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String in a LQG context

  1. Aug 9, 2009 #1

    marcus

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    Even if string should turn out to be a dead end as far as unification goes, and fail to produce a predictive theory of how the world works, it could still be interesting and constructive to formulate some version of string within the background independent LQG context. Here is a recent paper which addresses that challenge.
    http://arxiv.org/abs/0908.0953
    Canonical Analysis of the Algebraic String
    Winston J. Fairbairn, Karim Noui, Francesco Sardelli

    And here is a quote from the conclusions section of the paper:

    "One of the main aspects that strikes us with the algebraic string is its numerous similarities with the Ashtekar formulation of gravity. It is a first order formulation, it is of course diffeomorphism invariant, and admits an Immirzi type parameter. All these aspects makes the system a very nice arena to test the ideas and techniques of Loop Quantum Gravity (LQG). Indeed, the system is very interested in itself and simple enough to be completely quantised. The Fock quantisation already partially exists. Our aim, in future work, is to develop a background independent quantisation à la LQG in order to conclude on the equivalence or not with the Fock quantisation.

    This idea was in fact initiated by Thiemann [14] in the context of the Nambu-Goto string but we think that the algebraic formulation of the string is more suited for that specific problem. Furthermore, there is an Immirzi-type parameter and then we hope to understand its effects in the quantum theory. We hope the algebraic string helps us to understand some other fundamental aspects of LQG. For instance, we can ask the question if this formulation can lead us to a Spin-Foam [20] formulation of the string.

    If this is the case, we have a new arena, simpler than gravity, to understand the
    link between the covariant and canonical quantisations of background independent
    theories.

    In brief, the next step is the compare the quantisations of the algebraic string à la LQG vs. à la Fock. We are currently working in that direction."
     
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  3. Aug 9, 2009 #2

    marcus

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    In case anyone would like more information, here is a further quote from the Fairbairn, Noui, Sardelli paper.

    ==quote from the introduction==
    Few years ago, Thiemann reconsidered the Nambu-Goto string and proposed a quantisation of it using the techniques of Loop Quantum Gravity (LQG) [14]. He showed that the LQG techniques, based on background independent quantisation, provides in particular a quantisation of the bosonic string in any dimensions, i.e. there is no need of critical dimensions for the quantum theory to be consistent. This result has sparked off some discussions [15] and certainly deserves to be understood deeper.

    We think that the algebraic formulation of the bosonic string is a better starting point to test the LQG techniques than the Nambu-Goto string for it admits a lot of similarities with Ashtekar gravity [16]. It is a first order formulation and possesses an Immirzi-type parameter.

    In fact, the main motivation of this article is to open an arena for a background independent quantisation of the bosonic string and to compare it to the standard Fock quantisation. In that sense, we want to continue the work initiated by Thiemann from a quite different starting point in order to confirm or not his predictions and even go further.
    ==endquote==

    Here is the abstract, which however is couched in more technical language making it less transparent than the passages already quoted.
    http://arxiv.org/abs/0908.0953
    Canonical Analysis of the Algebraic String
    Winston J. Fairbairn, Karim Noui, Francesco Sardelli
    27 pages
    (Submitted on 6 Aug 2009)
    "We investigate the canonical aspects of the algebraic first order formulation of strings introduced two decades ago by Balachandran and collaborators. We slightly enlarge the Lagrangian framework and show the existence of a self-dual formulation and of an Immirzi-type parameter reminiscent of four-dimensional first order gravity. We perform a full Hamiltonian analysis of the self-dual case: we extract the first class constraints and construct the Dirac bracket associated to the second class constraints. The first class constraints contain the diffeomorphisms algebra on the world sheet, as expected; and the coordinates are shown to be non-commutative with respect to the Dirac bracket. Then, the Hamilton equations in a particular (but very natural) gauge are shown to reproduce the wave equation for the string coordinates. In the general, non-self-dual case, we also explicit the first class constraints of the system and show that, unlike the self-dual formulation, the theory admits an extra propagating degree of freedom than the two degrees of freedom of conventional string theory. This prevents the general algebraic string from being strictly equivalent to the Nambu-Goto string."
     
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