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5.18 Pre-classicality and boundedness

The high order of difference equations implies that there are in general many independent solutions, most of which are oscillating on small scales, i.e., when the labels change only slightly. One possibility to restrict the number of solutions then is to require suppressed or even absent oscillations on small scales [48Jump To The Next Citation Point]. Intuitively, this seems to be a prerequisite for semiclassical behavior and has therefore been called Pre-classicality. It can be motivated by the fact that a semiclassical solution should not be sensitive to small changes of, e.g., the volume by amounts of Planck size. However, even though the criterion sounds intuitively reasonable, there is so far not much justification through more physical arguments involving observables or measurement processes to extract information from wave functions. One possible mechanism is that group averaging the connection dependent constraint would have to result in almost constant physical states. The exponentiated constraint is not a precise translation operator because it is not linear in the connection and also depends on the triad and matter, and so states would certainly change on larger scales. In this way, Pre-classicality might be related to physical normalization. Overall, however, the status of Pre-classicality as a selection criterion is not final.

Moreover, Pre-classicality is not always consistent in all disjoint classical regimes or with other conditions. For instance, as discussed in the following section, there can be additional conditions on wave functions arising from the constraint equation at the classical singularity. Such conditions do not arise in classical regimes, but they nevertheless have implications for the behavior of wave functions there through the evolution equation [124Jump To The Next Citation Point122Jump To The Next Citation Point]. It may also not be possible to impose Pre-classicality in all disconnected classical regimes. If the evolution equation is locally stable – which is a basic criterion for constructing the constraint – choosing initial values in classical regimes, which do not have small-scale oscillations, guarantees that oscillations do not build up through evolution in a classical regime [79Jump To The Next Citation Point]. However, when the solution is extended through the quantum regime around a classical singularity, oscillations do arise and do not in general decay after a new supposedly-classical regime beyond the singularity is entered. It is thus not obvious that indeed a new semiclassical region forms, even if the quantum evolution for the wave function is nonsingular. On the other hand, evolution does continue to large volume and macroscopic regions, which is different from other scenarios, such as [179] where inhomogeneities have been quantized on a background.

A similar issue is the boundedness of solutions, which is also motivated intuitively by referring to the common-probability interpretation of quantum mechanics [174], but must be supported by an analysis of physical inner products. The issue arises, in particular, in classically forbidden regions where one expects exponentially growing and decaying solutions. If a classically forbidden region extends to infinite volume, as happens for models of re-collapsing universes, the probability interpretation would require that only the exponentially decaying solution is realized. As before, such a condition at large volume is in general not consistent in all asymptotic regions or with other conditions arising in quantum regimes. For a free, massless scalar as matter source one can compute the physical inner product with the result that physical solutions indeed decay beyond the collapse point [28Jump To The Next Citation Point].

Both issues, pre-classicality and boundedness, seem to be reasonable, but their physical significance has to be founded on properties of the physical inner product. They are rather straightforward to analyze in isotropic models without matter fields, where one is dealing with ordinary difference equations. However, other cases can be much more complicated such that conclusions drawn from isotropic models alone can be misleading. Moreover, numerical investigations have to be taken with care since, in particular for boundedness, an exponentially increasing contribution can easily arise from numerical errors and dominate the exact, potentially-bounded solution.

Thus, one needs analytical or at least semi-analytical techniques to deal with these issues. For pre-classicality one can advantageously use generating function techniques [124Jump To The Next Citation Point], if the difference equation is of a suitable form, e.g., has only coefficients with integer powers of the discrete parameter. The generating function G (x ) := ∑ ψ xn n n for a solution ψ n on an equidistant lattice then solves a differential equation equivalent to the difference equation for ψn. If G (x ) is known, one can use its pole structure to get hints for the degree of oscillation in ψn. In particular, the behavior around x = − 1 is of interest to rule out alternating behavior where ψn is of the form ψn = (− 1)nξn with ξn > 0 for all n (or at least all n larger than a certain value). At x = − 1 we then have G (− 1) = ∑ ξ n n, which is less convergent than the value for a non-alternating solution ψn = ξn resulting in ∑ n G (− 1) = n(− 1) ξn. One can similarly find conditions for the pole structure to guarantee boundedness of ψn, but the power of the method depends on the form of the difference equation. Generating functions have been used in several cases for isotropic and anisotropic models [122Jump To The Next Citation Point143Jump To The Next Citation Point123Jump To The Next Citation Point]. More general techniques are available for the boundedness issue, and also for alternating behavior, by mapping the difference equation to a continued fraction which can be evaluated analytically or numerically [101Jump To The Next Citation Point]. One can then systematically find initial values for solutions that are guaranteed to be bounded.


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