Barrett-Crane model

The Barrett-Crane model is a model in loop quantum gravity which was defined using the Plebanski action. <math>B<math> field in the action is supposed to be a <math>so(3, 1)<math>-valued 2-form. The term <math>B^{ij} \wedge B^{kl}<math> in the action has the same symmetries as it does to provide the Einstein-Hilbert action. But the form of <math>B^{ij}<math> is not unique and can be posed by the different forms:

• <math>\pm e^i \wedge e^j<math>
• <math>\pm \epsilon^{ijkl} e_k \wedge e_l<math>

where <math>e^i<math> field is tetrad's and <math>\epsilon^{ijkl}<math> is antisymmetric symbol of the <math>so(3, 1)<math>-valued 2-form fields.

Plenbanski action can be constrainted to produce BF model which is a theory of no local degrees of freedom. Barrett and Crane modeled the analogus constriant on the summation over spin foam.

The Barrett-Crane model on spin foam quantize the Plebanski theory, but its path integral amplitude responds to the degenerate <math>B<math> feld and not specific definition definition of <math>B^{ij} = e^i \wedge e^j<math>, which satisfied the Einstein's field equation of general relativity.