in the action has the same symmetries as it does to provide the Einstein–Hilbert action. But the form of
is not unique and can be posed by the different forms:
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.
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 , and so the model is not identical to loop quantum gravity.
- Barrett, John W.; Louis Crane (1998), "Relativistic spin networks and quantum gravity", J.Math.Phys. 39 39: 3296–3302, arXiv:gr-qc/9709028, Bibcode:1998JMP....39.3296B, doi:10.1063/1.532254
- Barrett, John W.; Louis Crane, "A Lorentzian signature model for quantum general relativity", Classical and Quantum Gravity 17, arXiv:gr-qc/9904025, Bibcode:2000CQGra..17.3101B, doi:10.1088/0264-9381/17/16/302
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