Testing the imposition of the Spin Foam Simplicity Constraints

Abstract

We introduce a three-dimensional Plebanski action for the gauge group SO(4). In this model, the $B$ field satisfies quadratic simplicity constraints similar to that of the four-dimensional Plebanski theory, but with the difference that the $B$ field is now a one-form. We exhibit a natural notion of "simple one-form", and identify a gravitational sector, a topological sector and a degenerate sector in the space of solutions to the simplicity constraints. Classically, in the gravitational sector, the action is shown to be equivalent to that of three-dimensional first order Riemannian gravity. This enables us to perform the complete spin foam quantization of the theory once the simplicity constraints are solved at the classical level, and to compare this result with the various models that have been proposed for the implementation of the constraints after quantization. In particular, we impose the simplicity constraints following the prescriptions of the so-called BC and EPRL models. We observe that the BC prescription cannot lead to the proper vertex amplitude. The EPRL prescription allows to recover the expected result when, in this three-dimensional model, it is supplemented with additional secondary second class constraints.

30 pages. 18 figures. References added. New paragraph in the conclusion

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• Super-Group Field Cosmology
• Testing the role of the Barbero-Immirzi parameter and the choice of connection in Loop Quantum Gravity
• Loop Quantum Cosmology with Complex Ashtekar Variables
• Noether's Charge in the Super-Group Field Cosmology
• A note on the secondary simplicity constraints in loop quantum gravity
• Spectra of geometric operators in three-dimensional LQG: From discrete to continuous
• Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators
• Hamiltonian dynamics of an exotic action for gravity in three dimensions
• Degenerate Plebanski Sector and Spin Foam Quantization
• Towards self dual Loop Quantum Gravity
• Simplicity constraints: a 3d toy-model for Loop Quantum Gravity
• Loop Quantization of a Model for D=1+2 (Anti)de Sitter Gravity Coupled to Topological Matter