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Classical and Quantum Gravity 19 (2002) 4953-5015
Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
E. Buffenoir1, 2, K. Noui3, P. Roche1, 2

We analyze the hamiltonian quantization of Chern-Simons theory associated to the universal covering of the Lorentz group SO(3,1). The algebra of observables is generated by finite dimensional spin networks drawn on a punctured topological surface. Our main result is a construction of a unitary representation of this algebra. For this purpose, we use the formalism of combinatorial quantization of Chern-Simons theory, i.e we quantize the algebra of polynomial functions on the space of flat SL(2,C)-connections on a topological surface with punctures. This algebra admits a unitary representation acting on an Hilbert space which consists in wave packets of spin-networks associated to principal unitary representations of the quantum Lorentz group. This representation is constructed using only Clebsch-Gordan decomposition of a tensor product of a finite dimensional representation with a principal unitary representation. The proof of unitarity of this representation is non trivial and is a consequence of properties of intertwiners which are studied in depth. We analyze the relationship between the insertion of a puncture colored with a principal representation and the presence of a world-line of a massive spinning particle in de Sitter space.
1 :  LPTA - Laboratoire de Physique Théorique et Astroparticules
2 :  PMT - Laboratoire de Physique Mathématique et Théorique
3 :  CGPG - Center for Gravitational Physics and Geometry
Mathématiques/Algèbres quantiques

Physique/Relativité Générale et Cosmologie Quantique

Physique/Physique des Hautes Energies - Théorie
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