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A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements

Marcus Gaul, Carlo Rovelli

Abstract

We study a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity. The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken. This leads to a quantization ambiguity and to a family of operators with the same classical limit. We calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of Temperley-Lieb recoupling theory. We discuss the relation between this generalization of the Hamiltonian constraint and crossing symmetry.

A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements

Abstract

We study a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity. The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken. This leads to a quantization ambiguity and to a family of operators with the same classical limit. We calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of Temperley-Lieb recoupling theory. We discuss the relation between this generalization of the Hamiltonian constraint and crossing symmetry.

Paper Structure

This paper contains 26 sections, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The generalized Hamiltonian
  3. Classical Theory
  4. Quantum Theory
  5. A brief note on the quantization ambiguity
  6. The action of $\mathcal{H}^m$ on trivalent vertices
  7. Remarks on the Computational Tools
  8. Evaluating the action of $\hat{\mathcal{H}}^m$ on a single 3-vertex
  9. The action of $\hat{V}$
  10. Completing the action of $\hat{\mathcal{H}}^m$
  11. The final result for the action of $\hat{\mathcal{H}}^m_{\Delta}$
  12. Conclusions and Outlook
  13. Acknowledgements.
  14. The $m=1$ case: Thiemann's Hamiltonian
  15. The volume operator for $m=1$
  16. ...and 11 more sections

Figures (3)

  • Figure 1: An elementary tetrahedron $\Delta \in T$ constructed by adapting it to a graph $\gamma$ which underlies a cylindrical function.
  • Figure 2: The graphical representation of the part of the spin network state containing the vertex $| v(p,q,r) \rangle$. Only the region around the vertex, i.e. its adjacent edges, are shown.
  • Figure 3: The vertex operator $\hat{W}_{[abc]}$ 'grasps' the indicated edges of a 4-valent vertex (the edges are labelled by their colors). To avoid sign confusion, the three graspings have to be performed 'on the same side' of the involved edges. The last term illustrates a virtual trivalent decomposition of the intertwiner.