Commuting Simplicity and Closure Constraints for 4D Spin Foam Models

TL;DR

This paper introduces a new Euclidean Spin Foam Model derived from the Plebanski-Holst path integral with commuting B fields, addressing simplicity constraints and closure issues, and relating it to existing models like FK and Barrett-Crane.

Contribution

It constructs a novel Spin Foam Model using standard discretisation methods from the Plebanski-Holst action with commuting B fields, clarifying the role of closure constraints.

Findings

Model differs from existing ones in handling closure constraints.

Large spin limit relates the model to FK Model.

Non-commutative deformation connects to Barrett-Crane Model.

Abstract

Spin Foam Models are supposed to be discretised 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 non standard 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 article, we construct a new Euclidian Spin Foam Model which is constructed by standard methods from the Plebanski-Holst path integral with commuting B fields discretised 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 FK Model but even then the face and edge amplitude differ. Curiously, an ad hoc non-commutative deformation of the $B^{IJ}$ variables leads from our new model to the Barrett-Crane Model in the case of Barbero-Immirzi parameter goes to infinity.