3.2 Dirac quantization

In Dirac’s approach to quantization one starts from a kinematical vector space V adapted (i.e., with the right dimensionality among other requirements) to the description of a physical system defined on the phase space Γ. The constraints CI = 0 are then represented as operators whose kernels define the physical states Ψ ∈ V of the quantum theory, ˆCIΨ = 0. Finally, to define probabilities, the physical subspace V phys is endowed with a suitable inner product ⟨⋅,⋅⟩ such that (V ,⟨⋅,⋅⟩) phys becomes a Hilbert space ℋphys. In order to make these ideas explicit, the following concrete points must be addressed:

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 V 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 [3Jump To The Next Citation Point]. 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.

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