On the causal Barrett--Crane model: measure, coupling constant, Wick rotation, symmetries and observables

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 space-like. 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 re-expressed 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.

22 pages, LaTeX 2e, 1 figure

References in corpus (8)

• Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity
• Diffeomorphisms and spin foam models
• Asymptotics of Relativistic Spin Networks
• Spin Foam Models of Riemannian Quantum Gravity
• Positivity of Spin Foam Amplitudes
• Dual variables and a connection picture for the Euclidean Barrett-Crane model
• Relating Covariant and Canonical Approaches to Triangulated Models of Quantum Gravity
• Exact duality transformations for sigma models and gauge theories

Cited by in corpus (6)

• Spin Foam Models for Quantum Gravity
• Spin foam models as energetic causal sets
• The Conformal Method and the Conformal Thin-Sandwich Method Are the Same
• Initial data for perturbed Kerr black holes on hyperboloidal slices
• Causal structure in spin-foams
• Finiteness of Lorentzian 10j symbols and partition functions