Positivity of Spin Foam Amplitudes

TL;DR

This paper proves that spin foam amplitudes in the Barrett-Crane model are nonnegative, enabling statistical methods for quantum gravity calculations, and provides evidence that Lorentzian models may share this positivity property.

Contribution

It establishes the positivity of Riemannian spin foam amplitudes and discusses potential positivity in Lorentzian models, facilitating computational approaches in quantum gravity.

Findings

Riemannian 10j symbols are nonnegative or have signs depending on spin sums.

Amplitudes for closed spin foams are always nonnegative.

Numerical evidence suggests Lorentzian 10j symbols may also be nonnegative.

Abstract

The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. As a corollary, all open spin foams going between a fixed pair of spin networks have real amplitudes of the same sign. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian Barrett-Crane model, as in statistical mechanics, even though this theory is based on a real-time (e^{iS}) rather than imaginary-time (e^{-S}) path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or half-integer. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative, which would imply similar results for the Lorentzian Barrett-Crane model.