Spin Foam in Loop Quantum Gravity
Loop Quantum Gravity has a covariant formulation that provides at present the best formulation of the dynamics of the theory. This is a Quantum Field Theory where the invariance under diffeomorphisms of general relativity is implemented. The resulting path integral represents a sum over all the possible configuration of the geometry, coded in the spinfoam. A spin network is defined as a diagram (like the Feynman diagram) that makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them. Spin networks provide a representation for computations of amplitudes between two different hypersurfaces of the manifold. Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam may be viewed as a quantum history.
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