Causal Barrett-Crane model: Measure, coupling constant, Wick rotation, symmetries, and observables

Hendryk Pfeiffer
Phys. Rev. D 67, 064022 – Published 28 March 2003
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Abstract

We discuss various features and details of two versions of the Barrett-Crane spin foam model of quantum gravity: first of the Spin(4)-symmetric Riemannian model and second of the SL(2,C)-symmetric Lorentzian version in which all tetrahedra are spacelike. Recently, Livine and Oriti proposed to introduce a causal structure into the Lorentzian Barrett-Crane model from which one can construct a path integral that corresponds to the causal (Feynman) propagator. We show how to obtain convergent integrals for the 10j symbols and how a dimensionless constant can be introduced into the model. We propose a “Wick rotation” which turns the rapidly oscillating complex amplitudes of the Feynman path integral into positive real and bounded weights. This construction does not yet have the status of a theorem, but it can be used as an alternative definition of the propagator and makes the causal model accessible by standard numerical simulation algorithms. In addition, we identify the local symmetries of the models and show how their four-simplex amplitudes can be reexpressed in terms of the ordinary relativistic 10j symbols. Finally, motivated by possible numerical simulations, we express the matrix elements that are defined by the model, in terms of the continuous connection variables and determine the most general observable in the connection picture. Everything is done on a fixed two complex.

  • Received 12 December 2002

DOI:https://doi.org/10.1103/PhysRevD.67.064022

©2003 American Physical Society

Authors & Affiliations

Hendryk Pfeiffer*

  • Perimeter Institute for Theoretical Physics, 35 King Street N, Waterloo, Ontario, N2J 2W9, Canada
  • Emmanuel College, St Andrew’s Street, Cambridge CB2 3AP, United Kingdom

  • *E-mail address: hpfeiffer@perimeterinstitute.ca

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Vol. 67, Iss. 6 — 15 March 2003

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