Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group

E. Buffenoir (1, 2) , K. Noui (3) , P. Roche (1, 2)

1 LPTA - Laboratoire de Physique Théorique et Astroparticules
2 PMT - Laboratoire de Physique Mathématique et Théorique
3 CGPG - Center for Gravitational Physics and Geometry

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.

Domaines: Algèbres quantiques [math.QA]; Relativité Générale et Cosmologie Quantique [gr-qc]; Physique des Hautes Energies - Théorie [hep-th]

E. Buffenoir, K. Noui, P. Roche. Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group. Classical and Quantum Gravity, 2002, 19, pp.4953-5015. ⟨10.1088/0264-9381/19/19/313⟩. ⟨hal-00178161⟩