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It is always possible, even if string does not give a good model of nature, that LQG can be used to give a background independent formulation of a string theory. To comprehend string within the background independent context of LQG would be interesting, if only as an academic exercise. Thiemann took some steps in that direction back in 2004 which at the time many people considered outrageous
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."
==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==
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