Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 008, 22 pages      arXiv:1207.4596
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

The Construction of Spin Foam Vertex Amplitudes

Eugenio Bianchi a and Frank Hellmann b
a) Perimeter Institute for Theoretical Physics, Canada
b) Max Planck Institute for Gravitational Physics (AEI), Germany

Received July 20, 2012, in final form January 27, 2013; Published online January 31, 2013

Spin foam vertex amplitudes are the key ingredient of spin foam models for quantum gravity. These fall into the realm of discretized path integral, and can be seen as generalized lattice gauge theories. They can be seen as an attempt at a 4-dimensional generalization of the Ponzano-Regge model for 3d quantum gravity. We motivate and review the construction of the vertex amplitudes of recent spin foam models, giving two different and complementary perspectives of this construction. The first proceeds by extracting geometric configurations from a topological theory of the BF type, and can be seen to be in the tradition of the work of Barrett, Crane, Freidel and Krasnov. The second keeps closer contact to the structure of Loop Quantum Gravity and tries to identify an appropriate set of constraints to define a Lorentz-invariant interaction of its quanta of space. This approach is in the tradition of the work of Smolin, Markopoulous, Engle, Pereira, Rovelli and Livine.

Key words: spin foam models; discrete quantum gravity; generalized lattice gauge theory.

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