Boucles et Mousses de Spin en Gravite Quantique

TL;DR

This paper reviews loop quantum gravity and spin foam theory, highlighting the discreteness of timelike intervals and the continuous nature of spacelike distances at the quantum level, with extensions to four dimensions and implications for general relativity.

Contribution

It provides a detailed analysis of the formalism in three and four dimensions, including the Barrett-Crane model and its connection to canonical loop gravity, emphasizing the quantum properties of spacetime geometry.

Findings

Discreteness of timelike intervals at quantum level

Continuity of spacelike distances even quantum mechanically

Connection between spin foam models and canonical loop gravity

Abstract

I review the formalism of loop quantum gravity, in both its real and complex formulations, and spin foam theory which is its path integral counterpart. Spin networks for non-compact groups are introduced (following hep-th/0205268) to deal with gauge invariant structures based on the Lorentz group. The whole formalism is studied in details in three dimensions in both its canonical formulation (loop gravity) and its spin foam formulation. The main output (following gr-qc/0212077) is the discreteness of timelike intervals and the continuous character of spacelike distances even at the quantum level. Then it is explained how to extend these considerations to the 4-dimensional case. I review the Barrett-Crane model, its geometrical interpretation, its link with general relativity and the role of causality. It is shown to be the history formulation of a covariant canonical formulation of loop gravity (following gr-qc/0209105), whose link with standard loop quantum gravity is discussed. Similarly to the 3d case, spacelike areas turn out continuous. Finally, ways of extracting informations from the non-perturbative spin foam structures are discussed.