### 4.4 Isotropy: Effective densities in phenomenological equations

The isotropic model is thus quantized in such a way that the operator has a discrete
spectrum containing zero. This immediately leads to a problem, since we need a quantization of
in order to quantize a matter Hamiltonian such as (24), where not only the matter fields
but also the geometry is quantized. However, an operator with zero in the discrete part of its
spectrum does not have a densely defined inverse and does not allow a direct quantization of
.
This leads us to the first main effect of the loop quantization: It turns out that despite
the non-existence of an inverse operator of one can quantize the classical to a
well-defined operator. This is not just possible in the model but also in the full theory, where it has
even been defined first [291]. Classically, one can always write expressions in many equivalent
ways, which usually results in different quantizations. In the case of , as discussed in
Section 5.3, there is a general class of ways to rewrite it in a quantizable manner [49], which
differ in details but all have the same important properties. This can be parameterized by a
function [55, 58], which replaces the classical and strongly deviates from it
for small , while being very close at large . The parameters and
specify quantization ambiguities resulting from different ways of rewriting. With the function

we have
which indeed fulfills for , but is finite with a peak around and
approaches zero at in a manner
as it follows from . Some examples displaying characteristic properties are shown in
Figure 9 in Section 5.3.
The matter Hamiltonian obtained in this manner will thus behave differently at small . At those
scales other quantum effects such as fluctuations can also be important, but it is possible to isolate the
effect implied by the modified density (26). We just need to choose a rather large value for the ambiguity
parameter , such that modifications become noticeable even in semiclassical regimes. This is mainly a
technical tool to study the behavior of equations, but can also be used to find constraints on the allowed
values of ambiguity parameters.

We can thus use classical equations of motion, which are corrected for quantum effects by using the
phenomenological matter Hamiltonian

(see Section 6 for details on the relationship to effective equations). This matter Hamiltonian changes the
classical constraint such that now
Since the constraint determines all equations of motion, they also change; we obtain the Friedmann
equation from ,
and the Raychaudhuri equation from ,
Matter equations of motion follow similarly as

which can be combined to form the Klein–Gordon equation