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

TL;DR

This paper analyzes two versions of the Barrett-Crane spin foam model for quantum gravity, introducing a causal structure, a Wick rotation, and exploring symmetries and observables to facilitate numerical simulations.

Contribution

It proposes a new Wick rotation and a causal structure for the Lorentzian Barrett-Crane model, enabling convergent integrals and standard numerical methods.

Findings

Convergent integrals for 10j-symbols are obtained.

A Wick rotation transforms oscillatory amplitudes into positive weights.

Local symmetries and observables are characterized in the model.

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.