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52
Three dimensional Loop Quantum Gravity: physical scalar product and spin foam models
 35 K. Noui, “Three dimensional Loop Quantum Gravity: particles and the Quantum
, 2005
"... 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 selfgravitating quantum field theory (massive spinless ..."
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Cited by 76 (14 self)
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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 selfgravitating quantum field theory (massive spinless noncausal scalar field) in three dimensional Riemannian space. We start by constructing the Fock space of the free selfgravitating field: the vacuum is the unique DSU(2) invariant state, oneparticle states correspond to DSU(2) unitary irreducible simple representations and any multiparticles 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 selfinteracting potential (the obtained model is a Group Field Theory) and compute explicitely the lowest order terms (in the selfinteraction coupling constant λ) of the propagator and of the threepoints function. Finally, we compute the lowest order quantum gravity corrections (in the Newton constant G) to the propagator and to the threepoints function.
Colored Tensor Models  a Review
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2012
"... Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor mo ..."
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Cited by 43 (6 self)
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Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating twodimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger–Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), nontrivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.
Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity
"... We show that the ⋆product for U(su2), group Fourier transform and effective action arising in [1] in an effective theory for the integer spin PonzanoRegge quantum gravity model are compatible with the noncommutative bicovariant differential calculus, quantum group Fourier transform and noncommutat ..."
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Cited by 33 (4 self)
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We show that the ⋆product for U(su2), group Fourier transform and effective action arising in [1] in an effective theory for the integer spin PonzanoRegge quantum gravity model are compatible with the noncommutative bicovariant differential calculus, quantum group Fourier transform and noncommutative scaler field theory previously proposed for 2+1 Euclidean quantum gravity using quantum group methods in [2]. The two are related by a classicalisation map which we introduce. We show, however, that noncommutative spacetime has a richer structure which already sees the halfinteger spin information. We argue that the anomalous extra ‘time ’ dimension seen in the noncommutative geometry should be viewed as the renormalisation group flow visible in the coarse graining in going from SU2 to SO3. Combining our methods we develop practical tools for noncommutative harmonic analysis for the model including radial quantum deltafunctions and Gaussians, the Duflo map and elements of ‘noncommutative sampling theory’. This allows us to understand the bandwidth limitation in 2+1 quantum gravity arising from the bounded SU2 momentum and to interpret the Duflo map as noncommutative compression. Our methods also provide a generalised twist operator for the ⋆product.
The PonzanoRegge model
, 2008
"... The definition of the PonzanoRegge statesum model of threedimensional quantum gravity with a class of local observables is developed. The main definition of the PonzanoRegge model in this paper is determined by its reformulation in terms of group variables. The regularisation is defined and a pro ..."
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Cited by 19 (5 self)
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The definition of the PonzanoRegge statesum model of threedimensional quantum gravity with a class of local observables is developed. The main definition of the PonzanoRegge model in this paper is determined by its reformulation in terms of group variables. The regularisation is defined and a proof is given that the partition function is welldefined only when a certain cohomological criterion is satisfied. In that case, the partition function may be expressed in terms of a topological invariant, the Reidemeister torsion. This proves the independence of the definition on the triangulation of the 3manifold and on those arbitrary choices made in the regularisation. A further corollary is that when the observable is a knot, the partition function (when it exists) can be written in terms of the Alexander polynomial of the knot. Various examples of observables in S 3 are computed explicitly. Alternative regularisations of the PonzanoRegge model by the simple cutoff procedure and by the limit of the TuraevViro model are discussed, giving successes and limitations of these approaches. 1
qDeformation and semidualisation in 3D quantum gravity
"... Abstract. We explore in detail the role in euclidean 3d quantum gravity of quantum Born reciprocity or ‘semidualisation’. The latter is an algebraic operation defined using quantum group methods that interchanges position and momentum. Using this we are able to clarify the structural relationships b ..."
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Cited by 14 (11 self)
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Abstract. We explore in detail the role in euclidean 3d quantum gravity of quantum Born reciprocity or ‘semidualisation’. The latter is an algebraic operation defined using quantum group methods that interchanges position and momentum. Using this we are able to clarify the structural relationships between the effective noncommutative geometries that have been discussed in the context of 3d gravity. We show that the spin model based on D(U(su2)) for quantum gravity without cosmological constant is the semidual of a quantum particle on a threesphere, while the bicrossproduct (DSR) model based on C[R 2>⊳R]◮⊳U(su2) is the semidual of a quantum particle on hyperbolic space. We show further how the different models are all specific limits of qdeformed models with q = e −�√−Λ/mp where mp is the Planck mass and Λ is the cosmological constant, and argue that semidualisation interchanges mp ↔ lc, where lc is the cosmological length scale lc = 1 / p Λ. We investigate the physics of semidualisation by studying representation theory. In both the spin model and its semidual we show that irreducible representations have a physical picture as solutions of a respectively noncommutative/curved wave equation. We explain, moreover, that the qdeformed model, at a certain algebraic level, is selfdual under semidualisation. 1.
Generic predictions of quantum theories of gravity,” [arXiv:hepth/0605052]; A. Ashtekar and
, 2004
"... I discuss generic consequences (sometimes called “soft predictions”) of a class of background independent quantum theories of spacetime called causal spin network theories. These are theories whose kinematics and dynamics is based on the evolution of labeled graphs, by local moves, such as in loop q ..."
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Cited by 13 (1 self)
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I discuss generic consequences (sometimes called “soft predictions”) of a class of background independent quantum theories of spacetime called causal spin network theories. These are theories whose kinematics and dynamics is based on the evolution of labeled graphs, by local moves, such as in loop quantum gravity and spin foam models. Some generic consequences are well known, including the discreteness of quantum geometry, the elimination of spacetime singularities, the entropy of black hole and cosmological horizons and the fact that positive cosmological constant spacetimes are hot. Within the last few years three possible generic consequences have come to light. These are 1) Deformed special relativity as the symmetry of the ground state, 2) Elementary particles as coherent excitations of quantum geometry, 3) Locality is disordered. I discuss some possible experimental consequences of each. For inclusion in ”Approaches to Quantum Gravity toward a new understanding of space, time, and matter”, edited by D. Oriti, to be published by Cambridge University Press.
Braided quantum field theories and their symmetries”, arXiv: 0704.0822 [hepth
"... Braided quantum field theories proposed by Oeckl can provide a framework for defining quantum field theories having Hopf algebra symmetries. In quantum field theories, symmetries lead to nonperturbative relations among correlation functions. We discuss Hopf algebra symmetries and such relations in ..."
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Cited by 11 (4 self)
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Braided quantum field theories proposed by Oeckl can provide a framework for defining quantum field theories having Hopf algebra symmetries. In quantum field theories, symmetries lead to nonperturbative relations among correlation functions. We discuss Hopf algebra symmetries and such relations in braided quantum field theories. We give the four algebraic conditions between Hopf algebra symmetries and braided quantum field theories, which are required for the relations to hold. As concrete examples, we apply our discussions to the Poincaré symmetries of two examples of noncommutative field theories. One is the effective quantum field theory of threedimensional quantum gravity coupled with spinless particles given by Freidel and Livine, and the other is noncommutative field theory on Moyal plane. We also comment on quantum field theory on κMinkowski spacetime.