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

Abstract

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 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 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 ) intertwiners 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.

Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity

“…Recent work on the combinatorial quantisation of Chern-Simons theory with gauge group SL(2, C) (corresponding to Lorentzian gravity with a cosmological positive constant) in [38] and [39] shows how these question can be addressed but also illustrates the level of technical difficulty involved.…”

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

“…This theory can be treated similarly, and many of the arguments below easily extend to this case. For work on quantization of this theory see [1,8,9].…”

Triply special relativity

“…In the case of 2 + 1 gravity, the result that the symmetry algebra is quantum deformed when the cosmological constant is turned on is rigorous, a complete argument is given in [15]. For the case of 3 + 1 there is good evidence that the local gauge symmetry of the spacetime connection is quantum deformed from SU (2) to SU q (2) [16,17,18].…”