Commuting simplicity and closure constraints for 4D spin-foam models

Abstract

Spin-foam models are supposed to be discretized path integrals for quantum gravity constructed from the Plebanski–Holst action. The reason for there being several models currently under consideration is that no consensus has been reached for how to implement the simplicity constraints. Indeed, none of these models strictly follows from the original path integral with commuting B fields, rather, by some nonstandard manipulations one always ends up with non-commuting B fields and the simplicity constraints become in fact anomalous which is the source for there being several inequivalent strategies to circumvent the associated problems. In this paper, we construct a new Euclidian spin-foam model which is constructed by standard methods from the Plebanski–Holst path integral with commuting B fields discretized on a 4D simplicial complex. The resulting model differs from the current ones in several aspects, one of them being that the closure constraint needs special care. Only when dropping the closure constraint by hand and only in the large spin limit can the vertex amplitudes of this model be related to those of the FKγ model but even then the face and edge amplitude differ. Interestingly, a non-commutative deformation of the BIJ variables leads from our new model to the Barrett–Crane model in the case of γ = ∞.