this just out http://arxiv.org/abs/0705.2388 The loop-quantum-gravity vertex-amplitude Jonathan Engle, Roberto Pereira, Carlo Rovelli 6 pages "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." important paper ==exerpt== However, the suspicion that something is wrong with the BC model has long been agitated. Its boundary state space is similar, but does not exactly match, that of loop quantum gravity; in particular the volume operator is ill-defined. Worse, recent calculations appear to indicate that some n-point functions fail to yield the correct low-energy limit . All these problems are related to the way the intertwiner quantum numbers (associated to the operators measuring angles between the faces bounding the elementary quanta of space) are treated: These quantum numbers are fully constrained in the BC model by imposing the simplicity constraints as strong operator equations (Cn psi = 0). But these constraints are second class and imposing such constraints strongly may lead to the incorrect elimination of physical degrees of freedom. It is therefore natural to try to implement in 4d the general picture discussed above by correcting the BC model[7, 15]. In this letter we show that this is possible, by properly imposing some of the constraints weakly (<phi, Cn psi> = 0), and that the resulting theory has remarkable features. First, its boundary quantum state space matches exactly the one of SO(3) loop quantum gravity: no degrees of freedom are lost. Second, as the degrees of freedom missing in BC are recovered, the vertex may yield the correct low-energy n-point functions. Third, the vertex can be seen as a vertex over SO(3) spin networks or SO(4) spin networks, and is both SO(3) and SO(4) covariant. Finally, the theory can be obtained as a bona fide quantization of a discretization of euclidean GR on a Regge triangulation... ==endquote== EDIT to reply to Jal without needing an extra post: Hi Jal, I agree with your point about these papers being stimulating reading. Not sure I understand what you mean about the 12 tetrahedra. However in any case the main thing is probably that 5 tetrahedra fit together to make a jacket around a foursimplex. The 3d surface of a foursimplex, in other words, consists of five tets. And 9 tets would form the surface for two adjacent foursimplices. Maybe the 12 tets you mention could be the surface tets of three adjacent 4d simplices. For me the exciting thing is that they may be on the way to fixing the BC vertex amplitude formula. People have been using the Barrett-Crane vertex formula for some 10 years as the provisional basis for SPINFOAM DYNAMICS. A spinfoam is like a "Feynman diagram" for the geometry of spacetime and just like a regular Feynman diagram for electrons and light you need a formula for calculating the amplitude at each vertex (like the probability that whatever the vertex says to happen actually happens). And the BC spinfoam approach has been working pretty well but there were signs that the formula for calculating the amplitude at any given vertex might not be quite right. So Rovelli and friends may have FIXED the formula so it works better. If this is true it would make this the QG Paper of the Year for 2007.