Go to previous page Go up Go to next page

5.17 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 [40Jump To The Next Citation Point]. Intuitively, this seems to be a pre-requisite for semiclassical behavior and has thus 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 no justification through more physical arguments involving observables or measurement processes to extract information from wave functions. The status of pre-classicality as a selection criterion is thus 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 [87Jump To The Next Citation Point86Jump To The Next Citation Point]. Pre-classicality also may not be possible to impose 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 [59]. 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 non-singular. On the other hand, evolution does continue to large volume and macroscopic regions, which is different from other scenarios such as [124] where inhomogeneities have been quantized on a background.

A similar issue is the boundedness of solutions, which also is motivated intuitively by referring to the common probability interpretation of quantum mechanics [119] 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 recollapsing 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.

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.

One thus needs analytical or at least semi-analytical techniques to deal with these issues. For pre-classicality one can advantageously use generating function techniques [87Jump 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 sum G(x) := nynxn for a solution yn on an equidistant lattice then solves a differential equation equivalent to the difference equation for y n. If G(x) is known, one can use its pole structure to get hints for the degree of oscillations in yn. In particular the behavior around x = - 1 is of interest to rule out alternating behavior where yn is of the form n yn = (- 1) qn with qn > 0 for all n (or at least all n larger than a certain value). At x = - 1 we then have sum G( - 1) = n qn, which is less convergent than the value for a non-alternating solution yn = qn resulting in sum G(- 1) = n(- 1)nqn. One can similarly find conditions for the pole structure to guarantee boundedness of y n, but the power of the method depends on the form of the difference equation. 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 [71Jump To The Next Citation Point]. One can then systematically find initial values for solutions that are guaranteed to be bounded.


  Go to previous page Go up Go to next page