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5.5 Dynamical refinements of the discreteness scale

Although basic holonomies depend only on the connection, the Hamiltonian constraint operator may in principle contain almost periodic functions of connection components with a coefficient δ depending on the scale μ. Since the step size of the difference equation is determined by this coefficient, the equation will no longer be of constant step size. While such behavior in general makes an analysis of the equation more complicated, there are advantages for the viability of specific solutions, if the discreteness becomes smaller for large volume. In some models, problems can indeed arise if the step size of the difference equation stays constant in the triad component, i.e., in μ, in particular in the presence of a cosmological constant [54Jump To The Next Citation Point]. Several other issues related to such effects have been discussed recently in isotropic models [27Jump To The Next Citation Point], anisotropic ones [75Jump To The Next Citation Point] and in phenomenological applications [240Jump To The Next Citation Point23990Jump To The Next Citation Point91Jump To The Next Citation Point].

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 [27Jump To The Next Citation Point] that one build in a scale dependence of the discreteness by hand, simply postulating that δ(μ) has a certain, non-constant form. The specific behavior, −1∕2 δ ∝ |μ|, proposed in [27Jump To The Next Citation Point], 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 δ ∝ |μ |x with 0 < x < − 1∕2. In this class, the case of constant step size in μ (x = 0) and the proposal of [27Jump To The Next Citation Point] (x = − 1 ∕2) appear as limiting cases. A value of x near − 1∕2 currently seems preferred by several independent arguments [75Jump To The Next Citation Point24091Jump To The Next Citation Point]. (Quantizations for different x 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 x = − 1∕2 from the outset, mainly based on the observation that this case makes holonomy corrections to effective equations independent of the parameter V0 used in the formulation of homogeneity (Section 4.2). This argument is flawed, however, because (i) effective densities will nevertheless depend on V0, (ii) the quantization is still not invariant because it depends on the coefficient in the proportionality δ ∝ |μ |− 1∕2 (sometimes called the area gap) and (iii) while for isotropic models a semiclassically suitable value for x 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 V0-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 μ and change the factor ordering of the constraint operator in such a way that one obtains an equidistant difference equation in the new variable [75Jump To The Next Citation Point]. Otherwise, if models are anisotropic or even inhomogeneous, this is possible only in special cases, which require new analytical or numerical tools. (See [262Jump To The Next Citation Point] for a numerical procedure.)


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