The quantisation of Poisson structures arising in Chern-Simons theory with gauge group $G \ltimes \mathfrac{g}^{*}$

Meusburger, Catherine; Schroers, Bernd J.

doi:10.4310/atmp.2003.v7.n6.a3

Cited by 57 publications

(178 citation statements)

References 28 publications

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“…13 This is different from the κ-deformation of the 3+1 Poincaré algebra. 14 This is exactly the Hopf algebra structure of the κ-deformation of the 2+1 Poincaré algebra, or equivalently of the quantum double D(SU (2)), which appears in the quantization of 2+1 gravity [4,17]. The infinitesimal deformation…”

Section: Non-commutative Field Theory As Effective Quantum Gravitymentioning

confidence: 99%

Ponzano–Regge model revisited: III. Feynman diagrams and effective field theory

Freidel

Livine

2006

Class. Quantum Grav.

227

534

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We study the no gravity limit G N → 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G N expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the effective dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagators.

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Section: Non-commutative Field Theory As Effective Quantum Gravitymentioning

confidence: 99%

Ponzano–Regge model revisited: III. Feynman diagrams and effective field theory

Freidel

Livine

2006

Class. Quantum Grav.

227

534

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“…* is not simply connected, the results of [14,15] can nevertheless be applied to this case 2 and are summarised in the following theorem.…”

Section: Phase Space and Poisson Structurementioning

confidence: 99%

“…3, we briefly review the Hamiltonian version of the Chern-Simons formulation of (2+1)-dimensional gravity. We discuss the role of holonomies and summarise the relevant results of [14,15], in which phase space and Poisson structure are characterised by a symplectic potential on the manifold (P ↑ 3 ) 2g with different copies of P ↑ 3 standing for the holonomies of a set of generators of the fundamental group π 1 (S g ).…”

Section: Introductionmentioning

confidence: 99%

Grafting and Poisson Structure in (2+1)-Gravity with Vanishing Cosmological Constant

Meusburger

2006

Commun. Math. Phys.

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We relate the geometrical construction of (2+1)-spacetimes via grafting to phase space and Poisson structure in the Chern-Simons formulation of (2+1)-dimensional gravity with vanishing cosmological constant on manifolds of topology R × S g , where S g is an orientable two-surface of genus g > 1. We show how grafting along simple closed geodesics λ is implemented in the Chern-Simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on S g . We prove that this action is generated via the Poisson bracket by a gauge invariant observable associated to the holonomy of λ. We deduce a symmetry relation between the Poisson brackets of observables associated to the Lorentz and translational components of the holonomies of general closed curves on S g and discuss its physical interpretation. Finally, we relate the action of grafting on the phase space to the action of Dehn twists and show that grafting can be viewed as a Dehn twist with a formal parameter θ satisfying θ 2 = 0.

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“…Defining a Poincaré element T ∈P ↑ 3 as in (4.10) and setting g = hw −1 , we can rewrite (4.13) as 14) which shows in particular that the kinetic term for the variables associated to the puncture does not depend on the way the coefficient of the curvature singularity (4.9) is parametrised. Moreover, the expression (4.14) has a simple geometric interpretation.…”

Section: The Boundary Condition At Spatial Infinitymentioning

confidence: 99%

Boundary conditions and symplectic structure in the Chern–Simons formulation of (2+1)-dimensional gravity

Meusburger¹,

Schroers²

2005

Class. Quantum Grav.

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We propose a description of open universes in the Chern-Simons formulation of (2+1)-dimensional gravity where spatial infinity is implemented as a puncture. At this puncture, additional variables are introduced which lie in the cotangent bundle of the Poincaré group, and coupled minimally to the Chern-Simons gauge field. We apply this description of spatial infinity to open universes of general genus and with an arbitrary number of massive spinning particles. Using results of [9] we give a finite dimensional description of the phase space and determine its symplectic structure. In the special case of a genus zero universe with spinless particles, we compare our result to the symplectic structure computed by Matschull in the metric formulation of (2+1)-dimensional gravity. We comment on the quantisation of the phase space and derive a quantisation condition for the total mass and spin of an open universe.

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The quantisation of Poisson structures arising in Chern-Simons theory with gauge group $G \ltimes \mathfrac{g}^{*}$

Cited by 57 publications

References 28 publications

Ponzano–Regge model revisited: III. Feynman diagrams and effective field theory

Ponzano–Regge model revisited: III. Feynman diagrams and effective field theory

Grafting and Poisson Structure in (2+1)-Gravity with Vanishing Cosmological Constant

Boundary conditions and symplectic structure in the Chern–Simons formulation of (2+1)-dimensional gravity

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