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 . 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 all continuous functions on a compact space, called its spectrum. This compactification can be imagined as being obtained from enlarging the classical configuration space by adding points, and thus more continuity conditions, until only functions of the given algebra survive as continuous ones. A well-known example is the one point compactification, which is the spectrum of the algebra of continuous functions for which exists. In this case, one just needs to add a single point at infinity.
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 . 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 , it becomes essential at small scales which are in particular realized in cosmology.
© Max Planck Society and the author(s)