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

E. Buffenoir, K. Noui, P. Roche

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.

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

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.

Paper Structure

This paper contains 24 sections, 45 theorems, 264 equations, 22 figures.

Table of Contents

  1. I. Introduction
  2. II. Summary of the Combinatorial Quantization Formalism: the compact group case.
  3. III Combinatorial Quantization in the $SL(2,\mathbb C)_{\mathbb R}$ case.
  4. III. Unitary Representations of the Moduli Algebra in the $SL(2,\mathbb C)_{\mathbb R}$ case.
  5. III.1 Unitary representation of the graph algebra.
  6. III.2 Unitary Representation of the moduli algebra
  7. IV. Discussion and Conclusion
  8. Appendix A: Quantum Lorentz group
  9. Appendix B: Graphical proofs

Key Result

Proposition 1

Let $P$ be a palette labelling a quantum spin network ${\cal N}_P$ associated to the standard graph. We will define an element of ${\cal L}_{n,p}$ where we have defined $v_{ I}^{1/2} = v_{{I}_1}^{1/2} \cdots v_{{I}_n}^{1/2}.$ The elements $\stackrel{ P}{\cal O}{}\!\!_{n,p}^{(\pm)}$ are gauge invariant elements and if $\epsilon \in\{ +,-\}$ is fixed the nonzero elements of the family $\stackrel{ P}

Figures (22)

  • Figure 1: Standard Graph.
  • Figure 2: Expression of $\stackrel{P}{\cal F}{}\!\!_{0,4}^{(+)}$.
  • Figure 3: Expression of $\stackrel{P}{K}{}\!\!_{0,4}^{(+)}$.
  • Figure 4: Expression of $\stackrel{P}{K}{}\!\!_{1,0}$.
  • Figure 5: Expression of $\stackrel{P}{F}{}\!\!_{3,0}^{(+)}$.
  • ...and 17 more figures

Theorems & Definitions (56)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Lemma 1
  • Definition 5
  • Proposition 2
  • Lemma 2
  • Lemma 3
  • ...and 46 more