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8.2 Mathematical development of models

The main open issue, requiring new insights at all levels, is that of inhomogeneities. While inhomogeneous models have been formulated and partly analyzed, the following tasks are still to be completed:
Exact models:
In particular the dynamics of inhomogeneous models is much more complicated to analyze than in homogeneous ones. Understanding may be improved by an interesting cross-relation with black holes. This allows one to see if the different ingredients and effects of a loop quantization fit together in a complete picture, which so far seems to be the case [141591376Jump To The Next Citation Point57]. Moreover, the dynamics can possibly be simplified and understood better around slowly evolving horizons [7976]. Other horizon conditions are also being studied in related approaches [132107].

 Not directly related to physical applications but equally important is the issue of consistency of the constraints. The constraint algebra trivializes in homogeneous models, but is much more restrictive with inhomogeneities. Here, the feasibility of formulating a consistent theory of quantum gravity can be tested in a treatable situation. Related to consistency of the algebra, at least at a technical level, is the question of whether or not quantum gravity can predict initial conditions for a universe, or at least restrict its set of solutions.

Relation between models and the full theory:
 By strengthening the relation between models and the full theory, ideally providing a complete derivation of models, physical applications will be put on a much firmer footing. This is also necessary to understand better effects of reductions such as degeneracies between different concepts or partial backgrounds. One aspect not realized in models so far is the large amount of non-Abelian effects in the full theory, which can be significant also in models [54].

Numerical quantum gravity:
 Most systems of difference equations arising in loop quantum gravity are too complicated to solve exactly or even to analyze. Special techniques, such as those in [72877124Jump To The Next Citation Point88] have to be developed so as to apply to more general systems. In particular for including inhomogeneities, both for solving equations and interpreting solutions, a new area of numerical quantum gravity has to be developed.

 If the relation between different models is known, as presently realized for isotropic within homogeneous models [63], one can formulate the less symmetric model perturbatively around the more symmetric one. This then provides a simpler formulation of the more complicated system, easing the analysis and uncovering new effects. In this context, also alternative methods to introduce approximate symmetries, based on coherent states as e.g., advocated in [84], exist.

Effective equations:
 Finding effective equations that capture the quantum behavior of basic difference equations, at least in some regimes, will be most helpful for a general analysis. However, their derivation is much more complicated for inhomogeneous systems owing to the consistency issue. On the other hand, trying to derive them will provide important tests for the framework, in addition to giving rise to new applications.

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