The canonical ``loop'' formulation of quantum gravity is a mathematically
well defined, background independent, non perturbative standard quantization of
Einstein's theory of General Relativity. Some among the most meaningful results
of the theory are: 1) the complete calculation of the spectrum of geometric
quantities like the area and the volume and the consequent physical predictions
about the structure of the space-time at the Plank scale; 2) a microscopical
derivation of the Bekenstein-Hawking black-hole entropy formula. Unfortunately,
despite recent results, the dynamical aspect of the theory (imposition of the
Wheller-De Witt constraint) remains elusive.
After a short description of the basic ideas and the main results of loop
quantum gravity we show in which sence the exponential of the super Hamiltonian
constraint leads to the concept of spin foam and to a four dimensional
formulation of the theory. Moreover, we show that some topological field
theories as the BF theory in 3 and 4 dimension admits a spin foam formulation.
We argue that the spin-foams/spin-networks formalism it is the natural
framework to discuss loop quantum gravity and topological field theory.Comment: 17 pages, LaTeX2e, 7 figures. To appear in the proceeding of the
XXIII SIGRAV conference, Monopoli (ITALY), September 21st-25th, 1998. Minor
correction
