While it was expected early on that inhomogeneous situations would provide a refinement
of the discreteness scale at larger volumes, a direct derivation from the full theory remains
out of reach and the overall status is still at a somewhat heuristic level. Nevertheless, within
models one can test different versions of refinements for their viability, which then also
provides valuable feedback for the full dynamical behavior. For such reasons it was suggested
in [27] that one build in a scale dependence of the discreteness by hand, simply postulating
that has a certain, non-constant form. The specific behavior, , proposed
in [27], was argued to have some aesthetic advantages, but other functional forms are possible,
too.^{2}
In fact, considerations of inhomogeneous constraint operators suggest a behavior of the form
with . In this class, the case of constant step size in () and the proposal
of [27] () appear as limiting cases. A value of near currently seems
preferred by several independent arguments [75, 240, 91]. (Quantizations for different are not
related by unitary transformations, and one can indeed expect physical arguments to distinguish
between different values. Some non-physical arguments have been proposed which aim to fix
from the outset, mainly based on the observation that this case makes holonomy
corrections to effective equations independent of the parameter used in the formulation of
homogeneity (Section 4.2). This argument is flawed, however, because (i) effective densities will
nevertheless depend on , (ii) the quantization is still not invariant because it depends on the
coefficient in the proportionality (sometimes called the area gap) and (iii) while for
isotropic models a semiclassically suitable value for results from this argument, the situation is
more complicated in anisotropic models. As we will see in the discussion of inhomogeneous
situations in Section 6.4, the -dependence results from the minisuperspace reduction, rather
than from a gauge artifact, as is sometimes claimed. It can only be understood properly in the
relation between homogeneous and inhomogeneous models. For this, only the algebra of basic
operators is needed, which is also under good control in inhomogeneous settings, as described in
Section 7.)

With a scale-dependent discreteness, the continuum limit can be approached dynamically rather than as a mathematical process. While the universe expands, the discreteness is being refined and thus wave functions at large volume are supported on much finer lattices than at small volume. It is clear that this is relevant for the semiclassical limit of the theory, which should be realized for a large universe. Moreover, the relation to dynamics ties in fundamental constraint operators with low-energy behavior and phenomenology. Thus, studying such refinements results in restrictions for admissible fundamental constraints and can reduce possible quantization ambiguities. On the other hand, a scale-dependent discreteness implies more complicated difference equations. In isotropic models one can always transform the single variable and change the factor ordering of the constraint operator in such a way that one obtains an equidistant difference equation in the new variable [75]. Otherwise, if models are anisotropic or even inhomogeneous, this is possible only in special cases, which require new analytical or numerical tools. (See [262] for a numerical procedure.)

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