Vector fields, flows and Lie groups of diffeomorphisms

Abstract. The freedom in choosing finite renormalizations in quantum field theories (QFT) is characterized by a set of parameters $\{c_i \}, i = 1 \ldots, n \ldots$, 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.