If we restrict ourselves to the 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
where depends on the parameter length of the curve and can be any real number (thanks to
homogeneity, path ordering is not necessary). Since knowing the values of
and
for
all
uniquely determines the value of
, which is the only gauge-invariant information
contained in the connection, these holonomies describe the configuration space of connections
completely.
This illustrates how symmetric configurations allow one to simplify the constructions behind the full
theory. But it also shows what effects the presence of a partial background can have on the formalism [16].
In the present case, the background enters through the left-invariant 1-forms
defined on the spatial
manifold, whose influence is condensed in the parameter
. All information about the edge used to
compute the holonomy is contained in this single parameter, which leads to degeneracies in comparison 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 role 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. One can see in detail how the degeneracy arises in the
process of inducing the representation of the symmetric model [66
]. 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 [16
]
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 functions that are continuous on both spaces are constants. 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 explicitly 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
,
Similar 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 [45]. Likewise, 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 [19], it becomes essential at small scales, which are found frequently in
cosmology.
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