The Renormalisation Group Equation As An Equation For Lie Transport Of Amplitudes

TL;DR

This paper presents a novel interpretation of the renormalisation group equation as a Lie transport equation for physical amplitudes, linking it to diffeomorphisms and the geometry of coupling space.

Contribution

It introduces a geometric perspective on the RG equation, interpreting it as Lie transport along flows generated by beta functions in coupling space.

Findings

RG equation as Lie transport of amplitudes

Anomalous dimensions from Lie transport of basis vectors

Connection between dilations and diffeomorphisms in coupling space

Abstract

It is shown that the renormalisation group (RG) equation can be viewed as an equation for Lie transport of physical amplitudes along the integral curves generated by the $\beta$-functions of a quantum field theory. The anomalous dimensions arise from Lie transport of basis vectors on the space of couplings. The RG equation can be interpreted as relating a particular diffeomorphism of flat space-(time), that of dilations, to a diffeomorphism on the space of couplings generated by the vector field associated with the $\beta$-functions.