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 $s o (3,1)$ -valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term

$B^{ij} \wedge B^{kl}$

in the action has the same symmetries as it does to provide the Einstein-Hilbert action. But the form of

B i j

is not unique and can be posed by the different forms:

$\pm e^i \wedge e^j$
$\pm \epsilon^{ijkl} e_k \wedge e_l$

where $e i$ field is tetrads and $ε i j k l$ is antisymmetric symbol of the $s o (3,1)$ -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

$B^{ij} = e^i \wedge e^j$ ,

which formally satisfies the Einstein's field equation of general relativity. However, the Barrett-Crane vertex is known to give an incorrect long-distance limit [1] and cannot be used a model of physics.