Hamiltonian quantization of Chern-Simons theory with SL(2,C) group

We analyse the Hamiltonian quantization of Chem-Simons theory associated with the real group SL (2, C)(R), universal covering group 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 Chem-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 E 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 a Hilbertspace which consists of wave packets of spin networks associated with 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 nontrivial and is a consequence of the properties of U-q (sl (2, C)(R)) intertwiners which are studied in depth. We analyse the relationship between the insertion of a puncture coloured with a principal representation and the presence of a worldline of a massive spinning particle in de Sitter space.