Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group

TL;DR

This paper develops a unitary representation of the algebra of observables in Chern-Simons theory with the SL(2,C) group, using combinatorial quantization and spin-network techniques, linking it to quantum Lorentz group representations.

Contribution

It constructs a novel unitary representation of the observable algebra in SL(2,C) Chern-Simons theory via combinatorial quantization and spin-networks, advancing the understanding of quantum Lorentz group applications.

Findings

Constructed a unitary representation on a Hilbert space of spin-networks.

Linked puncture insertions to massive spinning particles in de Sitter space.

Proved unitarity using properties of intertwiners and Clebsch-Gordan decomposition.

Abstract

We analyze the hamiltonian quantization of Chern-Simons theory associated to the 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)-connections on a topological surface with punctures. 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 the quantum Lorentz group. This representation is constructed using only Clebsch-Gordan decomposition of a tensor product of a finite dimensional representation with a principal unitary representation. The proof of unitarity of this representation is non trivial and is a consequence of properties of 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.