### 4.11 Inhomogeneities

Allowing for inhomogeneities inevitably means to take a big step from finitely many degrees of freedom
to infinitely many ones. There is no straightforward way to cut down the number of degrees of freedom to
finitely many ones while being more general than in the homogeneous context. One possibility would be to
introduce a small-scale cut-off such that only finitely many wave modes arise (e.g., through a lattice as is
indeed done in some coherent state constructions [184]). This is in fact expected to happen in a discrete
framework such as quantum geometry, but would at this stage of defining a model simply be introduced by
hand.
For the analysis of inhomogeneous situations there are several different approximation schemes:

- Use only isotropic quantum geometry and in particular its effective description, but couple
to inhomogeneous matter fields. Problems in this approach are that back-reaction effects are
ignored (which is also the case in most classical treatments) and that there is no direct way
how to check modifications used in particular for gradient terms of the matter Hamiltonian. So
far, this approach has led to a few indications of possible effects.
- Start with the full constraint operator, write it as the homogeneous one plus correction terms
from inhomogeneities, and derive effective classical equations. This approach is more ambitious
since contact to the full theory is realized. So far, there are not many results since a suitable
perturbation scheme has to be developed.
- There are inhomogeneous symmetric models, such as Einstein-Rosen waves or the spherically
symmetric one, which have infinitely many kinematical degrees of freedom but can be treated
explicitly. Also here, contact to the full theory is present through the symmetry reduction
procedure of Section 6. This procedure itself can be tested by studying those models between
homogeneous ones and the full theory, but results can also be used for physical applications
involving inhomogeneities. Many issues that are of importance in the full theory, such as the
anomaly problem, also arise here and can thus be studied more explicitly.