Most of us are familiar with the fact that in Loop gravity the area and volume observables have discrete spectrum. The discrete spectrum of the area operator, leading to a smallest positive measurable area, has lots of mathematical consequences that have been derived in the theory. It helps ensure that energy density and curvature are bounded, replacing the cosmological singularity by a rebound.(adsbygoogle = window.adsbygoogle || []).push({});

LENGTH operators may not have been as much studied and used as the area operator. But one was , for example, introduced and studied by Bianchi in 2008. And now it seems that Alesci and friends have used Bianchi's length operator, or something very much like it, to construct a

CURVATURE operator for the theory. This should be interesting. Others besides me may wish to keep track of this development, so I'll post the links:

http://arxiv.org/abs/0806.4710

The length operator in Loop Quantum Gravity

Eugenio Bianchi

(Submitted on 28 Jun 2008)

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity - the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.

33 pages, 12 figures; published inNuclear Physics B

http://arxiv.org/abs/1403.3190

A curvature operator for LQG

Emanuele Alesci, Mehdi Assanioussi, Jerzy Lewandowski

(Submitted on 13 Mar 2014)

We introduce a new operator in Loop Quantum Gravity - the 3D curvature operator - related to the 3-dimensional scalar curvature. The construction is based on Regge Calculus. We define it starting from the classical expression of the Regge curvature, then we derive its properties and discuss some explicit checks of the semi-classical limit.

20 pages.

To see how it goes, refer to SECTION III Construction of the Curvature Operator

and to see how Bianchi's length operator fits in, see page 7, subsection B "The Length Operator".

Angle operators are also defined in Loop gravity. A goodly kit of geometric observables has been developed. It seem that what Alesci, Assaniousi, Lewandowski are doing here is to straightforwardly implement a quantum version of Regge GR, which we know works.

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# Length and curvature operators in Loop gravity

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