Introduction

Suppose the usual description of spacetime as a 3 + 1 manifold breaks down at the scale where a quantum theory of gravity is expected to describe the world, generally agreed to be the Planck scale, lp = 10–35m. Can we still construct sensible theoretical models of the universe? Are they testable? Do they lead to a consistent quantum cosmology? Is this cosmology different than the standard one? The answer is yes, to all these questions, assuming that quantum theory is still valid. After 80 years work on quantum gravity, we do have the first detailed models for the microscopic structure of spacetime: spin foams.

The first spin foam models (Reisenberger 1994, 1997; Markopoulou and Smolin 1997; Reisenberger and Rovelli 1997; Baez 1998) were based on the predictions of loop quantum gravity, namely the quantization of general relativity, for the quantum geometry at Planck scale. A main result of loop quantum gravity is that the quantum operators for spatial areas and volumes have discrete spectra. (Rovelli and Smolin 1995; for a recent detailed review of loop quantum gravity see Thiemann (2001), and for a nontechnical review of the field see Smolin (2001)). Discreteness is central to spin foams, which are discrete models of spacetime at Planck scale.