Spin-foam models and the physical scalar product

Emanuele Alesci, Karim Noui, and Francesco Sardelli
Phys. Rev. D 78, 104009 – Published 7 November 2008

Abstract

This paper aims at clarifying the link between loop quantum gravity and spin-foam models in four dimensions. Starting from the canonical framework, we construct an operator P acting on the space of cylindrical functions Cyl(Γ), where Γ is the four-simplex graph, such that its matrix elements are, up to some normalization factors, the vertex amplitude of spin-foam models. The spin-foam models we are considering are the topological model, the Barrett-Crane model, and the Engle-Pereira-Rovelli model. If one of these spin-foam models provides a covariant quantization of gravity, then the associated operator P should be the so-called “projector” into physical states and its matrix elements should give the physical scalar product. We discuss the possibility to extend the action of P to any cylindrical functions on the space manifold.

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  • Received 24 July 2008

DOI:https://doi.org/10.1103/PhysRevD.78.104009

©2008 American Physical Society

Authors & Affiliations

Emanuele Alesci1,2,*, Karim Noui3,†, and Francesco Sardelli3,4,‡

  • 1Laboratoirede Physique, ENS Lyon, CNRS UMR 5672, 46 Allée d’Italie, 69007 Lyon, France EU
  • 2Centre de Physique Théorique de Luminy§, Université de la Méditerranée, F-13288 Marseille EU
  • 3Laboratoire de Mathématiques et de Physique Théorique∥, Parc de Grammont, 37200 Tours EU
  • 4Dipartimento di Fisica Enrico Fermi, Università di Pisa, 56127 Pisa EU

  • *alesci@fis.uniroma3.it
  • noui@lmpt.univ-tours.fr
  • francesco.sardelli@yahoo.it
  • §Unité mixte de recherche (UMR 6207) du CNRS et des Universités de Provence (Aix-Marseille I), de la Mediterranée (Aix-Marseille II) et du Sud (Toulon-Var); laboratoire affilié à la FRUMAM (FR 2291).
  • UMR/CNRS 6083, Fédération Denis Poisson

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Issue

Vol. 78, Iss. 10 — 15 November 2008

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