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 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.