Hamiltonian quantization of Chern Simons theory with <i>SL</i> (2, ) group

Abstract

We analyze the hamiltonian quantization of Chern-Simons theory associated to\nthe universal covering of the Lorentz group SO(3,1). The algebra of observables\nis generated by finite dimensional spin networks drawn on a punctured\ntopological surface. Our main result is a construction of a unitary\nrepresentation of this algebra. For this purpose, we use the formalism of\ncombinatorial quantization of Chern-Simons theory, i.e we quantize the algebra\nof polynomial functions on the space of flat SL(2,C)-connections on a\ntopological surface with punctures. This algebra admits a unitary\nrepresentation acting on an Hilbert space which consists in wave packets of\nspin-networks associated to principal unitary representations of the quantum\nLorentz group. This representation is constructed using only Clebsch-Gordan\ndecomposition of a tensor product of a finite dimensional representation with a\nprincipal unitary representation. The proof of unitarity of this representation\nis non trivial and is a consequence of properties of intertwiners which are\nstudied in depth. We analyze the relationship between the insertion of a\npuncture colored with a principal representation and the presence of a\nworld-line of a massive spinning particle in de Sitter space.\n