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 . Thus, the second “degree of freedom” here was coordinate time. A numerical analysis with a scalar field as internal time was done in . 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 . Stability, in this case, requires non-trivial refinement models, which necessarily lead to non-equidistant difference equations . This again poses new numerical issues, on which there has recently been some progress . 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 . The same model is discussed from the perspective of computational physics in .
Here also, long-term evolution can be used to analyze refinement models , 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 . 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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