This prescription is rooted in quantum mechanics, which, despite its formal similarity, is physically very
different from cosmology. The procedure looks innocent, but one should realize that there are already basic
choices involved. Choosing the factor ordering is harmless, even though results can depend on it [206]. More
importantly, one has chosen the Schrödinger representation of the classical Poisson algebra,
which immediately implies the familiar properties of operators such as the scale factor with a
continuous spectrum. There are inequivalent representations with different properties, and it is
not clear that this representation, which works well in quantum mechanics, is also correct for
quantum cosmology. In fact, quantum mechanics is not very sensitive to the representation
chosen [19] and one can use the most convenient one. This is the case because energies and thus
oscillation lengths of wave functions, described usually by quantum mechanics, span only a limited
range. Results can then be reproduced to arbitrary accuracy in any representation. Quantum
cosmology, in contrast, has to deal with potentially infinitely-high matter energies, leading to small
oscillation lengths of wave functions, such that the issue of quantum representations becomes
essential.
That the Wheeler–DeWitt representation may not be the right choice is also indicated by the fact that
its scale factor operator has a continuous spectrum, while quantum geometry, which at least kinematically is
a well-defined quantization of the full theory, implies discrete volume spectra. Indeed, the Wheeler–DeWitt
quantization of full gravity exists only formally, and its application to quantum cosmology simply quantizes
the classically-reduced isotropic system. This is much easier, and also more ambiguous, and leaves open
many consistency considerations. It would be more reliable to start with the full quantization and introduce
the symmetries there, or at least follow the same constructions of the full theory in a reduced model. If
this is done, it turns out that indeed we obtain a quantum representation inequivalent to the
Wheeler–DeWitt representation, with strong implications in high-energy regimes. In particular, like the full
theory, such a quantization has a volume or operator with a discrete spectrum, as derived in
Section 5.2.
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