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9.2 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 difficult to analyze than that of 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 [142338313108Jump To The Next Citation Point65]. Moreover, the dynamics can possibly be simplified and understood better around slowly evolving horizons [113108]. Other horizon conditions are also being studied in related approaches [190155].
Consistency:
 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. Effective treatments of constrained systems are beginning to shed valuable light on the anomaly issue and possible restrictions of quantization ambiguities by the condition of anomaly freedom. 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.
Relationship between models and the full theory:
 By strengthening the relationship 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 even in models [62].
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 [10312410126Jump To The Next Citation Point130] have to be developed so as to apply them to more general systems. Isotropic models with a free massless scalar can be analyzed by efficient numerical techniques, but this is much more complicated for interacting matter or when several gravitational degrees of freedom are present. In the latter case, possible non-equidistancy may further complicate the analysis. In particular for inhomogeneities, both for solving equations and interpreting solutions, a new area of numerical quantum gravity has to be developed.
Perturbations:
 If the relationship between different models is known, as presently realized for isotropic models within homogeneous ones [88], 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, alternative methods for introducing approximate symmetries, based on coherent states as, e.g., advocated in [119], also exist. Inhomogeneous perturbations have been formulated in [66] and are being developed for cosmological perturbation theory.
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. This is the place where probably the most unexpected progress has taken place over the last two years, leading to an exactly solvable isotropic model, which can be used as the “free” basis for a systematic perturbation analysis of more general systems. A derivation of effective equations is much more complicated for inhomogeneous systems, where it also has to be combined with an analysis of the consistency issue. On the other hand, such a derivation will provide important tests for the framework, in addition to giving rise to new applications.

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