There are a lot of open problems in the twistor approach to Loop QG. They take the time to point them out. It's a fairly new approach so there are a lot of different things to work on. It's a 39 page paper, and at least from my perspective quite interesting.(adsbygoogle = window.adsbygoogle || []).push({});

The authors are based in Marseille and part of the work was done in Warsaw and Erlangen. They have spent postdoc and/or visitor time at Perimeter. So the influences on this line of research are from pretty much all over.

It seemed as if the paper should have its own thread, in case anyone else is reading it and wishes to comment or explain. Here's the paper that just came out:

http://arxiv.org/abs/1207.6348

The twistorial structure of loop-gravity transition amplitudes

Simone Speziale, Wolfgang M. Wieland

(Submitted on 26 Jul 2012)

The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many open questions are pressing on. In this paper we address some of them using a twistorial description, which brings new light on both classical and quantum aspects of the theory. At the classical level, we clarify the covariant properties of the discrete geometries involved, and the role of the simplicity constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas and Lorentzian dihedral angles in twistor space, and show that they form acanonical pair. The primarysimplicity constraintsare solved by simple twistors, parametrized by SU(2) spinors and the dihedral angles. We construct an SU(2) holonomy and prove it to correspond to the Ashtekar-Barbero connection. We argue that the role of secondary constraints is to provide a non trivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schroedinger representation leads to a spinorial version of simple projected spin networks, where the argument of the wave functions is aspinor instead of a group element. We rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral in twistor space. Using these tools, we show that the Engle-Pereira-Rovelli-Livinetransition amplitudescan be derived from a path integral in twistor space. We construct a curvature tensor, show that it carries torsion off-shell, and that its Riemann part is of Petrov type D. Finally, we make contact between the semiclassical asymptotic behaviour of the model and our construction, clarifying the relation of the Regge geometries with the original phase space.

39 pages

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# Speziale-Wieland Twistor LQG paper, July 2012

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