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
- 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
[14, 159, 13, 76, 57]. Moreover, the dynamics can possibly be simplified and understood better
around slowly evolving horizons [79, 76]. Other horizon conditions are also being studied in
related approaches [132, 107].
- 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 .
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 [72, 87, 71, 24, 88] 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 , 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 , exist.
- 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.