Canonical “Loop” Quantum Gravity and Spin Foam Models

Abstract

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.KeywordsHilbert SpaceQuantum GravityLoop Quantum GravityHamiltonian ConstraintSpin FoamThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.