Chen Integrals , Generalized Loops and Loop Calculus

We use Chen iterated line integrals to construct a topological algebra Ap of separating functions on the Group of Loops LMp. Ap has an Hopf algebra structure which allows the construction of a group structure on its spectrum. We call this topological group, the group of generalized loops L̃Mp. Then we develope a Loop Calculus, based on the Endpoint and Area Derivative Operators, providing a rigorous mathematical treatment of early heuristic ideas of Gambini, Trias and also Mandelstam, Makeenko and Migdal. Finally we define a natural action of the ”pointed” diffeomorphism group Diffp(M) on L̃Mp, and consider a Variational Derivative which allows the construction of homotopy invariants. This formalism is useful to construct a mathematical theory of Loop Representation of Gauge Theories and Quantum Gravity.