Positivity of Spin Foam Amplitudes
John C. Baez, J. Daniel Christensen
TL;DR
This work investigates whether spin foam amplitudes in the Barrett–Crane model are nonnegative, addressing the challenge of oscillatory real-time path integrals of the form $e^{iS}$ and enabling Metropolis sampling by leveraging positivity. Focusing on the Riemannian Barrett–Crane model, the vertex amplitudes are given by products of $10j$ symbols, and the authors prove that closed spin foams have nonnegative amplitudes because the sign of each $10j$ symbol is fixed by parity and cancels across vertices; a corollary extends to open foams between fixed spin networks having amplitudes of a uniform sign. They introduce a sign-free modified Barrett–Crane intertwiner to obtain a sum-of-squares representation via a real $15j$ symbol, ensuring positivity whenever nonzero and yielding a robust open-foam corollary; they also discuss conjectural Lorentzian positivity supported by numerics. For the Lorentzian model, the lack of the Riemannian miracles prevents a proof, but numerical evidence using Monte Carlo and Vegas integrators for the Lorentzian $10j$ symbols suggests nonnegativity in tested regimes, motivating further analytic progress. Overall, the results make the Riemannian Barrett–Crane model accessible to stochastic methods and motivate deeper exploration of positivity and its physical implications in quantum gravity.
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.