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5.21 Summary

There is a general construction of a loop representation in the full theory and its models, which is characterized by compactified connection spaces and discrete triad operators. Strong simplifications of some technical and conceptual steps occur in diverse models. Such a general construction allows a view not only in the simplest case, isotropy, but in essentially all representative systems for gravity.

Most important is the dynamics, which in the models discussed here can be formulated by a difference equation on superspace. A general scheme for a unique extension of wave functions through classical singularities is realized, such that the quantum theory is non-singular. This general argument, which has been verified in many models, is quite powerful since it does not require detailed knowledge of, or assumptions about, matter. It is independent of the availability of a global internal time, and so the problem of time does not present an obstacle. Moreover, a complicated discussion of quantum observables can be avoided since once it is known that a wave function can be continued uniquely, one can extract relational information at both sides of the classical singularity. (If observables would distinguish both sides with their opposite orientations, they would strongly break parity even on large scales in contradiction with classical gravity.) Similarly, information on the physical inner product is not required since there is a general statement for all solutions of the constraint equation. The uniqueness of an extension through the classical singularity thus remains, even if some solutions have to be excluded from physical Hilbert space or factored out if they have zero norm.

This is far from saying that observables or the physical inner product are irrelevant for an understanding of dynamical processes. Such constructions can, fortunately, be avoided for a general statement of non-singular evolution in a wide class of models. For details of the transition and to get information on the precise form of spacetime at the other side of classical singularities, however, all those objects are necessary and conceptual problems in their context have to be understood.

So far, the transition has often been visualized by intuitive pictures such as a collapsing universe turning inside out when the orientation is reversed. An hourglass is a good example of the importance of discrete quantum geometry close to the classical singularity and the emergence of continuous geometry on large scales: away from the bottleneck of the hourglass, its sand seems to be sinking down almost continuously. Directly at the bottleneck with its small circumference, however, one can see that time measured by the hourglass proceeds in discrete steps – one grain at a time.

Two issues remain: one would like to derive more geometrical or intuitive pictures of the non-singular behavior, such as those discussed in Section 4, but on a firmer basis. This can be done by effective equations as discussed in Section 6. Then there is the question of how models are related to the full theory and to what extent they are characteristic of full quantum geometry, which is the topic of Section 7.


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