### 4.12 Inhomogeneous matter with isotropic quantum geometry

Inhomogeneous matter fields cannot be introduced directly to isotropic quantum geometry since after
the symmetry reduction there is no space manifold left for the fields to live on. There are then two
different routes to proceed: One can simply take the classical field Hamiltonian and introduce
effective modifications modeled on what happens to the isotropic Hamiltonian, or perform a
mode decomposition of the matter fields and just work with the space-independent amplitudes.
The latter is possible since the homogeneous geometry provides a background for the mode
decomposition.
The basic question, for the example of a scalar field, then is how the metric coefficient in the gradient
term of Equation (12), , would be replaced effectively. For the other terms, one
can simply use the isotropic modification, which is taken directly from the quantization. For
the gradient term, however, one does not have a quantum expression in this context and a
modification can only be guessed. The problem arises since the inhomogeneous term involves
inverse powers of , while in the isotropic context the coefficient just reduces to ,
which would not be modified at all. There is thus no obvious and unique way to find a suitable
replacement.

A possible route would be to read off the modification from the full quantum Hamiltonian, or at least
from an inhomogeneous model, which requires a better knowledge of the reduction procedure. Alternatively,
one can take a more phenomenological point of view and study the effects of possible replacements. If the
robustness of these effects to changes in the replacements is known, one can get a good picture of possible
implications. So far, only initial steps have been taken and there is no complete programme in this
direction.

Another approximation of the inhomogeneous situation has been developed in [70] by patching isotropic
quantum geometries together to support an inhomogeneous matter field. This can be used to study
modified dispersion relations to the extent that the result agrees with preliminary calculations
performed in the full theory [115, 3, 4, 181, 182] even at a quantitative level. There is thus further
evidence that symmetric models and their approximations can provide reliable insights into the full
theory.