Canonical "Loop" quantum gravity and spin foam models

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 of the most meaningful results of the theory are: 1) the complete calculation of the spectrum of geometric quantities like area and volume and the consequent physical predictions about the structure of space-time at the Planck 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 Weller-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 sense 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 dimensions admits a spin foam formulation. We argue that the spin-foam/spin-network formalism it is the natural framework in which to discuss loop quantum gravity and topological field theory.