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