On the Quantization of Polygon Spaces

Charles, Laurent

doi:10.4310/ajm.2010.v14.n1.a6

Cited by 19 publications

(26 citation statements)

References 19 publications

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“…(i ) Intertwiners are the building blocks of spin-network states, an orthonormal basis of the Hilbert space of loop quantum gravity [20,21] (ii ) Intertwiners are the quantization of the phase space of Kapovich and Millson [22,9,23] (see also [24,25]), i.e. of the space of shapes of polyhedra with fixed areas discussed in the previous sections.…”

Section: Relation To Loop Quantum Gravitymentioning

confidence: 99%

Polyhedra in loop quantum gravity

2011

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Interwiners are the building blocks of spin-network states. The space of intertwiners is the quantization of a classical symplectic manifold introduced by Kapovich and Millson. Here we show that a theorem by Minkowski allows us to interpret generic configurations in this space as bounded convex polyhedra in Ê 3 : a polyhedron is uniquely described by the areas and normals to its faces. We provide a reconstruction of the geometry of the polyhedron: we give formulas for the edge lengths, the volume and the adjacency of its faces. At the quantum level, this correspondence allows us to identify an intertwiner with the state of a quantum polyhedron, thus generalizing the notion of quantum tetrahedron familiar in the loop quantum gravity literature. Moreover, coherent intertwiners result to be peaked on the classical geometry of polyhedra. We discuss the relevance of this result for loop quantum gravity. In particular, coherent spin-network states with nodes of arbitrary valence represent a collection of semiclassical polyhedra. Furthermore, we introduce an operator that measures the volume of a quantum polyhedron and examine its relation with the standard volume operator of loop quantum gravity. We also comment on the semiclassical limit of spinfoams with non-simplicial graphs.

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Section: Relation To Loop Quantum Gravitymentioning

confidence: 99%

Polyhedra in loop quantum gravity

2011

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“…The second way to understand the relation between the two systems is to consider the quantization of the classical system (a). This has been done by Kapovich, Millson 2 [20] and Charles [21], and shown to be the Hilbert space of intertwiners (b).…”

Section: The Lqg Physical Cut-off: a Tessellated Horizonmentioning

confidence: 99%

Black hole entropy, loop gravity, and polymer physics

Bianchi

2011

Class. Quantum Grav.

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Loop Gravity provides a microscopic derivation of Black Hole entropy. In this paper, I show that the microstates counted admit a semiclassical description in terms of shapes of a tessellated horizon. The counting of microstates and the computation of the entropy can be done via a mapping to an equivalent statistical mechanical problem: the counting of conformations of a closed polymer chain. This correspondence suggests a number of intriguing relations between the thermodynamics of Black Holes and the physics of polymers. *

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“…It was conjectured in [73] based on numerical evidence and some very clever consistency checks, and it was rigorously proved in [76]. Other work related to this asymptotic formula can be found in [80,13,40,22,50,44]. Note that [40,13] consider the square of the 6-j symbol which is also suitable for the needs of this section.…”

Section: The Lower Boundmentioning

confidence: 95%

On the Volume Conjecture for Classical Spin Networks

Abdesselam

2012

J. Knot Theory Ramifications

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Abstract. We prove an upper bound for the evaluation of all classical SU 2 spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.Mathematics Subject Classification (2000): 13A50; 22E70; 57M15; 57M25; 57N10;

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On the Quantization of Polygon Spaces

Cited by 19 publications

References 19 publications

Polyhedra in loop quantum gravity

Polyhedra in loop quantum gravity

Black hole entropy, loop gravity, and polymer physics

On the Volume Conjecture for Classical Spin Networks

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