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
- 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 [14, 233, 83, 13, 108, 65]. Moreover, the dynamics can possibly be simplified and
understood better around slowly evolving horizons [113, 108]. Other horizon conditions are
also being studied in related approaches [190, 155].
- 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 .
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 [103, 124, 101, 26, 130] 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.
- If the relationship between different models is known, as presently realized for
isotropic models within homogeneous ones , 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 , also exist. Inhomogeneous perturbations have been formulated in  and
are being developed for cosmological perturbation theory.
- 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.