Diffeomorphism invariant Quantum Field Theories of Connections in terms of webs

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In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise smooth finite paths and loops. In particular, we (i) characterize the spectrum of the Ashtekar-Isham configuration space, (ii) introduce spin-web states, a generalization of the spin-network states, (iii) extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism invariant states and finally (iv) extend the 3-geometry operators and the Hamiltonian operator.