Consistent Discretization and Loop Quantum Geometry

Rodolfo Gambini and Jorge Pullin
Phys. Rev. Lett. 94, 101302 – Published 18 March 2005

Abstract

We apply the “consistent discretization” approach to general relativity leaving the spatial slices continuous. The resulting theory is free of the diffeomorphism and Hamiltonian constraints, but one can impose the diffeomorphism constraint to reduce its space of solutions and the constraint is preserved exactly under the discrete evolution. One ends up with a theory that has as physical space what is usually considered the kinematical space of loop quantum geometry, given by diffeomorphism invariant spin networks endowed with appropriate rigorously defined diffeomorphism invariant measures and inner products. The dynamics can be implemented as a unitary transformation and the problem of time explicitly solved or at least reduced to a numerical problem. We exhibit the technique explicitly in (2+1)-dimensional gravity.

  • Received 20 September 2004

DOI:https://doi.org/10.1103/PhysRevLett.94.101302

©2005 American Physical Society

Authors & Affiliations

Rodolfo Gambini1 and Jorge Pullin2

  • 1Instituto de Física, Facultad de Ciencias, Universidad de la República Iguá esq. Mataojo, CP 11400 Montevideo, Uruguay
  • 2Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA

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Issue

Vol. 94, Iss. 10 — 18 March 2005

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