Relating spin foams and canonical quantum gravity: (n-1)+1 formulation of nD spin foams

In this article we lay foundations for a formal relationship of spin foam models of gravity and BF theory to their continuum canonical formulations. First the derivation of the spin foam model of the BF theory from the discrete BF theory action in n dimensions is reviewed briefly. By foliating the underlying n dimensional simplicial manifold using 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. Quantum Transition amplitudes are defined. We calculate the transition amplitudes related to 2D BF theory explicitly. The application to the spin foam models of gravity is discussed briefly.