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 [87, 86
]. 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 [87] if the difference equation is
of a suitable form, e.g., has only coefficients with integer powers of the discrete parameter. The generating
function
for a solution
on an equidistant lattice then solves a differential equation
equivalent to the difference equation for
. If
is known, one can use its pole structure to get hints
for the degree of oscillations in
. In particular the behavior around
is of interest to rule out
alternating behavior where
is of the form
with
for all
(or at least all
larger than a certain value). At
we then have
, which is less convergent
than the value for a non-alternating solution
resulting in
.
One can similarly find conditions for the pole structure to guarantee boundedness of
,
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
[71
]. One can then systematically find initial values for solutions that are guaranteed to be
bounded.
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