Immirzi parameter in the Barrett-Crane model?

TL;DR

The paper demonstrates that incorporating the Immirzi parameter into a generalized constrained BF formulation of spin foams does not alter the underlying simple representation content of the Barrett–Crane model. By reexpressing the bivector B with a suitable change of basis in the SO(4) generators, the simplicity constraint remains compatible with simple representations, and the area spectrum can be written without an explicit Immirzi-dependence in the covariant setting. The Immirzi parameter can enter only through a particular canonical quantization choice, not through the covariant spin foam geometry itself. The Lorentzian and Euclidean analyses yield consistent simple representations and show that the Barrett–Crane framework remains robust under this generalization, with the area spectrum either gamma-free (covariant) or gamma-corrected (canonical Holst-type quantization). Overall, the work clarifies how Immirzi-like parameters can be reconciled with covariant spin foams without destabilizing their geometric content.

Abstract

We study the generalised constrained BF theory described in gr-qc/0102073 in order to introduce the Immirzi parameter in spin foam models. We show that the resulting spin foam model is still based on simple representations and that the generalised BF action is simply a deformation of the Barrett-Crane model. The Immirzi parameter doesn't change the representations used in the spin foam model, so it doesn't affect the geometry of the model. However we show how it may still appear as a factor in the area spectrum.