The loop-quantum-gravity vertex-amplitude

TL;DR

This paper introduces a new SO(4)-covariant vertex amplitude for loop quantum gravity, derived from a Regge discretization, addressing limitations of the Barrett-Crane model and improving the boundary state space.

Contribution

It proposes an alternative spin foam model with constraints imposed weakly, enhancing the boundary state space and covariance in loop quantum gravity.

Findings

Provides a covariant vertex amplitude consistent with Euclidean general relativity.

Addresses issues in the Barrett-Crane model related to boundary states.

Aligns the spin network space with the SO(3) Hamiltonian framework.

Abstract

Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy $n$-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity.