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Speziale-Wieland Twistor LQG paper, July 2012

  1. Jul 29, 2012 #1


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    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.

    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:

    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 a canonical pair. The primary simplicity constraints are 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 a spinor 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-Livine transition amplitudes can 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
    Last edited: Jul 29, 2012
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  3. Jul 29, 2012 #2


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    Here is a 4-page tutorial on twistors with 15 transparencies sketched by Penrose.
    Here is a PIRSA video talk by Wieland that can also serve as introduction to the paper.
    Spinor Quantisation for Complex Ashtekar Variables
    Speaker(s): Wolfgang Wieland
    Abstract: During the last couple of years Dupuis, Freidel, Livine, Speziale and Tambornino developed a twistorial formulation for loop quantum gravity.
    Constructed from Ashtekar--Barbero variables, the formalism is restricted to SU(2) gauge transformations. In this talk, I perform the generalisation to the full Lorentzian case, that is the group SL(2,C).
    The phase space of SL(2,C) (i.e. complex or selfdual) Ashtekar variables on a spinnetwork graph is decomposed in terms of twistorial variables. To every link there are two twistors---one to each boundary point---attached. The formalism provides a clean derivation of the solution space of the reality conditions of loop quantum gravity.
    Key features of the EPRL spinfoam model are perfectly recovered.
    If there is still time, I'll sketch my current project concerning a twistorial path integral for spinfoam gravity as well.
    Date: 29/02/2012 - 4:00 pm

    Wieland is using pairs of spinors to label the links of his spinnetworks, and two spinors make a twistor. (Just saying ℂ2 x ℂ2 = ℂ4.)
    In this context, a twistor can be called a "bi-spinor".
    When an oriented link gets two twistor labels (source & target labels) that's 8 complex numbers but relations they satisfy can bring down the actual number of d.o.f.
    Last edited: Jul 29, 2012
  4. Jul 29, 2012 #3


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    Here is the Speziale-Wieland paper's reference [2], I think it's an important precursor:
    Holomorphic Lorentzian Simplicity Constraints
    Maité Dupuis, Laurent Freidel, Etera R. Livine, Simone Speziale
    (Submitted on 26 Jul 2011 (v1), last revised 20 Feb 2012 (this version, v2))
    We develop an Hamiltonian representation of the sl(2,C) algebra on a phase space consisting of N copies of twistors, or bi-spinors. We identify a complete set of global invariants, and show that they generate a closed algebra including gl(N,C) as a subalgebra. Then, we define the linear and quadratic simplicity constraints which reduce the spinor variables to (framed) 3d spacelike polyhedra embedded in Minkowski spacetime. Finally, we introduce a new version of the simplicity constraints which
    (i) are holomorphic and
    (ii) Poisson-commute with each other,
    and show their equivalence to the linear and quadratic constraints.
    20 pages (explicit counting of the holomorphic constraints added)

    If you follow Loop research you know that the usual graph Hilbert space HΓ containing the spinnetwork quantum states of geometry consists of square-integrable functions on L copies of the group G where L is the number of links in the graph. Typically G = SU(2).
    This reflects the fact that the links of the graph are to be labeled with SU(2) representations.

    In this twistorial approach I think the intent is that in the quantum theory the Hilbert space functions are defined on N copies of ℂ4: the twistors , or "bi-spinors". In the Speziale-Wieland paper they seem to be using pairs of twistors to lable oriented links: one for the source node and one for the target node of the link.
    This corresponds to the highlighted phrase in the abstract in post #1:
    where the argument of the wave functions is a spinor instead of a group element
    I would expect that the argument of the wave functions is an N-tuple of spinors where N is related to the number of links in the graph.
    Last edited: Jul 29, 2012
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