### 4.16 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 phenomenological
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 to be
defined.
The basic question then, for the example of a scalar field, is how to replace the metric coefficient
in the gradient term of Equation (12). 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 because the inhomogeneous term involves inverse powers of , while in the isotropic context the
coefficient just reduces to , which would not receive corrections at all. There is thus no obvious and
unique way to find a suitable replacement.

A possible route would be to read off the corrections 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 different
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
(see [182] for scalar modes and [226] for tensor modes) and there is no complete program in this
direction.

Another approximation of the inhomogeneous situation has been developed in [100] 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 [168, 3, 4, 267, 268] even at a quantitative level. There is thus further
evidence that symmetric models and their approximations can provide reliable insights into the full
theory.