Twisted geometries, twistors, and conformal transformations

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Långvik , M & Speziale , S 2016 , ' Twisted geometries, twistors, and conformal transformations ' , Physical Review D , vol. 94 , no. 2 , 024050 .

Title: Twisted geometries, twistors, and conformal transformations
Author: Långvik, Miklos; Speziale, Simone
Contributor organization: Department of Physics
Date: 2016-07-26
Language: eng
Number of pages: 18
Belongs to series: Physical Review D
ISSN: 2470-0010
Abstract: The twisted geometries of spin network states are described by simple twistors, isomorphic to null twistors with a timelike direction singled out. The isomorphism depends on the Immirzi parameter gamma and reduces to the identity for gamma = 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 one-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 discretization 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 nonlinear 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 [Classical Quantum Gravity 32, 195015 (2015)].
114 Physical sciences
Peer reviewed: Yes
Rights: cc_by
Usage restriction: openAccess
Self-archived version: publishedVersion

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