At this point, one has to find an alternative route to the quantization of , or else one
could only conclude that there is no well-defined quantization of matter Hamiltonians as a manifestation
of the classical divergence. In the case of loop quantum cosmology it turns out, following a
general scheme of the full theory [193], that one can reformulate the classical expression in an
equivalent way such that quantization becomes possible. One possibility is to write, similarly to
(13
)
where we use holonomies of isotropic connections and the volume . In this expression we can
insert holonomies as multiplication operators and the volume operator, and turn the Poisson bracket into a
commutator. The result
Rewriting a classical expression in such a manner can always be done in many equivalent ways, which in
general all lead to different operators. In the case of , we highlight the choice of the representation
in which to take the trace (understood as the fundamental representation above) and the power of
in the Poisson bracket (
above). This freedom can be parameterized by
two ambiguity parameters
for the representation and
for the power such
that
Following the same procedure as above, we obtain eigenvalues [47, 50]
which, for larger , can be approximated by (25
), see also Figure 9
. This provides the basis for loop
cosmology as described in Section 4.
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