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Length and curvature operators in Loop gravity

  1. Mar 13, 2014 #1


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    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.

    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:

    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 in Nuclear Physics B

    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.
    Last edited: Mar 14, 2014
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  3. Mar 14, 2014 #2
    Thanks Marcus, enjoyed Bianchia's 2008 paper and just finished reviewing Alesci' s PhD thesis, So looking forward to reading the curvature paper.
  4. Mar 14, 2014 #3


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    Thanks for your comment, David! It's encouraging to know that a PhD theoretical physicist (member of the Royal Astronomical Society too) who is doing research in quantum gravity has time to drop in here at PF and check out Loop gravity news at BtSM.

    I was impressed by the reviewing work you do for QG community simply by commenting on papers as you read them, on your blog. For example this review of Alesci's thesis on the graviton propagator in LQG:

    It seems that when you read a research paper and it interests you, you take the time to type out some review comments and post them on blog. I wish more QG researchers did this, wouldn't the whole community benefit?
  5. Mar 27, 2014 #4


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    David Horgan recently (25 March) posted something relevant to this on his blog.http://quantumtetrahedron.wordpress...or-lqg-by-alesci-assanioussi-and-lewandowski/
    It's potentially useful to read it along with the LQG Curvature Operator paper because it explains some things you have to know to understand the paper likeRegge calculus essentials and how curvature is defined in Regge calculus.
    Hi David! I think it is remarkably constructive what you are doing with your personal blog "quantum tetrahedron". I hope the blog will become more ever more widely known and used. I would suggest to anybody who wants to read the LQG Curvature paper (of Alesci et al) that they access your blog so as to have the convenient supplemental material for that paper ready at hand to consult as needed.
    Last edited: Mar 27, 2014
  6. Mar 27, 2014 #5
    Hi Marcus, thanks for your feedback. I'll be doing numerical work on the curvature operator during the next week or so. I've posted a number of sagemath and python programs related to LQG on my blog (http://quantumtetrahedron.wordpress.com/category/personal-research/). I've also posted an interactive sagemath page for LQG at http://wiki.sagemath.org/interact/Loop Quantum Gravity. This mainly looks at eigenvalues for the volume, area and length operators. Anyone interested in these is welcome to contact me.
    Last edited: Mar 27, 2014
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