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  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.
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