Lagrangian approach to the Barrett-Crane spin foam model
- Centre de Physique Theorique, CNRS UMR 6207, Luminy Case 907, 13007 Marseille (France)
- Laboratoire de Physique, ENS Lyon, CNRS UMR 5672, 46 Allee d'Italie, 69007 Lyon (France)
We provide the Barrett-Crane spin foam model for quantum gravity with a discrete action principle, consisting in the usual BF term with discretized simplicity constraints which in the continuum turn topological BF theory into gravity. The setting is the same as usually considered in the literature: space-time is cut into 4-simplices, the connection describes how to glue these 4-simplices together and the action is a sum of terms depending on the holonomies around each triangle. We impose the discretized simplicity constraints on disjoint tetrahedra and we show how the Lagrange multipliers distort the parallel transport and the correlations between neighboring simplices. We then construct the discretized BF action using a noncommutative * product between SU(2) plane waves. We show how this naturally leads to the Barrett-Crane model. This clears up the geometrical meaning of the model. We discuss the natural generalization of this action principle and the spin foam models it leads to. We show how the recently introduced spin foam fusion coefficients emerge with a nontrivial measure. In particular, we recover the Engle-Pereira-Rovelli spin foam model by weakening the discretized simplicity constraints. Finally, we identify the two sectors of Plebanski's theory and we give the analog of the Barrett-Crane model in the nongeometric sector.
- OSTI ID:
- 21260168
- Journal Information:
- Physical Review. D, Particles Fields, Vol. 79, Issue 6; Other Information: DOI: 10.1103/PhysRevD.79.064034; (c) 2009 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 0556-2821
- Country of Publication:
- United States
- Language:
- English
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