Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity

TL;DR

This paper proves that Lorentzian Barrett-Crane quantum gravity models have non-negative amplitudes for all closed 4-manifold triangulations, enabling more efficient numerical simulations and suggesting no interference between different triangulations.

Contribution

It demonstrates the non-negativity of amplitudes in Lorentzian Barrett-Crane models by transforming them into a dual variables formulation involving hyperbolic space integrals.

Findings

Amplitudes are non-negative for various model choices.

The dual variables formulation reveals non-negativity of the integrand.

Facilitates the use of statistical methods for numerical computations.

Abstract

The Barrett-Crane models of Lorentzian quantum gravity are a family of spin foam models based on the Lorentz group. We show that for various choices of edge and face amplitudes, including the Perez-Rovelli normalization, the amplitude for every triangulated closed 4-manifold is a non-negative real number. Roughly speaking, this means that if one sums over triangulations, there is no interference between the different triangulations. We prove non-negativity by transforming the model into a ``dual variables'' formulation in which the amplitude for a given triangulation is expressed as an integral over three copies of hyperbolic space for each tetrahedron. Then we prove that, expressed in this way, the integrand is non-negative. In addition to implying that the amplitude is non-negative, the non-negativity of the integrand is highly significant from the point of view of numerical computations, as it allows statistical methods such as the Metropolis algorithm to be used for efficient computation of expectation values of observables.