Convolutions and Gaussians in Renormalization

TL;DR

This paper offers an algebraic perspective on the renormalization process in quantum field theory, using convolutions and Gaussian-based Lie group actions to analyze the renormalization transform.

Contribution

It introduces an alternative algebraic framework for renormalization, replacing traditional path integral and perturbation methods with convolution semigroups and Lie group actions.

Findings

Provides a new algebraic description of the renormalization transform.

Connects renormalization with Lie group actions related to the quantum harmonic oscillator.

Offers potential for more abstract and possibly simplified analysis of renormalization processes.

Abstract

The Kadanoff-Wilson-Fisher approach to renormalization is based upon studying the renormalization transform, which may be described as an action of the monoid $\mathbb{R}^{\times}_{\geq 1}$ on a suitable space of interactions. It is typically computed by manipulating the path integral or the perturbation series. Here we will present an alternative algebraic description of the renormalization transform. We treat the space of interactions as a semigroup under convolution and act on it with a Lie group associated with the quantum harmonic oscillator.