Abstract
The Lasserre’s reconstruction algorithm is extended to the D-polytopes with the construction of their shape space. Thus, the areas of d-skeletons (1le dle D) can be expressed as functions of the areas and normal bi-vectors of the (D-1)-faces of D-polytopes. As weak solutions of the simplicity constraints in all dimensional loop quantum gravity, the simple coherent intertwiners are employed to describe semiclassical D-polytopes. New general geometric operators based on D-polytopes are proposed by using the Lasserre’s reconstruction algorithm and the coherent intertwiners. Such kind of geometric operators have expected semiclassical property by the definition. The consistent semiclassical limit with respect to the semiclassical D-polytopes can be obtained for the usual D-volume operator in all dimensional loop quantum gravity by fixing its undetermined regularization factor case by case.
Highlights
The states of coherent intertwiners in HF are labelled with the points in SF and comprise an overcomplete basis in the kinematical Hilbert space of loop quantum gravity (LQG) based on the F-valent vertex
The idea that the coherent intertwiners can be regarded as the semiclassical states of certain spatial geometry in large quantum number limit was extended to all dimensional LQG [11,12]
The geometric operators of quantum polytopes involving coherent states can be constructed based on following two facts: (i) A simple coherent intertwiner state is labelled by a point of the shape space of D-polytopes with fixed (D-1)-faces’ areas, and the corresponding wave function is peaked at this point. (ii) A point in the shape space of D-polytopes gives full geometric information of the corresponding D-polytope
Summary
It provides a way to deal with the anomalous quantum simplicity constraints in all dimensional LQG with certain geometrical meaning In this way, the non-commutative quantum vertex simplicity constraints [13] were imposed weakly [14], so that some kinds of weak solutions were constructed as well-behaved semiclassical states. The non-commutative quantum vertex simplicity constraints [13] were imposed weakly [14], so that some kinds of weak solutions were constructed as well-behaved semiclassical states These weak solutions are composed of S O(D + 1) coherent states [15] and comprise the so-called simple coherent intertwiner space in all dimensional LQG. The construction of Kapovich and Millson’s phase space can be extended to higher dimensional cases but with some different characters as introduced in [14] In this description, one considers the D-polytopes embedded into a (D+1)-dimensional Euclidean space R(D+1). The simplicity constraint leads to the fact that the shape space
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