It is convenient to absorb factors of into the basic variables, which is also suggested by the integrations in holonomies and fluxes on which background-independent quantizations are built . We thus define1 Note that here seems to differ from in (18) by a factor of 8. This happens due to different normalizations of the coordinate volume , which is a unit 3-sphere for a closed model in Friedmann–Robertson–Walker form, while isotropic connection variables, as reduced in Appendix B.2, are based on a 2-fold covering space obtained from isotropizing a Bianchi IX model.)
The part of phase space where we have and thus plays a special role since this is where isotropic classical singularities are located. On this subset the evolution equation (17) with standard matter choices is singular in the sense that , e.g.,[178, 114], any classical trajectory must intersect the subset for the matter we need in our universe, the classical theory is incomplete.
This situation, certainly, is not changed by introducing triad variables instead of metric variables. However, the situation is already different since is a sub-manifold in the classical phase space of triad variables, where can have both signs (the sign determining whether the triad is left or right handed, i.e., the orientation). This is in contrast to metric variables where is a boundary of the classical phase space. There are no implications in the classical theory since trajectories end there anyway, but it will have important ramifications in the quantum theory (see Sections 5.14, ??, 5.17 and 5.19).
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