According to the principle of locality in physics, events taking place at different locations should behave independently, a feature expected to be reflected in the measurements. In quantum field theory, measuring often requires to renormalise and one expects renormalisation to preserve locality. However, a “naïve finite part procedure” based on commonly used regularisation methods, such as dimensional regularisation, does not do the job, so a “naïve” regularisation is not a priori compatible with the locality principle. In order to preserve locality while renormalising, physicists have developped sophisticated methods such as the forest formula which was later revisited by Connes and Kreimer using a coalgebraic approach.
We shall show how one can implement a “naïve finite part procedure” provided one uses a multivariate regularisation instead of a univariate one such as the commonly used dimensional regularisation. A multivariate regularisation yields a way to keep track of independence of events reflected in the fact that the corresponding meromorphic functions in several variables involve independent sets of variables. This multivariate renormalisation scheme, which takes place on the target space of meromorphic germs, can be applied to various situations involving renormalisation, such as Feynman integrals, multizeta functions and their generalisations, namely discrete sums on cones and discrete sums associated with trees.
This talk is based on joint work with Pierre Clavier, Li Guo and Bin Zhang.