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Linking Connes NCG to LQG (new papers by Aastrup et al)

  1. Feb 13, 2008 #1


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    We have had some discussion of this in earlier threads.
    Alain Connes et al have been able to derive the standard menu of particles from a certain spectral triple, in Noncommutative Geometry (NCG).

    On the other hand, Loop Quantum Gravity (LQG) as developed by Rovelli, Smolin and many others is primarily a quantum theory of spacetime geometry---thus it contains gravity but not the matter fields of the standard particle model.

    For over two years now, Johannes Aastrup and Jesper Grimstrup have been investigating a way of combining NCG with LQG, so that LQG would not only constitute a quantized version of General Relativity---a quantum theory of gravity, in other words----but would also contain the Standard Model. It is an approach to unification.

    The two original researchers have been joined by a third, Ryszard Nest. Now Aastrup, Grimstrup and Nest have brought out two new papers as part of this program:

    On Spectral Triples in Quantum Gravity I
    Johannes Aastrup, Jesper M. Grimstrup, Ryszard Nest
    84 pages, 8 figures
    (Submitted on 13 Feb 2008)

    "This paper establishes a link between Noncommutative Geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac type operator which resembles a global functional derivation operator. The interaction between the Dirac operator and the algebra reproduces the Poisson structure of General Relativity. Moreover, the associated Hilbert space corresponds, up to a discrete symmetry group, to the Hilbert space of diffeomorphism invariant states known from Loop Quantum Gravity. Correspondingly, the square of the Dirac operator has, in terms of canonical quantum gravity, the form of a global area-squared operator. Furthermore, the spectral action resembles a partition function of Quantum Gravity. The construction is background independent and is based on an inductive system of triangulations. This paper is the first of two papers on the subject."

    On Spectral Triples in Quantum Gravity II
    Johannes Aastrup, Jesper M. Grimstrup, Ryszard Nest
    43 pages, 1 figure
    (Submitted on 13 Feb 2008)

    "A semifinite spectral triple for an algebra canonically associated to canonical quantum gravity is constructed. The algebra is generated by based loops in a triangulation and its barycentric subdivisions. The underlying space can be seen as a gauge fixing of the unconstrained state space of Loop Quantum Gravity. This paper is the second of two papers on the subject."
    Last edited: Feb 13, 2008
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  3. Feb 13, 2008 #2


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    I suppose the next thing to look out for is whether this NCG+LQG bid for unification is presented at the May QG school in Denmark.

    The school (at Roskilde, May 12-18) is supported by the ESF through John Barrett's QG-squared organization.
    This is a page about the school

    This is a page about the European Science Foundation QG2 network.
    It lists some of the schools and conferences they have sponsored or will sponsor

    One of the NCG+LQG authors, namely Ryszard Nest, is helping to direct the Roskilde QG School. So it seems possible that this new approach to unification will make a showing at the school.

    The school is called New Paths to Quantum Gravity and it has a strong NCG representation in the organizers and advisory committee. A number of the participants are involved in Noncommutative Geometry. So it seems like a natural venue.

    The next event where one might expect to hear about the Aastrup et al work would be at the July 2008 QG2 conference to be hosted by John Barrett et al at Nottingham.
    http://www.maths.nottingham.ac.uk/conferences/qgsquared-2008/ [Broken]

    QG2 2008 - Quantum Geometry and Quantum Gravity Conference

    Monday June 30th - Friday July 4th 2008
    University of Nottingham, UK


    * Quantum gravity, including loop quantum gravity, spin foam models, 1+1 and 2+1 quantum gravity, pertubative and discrete approaches. Loop quantum cosmology.

    * Quantum geometry, including physical aspects of non-commutative geometry, quantum groups and quantum topology. Non-commutative field theory and deformed special relativity.
    Last edited by a moderator: May 3, 2017
  4. Feb 18, 2008 #3


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    More on connecting LQG with NCG

    Today Abhay Ashtekar weighed in on this.
    His new paper with Engle and Sloan has the LQG-NCG link as the main subject of its final paragraph, on page 16.

    "We conclude with a comment. In non-commutative geometry, in place of Riemannian geometry, one introduces a spectral triplet (A,H,D) consisting of a non-commutative C* algebra A, a representation of it on a Hilbert space H and a Dirac operator D acting on H. A certain choice of the triplet is made to describe (a generalization of) the standard model of particle physics together with Einstein gravity.

    Rather general symmetry considerations then lead to a so-called ‘spectral action’ from which dynamics can be derived [17]. It has been known for some time that an asymptotic expansion of this action can be performed to make contact with the low energy physics and the first terms reproduce the Einstein-Hilbert action with a cosmological constant. Recently it was realized [18] that the spectral action can be naturally extended to incorporate the presence of boundaries and the asymptotic expansion of the new action produces precisely the Einstein Hilbert action with
    the Gibbons-Hawking counter term for C = 0 (see (1.1)). This is an exciting development.

    However, as the discussion of section I shows, in the asymptotically flat context this action has severe limitations and the extension of the spectral action to incorporate the boundary term [18] was motivated using precisely the Hamiltonian formulation in the asymptotically flat context. More generally, the non-commutative framework has been developed primarily for the Riemannian signature and passage to the Lorentz signature is contemplated via a Wick transform in the asymptotically flat context. Therefore asymptotic considerations of [3] and this paper are directly relevant to the spectral action approach.

    The natural question then is: Can the spectral action framework be further generalized so that the leading terms in the asymptotic expansion yields an action which is free from the drawbacks of (1.1)? The first order framework discussed in this paper presents a natural avenue for such a generalization. Indeed, the gravitational sector of the non-commutative geometry requires a spin-bundle —and hence a frame field e— as well a Dirac operator —i.e., a spin connection A. However, in the non-commutative framework the two are in essence compatible with one another from the beginning. The question is whether one can extend the framework so that they are independent to begin with and made compatible only by equations of motion. The spectral action in such a generalization could then descend to (3.6) upon a suitable asymptotic expansion. Quite apart from this specific application, such a ‘first order’ framework in non-commutative geometry appears also to lead to mathematical structures which are interesting in their own right, and could provide a technical bridge between non commutative geometry and loop quantum gravity."

    Last edited: Feb 18, 2008
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