ep - t h / 02 02 12 1 v 1 1 9 Fe b 20 02 Hamiltonian Quantization of Chern - Simons theory with SL ( 2 , C )

We analyze the hamiltonian quantization of Chern-Simons theory associated to the real group SL(2, C) R , 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 combinato-rial quantization of Chern-Simons theory, i.e we quantize the algebra of polynomial functions on the space of flat SL(2, C) R −connections on a topological surface Σ with punctures. This algebra, the so called moduli algebra, is constructed along the lines of Fock-Rosly, Alekseev-Grosse-Schomerus, Buffenoir-Roche using only finite dimensional representations of U q (sl(2, C) R). It is shown that 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 U q (sl(2, C) R). The representation of the moduli algebra is constructed using only Clebsch-Gordan decomposition of a tensor product of a finite dimensional representation with a principal unitary representation of U q (sl(2, C) R). The proof of unitarity of this representation is non trivial and is a consequence of properties of U q (sl(2, C) R) in-tertwiners 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.