Barrett-Crane model
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The Barrett-Crane model is a model in loop quantum gravity which was defined using the Plebanski action.
The B field in the action is supposed to be a so(3,1)-valued 2-form. The term
in the action has the same symmetries as it does to provide the Einstein-Hilbert action. But the form of Bij is not unique and can be posed by the different forms:
where ei field is tetrads and εijkl is antisymmetric symbol of the so(3,1)-valued 2-form fields.
The Plebanski action can be constrained to produce the BF model which is a theory of no local degrees of freedom. John W. Barrett and Louis Crane modeled the analogous constraint on the summation over spin foam.
The Barrett-Crane model on spin foam quantizes the Plebanski theory, but its path integral amplitude corresponds to the degenerate B field and not the specific definition
,
which satisfies the Einstein's field equation of general relativity.
