Considerations on some algebraic properties of Feynman integrals - Lucian M. Ionescu (Surveys in Mathematics and its Applications, 2008)
Some algebraic properties of integralsover configuration spaces are investigated in order to better understandquantization and the Connes-Kreimer algebraic approach to renormalization. In order to isolate the mathematical-physics interface toquantum field theory independent from the specifics of the variousimplementations, the sigma model of Kontsevich is investigated in moredetail. Due to the convergence of the configuration space integrals, themodel allows to study the Feynman rules independently, from an axiomaticpoint of view, avoiding the intricacies of renormalization, unavoidablewithin the traditional quantum field theory. As an application, a combinatorial approach to constructingthe coefficients of formality morphisms is suggested, as an alternative tothe analytical approach used by Kontsevich. These coefficients are "Feynman integrals", although not quite typical since they do converge. A second example of "Feynman integrals", defined asstate-sum model, is investigated. Integration is understood here as formalcategorical integration, or better as a duality structure on thecorresponding category. The connection with a related TQFT is mentioned,supplementing the Feynman path integral interpretation of Kontsevichformula. A categorical formulation for the Feynman path integralquantization is sketched, towards Feynman Processes, i.e. representations ofdg-categories with duality, thought of as complexified Markov processes.