### 4.3 Isotropy: Implications of a loop quantization

We are now dealing with a simple system with finitely many degrees of freedom, subject to a
constraint. It is well known how to quantize such a system from quantum mechanics, which has been
applied to cosmology starting with DeWitt [104]. Here, one chooses a metric representation for
wave functions, i.e., , on which the scale factor acts as multiplication operator and its
conjugate , related to , as a derivative operator. These basic operators are then used to
form the Wheeler-DeWitt operator quantizing the constraint (17) once a factor ordering is
chosen.
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 [142]. 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 [18] 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 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, just as the full
theory such a quantization has a volume or operator with a discrete spectrum, as derived in
Section 5.2.