside the domain of definition of the classicalhypersurface deformation algebra. Various suggestions for addressing the issuesuch as non-standard operator topologies, dual spaces (habitats) and densityweights have been propposed to address this issue with respect to the quantumdynamics of LQG. In this article we summarise these developments and argue thatinsisting on a dense domain of non-degenerate states within the LQGrepresentation may provide a natural resolution of the issue thereby possiblyavoiding the above mentioned non-standard constructions.

Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity