Spin Foams and Canonical Quantization

TL;DR

This review examines the consistency between spin foam and canonical loop quantum gravity approaches in 3D and 4D, highlighting their relations, differences, and proposing criteria for compatible models.

Contribution

It provides a comprehensive comparison of spin foam and canonical quantization methods, introduces a new model, and clarifies their interrelations in various dimensions.

Findings

In 3D, the approaches are in good agreement and related to known models.

In 4D, the paper compares covariant and canonical formulations, identifying key relations.

Proposes criteria for spin foam models to be consistent with canonical quantization.

Abstract

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 \`a 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.