# Papers that combine LQG with twistor theory

## Main Question or Discussion Point

marcus recently posted this paper
Twisted geometries, twistors and conformal transformations
Miklos Långvik, Simone Speziale
(Submitted on 4 Feb 2016)
The twisted geometries of spin network states are described by simple twistors, isomorphic to null twistors with a time-like direction singled out. The isomorphism depends on the Immirzi parameter, and reduces to the identity when the parameter goes to infinity. Using this twistorial representation we study the action of the conformal group SU(2,2) on the classical phase space of loop quantum gravity, described by twisted geometry. The generators of translations and conformal boosts do not preserve the geometric structure, whereas the dilatation generator does. It corresponds to a 1-parameter family of embeddings of T*SL(2,C) in twistor space, and its action preserves the intrinsic geometry while changing the extrinsic one - that is the boosts among polyhedra. We discuss the implication of this action from a dynamical point of view, and compare it with a discretisation of the dilatation generator of the continuum phase space, given by the Lie derivative of the group character. At leading order in the continuum limit, the latter reproduces the same transformation of the extrinsic geometry, while also rescaling the areas and volumes and preserving the angles associated with the intrinsic geometry. Away from the continuum limit its action has an interesting non-linear structure, but is in general incompatible with the closure constraint needed for the geometric interpretation. As a side result, we compute the precise relation between the extrinsic geometry used in twisted geometries and the one defined in the gauge-invariant parametrization by Dittrich and Ryan, and show that the secondary simplicity constraints they posited coincide with those dynamically derived in the toy model of [1409.0836].
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)
Cite as: arXiv:1602.01861 [gr-qc]
(or arXiv:1602.01861v1 [gr-qc] for this version)
Submission history
From: Simone Speziale [view email]
[v1] Thu, 4 Feb 2016 21:52:05 GMT (31kb)

Penrose originated both spin networks and twistor theory.

LQG makes use of spin networks.

there has been reseach papers that combine twistor theory with LQG. what are the advantages of combining the 2? it's my understanding they are attempting to quantize twistor space.

what would be the advantages of a physical theory of quantum gravity that combines LQG spin networks with twistor theory?

what problems does twistor theory in combination with LQG is supposed to address? and how successful has the program been?

there are actually many other papers that discuss twistor theory with LQG. is this a separate program from mainstream LQG?

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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

Twistor theory was originally designed to obtain an effective spinor representation of the conformal group SO(4,2) of the Minkowski space, i.e. the representation of SU(2,2). All constructions work only for the flat Minkowski space and Penrose (around 1976) tried to extend it to the curved space. As far as I can remember this space is not defined at all but the construction in the 1976 paper is known as non-linear graviton. Soon after, Hitchin wrote a paper (Graviton and polygons) where he described this non-linear graviton construction completely.
In the paper above, the author used only this twistor construction as a description of twisted geometries and T*SL(2,C) is embedded into the twistor space. It seems to me that these twisted geometries must be related to the non-linear graviton construction. But the whole construction depends strongly on this embedding.

Twistor theory was originally designed to obtain an effective spinor representation of the conformal group SO(4,2) of the Minkowski space, i.e. the representation of SU(2,2). All constructions work only for the flat Minkowski space and Penrose (around 1976) tried to extend it to the curved space. As far as I can remember this space is not defined at all but the construction in the 1976 paper is known as non-linear graviton. Soon after, Hitchin wrote a paper (Graviton and polygons) where he described this non-linear graviton construction completely.
In the paper above, the author used only this twistor construction as a description of twisted geometries and T*SL(2,C) is embedded into the twistor space. It seems to me that these twisted geometries must be related to the non-linear graviton construction. But the whole construction depends strongly on this embedding.
why does this research program interest both LQG and string i.e witten? is it in strings to reduce the dimensions ?

I would be suspicious if it was April 1.

why does this research program interest both LQG and string i.e witten? is it in strings to reduce the dimensions ?
As far as I know the motivations are different: Witten found twistor-like relations between string amplitudes. With the help of these ideas, some people were able to express loop amplitudes in some QFT (mostly supersymmetric) by tree amplitudes.
As I explained above, the LQG use of twistor theory is mainly related to twisted geometries.

As far as I know the motivations are different: Witten found twistor-like relations between string amplitudes. With the help of these ideas, some people were able to express loop amplitudes in some QFT (mostly supersymmetric) by tree amplitudes.
As I explained above, the LQG use of twistor theory is mainly related to twisted geometries.
what is your evaluation of twisted geometries?

marcus
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what is your evaluation of twisted geometries?
AFAICS it is too early to be pressing people for an evaluation in the sense of guessing as to the validity. But maybe T. can give an evaluation in the sense of how interested people are in investigating twisted geometries in LQG and why it's interesting to pursue. There are dozens of papers that use the twisted geometries idea. Laurent Freidel would be a good author to look up in conjunction with that keyword, I think. Also possibly Wolfgang Wieland, and Speziale himself.

Unfortunately I can't do the search right now, but as I recall there is a lot of activity in that direction, people getting interesting results using the twisted geometries idea is an indicator of why it could be a good direction in which to investigate. Hopefully Torsten has something more concrete and definite to say about it.

I agree with marcus: I can only describe my own view. A real evaluation is not possible, even experimental verifications are missed.
But back to twisted geometries: I want to compare it with the two approaches to geometry: Riemann or Cartan.
The discrete version of Riemannian geometry is Regge calculus and the discrete version of Cartan geometry is the twisted geometry. The limes of Regge calculus is problematic: one measures only deviation from flat space. In contrast, Cartan geometry measures the deviation from a space of constant curvature. Twisted geometries must be admit a limit of a Cartan geometry (but I don't found in the literature).
I think the whole approach is a promising way to get the smooth limit of the discrete approach via LQG.

does twisted geometries offer the prospect of better semi-classical limit or better contact with QFT-SM fields or even twistor strings?

does twisted geometries offer the prospect of better semi-classical limit or better contact with QFT-SM fields or even twistor strings?
In my opinion, twisted geometries will give a better semi-classical limit. The problem is viewable in perturbative quantum gravity. Every graviton gives only a (in principle) beglectable contribution and you need uncountable infinite many gravitons to get measurable contribution. Regge calculus is comparable to this picture. In contrast, if you approximate the curved space by small homogenous geometries (a la Cartan) then you need only a finite portion of these parts to get a measurable effect. That is the main idea of the non-linear graviton of Penrose. Twisted geometries are a discrete version of this idea, I expect (modulo technical problems) a better semi-classical limit.
I cannot see a direct connection to twistor strings (but I'm also not an expert of this topic). Maybe some relation to QFT-SM fields but then one has to get in contact to the Bilson-Thompson model. But it is pure speculation...