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5.12 Loop inspired quantum cosmology

The constructions described so far in this section follow all the steps in the full theory as closely as possible. Most importantly, one obtains quantum representations inequivalent to those used in a Wheeler-DeWitt quantization, which results in many further implications. This has inspired investigations where not all the steps of loop quantum gravity are followed, but only the same type of representation, i.e., the Bohr Hilbert space in an isotropic model, is used. Other constructions, based on ADM rather than Ashtekar variables, are then done in the most straightforward way rather than a way suggested by the full theory [131].

In isotropic models the results are similar, but already here one can see conceptual differences. Since the model is based on ADM variables, in particular using the metric and not triads, it is not clear what the additional sign factor sgn(m), which is then introduced by hand, means geometrically. In loop quantum cosmology it arose naturally as orientation of triads, even before its role in removing the classical singularity, to be discussed in Section 5.15, had been noticed. (The necessity of having both signs available is also reinforced independently by kinematical consistency considerations in the full theory [117].) In homogeneous models the situation is even more complicated since sign factors are still introduced by hand, but not all of them are removed by discrete gauge transformations as in Section 5.7 (see [158] as opposed to [14Jump To The Next Citation Point]). Those models are useful to illuminate possible effects, but they also demonstrate how new ambiguities, even with conceptual implications, arise if guidance from a full theory is lost.

In particular the internal time dynamics is more ambiguous in those models and thus not usually considered. There are then only arguments that the singularity could be avoided through boundedness of relevant operators, but those statements are not generic in anisotropic models [62Jump To The Next Citation Point] or even the full theory [85Jump To The Next Citation Point]. Moreover, even if all curvature quantities could be shown to be bounded, the evolution could still stop (as happens classically where not any singularity is also a curvature singularity).


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