Isotropic difference equations for the gravitational degree of freedom alone are straightforward to
implement, but adding a second degree of freedom already makes some of the questions quite
involved. This second degree of freedom might be a conventionally quantized scalar, in which case
the equation may be of mixed difference/differential type, or an anisotropy parameter. The
first non-trivial numerical analysis was done for an isotropic model, but for a solution of the
gauge evolution of non-physical states generated by the Hamiltonian constraint [103]. Thus, the
second “degree of freedom” here was coordinate time. A numerical analysis with a scalar field as
internal time was done in [280]. Numerical investigations for the first models with two true
gravitational degrees of freedom were performed in the context of anisotropic and especially black hole
interior models [130
, 123], which in combination with analytical tools have provided valuable
insights into the stability issue [250]. Stability, in this case, requires non-trivial refinement
models, which necessarily lead to non-equidistant difference equations [75
]. This again poses new
numerical issues, on which there has recently been some progress [262]. Variational and other
methods have been suggested [275, 274], but not yet developed into a systematic numerical
tool.
A scalar as a second degree of freedom in an isotropic model does not lead to severe stability issues. For
a free and massless scalar, moreover, one can parameterize the model with the scalar as internal time and
solve the evolution it generates. There is a further advantage because one can easily derive the physical
inner product, and then numerically study physical states [26, 28
]. This has provided the first geometrical
pictures of a bouncing wave function [25]. The same model is discussed from the perspective of
computational physics in [211].
Here also, long-term evolution can be used to analyze refinement models [27], although the emphasis on
isotropic models avoids many issues related to potentially non-equidistant difference equations. Numerical
results for isotropic free-scalar models also indicate that they are rather special, because wave functions do
not appear to spread or deform much, even during long-term evolution. This suggests that there is a hidden
solvability structure in such models, which was indeed found in [70
]. As described in Section 6, the
solvability provides new solution techniques as well as generalizations to non-solvable models
by the derivation of effective equations in perturbation theory. Unless numerical tools can be
generalized considerably beyond isotropic free-scalar models, effective techniques seem much more
feasible to analyze, and less ambiguous to interpret, than results from this line of numerical
work.
The derivation and analysis of such effective systems has, by the existence of an exactly solvable model,
led to a clean mathematical analysis of dynamical coherent states suitable for quantum cosmology. In
particular, the role of squeezed states has been highlighted [69, 74
]. Further mathematical issues show up in
possible extensions of these results to more realistic models. Independently, some numerical studies of loop
quantum cosmology have suggested a detailed analysis of self-adjointness properties of Hamiltonian
constraint operators used. In some cases, essential self-adjointness can strictly be proven [286, 285, 197],
but there are others where the constraint is known not to be essentially self-adjoint. Possible
implications of the choice of a self-adjoint extension are under study. Unfortunately, however, the
theorems used so far to conclude essential self-adjointness only apply to a specific factor ordering
of the constraint, which splits the two sine factors of
in Equation (52
) arising from
the holonomy
around a square loop evaluated in an isotropic connection; see
Section 5.4. This splitting with both factors sandwiching the commutator
could not
be done under general circumstances, where one only has the complete
along some loop
as a single factor in the constraint. Thus the e mathematical techniques used need to be
generalized.
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