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

Abstract

We analyse the Hamiltonian quantization of Chern–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 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 Uq(sl(2, ℂ)ℝ). It is shown that this algebra admits a unitary representation acting on a Hilbert space which consists of wave packets of spin networks associated with principal unitary representations of Uq(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 Uq(sl(2, ℂ)ℝ). The proof of unitarity of this representation is nontrivial and is a consequence of the properties of Uq(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.