2002/02/19 by E. Buffenoir, Karim Noui, K. Noui +1 · 103 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Chern–Simons theory #Gauge theory #Hamiltonian (control theory) #Hilbert space #Homotopy and Cohomology in Algebraic Topology #Lie group #Lorentz group #Lorentz transformation #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #Unitarity #Unitary representation #gr-qc #hep-th #math.QA
paper · pdf · doi:10.1088/0264-9381/19/19/313
published in Classical and Quantum Gravity 19(19), 4953-5015 (IOP Publishing) · 78 pages. Packages included
arxiv created 2002/02/19 · openalex publication_date 2002/09/17 · arxiv updated 2014/11/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/08
We analyse the Hamiltonian quantization of Chern–Simons theory associated with the real group SL (2, ) , 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, ) 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, ) ). 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 U q ( sl (2, ) ). 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, ) ). The proof of unitarity of this representation is nontrivial and is a consequence of the properties of U q ( sl (2, ) ) 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.