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 even
has been defined first [193
]. Classically, one can always write expressions in many equivalent
ways, which usually result 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 [41
] which
differ in details but have all the same important properties. This can be parameterized by a
function
[47
, 50
] 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
The matter Hamiltonian obtained in this manner will thus behave differently at small . At those
scales also other quantum effects such as fluctuations can be important, but it is possible to isolate the
effect implied by the modified density (25
). We just need to choose a rather large value for the ambiguity
parameter
such that modifications become noticeable already 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 effective matter Hamiltonian
(see Section 5.5 for details on 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 effective Friedmann equation fromMatter equations of motion follow similarly as
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