If he uses a model at all, he is always aware that it pictures only certain aspects of the situation and leaves out other aspects. The total system of physics is no longer required to be such that all parts of its structure can be clearly visualized. …

A physicist must always guard against taking a visual model as more than a pedagogical device or makeshift help. At the same time, he must also be alert to the possibility that a visual model can, and sometimes does, turn out to be literally accurate. Nature sometimes springs such surprises.

RUDOLF CARNAP

An Introduction to the Philosophy of Science

In the preceding section, the link between models and the full theory was given through the same basic variables and the same kind of representation used, as well as a general construction scheme for the Hamiltonian constraint operator. The desired simplifications were realized thanks to the symmetry conditions, but not too strongly since basic features of the full theory are still recognizable in models. For instance, even though possible in many ways and often made use of, we did not employ special gauges or coordinate or field dependent transformations obscuring the relation. The models are thus as close to the full theory as possible while making full use of simplifications in order to have explicit applications.

Still, there are always some differences not all of which are easy to disentangle. For instance, we have discussed possible degeneracies between spin labels and edge lengths of holonomies, which can arise in the presence of a partial background and lead to new ambiguity parameters not present in the full theory. The question thus arises what the precise relation between models and the full theory is, or even how and to what extent a model for a given symmetry type can be derived from the full theory.

This is possible for the basic representation: The symmetry and the partial background it provides can be used to define natural subalgebras of the full holonomy/flux algebra by using holonomies and fluxes along symmetry generators and averaging in a suitable manner. Since the full representation is unique and cyclic, it induces uniquely a representation of models that is taken directly from the full theory. This will now be described independently for states and basic operators to provide the idea and to demonstrate the role of the extra structure involved. See also [57] and [63] for illustrations in the context of spherical symmetry and anisotropy, respectively.

6.1 Symmetric states

6.2 Basic operators

6.3 Quantization before reduction

6.4 Minisuperspace approximation

6.5 Quantum geometry from models to the full theory

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