Spin Foams and Canonical Quantization

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2012
• ISSN: 1815-0659
• Main Authors: Alexandrov, S., Geiller, M., Noui, K.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2012
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/148408
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Spin Foams and Canonical Quantization / S. Alexandrov, M. Geiller, K. Noui // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 150 назв. — англ.

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.