From Riemann to Feynman geometry in Feynman approach to QFT

In the Feynmann approach to QFT correlators are given as an integrals over the space of fields. Since the space of fields is infinite-dimensional the integral has no mathematical sense, and in practical applications this definition is accompanied by the special rules of computation of the integral. These rules (called renormalization procedures) differ from theory to theory, and people are not even trying to prove that different rules lead to equivalent definitions.
We propose the following radical change of the situation. The Riemannian geometry with infinite-dimensional space of fields should be replaced by so-called Feymnan geometry where the space is replaced by an A-infinity structure that is either finite dimensional(strong Feymnan geometry) or infinite-dimensional with operations belong to the trace-class (weekly Feynman) geometry. We give an example of such geometries: lattice A-infinity geometry and noncommutative fuzzy geometry as examples of strongly Feynmann geometries, and Costello A-infinity geometry as an example of weekly Feynman geometry, and discuss a program of reformulation of QFT, in which standard infinite dimensional integral should be replaced by a limit of the existing integrals over fields in Feynman geometries.
Joint work with Sen Hu, USTC