Twistors to Twisted Geometries: Exploring Loop Quantum Gravity

[marcus]

http://arxiv.org/abs/1006.0199
From twistors to twisted geometries
Laurent Freidel, Simone Speziale
9 pages
(Submitted on 1 Jun 2010)
"In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic discrete geometry of a cellular decomposition dual to the graph. Here we unravel the origin of the phase space from a geometric interpretation of twistors."

Here is the earlier paper mentioned above:
http://arxiv.org/abs/1001.2748
Twisted geometries: A geometric parametrisation of SU(2) phase space
Laurent Freidel, Simone Speziale
28 pages
(Submitted on 15 Jan 2010)
"A cornerstone of the loop quantum gravity program is the fact that the phase space of general relativity on a fixed graph can be described by a product of SU(2) cotangent bundles per edge. In this paper we show how to parametrize this phase space in terms of quantities describing the intrinsic and extrinsic geometry of the triangulation dual to the graph. These are defined by the assignment to each triangle of its area, the two unit normals as seen from the two polyhedra sharing it, and an additional angle related to the extrinsic curvature. These quantities do not define a Regge geometry, since they include extrinsic data, but a looser notion of discrete geometry which is twisted in the sense that it is locally well-defined, but the local patches lack a consistent gluing among each other. We give the Poisson brackets among the new variables, and exhibit a symplectomorphism which maps them into the Poisson brackets of loop gravity. The new parametrization has the advantage of a simple description of the gauge-invariant reduced phase space, which is given by a product of phase spaces associated to edges and vertices, and it also provides an abelianisation of the SU(2) connection. The results are relevant for the construction of coherent states, and as a byproduct, contribute to clarify the connection between loop gravity and its subset corresponding to Regge geometries.".

To put this development provisionally into context, these two papers are references [6] and [17] of Rovelli's April 2010, which is basically a summary and status report of what's happening in Loop gravity research (called "A new look at LQG"). It is a short paper, only 15 pages. If you look there you can find Rovelli's current assessment of the twist-LQG gambit and how he thinks it might fit into the program.
http://arxiv.org/abs/1004.1780

 

[marcus]

Sample excerpt
http://arxiv.org/abs/1006.0199
From twistors to twisted geometries

===quote===
...the space of closed twisted geometries can be related to the phase space of Regge calculus when one further imposes the gluing or shape matching conditions [10]. For more discussions on the relation between loop gravity/twisted geometries and discrete gravity, see discussions in [1, 2, 3].

The various phase spaces that can be associated to a graph, and their relations, are summarized by the following scheme:

Twistor space

↓ area matching reduction

Twisted geometries ⇔ loop gravity

↓ closure reduction

Closed twisted geometries ⇔ gauge-invariant loop gravity

↓ shape matching reduction

Regge phase space ⇔ Regge calculus

This scheme shows how twisted geometries fit into a larger hierarchy. From top to bottom, we move from larger and simpler spaces, with less intuitive geometrical meaning, to smaller and more constrained spaces, with clearer geometrical meaning.

The results establish a path between twistors and Regge geometries, via loop gravity. Furthermore, notice also that each phase space but the twistor one is related to a well-known representation of general relativity on a given graph, be it loop gravity or Regge calculus. This raises the intriguing question of whether such a representation can be given directly in terms of twistors. The possibility of defining a “twistor gravity” is a fascinating new direction opened by this new way of looking at loop quantum gravity.

==endquote==

The Freidel Speziale paper cites the original Penrose Rindler source that introduced twistors. Apparently the Witten twistor paper was not relevant, however, since it is not mentioned.

 

[MTd2]

Apparently the Witten twistor paper was not relevant, however, since it is not mentioned.

Witten's paper did not bring anything new to twistors or even super twistors (known since 1978). Just new ways to calculate SYM amplitudes.

 

[MTd2]

Marcus, now going on topic. I've been making some research on spinors, and what is used in this paper is not Twistor proper, that is, defined as a CP3. What we have here is a generalization of it and probably with a non local character given that Twistors are commonly defined to be on shell. Any idea?