  The freedom in choosing finite renormalizations in quantum field theories
(QFT) is characterized by a set of parameters $\{c_i \}, i = 1 ..., n >...$,
which specify the renormalization prescriptions used for the calculation of
physical quantities. For the sake of simplicity, the case of a single $c$ is
selected and chosen mass-independent if masslessness is not realized, this with
the aim of expressing the effect of an infinitesimal change in $c$ on the
computed observables. This change is found to be expressible in terms of an
equation involving a vector field $V$ on the action's space $M$ (coordinates
x). This equation is often referred to as ``evolution equation'' in physics.
This vector field generates a one-parameter (here $c$) group of diffeomorphisms
on $M$. Its flow $\sigma_c (x)$ can indeed be shown to satisfy the functional
equation $$ \sigma_{c+t} (x) = \sigma_c (\sigma_t (x)) \equiv \sigma_c \circ
\sigma_t $$ $$\sigma_0 (x) = x,$$ 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 trivial consequence of the existence of a vector
field on the action's space of renormalized QFT.
