Conformal variations and quantum fluctuations in discrete gravity

Marzuoli, Annalisa; Merzi, Dario

doi:10.1142/s0219887816500845

Cited by 1 publication

(2 citation statements)

References 26 publications

(51 reference statements)

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“…The transformations are nonlinear, 17 as expected, and can not be visualised as symplectic dilatations in the continuum sense (61). Nonetheless, they have some interesting properties.…”

Section: Some Dynamical Considerationsmentioning

confidence: 77%

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Twisted geometries, twistors, and conformal transformations

Långvik

Speziale

2016

Phys. Rev. D

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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 γ, and reduces to the identity for γ = ∞. 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 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 [2].

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“…The transformations are nonlinear, 17 as expected, and can not be visualised as symplectic dilatations in the continuum sense (61). Nonetheless, they have some interesting properties.…”

Section: Some Dynamical Considerationsmentioning

confidence: 77%

“…Furthermore, it is not possible to extend this transformation to a a finite group action, because composition of two finite transformations generically breaks the triangle inequalities. Nonetheless, this definition has useful applications to discrete Ricci flow [59,60] and quantum Regge calculus [61].…”

Section: Some Dynamical Considerationsmentioning

confidence: 99%

Twisted geometries, twistors, and conformal transformations

Långvik

Speziale

2016

Phys. Rev. D

View full text Add to dashboard Cite

show abstract

Conformal variations and quantum fluctuations in discrete gravity

Cited by 1 publication

References 26 publications

Twisted geometries, twistors, and conformal transformations

Twisted geometries, twistors, and conformal transformations

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