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

SIGMA 8 (2012), 055, 79 pages      arXiv:1112.1961
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Spin Foams and Canonical Quantization

Sergei Alexandrov a, b, Marc Geiller c and Karim Noui d, c
a) Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095, Montpellier, France
b) CNRS, Laboratoire Charles Coulomb UMR 5221, F-34095, Montpellier, France
c) Laboratoire APC, Université Paris Diderot Paris 7, 75013 Paris, France
d) LMPT, Université François Rabelais, Parc de Grandmont, 37200 Tours, France

Received January 30, 2012, in final form August 12, 2012; Published online August 19, 2012

This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the three-dimensional context, where the two approaches are in good agreement, we show how the canonical quantization à la Witten of Riemannian gravity with a positive cosmological constant is related to the Turaev-Viro spin foam model, and how the Ponzano-Regge amplitudes are related to the physical scalar product of Riemannian loop quantum gravity without cosmological constant. In the four-dimensional case, we recall a Lorentz-covariant formulation of loop quantum gravity using projected spin networks, compare it with the new spin foam models, and identify interesting relations and their pitfalls. Finally, we discuss the properties which a spin foam model is expected to possess in order to be consistent with the canonical quantization, and suggest a new model illustrating these results.

Key words: spin foam models; loop quantum gravity; canonical quantization.

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