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5.2 Isotropy

Isotropic models are described purely in terms of the scale factor a(t) such that there is only a single kinematical degree of freedom. In connection variables, this is parameterized by the triad component p conjugate to the connection component c.

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 Aia = &tidle;cΛiIωIa (see Appendix B.2) we can choose holonomies along one integral curve of a symmetry generator XI. They are of the form

i a 1 i 1 hI = exp (∫ A aX Iτi) = cos(2μc ) + 2Λ Iτisin (2μc), (44 )

where μ 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 1 cos(2μc) and 1 sin(2μc ) for all μ uniquely determines the value of c, 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 [16Jump To The Next Citation Point]. In the present case, the background enters through the left-invariant 1-forms ωI defined on the spatial manifold, whose influence is condensed in the parameter μ. 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 μ, 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 [66Jump To The Next Citation Point]. 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 c but only almost periodic ones of the form [16Jump To The Next Citation Point]

∑ f(c) = fμexp (iμc ∕2) (45 ) μ
where the sum is over a countable subset of ℝ. This is analogous to the full situation, reviewed in Section 3.4, where matrix elements of holonomies define a special algebra of continuous functions of connections. As in this case, the algebra is represented as the set of all continuous functions on a compact space, called its spectrum. This compactification can be imagined as being obtained from enlarging the classical configuration space ℝ 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 f for which lim f (x ) = lim f (x) x→ −∞ x→∞ 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 -- ℝBohr, which contains ℝ 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 ℝ itself. Moreover, there is a one-to-one correspondence between irreducible representations of ℝ and irreducible representations of -- ℝBohr, which can also be used as the definition of the Bohr compactification. Representations of -- ℝBohr are thus labeled by real numbers and given by -- ρμ :ℝBohr → ℂ,c ↦→ eiμc. As with any compact group, there is a unique normalized Haar measure dμ(c) given explicitly by

∫ 1 ∫ T -- f (c)dμ(c) = lim --- f(c)dc , (46 ) ℝBohr T→ ∞ 2T −T
where on the right-hand side the Lebesgue measure on ℝ is used. (For further details of the Bohr compactification and the Hilbert space representation it defines, see ,e.g., [165Jump To The Next Citation Point].)

The Haar measure defines the inner product for the Hilbert space 2 -- L (ℝBohr,dμ(c)) of square integrable functions on the quantum configuration space. As one can easily check, exponentials of the form ⟨c|μ⟩ = eiμc∕2 are normalized and orthogonal to each other for different μ,

⟨μ1|μ2⟩ = δμ ,μ , (47 ) 1 2
which demonstrates that the Hilbert space is not separable.

Similar to holonomies, one needs to consider fluxes only for special surfaces, and all information is contained in the single number p. Since it is conjugate to c, it is quantized to a derivative operator

1 2 d pˆ= − 3iγ ℓP dc-, (48 )
whose action
ˆp|μ ⟩ = 1γℓ2μ |μ ⟩ =: pμ|μ⟩ (49 ) 6 P
on basis states |μ⟩ can easily be determined. In fact, the basis states are eigenstates of the flux operator, which demonstrates that the flux spectrum is discrete (all eigenstates are normalizable).

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 [45Jump To The Next Citation Point]. 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 c, and one can see that these operators are not continuous in the parameter μ. Thus, an operator quantizing c 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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