6j symbols for $$U_q (\mathfrak{s}\mathfrak{l}_2 )$$ and non-Euclidean tetrahedra

Taylor, Yuka U.; Woodward, Chris

doi:10.1007/s00029-005-0014-9

Cited by 47 publications

(80 citation statements)

References 34 publications

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“…Thus equipping the space of normal vectors with the Poisson bracket on R 3 ∼ su (2), that is {x i , x j } = ijk x k for a 3d vector x, we define: (13) where N However so far, we have a degenerate Poisson structure. It is possible to remedy this.…”

Section: The Flat Tetrahedron Phase Spacementioning

confidence: 99%

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Closure constraints for hyperbolic tetrahedra

Charles

Livine

2015

Class. Quantum Grav.

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We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R 3 . One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Λ = 0, and in particular that deforming the loop gravity phase space with real parameter q ∈ R+ would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in SU(2) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in SB(2, C)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.

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Section: The Flat Tetrahedron Phase Spacementioning

confidence: 99%

“…We define the 3-vectors by projecting the spinors onto the Pauli matrices, N i a = z a |σ i |z a , which leads to the exact same Poisson bracket as before in (13).…”

Section: The Flat Tetrahedron Phase Spacementioning

confidence: 99%

Closure constraints for hyperbolic tetrahedra

Charles

Livine

2015

Class. Quantum Grav.

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“…has been shown to be related to the action of 2 + 1 gravity with non vanishing cosmological constant in the WKB approximation [11,12].…”

Section: Jhep10(2011)036mentioning

confidence: 99%

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Noui

Pérez

Pranzetti

2011

J. High Energ. Phys.

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In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant Λ > 0 in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection A and the canonically conjugate triad field e. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of A ± = A± √ Λe. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A λ = A + λe (for λ ∈ R) on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity (with q = exp(i λ/2)). The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.

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“…This much is clear already from the work of Mizoguchi and Tada (1992) and of Taylor and Woodward (2005), both of whom used one-dimensional WKB techniques along the lines of Schulten and Gordon (1975ab).…”

Section: Discussionmentioning

confidence: 99%

Symplectic and semiclassical aspects of the Schläfli identity

Hedeman

Kur

Littlejohn

et al. 2015

J. Phys. A: Math. Theor.

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Abstract. The Schläfli identity, which is important in Regge calculus and loop quantum gravity, is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space. In this case a proof is given, based on symplectic geometry. A series of symplectic and Lagrangian manifolds related to the Schläfli identity, including several versions of a Lagrangian manifold of tetrahedra, are discussed. Semiclassical interpretations of the various steps are provided. Possible generalizations to 3-dimensional spaces of constant (nonzero) curvature, involving Poisson-Lie groups and q-deformed spin networks, are discussed.

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6j symbols for $$U_q (\mathfrak{s}\mathfrak{l}_2 )$$ and non-Euclidean tetrahedra

Abstract: We relate the semiclassical asymptotics of the 6j symbols for the quantized enveloping algebra U q (sl 2 ) at q a root of unity (resp. q real positive) to the geometry of spherical (resp. hyperbolic) tetrahedra. (2000). 17B37, 33C80, 51M20. Mathematics Subject Classification

Cited by 47 publications

References 34 publications

Closure constraints for hyperbolic tetrahedra

Closure constraints for hyperbolic tetrahedra

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Symplectic and semiclassical aspects of the Schläfli identity

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