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Coherent states, quantum gravity, and the Born-Oppenheimer approximation. III.: Applications to loop quantum gravity
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An elementary introduction to the semi-analytic category can be found in Ref. 12.
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Unfortunately, this excludes fractal graphs, which are sometimes assumed to be of relevance for loop quantum gravity.
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The partial order ≤ is essentially the one, ≺, defined in Ref. 5.
It seems that this non-trivial condition has been overlooked in the main part of the literature with the exception of Ref. 6, where the partial orders ⋖L and ⋖R are defined making the use of edge inversions obsolete at the phase space level.
We consider only *-morphisms corresponding to and , because generic are not Poisson maps. Furthermore, we have compatibility with transitivity of ≤, and the commutative diagram (2.24) with respect to edge inversion for and .
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This means, that l should contain as least one pair of subgraphs, which are changed into one another.
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In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the proposed Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity.
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