Hilbert space structure of covariant loop quantum gravity

Sergei Alexandrov
Phys. Rev. D 66, 024028 – Published 22 July 2002
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Abstract

We investigate Hilbert space in the Lorentz covariant approach to loop quantum gravity. We restrict ourselves to the space where all area operators are simultaneously diagonalizable, assuming that it exists. In this sector, quantum states are realized by a generalization of spin network states based on Lorentz Wilson lines projected on irreducible representations of an SO(3) subgroup. The problem of infinite dimensionality of the unitary Lorentz representations is absent due to this projection. Nevertheless, the projection preserves the Lorentz covariance of the Wilson lines so that the symmetry is not broken. Under certain conditions, the states can be thought of as functions on a homogeneous space. We define the inner product as an integral over this space. With respect to this inner product, the spin networks form an orthonormal basis in the investigated sector. We argue that it is the only relevant part of a larger state space arising in the approach. The problem of the noncommutativity of the Lorentz connection is solved by restriction to the simple representations. The resulting structure shows similarities with the spin foam approach.

  • Received 1 February 2002

DOI:https://doi.org/10.1103/PhysRevD.66.024028

©2002 American Physical Society

Authors & Affiliations

Sergei Alexandrov*

  • Service de Physique Théorique, C.E.A.–Saclay, 91191 Gif-sur-Yvette Cedex, France
  • Laboratoire de Physique Théorique de l’École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France

  • *Also at V.A. Fock Department of Theoretical Physics, St. Petersburg University, Russia. Email address: alexand@spht.saclay.cea.fr

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Vol. 66, Iss. 2 — 15 July 2002

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