Cern-th/99-389 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 {ci}, 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 σc(x) can indeed be shown to satisfy the functional equation σc+t(x) = σc(σt(x)) ≡ σc ◦ σt σ0(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. CERN-TH/99-389 December 1999 To the memory of G. de Rham, my teacher in mathematics. The so-called “Renormalization Group” (RG) in physical science was, from its early beginnings [1], the theory that describes the geometry of action space. In this space the covariance of physical quantities turns out to be manifest. This paper is intended to be an overview of the RG as it is used by physicists, especially in Quantum Field Theory (QFT). However, the emphasis will be put on the geometry (as said before) of the space of actions, with a view, among other things, on how a functional equation, derived in [2], becomes a simple and trivial consequence of the geometrical set-up developed below by means of QFT manipulations. Of paramount importance has been the discovery [1] that renormalized physical quantities satisfy equations in which the basic geometrical object is a vector field defined in the action space M . Let V be this vector field. Then the theory of differentiable manifolds [3] implies, due to the very existence of a vector field, a set of theorems, lemmas and corollaries which exhausts all that can be said about the renormalization group in physical applications. In general, one deals with a set of parameters {ci} but, in the following, we shall restrict this set to a single parameter, denoted t for practical reasons. This restriction is mainly dictated by the fact that one wants to be able to make direct comparisons with Ref. [2] which uses a single parameter in its fixing of renormalization prescriptions. Thus, in the single parameter case, we have amongst others the following theorems. Theorem I. A smooth vector field V on a compact manifold M generates a one-parameter group of diffeomorphisms of M . Theorem II. Suppose V is a C vector field on the manifold M , then for every x ∈ M , there exists an integral curve of V , t → σ(t, x) such that 1. σ(t, x) is defined for t belonging to an interval I(x)cR, containing t = 0 and is of class C there. 2. σ(0, x) = x for every x ∈ M 3. Given x ∈ M , there is no C integral curve of V defined on an interval properly containing I(x), and passing through x (i.e. such that σ(0, x) = x). From the uniqueness property 3, follows at once Theorem III. If s, t and s + t ∈ I(x), then we have the functional equation σ(s + t, x) = σ(t, σ(s, x)) . (1)