Complex Ashtekar Variables and Reality Conditions for Holst’s Action

Wieland, Wolfgang

doi:10.1007/s00023-011-0134-z

Cited by 41 publications

(81 citation statements)

References 35 publications

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“…The boundary term for the outer and timelike cylinder B is an extension of the Gibbons -Hawking -York boundary term for tetrad connection variables. This term has been first introduced by Obukhov [24], and it can be written in the following form (see [25,26] for references)…”

Section: Hilbert -Palatini Actionmentioning

confidence: 99%

New boundary variables for classical and quantum gravity on a null surface

Wieland

2017

Class. Quantum Grav.

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The covariant Hamiltonian formulation for general relativity is studied in terms of self-dual variables on a manifold with an internal and lightlike boundary. At this inner boundary, new canonical variables appear: a spinor and a spinor-valued two-form that encode the entire intrinsic geometry of the null surface. At a two-dimensional cross-section of the boundary, quasilocal expressions for the generators of two-dimensional diffeomorphisms, time translations, and dilatations of the null normal are introduced and written in terms of the new boundary variables. In addition, a generalisation of the first-law of black-hole thermodynamics for arbitrary null surfaces is found, and the relevance of the framework for non-perturbative quantum gravity is stressed and explained.

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Section: Hilbert -Palatini Actionmentioning

confidence: 99%

New boundary variables for classical and quantum gravity on a null surface

Wieland

2017

Class. Quantum Grav.

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“…The bivector B is related to the Lorentz generators via B = (½−γ⋆)J, where γ ∈ R is the Immirzi parameter. See [8,22,23] for details, and [24] for extensions to the case of a null hypersurface. It is customary the fix the time gauge N I = (1, 0, 0, 0), but the construction extends to an arbitrary gauge [22].…”

Section: Twistors and Twisted Geometriesmentioning

confidence: 99%

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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“…while δ f k a = −δ f ∂ a u = 0 without loss of generality see section 2.1. In the last section, we have already considered the field variations of all other configuration variables on the space of histories (46). We are therefore free to declare how the vector field δ f on field space (46) acts on the spin bases (k A ± , ℓ A ± ).…”

Section: Conformal Transformationsmentioning

confidence: 99%

Generating functional for gravitational null initial data

Wieland

2019

Class. Quantum Grav.

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A field theory on a three-dimensional manifold is introduced, whose field equations are the constraint equations for general relativity on a three-dimensional null hypersurface. The underlying boundary action consists of two copies of the dressed Chern -Simons term for self-dual Ashtekar variables, a kinetic term for the null flag at the boundary plus additional junction conditions for the spin coefficients across the interface. In fact, there is a doubling of the field content, because the null hypersurface will be considered as an internal boundary between two adjacent slabs of spacetime. The paper concludes with a proposal for a construction of the gravitational transition amplitudes in the bulk via the auxiliary boundary field theory alone, namely by gluing amplitudes for edge states across two-dimensional corners, thus providing a proposal for a quasi-local realisation of the holographic principle at the light front.

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Complex Ashtekar Variables and Reality Conditions for Holst’s Action

Cited by 41 publications

References 35 publications

New boundary variables for classical and quantum gravity on a null surface

New boundary variables for classical and quantum gravity on a null surface

Twisted geometries, twistors, and conformal transformations

Generating functional for gravitational null initial data

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