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, 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

Bij

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 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

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.

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