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.