Lorentz covariance of loop quantum gravity

Carlo Rovelli and Simone Speziale
Phys. Rev. D 83, 104029 – Published 17 May 2011

Abstract

The kinematics of loop gravity can be given a manifestly Lorentz-covariant formulation: the conventional SU(2)-spin-network Hilbert space can be mapped to a space K of SL(2,C) functions, where Lorentz covariance is manifest. K can be described in terms of a certain subset of the projected spin networks studied by Livine, Alexandrov and Dupuis. It is formed by SL(2,C) functions completely determined by their restriction on SU(2). These are square-integrable in the SU(2) scalar product, but not in the SL(2,C) one. Thus, SU(2)-spin-network states can be represented by Lorentz-covariant SL(2,C) functions, as two-component photons can be described in the Lorentz-covariant Gupta-Bleuler formalism. As shown by Wolfgang Wieland in a related paper, this manifestly Lorentz-covariant formulation can also be directly obtained from canonical quantization. We show that the spinfoam dynamics of loop quantum gravity is locally SL(2,C)-invariant in the bulk, and yields states that are precisely in K on the boundary. This clarifies how the SL(2,C) spinfoam formalism yields an SU(2) theory on the boundary. These structures define a tidy Lorentz-covariant formalism for loop gravity.

  • Figure
  • Received 7 January 2011

DOI:https://doi.org/10.1103/PhysRevD.83.104029

© 2011 American Physical Society

Authors & Affiliations

Carlo Rovelli and Simone Speziale

  • Centre de Physique Théorique de Luminy*, Case 907, F-13288 Marseille, France

  • *Unité mixte de recherche du CNRS et des Universités de Aix-Marseille I, Aix-Marseille II et Toulon-Var; affilié à la FRUMAM

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Issue

Vol. 83, Iss. 10 — 15 May 2011

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