If we restrict ourselves to invariant connections of a given form, it suffices to probe them with
only special holonomies. For an isotropic connection (see Appendix B.2) we can
choose holonomies along one integral curve of a symmetry generator
. They are of the form
This illustrates how symmetric configurations allow one to simplify the constructions behind the full
theory. But it also shows which effects the presence of a partial background can have on the formalism [15].
In the present case, the background enters through the left-invariant 1-forms
defined on the spatial
manifold whose influence is contained in the parameter
. All information about the edge used to
compute the holonomy is contained in this single parameter, which leads to degeneracies compared to the
full theory. Most importantly, one cannot distinguish between the parameter length and the
spin label of an edge: Taking a power of the holonomy in a non-fundamental representation
simply rescales
, which could just as well come from a longer parameter length. That this is
related to the presence of a background can be seen by looking at the roles of edges and spin
labels in the full theory. There, both concepts are independent and appear very differently.
While the embedding of an edge, including its parameter length, is removed by diffeomorphism
invariance, the spin label remains well-defined and is important for ambiguities of operators. In the
model, however, the full diffeomorphism invariance is not available such that some information
about edges remains in the theory and merges with the spin label. Issues like that have to
be taken into account when constructing operators in a model and comparing with the full
theory.
The functions appearing in holonomies for isotropic connections define the algebra of functions on the
classical configuration space which, together with fluxes, is to be represented on a Hilbert space. This
algebra does not contain arbitrary continuous functions of but only almost periodic ones of the form [15
]
In the present case, the procedure is more complicated and leads to the Bohr compactification ,
which contains
densely. It is very different from the one point compactification, as can be seen from the
fact that the only function which is continuous on both spaces is the zero function. In contrast to the one
point compactification, the Bohr compactification is an Abelian group, just like
itself.
Moreover, there is a one-to-one correspondence between irreducible representations of
and
irreducible representations of
, which can also be used as the definition of the Bohr
compactification. Representations of
are thus labeled by real numbers and given by
. As with any compact group, there is a unique normalized Haar measure
given by
The Haar measure defines the inner product for the Hilbert space of square integrable
functions on the quantum configuration space. As one can easily check, exponentials of the form
are normalized and orthogonal to each other for different
,
Similarly to holonomies, one needs to consider fluxes only for special surfaces, and all information is
contained in the single number . Since it is conjugate to
, it is quantized to a derivative operator
This property is analogous to the full theory with its discrete flux spectra, and similarly it implies
discrete quantum geometry. We thus see that the discreteness survives the symmetry reduction in this
framework [37]. Similarly, the fact that only holonomies are represented in the full theory but not
connection components is realized in the model, too. In fact, we have so far represented only exponentials of
, and one can see that these operators are not continuous in the parameter
. Thus, an operator
quantizing
directly does not exist on the Hilbert space. These properties are analogous to the full
theory, but very different from the Wheeler-DeWitt quantization. In fact, the resulting representations in
isotropic models are inequivalent. While the representation is not of crucial importance when only small
energies or large scales are involved [18], it becomes essential at small scales which are in particular realized
in cosmology.
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