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http://arxiv.org/abs/gr-qc/0612144
Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double
Karim Noui
36 pages, published in J. Math. Phys. 47, 102501 (2006)
"It is well known that the quantum double structure plays an important role in three dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three dimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surface Sigma and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where Sigma is the sphere S^2, we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface Sigma.
http://arxiv.org/abs/gr-qc/0612145
Three dimensional Loop Quantum Gravity: towards a self-gravitating Quantum Field Theory
Karim Noui
36 pages, published in Class. Quant. Grav. 24 (2007)
In a companion paper, we have emphasized the role of the Drinfeld double DSU(2) in the context of three dimensional Riemannian Loop Quantum Gravity coupled to massive spinless point particles. We make use of this result to propose a model for a self-gravitating quantum field theory (massive spinless non-causal scalar field) in three dimensional Riemannian space. We start by constructing the Fock space of the free self-gravitating field: the vacuum is the unique DSU(2) invariant state, one-particle states correspond to DSU(2) unitary irreducible simple representations and any multi-particles states is obtained as the symmetrized tensor product between simple representations. The associated quantum field is defined by the usual requirement of covariance under DSU(2). Then, we introduce a DSU(2)-invariant self-interacting potential (the obtained model is a Group Field Theory) and compute explicitely the lowest order terms (in the self-interaction coupling constant lambda) of the propagator and of the three-points function. Finally, we compute the lowest order quantum gravity corrections (in the Newton constant G) to the propagator and to the three-points function.
http://arxiv.org/abs/gr-qc/0612147
Spectral Analysis of the Volume Operator in Loop Quantum Gravity
J. Brunnemann, D. Rideout
5 pages, 1 figure. Talk given by D. Rideout at the Eleventh Marcel Grossmann Meeting on General Relativity at the Freie U. Berlin, July 23 - 29, 2006
Imperial/TP/2006/DR/02
"We describe preliminary results of a detailed numerical analysis of the volume operator as formulated by Ashtekar and Lewandowski. Due to a simplified explicit expression for its matrix elements, it is possible for the first time to treat generic vertices of valence greater than four. It is found that the vertex geometry characterizes the volume spectrum."
Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double
Karim Noui
36 pages, published in J. Math. Phys. 47, 102501 (2006)
"It is well known that the quantum double structure plays an important role in three dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three dimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surface Sigma and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where Sigma is the sphere S^2, we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface Sigma.
http://arxiv.org/abs/gr-qc/0612145
Three dimensional Loop Quantum Gravity: towards a self-gravitating Quantum Field Theory
Karim Noui
36 pages, published in Class. Quant. Grav. 24 (2007)
In a companion paper, we have emphasized the role of the Drinfeld double DSU(2) in the context of three dimensional Riemannian Loop Quantum Gravity coupled to massive spinless point particles. We make use of this result to propose a model for a self-gravitating quantum field theory (massive spinless non-causal scalar field) in three dimensional Riemannian space. We start by constructing the Fock space of the free self-gravitating field: the vacuum is the unique DSU(2) invariant state, one-particle states correspond to DSU(2) unitary irreducible simple representations and any multi-particles states is obtained as the symmetrized tensor product between simple representations. The associated quantum field is defined by the usual requirement of covariance under DSU(2). Then, we introduce a DSU(2)-invariant self-interacting potential (the obtained model is a Group Field Theory) and compute explicitely the lowest order terms (in the self-interaction coupling constant lambda) of the propagator and of the three-points function. Finally, we compute the lowest order quantum gravity corrections (in the Newton constant G) to the propagator and to the three-points function.
http://arxiv.org/abs/gr-qc/0612147
Spectral Analysis of the Volume Operator in Loop Quantum Gravity
J. Brunnemann, D. Rideout
5 pages, 1 figure. Talk given by D. Rideout at the Eleventh Marcel Grossmann Meeting on General Relativity at the Freie U. Berlin, July 23 - 29, 2006
Imperial/TP/2006/DR/02
"We describe preliminary results of a detailed numerical analysis of the volume operator as formulated by Ashtekar and Lewandowski. Due to a simplified explicit expression for its matrix elements, it is possible for the first time to treat generic vertices of valence greater than four. It is found that the vertex geometry characterizes the volume spectrum."
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