General geometric operators in all dimensional loop quantum gravity

We propose a new kind of coherent state for the general SO(D + 1) formulation of loop quantum gravity in the (1 + D)-dimensional space-time. Instead of Thiemann's coherent state for SO(D+1) gauge theory, our coherent spin-network state is given by constructing proper superposition over quantum numbers of the spin-networks with vertices labelled by the coherent intertwiners. Such superposition type coherent states are labelled by the so-called generalized twisted geometric variables which capture the geometric meaning of discretized general relativity. We study the basic properties of this kind of coherent states, i.e., the completeness and peakedness property. Moreover, we show that the superposition type coherent states are consistent with Thiemann's coherent state for SO(D + 1) gauge theory in large η limit.

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“…we find that the large η e limit of the heat kernel coherent spin-networks (23), which is given by the super-position type coherent statẽ…”

Section: Relation Between Two Kinds Of Coherent Statesmentioning

confidence: 77%

Perelomov-type coherent states of SO(D+1) in all-dimensional loop quantum gravity

Long

Bodendorfer

2020

Phys. Rev. D

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“…The simple coherent intertwiner space is regarded as the quantum space of D-polytopes, and the geometric properties of these quantum polytopes should be given by geometric operators defined in this space. Two kinds of general spatial geometric operators based on the basic holonomy and flux operators in all dimensional LQG have been defined in [21]. Notice that the shape space of D-polytopes is the constraint surface of simplicity constraint in P A .…”

Section: General Geometric Operatorsmentioning

confidence: 99%

“…Certain "triad tests" were taken in the (1+3)-dimensional theory to fixe this pre-factor and a consistency result has been obtained [42,43]. However, it is difficult to extend such "triad test" to all dimensional case [21], because of the ambiguity introduced by the anomalous quantum simplicity constraint. By requiring the semiclassical consistency based on the semiclassical D-polytopes, the pre-factor can be fixed case by case.…”

Section: The Usual D-volume Operatormentioning

confidence: 99%

“…Since the geometrical quantities of the D-polytopes can be expressed as functions on their shape space, it is possible to define certain geometrical operators of the D-polytopes by the coherent intertwiners which are regarded as the coherent states of quantum D-polytopes. In fact, two kinds of general spatial geometrical operators in all dimensional LQG have been constructed in [21]. One of them was constructed with the flux operators by the so-called internal regularization procedure.…”

Section: Introductionmentioning

confidence: 99%

“…Note that in one of the two strategies proposed in Ref. [21] the geometric operators are totally composed of flux and volume operators. They are also expected to have well semiclassical behaviours for simple coherent intertwiners.…”

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confidence: 99%

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Polytopes in all dimensional loop quantum gravity

Long

2022

Eur. Phys. J. C

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The Lasserre’s reconstruction algorithm is extended to the D-polytopes with the construction of their shape space. Thus, the areas of d-skeletons $$(1\le d\le D)$$ ( 1 ≤ d ≤ 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.

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Coherent intertwiner solution of simplicity constraint in all dimensional loop quantum gravity

Long

Lin

2019

Phys. Rev. D

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We propose a new treatment of the quantum simplicity constraints appearing in the general SO(D + 1) formulation of loop quantum gravity for the (1 + D)-dimensional space-time. Instead of strongly imposing the constraints, we construct a specific form of weak solutions by employing the spin net-work states with specific SO(D + 1) coherent intertwiners. These states weakly satisfy the quantum simplicity constraint via the vanishing expectation values, and the quantum Gaussian constraints can be imposed strongly. Remarkably, those specific SO(D + 1) coherent intertwiners used to construct our solutions have natural interpretations of the D-dimensional polytopes, commonly viewed as basic units of the discrete spatial geometry. Therefore, while the strong imposition of the quantum simplicity constraints leads to an over-constrained solution space, our weak solution space for the constraints may contain the correct semiclassical degrees of freedom for intrinsic geometry of the spatial hypersurfaces. Moreover, some concrete relations are established between our construction and other existing approaches in solving the simplicity constraints in all dimensional loop quantum gravity, providing valuable insights into this unresolved important issue. *

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General geometric operators in all dimensional loop quantum gravity

Cited by 17 publications

References 38 publications

Perelomov-type coherent states of SO(D+1) in all-dimensional loop quantum gravity

Perelomov-type coherent states of SO(D+1) in all-dimensional loop quantum gravity

Polytopes in all dimensional loop quantum gravity

Coherent intertwiner solution of simplicity constraint in all dimensional loop quantum gravity

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