Vector fields, flows and Lie groups of diffeomorphisms

The freedom in choosing finite renormalizations in quantum field theories ( QFT) is characterized by a set of parameters {q}, i = 1..., n...., which sp ecify the renormalization prescriptions used for the calculation of physica l quantities. For the sake of simplicity, the case of a single c is selecte d and chosen mass-independent if masslessness is not realized, this with th e aim of expressing the effect of an infinitesimal change in c on the compu ted observables. This change is found to be expressible in terms of an equa tion involving a vector field V on the action's space M (coordinates x). Th is equation is often referred to as "evolution equation" in physics. This v ector field generates a one-parameter (here c) group of diffeomorphisms on M. Its how sigma(c)(x) can indeed be shown to satisfy the functional equati on sigma(c+t)(x) = sigma(c)(sigma(t)(x)) = sigma(c) o sigma(t) sigma(v)(x) = x, so that the very appearance of V in the evolution equation implies at once the Gell-Mann-Low functional equation. The latter appears therefore as a tr ivial consequence of the existence of a vector field on the action's space of renormalized QFT.