Spin foam models

Abstract

While the use of spin networks has greatly improved our understanding of the\nkinematical aspects of quantum gravity, the dynamical aspects remain obscure.\nTo address this problem, we define the concept of a `spin foam' going from one\nspin network to another. Just as a spin network is a graph with edges labeled\nby representations and vertices labeled by intertwining operators, a spin foam\nis a 2-dimensional complex with faces labeled by representations and edges\nlabeled by intertwining operators. Spin foams arise naturally as\nhigher-dimensional analogs of Feynman diagrams in quantum gravity and other\ngauge theories in the continuum, as well as in lattice gauge theory. When\nformulated as a `spin foam model', such a theory consists of a rule for\ncomputing amplitudes from spin foam vertices, faces, and edges. The product of\nthese amplitudes gives the amplitude for the spin foam, and the transition\namplitude between spin networks is given as a sum over spin foams. After\nreviewing how spin networks describe `quantum 3-geometries', we describe how\nspin foams describe `quantum 4-geometries'. We conclude by presenting a spin\nfoam model of 4-dimensional Euclidean quantum gravity, closely related to the\nstate sum model of Barrett and Crane, but not assuming the presence of an\nunderlying spacetime manifold.\n