**14**, 705-708

DOI 10.1007/s100520000375

## Vector fields, flows and Lie groups of diffeomorphisms

**A. Peterman**

Theoretical Physics Division, CERN, 1211 Geneva 23, Switzerland

Received: 25 February 2000 / Published online: 18 May 2000
- © Springer-Verlag 2000

*To the memory of G. de Rham, my teacher in mathematics.
*

**Abstract**

The freedom in choosing finite renormalizations in quantum field theories (QFT) is
characterized by a set of parameters
,
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
can indeed be shown to satisfy the functional equation

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.

Copyright Società Italiana di Fisica, Springer-Verlag 2000