The main problems of canonical quantum cosmology are the following:
The singularity. The Wheeler-DeWitt equation does not remove the singularity, although in some cases the solutions avoid the singularity due to the effective potential as we have seen. In short, canonical quantum gravity is not well defined. The models of canonical quantum cosmology do only apply on a semiclassical approximation and there is no way to make predictions about the dynamics of the singularity.
Decoherence. If the wavefunction of the universe is a linear combination of basic states, then these can interfere with each other. However, we observe a classical universe without interference on the macrocospic world. This requires of a mechanism to allow for decoherence of the wavefunction of the universe.
The problem of time. This is more a general problem in the canonical approach of quantum gravity, but its implications are important in cosmology. The Hamiltonian constraint indicates that the theory does not have a predefined notion of time. Reparametrizations of the time evolution are gauge transformations without physical content. To single out the real time evolution of the universe one has to fix a specific gauge. This breaks the original symmetry of the classical equations under diffeomorphisms and, moreover, different choices of different time variables lead to different quantum theories. The question that arises is related to the next issue: are we allowed to select the scale factor as a parameter that scans time evolution already in the quantum regime?
The minisuperspace approximation. The minisuperspace approximation does fix simultaneously canonically conjugate variables violating the uncertainty principle (due to symmetry the dynamic variable and its conjugate momentum are both zero). It leaves the scale factor as the single degree of freedom in the models. Although our universe is currently homogenous and isotropic, this assumption need not to be valid at the beginning. Note that the inflationary phase, already in the classical regime, might have created homogeneity out of an inhomogeneous and random initial state.
Inhomogeneities. There exist models that describe inhomogeneities, but they are poorly understood.
Initial conditions. Initial conditions that are imposed to the universe cannot be derived from any principle and become the same status than fundamental laws. Bryce DeWitt envisioned a theory in which the requirement of mathematical consistency should be sufficient to guarantee a unique solution to the Wheeler-deWitt equation. However, this cannot be realized in canonical quatum cosmology.
Formalism. The theory has diverse mathematical and consistency problems, like factor ordering and other ambiguities, and the definition of a proper notion of probabilites in a single universe.
Nevertheless, one would expect that some aspects of the mentioned canonical quantum cosmology models should be present in a complete and consistent model of quantum cosmology. There is no clear reason to expect the description of the origin of the universe beyond the singularity to be correct, but the calculations for transition probabilities and set-up of inflation near the classical regime may be a good approximation to reality.
(Sharing Credits with Herr von Bradford)
