- The identification of a set of elementary phase-space variables for the full (non-constrained) phase space of the classical system.
- The selection of a suitable Poisson algebra on the full phase space generated by the elementary variables.
- The construction of a representation for this Poisson algebra on the complex vector space .
- The implementation of the first class constraints as operators acting on the representation space.
- The characterization of the physical states, i.e., the space spanned by those vectors in the kernel of all the constraint operators.
- The identification of physical observables (the operators that leave invariant).
- Finally, if we want to answer physical questions – such as probability amplitudes or expectation values – we need to endow with an Hermitian inner product.

The Wheeler–DeWitt approach and LQG both follow the spirit of the Dirac quantization of constrained systems mentioned here. In LQG [12], the kinematical vector space is endowed with a Hilbert space structure defined in terms of the Ashtekar-Lewandowski measure. However, the identification of the inner product in the space of physical states is not as simple as the restriction of the kinematical Hilbert structure to the physical subspace because the spectrum of the constraint operators may have a complicated structure. In particular, it may happen that the kernel of these operators consists only of the zero vector of the kinematical Hilbert space. The Wheeler–DeWitt approach is less developed from the mathematical point of view but many constructions and ideas considered during the mathematical development of LQG can be exported to that framework. It is important to mention that under mathematical restrictions similar to the ones imposed in LQG some crucial uniqueness results (specifically the LOST [146] and Fleischack [88] theorems on the uniqueness of the vacuum state) do not hold [3]. Though the approach can probably be developed with the level of mathematical rigor of LQG this result strongly suggests that LQG methods are better suited to reach a complete and fully consistent quantum gravity theory. In any case we believe that it could be interesting to explore if suitable changes in the mathematical formulation of the Wheeler–DeWitt formalism could lead to uniqueness results of the type already available for LQG.

Living Rev. Relativity 13, (2010), 6
http://www.livingreviews.org/lrr-2010-6 |
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