Barrett-Crane model

The Barrett-Crane model is a model in quantum gravity which was defined using the Plebanski action. citation
author = Barrett, John W.; Louis Crane
year = 1998
title = Relativistic spin networks and quantum gravity
journal = J.Math.Phys. 39
volume = 39
pages = 3296–3302
doi = 10.1063/1.532254
date = 1998] citation
author = Barrett, John W.; LouisCrane,
title = A Lorentzian signature model for quantum general relativity
journal = Classical and Quantum Gravity
volume = 17
doi = 10.1088/0264-9381/17/16/302]

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

:$B^\{ij\}$

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 $epsilon^\{ijkl\}$ 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 action, 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, if analysed with the tools of loop quantum gravity the Barrett-Crane model gives an incorrect long-distance limit [http://arxiv.org/abs/0708.0883] , and so the model is not identical to loop quantum gravity.

ee also

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