CHEN INTEGRALS, GENERALIZED LOOPS AND LOOP CALCULUS

We use Chen iterated line integrals to construct a topological algebra A(p) of separating functions on the group of loops LM(p). A(p) has a Hopf algebra structure which allows the construction of a group struct ure on its spectrum. We call this topological group the group of gener alized loops LM(p). Then we develop a loop calculus, based on the end point and area derivative operators, providing a rigorous mathematical treatment of the early heuristic ideas of Gambini, Trias and also Man delstam, Makeenko and Migdal. Finally, we define a natural action of t he ''pointed'' diffeomorphism group Diff(p)(M) on LM(p), and consider a variational derivative which allows the construction of homotopy inv ariants. This formalism is useful for constructing a mathematical theo ry of loop representation of gauge theories and quantum gravity.