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In this article a discrete evolution formulation of the spin foam models of gravity and BF theory is presented. This work motivated by a desire to relate spin foams to their canonical formulation . We have tried to make this article as self-contained as possible. First the derivation of the spin foam model of BF theory from the discrete BF theory action in n dimensions is reviewed brielfy. By foliating the underlying n dimensional simplicial manifold using the n-1 dimensional simplicial hypersurfaces, the spin foam model is reformulated. Then it is shown that spin network functionals arise naturally on the foliations. The graphs of these spin network functionals are dual to the triangulations of the foliating hypersurfaces. A transition amplitude is defined in a discrete connection picture using path integral formulation. An elementary transition amplitude is defined in the spin-network picture. We calculate the elementary transition amplitudes in 2D BF theory explicitly. The application to the spin foam models of gravity is discussed briefly. The main result is that, we formulate an approach that is intermediate between the canonical and the spin foam formulations. We believe our formulation brings the spin foam models as close as possible to the canonical quantum formulation without introducing any approximations

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

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Relating spin foams and canonical quantum gravity:(n−1)+1formulation ofnDspin foams