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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].

Comments: 20 pages

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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# Papers that combine LQG with twistor theory

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