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The projector on physical states in loop quantum gravity

Carlo Rovelli

Abstract

We construct the operator that projects on the physical states in loop quantum gravity. To this aim, we consider a diffeomorphism invariant functional integral over scalar functions. The construction defines a covariant, Feynman-like, spacetime formalism for quantum gravity and relates this theory to the spin foam models. We also discuss how expectation values of physical quantity can be computed.

The projector on physical states in loop quantum gravity

Abstract

We construct the operator that projects on the physical states in loop quantum gravity. To this aim, we consider a diffeomorphism invariant functional integral over scalar functions. The construction defines a covariant, Feynman-like, spacetime formalism for quantum gravity and relates this theory to the spin foam models. We also discuss how expectation values of physical quantity can be computed.

Paper Structure

This paper contains 14 sections, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Loop quantum gravity
  3. Hilbert space and spin networks basis
  4. Diffeomorphisms
  5. A diffeomorphism invariant measure
  6. Dynamics: the regularized propagator
  7. The Hamiltonian constraint: first version
  8. The Hamiltonian constraint: second version
  9. Expansion
  10. Interpretation: spin foam
  11. Physical observables
  12. Quantum ADM surfaces
  13. 4d diff-invariant observables can be localized in time
  14. Conclusions

Figures (3)

  • Figure 1: Action of the hamiltonian constraint on a trivalent node.
  • Figure 2: The elementary vertex.
  • Figure 3: A term of second order.