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 . 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 , 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
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
Matter equations of motion follow similarly as
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