Continuous formulation of the Loop Quantum Gravity phase space

Freidel, Laurent; Geiller, Marc; Ziprick, Jonathan

doi:10.48550/arxiv.1110.4833

Cited by 22 publications

(48 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In many aspects, this new (BF or flux) representation can be understood as being dual to the AL representation 2 . Although the fact that such an alternative representation should exist has been suggested early on [29][30][31][32][33][34], the need to dualize every ingredient of the usual AL representation has been realized only recently in [27].…”

Section: Introductionmentioning

confidence: 99%

Flux formulation of loop quantum gravity: classical framework

Dittrich¹,

Geiller²

2015

Class. Quantum Grav.

Self Cite

140

View full text Add to dashboard Cite

We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gravity, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other.

show abstract

Section: Introductionmentioning

confidence: 99%

Flux formulation of loop quantum gravity: classical framework

Dittrich¹,

Geiller²

2015

Class. Quantum Grav.

Self Cite

140

View full text Add to dashboard Cite

show abstract

“…These cylindrical wave-functions realize a sampling of the geometry at the classical level (see e.g. [23]), but they will entirely define the state of geometry at the quantum level. Considering holonomies of the Ashtekar-Barbero connection for a fixed real Immirzi parameter irremediably introduces a periodicity in the extrinsic curvature K, which erases all the information about its high momentum fluctuations.…”

Section: Introductionmentioning

confidence: 99%

Ashtekar-Barbero holonomy on the hyperboloid: Immirzi parameter as a cutoff for quantum gravity

Charles

Livine

2015

Phys. Rev. D

View full text Add to dashboard Cite

In the context of the geometrical interpretation of the spin network states of Loop Quantum Gravity, we look at the holonomies of the Ashtekar-Barbero connection on loops embedded in spacelike hyperboloids. We use this simple setting to illustrate two points. First, the Ashtekar-Barbero connection is not a space-time connection, its holonomies depend on the spacetime embedding of the canonical hypersurface. This fact is usually interpreted as an inconvenience, but we use it to extract the extrinsic curvature from the holonomy and separate it from the 3d intrinsic curvature. Second, we show the limitations of this reconstruction procedure, due to a periodicity of the holonomy in the Immirzi parameter, which underlines the role of a real Immirzi parameter as a cut-off for general relativity at the quantum level in contrast with its role of a mere coupling constant at the classical level.

show abstract

“…The discrete change in triangulation that occurs when a vertex meets an edge is well-defined. This construction yields spatial geometries that are the two-dimensional analog of those in [1]. Having (a ∆ , χ ∆ ) in each triangle allows us to immediately write this data in terms of holonomy-flux variables.…”

Section: Discussionmentioning

confidence: 99%

“…Within each triangle ∆, we must choose (A ∆ , e ∆ ) such that G i = F i = 0. The general solution [1] is given by an SU(2) group element a ∆ and a closed one-form χ ∆ :…”

Section: Gauge Fixingmentioning

confidence: 99%

“…Three dimensional gravity has often been used as a toy model for the four dimensional theory. Here we study point particles in 2+1 dimensions using spatial geometries analogous to those in [1]. These geometries are isomorphic to a gauge-reduced holonomyflux phase space which descends from the → 0 limit of a loop quantum gravity Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation