Vector Fields, Flows and Lie Groups of Diffeomorphisms

TL;DR

This paper explores the mathematical structure of renormalization in quantum field theories, showing how vector fields generate diffeomorphisms that underpin the Gell-Mann-Low functional equation.

Contribution

It demonstrates that the Gell-Mann-Low equation naturally arises from the existence of a vector field on the action space in renormalized QFT.

Findings

Vector fields generate one-parameter groups of diffeomorphisms.

The Gell-Mann-Low equation is a consequence of these diffeomorphisms.

The evolution of observables can be described by these geometric transformations.

Abstract

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.